From d449b822c8a74ffa01b22058cfc042374b1a45cd Mon Sep 17 00:00:00 2001 From: Vlado Vojdanovski Date: Tue, 13 Oct 2026 23:13:40 -0700 Subject: [PATCH 1/3] Add capped phase fusion transfer at conditional kappa 0.001677160151714381 --- .../ALGEBRAIC-STABILITY-INDEPENDENT.md | 190 + research/capped-phase-fusion/BILATERAL-LDU.md | 131 + .../capped-phase-fusion/CAPPED-GAUGE-AUDIT.md | 123 + .../capped-phase-fusion/CAPPED-INTERIORS.md | 72 + .../CAPPED-OUTER-ARITHMETIC.json | 388 + .../capped-phase-fusion/CAPPED-SIGNOFF.md | 89 + .../FINAL-PACKAGE-REVIEW.json | 18 + .../FORWARD-SMALL-A-AUDIT.md | 109 + research/capped-phase-fusion/HALO-THEOREM.md | 271 + .../INDEPENDENT-FINAL-REVIEW.json | 24 + .../LOG-A-PHASE-SCHUR-AUDIT.md | 140 + .../capped-phase-fusion/MANIFEST-UPDATE.json | 172 + research/capped-phase-fusion/MANIFEST.json | 609 ++ .../MATRIX-COROLLARY-ARITHMETIC.json | 414 + .../MATRIX-COROLLARY-REVIEW.json | 164 + .../capped-phase-fusion/MATRIX-COROLLARY.md | 72 + 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a/research/capped-phase-fusion/ALGEBRAIC-STABILITY-INDEPENDENT.md b/research/capped-phase-fusion/ALGEBRAIC-STABILITY-INDEPENDENT.md new file mode 100644 index 0000000000..abdd30cc1d --- /dev/null +++ b/research/capped-phase-fusion/ALGEBRAIC-STABILITY-INDEPENDENT.md @@ -0,0 +1,190 @@ +# Independent algebraic-coefficient and stability transfer lemma + +This note uses no claimed finite certificate from any particular matrix multiplication paper. It isolates what follows from an exponent bound alone, including effectiveness and ordinary bit complexity. + +## Statement + +Let `omega_C` be the matrix multiplication exponent over the complex numbers. Suppose + +\[ +\omega_{\mathbb C}<\tau, +\] + +where `tau > 2` is a fixed rational number. There is a single, uniform algorithm, with a fixed finite description, with the following properties. + +* Its recursive block multiplication uses a fixed finite bilinear formula whose coefficients are computable algebraic complex numbers. +* Given two `W` by `W` complex dyadic matrices with component magnitudes at most `2^h`, it computes their product with maximum entrywise absolute error at most `2^{-p}` in + +\[ +O\!\left(W^\tau M(p+h+\log(W+1))\right) +\] + +bit operations. Here `M(t)` is a regular integer multiplication bound: `M(t) >= t`, `M(Ct)=O_C(M(t))` for fixed `C`, and precision doubling sums to `O(M(t))`. All constants can depend on the chosen fixed block formula. +* If the input encoding has `h = O(p)`, this is + +\[ +O\!\left(W^\tau M(p+\log(W+1))\right). +\] + +If also `log W = O(p)`, it is exactly `O(W^tau M(p))`, without a suppressed logarithmic factor. In particular this holds for `W = Theta(sqrt(p))`. + +The strict exponent slack is essential to this statement: an exponent infimum does not by itself supply a finite scheme attaining that infimum. If a source proves `omega_C <= 9/4`, one may take any fixed rational `tau` with `9/4 < tau < 5/2`, for example `tau = 19/8`. + +The proof below also gives an exponent strictly below a specified threshold from only the promise that `omega_C` is below that threshold. It does not give useful numerical constants or an efficient way to discover the block formula. + +## 1. A finite exact bilinear formula exists with exponent slack + +Write `R(b)` for the exact complex tensor rank of `b` by `b` matrix multiplication. Use the usual equivalent definition + +\[ +\omega_{\mathbb C}=\inf_{b\ge2}\log_b R(b). +\] + +Thus some fixed integers `b >= 2` and `r` satisfy + +\[ +R(b)\le r= 2`, `r >= 1` satisfying `r < b^tau`; +2. finite encoded algebraic tuples of the required length; +3. exact tests of all rank equations. + +For rational `tau=a/d`, the first comparison is the integer comparison `r^d < b^a`. The search halts under the stated exponent promise, by Sections 1 and 2. Its output supplies the single fixed block formula. This search may be extremely expensive, but its cost is an absolute constant independent of `W,p,h`, so including it does not change the asymptotic bound. The same result can be phrased by fixing the first successful tuple once and for all. + +For just the promise `omega_C < 5/2`, search using `r^2 < b^5`. Upon success choose a rational `sigma` with + +\[ +\max\{2,\log_b r\}<\sigma<5/2. +\] + +The resulting algorithm has exponent at most `sigma < 5/2`. Thus one need not know a numerical gap in advance to obtain a uniform algorithm below the threshold. A practical certificate or practical constant is a separate issue. + +## 4. Each fixed algebraic coefficient has an ordinary `O(M(q))` precision algorithm + +Every chosen coefficient has a fixed minimal polynomial and a fixed isolating region. In characteristic zero its minimal-polynomial root is simple. Fix a rational starting approximation in a sufficiently small Newton convergence neighborhood. Newton iteration with successively doubled precision computes the root to absolute error `2^{-q}` with `O(M(q))` bit operations. + +The polynomial degree, coefficient heights, root separation, initial precision, and bounds on the derivative and its reciprocal are all fixed constants here. Polynomial evaluation, complex division, and the finite number of operations per Newton step cost `O(M(t))` at precision `t`; the usual precision-doubling sum is `O(M(q))`. This is not a claim uniform in algebraic degree or height. + +There are only finitely many block coefficients. Compute their dyadic approximations once at the precision needed for the current multiplication, and reuse them at all recursive nodes. This costs `O(M(q))` total, with a formula-dependent constant, rather than a logarithmic factor per scalar operation. + +## 5. Recursive arithmetic work + +Pad the matrices with zeros to size `N=b^L`, where `W <= N < bW` (and handle `W=1` directly). At each node, form the fixed `r` input linear combinations, recursively multiply the corresponding submatrices, and form the fixed output linear combinations. + +The arithmetic operation count satisfies + +\[ +T(N)\le rT(N/b)+O(N^2). +\] + +Let `rho=log_b r`. If `rho>2`, this is `O(N^rho)`; if `rho=2`, it is `O(N^2 log N)`; and if `rho<2`, it is `O(N^2)`. Every case is `O(N^tau)` because `rho2`. In particular the possible equality case `r=b^2` creates no extra factor in the advertised bound: its logarithm is absorbed by the fixed exponent slack. + +One could instead tensor a fixed formula a fixed number of times and add a zero product to arrange `b^2 < r < b^tau`, obtaining an exact recursion exponent `rho` strictly between two and `tau`. This refinement is unnecessary for the advertised `O(W^tau)` conclusion. + +## 6. Fixed-point roundoff grows only polynomially in the dimension + +Use fixed-point complex arithmetic with `q` fractional bits, round after scalar products, and use dyadic block coefficients whose absolute errors are at most + +\[ +\epsilon=2^{-q}. +\] + +Complex arithmetic introduces only absolute constant factors. Additions may be done exactly on this fixed grid; allowing an additional absolute error `O(epsilon)` after an addition also satisfies the following bounds. + +For a multiplication node with remaining depth `d`, let `a,beta >= 1` bound the maximum absolute entries of its two inputs. There is a fixed constant `K`, depending only on the block formula, such that the computed result obeys + +\[ +\|\widehat T_d(A,B)-AB\|_{\max} +\le \epsilon E_d a\,\beta, +\qquad +E_d\le K E_{d-1}+K b^{d-1},\quad E_0\le K. +\] + +Here the symbol `b` in `b^{d-1}` is the fixed block dimension; `beta` is the second input bound. + +To verify the recurrence, every local input linear combination has fixed length and fixed coefficients. Its rounded value differs from its exact value by at most `c epsilon a` or `c epsilon beta`, and its size is at most `S a` or `S beta`, for fixed `c,S`. Applying the inductive child bound therefore contributes at most + +\[ +c\epsilon E_{d-1}a\beta. +\] + +Perturbing the two child inputs in their *exact* matrix product contributes at most + +\[ +c\epsilon b^{d-1}a\beta, +\] + +because each entry of that child product sums `b^{d-1}` scalar products. Finally, the output reconstruction has fixed length and fixed coefficients. Propagation through its exact coefficients is a fixed factor; coefficient approximation and reconstruction rounding contribute at most + +\[ +c\epsilon\bigl(b^{d-1}+\epsilon E_{d-1}+1\bigr)a\beta. +\] + +For `epsilon <= 1`, all these terms have the stated recurrence after enlarging `K`. Taking `K >= 2b` gives `E_d <= C K^d` for a fixed `C`. Thus at the root + +\[ +\|\widehat T_L(A,B)-AB\|_{\max} +\le C\,2^{-q}K^L a\beta +=2^{-q}N^{O(1)}a\beta. +\] + +For root input magnitude at most `2^h`, take `a=beta=max(1,2^h)` and choose + +\[ +q=p+2\max(h,0)+\lceil L\log_2K\rceil+O(1). +\] + +The final absolute error is then at most `2^{-p}`. + +There is also no hidden exponential-in-precision magnitude growth. Along a path, each preprocessing step multiplies a magnitude envelope by a fixed `S`. A node at depth `t` therefore receives inputs bounded by `S^t a` and `S^t beta`. Its computed output is bounded by its exact product size plus the proved error: + +\[ +\bigl(b^{L-t}+\epsilon C K^{L-t}\bigr)S^{2t}a\beta. +\] + +Local scalar products and partial sums obey the same bound up to fixed factors. All intermediate magnitudes are consequently `N^{O(1)} max(1,2^{2h})`. Only `O(h+log N)` integer bits are needed. The total working bit width is therefore `O(p+h+log N)`. + +The polynomial error estimate exploits the structure of a bilinear recursion: linear preprocessing, one variable-by-variable scalar product at a leaf, and linear reconstruction. Bounding an arbitrary multiplication by the largest intermediate magnitude at every level would give an unnecessarily pessimistic estimate and is not the argument used here. + +## 7. Bit complexity and scope + +Every scalar operation uses `O(p+h+log W)` bits and costs `O(M(p+h+log W))`. Multiplying by the `O(W^tau)` arithmetic operation count proves the statement. Padding, input conversion, output rounding, and storage of the fixed coefficient approximations fit the same bound. + +The result supplies absolute accuracy. It does not promise small relative error in entries canceled nearly to zero. It applies to arbitrary fixed finite complex bilinear formulas; large fixed coefficient heights or norms worsen only the constants and the fixed multiple of `log W` in the precision requirement. + +An exponent theorem alone therefore suffices for computable, stable approximate multiplication with any fixed strictly larger exponent. It does not justify using the exact infimum exponent without slack, treating a varying family of coefficients as fixed, or claiming a concrete efficient implementation without constructing its constants. diff --git a/research/capped-phase-fusion/BILATERAL-LDU.md b/research/capped-phase-fusion/BILATERAL-LDU.md new file mode 100644 index 0000000000..5cad6ee19f --- /dev/null +++ b/research/capped-phase-fusion/BILATERAL-LDU.md @@ -0,0 +1,131 @@ +# Bilateral Gaussian LDU and a fixed-width periodic inverse bound + +Independent audit of the parent's infinite-LDU observation. This is an analytic norm theorem. It does **not** assert that the bilateral factors already have a fully priced fast implementation. + +## Statement + +Let `1=1`, define on Z + +`G_ij=exp(-a[(rho*j-q_i)^2-beta_j^2])`. + +Let `r=exp(2a rho)` and `c=prod_(m>=1)(1-r^-m)>0`. Then G has an exact factorization on bounded bilateral sequences + +`G=L C U`, + +where L is unit lower triangular, U is unit upper triangular, C is diagonal, and all factors and their inverses are bounded. Moreover all six factors are invariant under simultaneous index translation by s. Their periodizations therefore factor the periodic Gaussian matrix N, and + +`||N^-1||_infinity <= c^-4 exp(a/2)/(1-exp(-a rho theta))^2`. + +In particular, at the fixed choice `a=1`, + +`||N^-1||_infinity < 16 theta^-2`. + +Any fixed a>=1, including a=4pi if an integer alpha>=2 is retained, gives a constant times theta^-2. No lower bound proportional to log(1/theta) on a is needed for this norm theorem. + +## 1. Finite-origin factorization and the shift correction + +Write `x_i=r^q_i`, `d_i=exp(-a q_i^2)`, and `e_i=exp(-a rho^2 i^2+a beta_i^2)`. Then `G_ij=d_i x_i^j e_j`, including negative indices, and `d_i x_i^i e_i=1`. + +For a finite square principal interval `[l,r0]`, the Vandermonde powers must begin at zero. The correct shifted row diagonal is **d_i x_i^l**, and the power is `j-l`. This removes a possible erroneous origin-dependent diagonal from the upper factor. + +Put + +`C_i(l)=prod_(l<=k=j`, + +`U^l_ij=(e_j/e_i) h_(j-i)(x_l,...,x_i)`, `j>=i`. + +Here h is the complete symmetric polynomial, h_0=1. Upper and lower inverses are + +`(U^l)^-1_ij=(-1)^(j-i)(e_j/e_i) e_(j-i)(x_l,...,x_(j-1))`, + +`(L^l)^-1_ij=(-1)^(i-j) exp(E_ij) C_i(l)/(C_j(l) D_ij)`, + +where + +`D_ij=prod_(j=j`, + +`|(U^l)^-1_ij|<=c^-1 exp(a/4) exp(-a rho theta(j-i))`, `j>=i`. + +Both use only `-1/2<=beta_k<1/2`. Thus their row norms are at most the corresponding constants divided by `1-exp(-a rho theta)`, uniformly in l and the finite size. + +The row sum of G is uniformly finite, since + +`G_i,i+h <= exp(a/4) exp(-a(rho*|h|-1/2)^2)`. + +The same positive Gaussian bound controls all finite-origin L and U factors. + +## 3. Letting the left origin tend to minus infinity + +For fixed i,j, C_i(l) converges to + +`C_i=prod_(k(i+s,j+s)`. Every displayed factor has the same invariance: ratios `x_k/x_i` are preserved, and the Gaussian/prefix expressions shift together. C_i is s-periodic. + +For an invariant matrix A with bounded absolute row sums, set + +`Pi(A)_ij=sum_(n in Z) A_(i,j+ns)`, `0<=i,j=e^2>7, and the elementary product bound gives `c>=1-sum_(m>=1)r^-m>5/6`. Also `1-exp(-rho theta)>=theta/2` for 04W, choose + +`B=min(Bcap, 2^floor(log2(R-W)))`, + +retain the next B vertices as an interior, then the next W vertices as boundary, and decrement R by B+W. Declare the final R<=4W vertices boundary. + +There are `O(L/Bcap+log Bcap)` interiors: while the cap is active, each step removes at least Bcap vertices; after it ceases to be active, the remaining length halves at each step. Every interior length is dyadic and at most Bcap. Every interior produced by the loop is longer than W; two successive interiors are separated by at least W retained coordinates. + +Total boundary count per original cell is + +`O(W(L/Bcap+log Bcap))`. + +Since original cell lengths are Theta(d), total boundary density is + +`O(W/Bcap + W log Bcap/d)=O((W/Bcap) polylog p)` + +when Bcap<=d. The same selected-input and target-output physical period cuts must be preserved. A clipped endpoint cell of O(W) length is wholly boundary. + +Interior elimination only fills between the adjacent boundary groups of one interior. Each group has O(W) vertices: inserted groups have W; an end remainder has at most 5W after joining the reserved end. Hence scalar Schur half-bandwidth remains O(W), independently of the cap and the number of pieces. No retained matrix coefficient is changed by this new partition. + +## 2. Narrow Gaussian: exact recentering of the existing rounded weights + +In one original phase cell, beta_i=theta*i-c and + +`log a_i=-a(1/4-beta_i^2)/theta`. + +For any two coordinates i,j in that same cell, + +`log(a_i/a_j)=a(i-j)(beta_i+beta_j)`. + +Since both beta values lie in [-1/2,1/2), + +`|log(a_i/a_j)|<=a|i-j|`. + +Thus a capped interior of at most Bcap vertices has ideal weight range at most exp(a Bcap), even when the absolute full-cell gauge has range exp(O(ad)). + +The actual matrix uses already-rounded positive weights ahat_i. Suppose, as ensured by the retained high-precision setup, + +`(1-epsilon) a_i <= ahat_i <= a_i`, with epsilon<=1/2. + +Then for an anchor m in the capped interior, + +`(1-epsilon) exp(-a Bcap) <= ahat_i/ahat_m` + +` <= exp(a Bcap)/(1-epsilon)`. + +Replacing ahat_i by `ahat_i/ahat_m` in both diagonal factors of + +`A=diag(ahat) Ghat diag(ahat)^-1` + +is **exact**. It changes neither A nor the fixed globally rounded H. These ratios must be formed in coefficient setup; the data circuit must not apply the full-cell diagonal and hope to cancel it later. One may anchor at the maximum ahat in the piece to make the direct diagonal contractive and its inverse at most `2 exp(a Bcap)`. + +The old setup provides absolute error below `4*2^-P` and ideal minimum at least exp(-a/(4theta)). It suffices that + +`4*2^-P exp(a/(4theta)) <= epsilon`. + +In the retained family a=Theta(log p), theta^-1=O(d), d=p^epsilon0 with fixed epsilon0<1, so ad=o(p). The existing P=Cp setup precision easily gives epsilon<=2^-cp. The **ad cost occurs in offline coefficient accuracy**, not in the runtime norm of the recentered diagonal. It is not multiplied by the number g of simultaneously executed axes. More explicitly, if the raw relative error is delta0<=4*exp(a/(4theta))*2^-P, then an anchored ratio has relative error at most 2delta0/(1-delta0). A g-fold product of anchored diagonals has error bounded by a polynomial factor times g*delta0*exp(O(g*a*Bcap)). Thus the needed precision is P>=p+O(ad+g*a*Bcap+log g+polylog p), rather than p+O(g*a*d). Both added power terms are o(p) in the proposed parameter family. + +## 3. Broad forward Gaussian: local gauges + +Use the target-grid notation from FORWARD-SMALL-A-AUDIT.md. With lambda=1/rho, b0=pi/alpha^2, original phase c, and an anchor m in the capped interior, + +`delta=(1-lambda)m-c`, `x=i-m`, `y=k-m`. + +The exact split is + +`Dcol(i)=exp[-b0((1-lambda)x^2+2delta*x)]`, + +`Drow(k)=alpha^-1 exp[b0 lambda(1-lambda)y^2+2b0 delta lambda*y-b0 delta^2]`, + +with a fixed Toeplitz kernel `exp[-b0 lambda(i-k)^2]`. + +Throughout the original selected-target phase cell, `beta'_i=theta*i-rho*c` lies in [-1/2,1/2). Therefore + +`d(log Dcol)/di = -2 b0 lambda beta'_i`, + +`d(log Drow)/dk = 2 b0 lambda^2 beta'_k`. + +Their absolute slopes are at most b0 and b0 respectively. Both diagonal logarithmic ranges on the capped block are at most `b0 Bcap`. At the anchor, Dcol=1 and Drow=alpha^-1 exp(-b0 delta^2), so their absolute values also have logarithmic size `O(b0 Bcap+log alpha)`. + +If a globally rounded broad block was already defined using a different anchor, use exact constant ratios to recenter its two diagonals: divide the column diagonal by its value at m and multiply the row diagonal by the same value. Ideal changes of anchor differ only by reciprocal constants, since the derivatives above are independent of m. Uniform relative coefficient error changes the range by only O(1), just as in the narrow case. This preserves the already-defined rounded block exactly. Alternatively the broad block can be initially defined with the capped anchor, using one globally specified rounding construction. + +Holes remain boundary. The completed narrow and broad interior inverse bounds and the completed global/Schur norm bounds proved earlier do not depend on the cap: every new interior is still a contiguous principal block of the same original phase matrix. + +## 4. Group guard and circuit count + +Each capped narrow diagonal contributes at most `a Bcap+O(1)` to the logarithmic norm; the broad contribution `O(Bcap/alpha^2+log alpha)` is smaller. Shallow Toeplitz circuit amplification contributes polylog(p) per axis. The number of capped blocks per line does not multiply these bounds: they are disjoint direct-sum blocks and their circuits run in parallel, padding completed smaller circuits with identity gates. + +Consequently g simultaneously executed axes require + +`O(g a Bcap + g polylog p)` + +extra logarithmic precision. With `Bcap<=p^b`, `g<=p^beta`, a=Theta(log p), and beta+b=1-r-zeta with r,zeta>0, + +`g a Bcap=O(p^(1-r-zeta) log p)=o(p)`. + +In particular no remaining full-cell term g*a*d may be charged after exact recentering. Offline accuracy still needs to dominate ad=o(p), as above. + +## 5. Schur solves and whole-group ordering + +For the narrow matrix, strict row dominance is inherited by any interior elimination, so all completed Schur and inverse norms remain polynomial under the cap. + +For the broad matrix, completed Schur/inverse norms follow from the square identity-hole completion and polynomial principal-interior bounds in FORWARD-SMALL-A-AUDIT.md. To obtain uniform norms for a banded online factorization, eliminate **whole boundary groups** of W to O(W) vertices, rather than appealing without proof to scalar prefixes inside one group. + +After all interiors and a prefix of whole boundary groups are eliminated, the eliminated set is a direct sum of one physical contiguous prefix and disjoint future interiors: each remaining full group contains at least W uneliminated coordinates, blocking every width-W connection across it. The same holds after adding the next whole group. A physical prefix reduces, under selected/hole ordering, to the small-b normalized Gaussian block plus identity hole columns, so it has polynomial inverse norm. Each future interior also has polynomial inverse norm. Thus every required eliminated-set inverse is polynomial, uniformly in the number of pieces. + +At the final group on a full periodic line, use the already-established full periodic H inverse bound instead of a nonperiodic-prefix bound. For a clipped principal window the complement-gap condition excludes wrap edges, so the prefix statement applies directly. Each block pivot inverse is a principal block of the inverse after adding that group. Block Schur entries are bounded by `H+H H_EE^-1 H`, hence polynomial. There are only O(1) neighboring boundary groups in a band row. The resulting block-L and block-U norms are polynomial; their full inverses are polynomial as well, from `L^-1=U S^-1` and `U^-1=S^-1 L`. A scalar expansion inside a group is unnecessary: precompute its dense O(W)-square inverse and apply the block updates to batches by Strassen. + +This whole-group order prevents the subtle gap in which a partially consumed separator of fewer than W vertices permits a connection to the next interior. It preserves the claimed batched boundary exponent and is unaffected by capped subdivision. + +## Scope + +The capped local-gauge change is valid. The listed exact controls check carving, analytic slope identities and preservation of the already-rounded narrow matrix. The final recurrence arithmetic and tensor routing remain separate reviewed components. diff --git a/research/capped-phase-fusion/CAPPED-INTERIORS.md b/research/capped-phase-fusion/CAPPED-INTERIORS.md new file mode 100644 index 0000000000..62c3fa27b6 --- /dev/null +++ b/research/capped-phase-fusion/CAPPED-INTERIORS.md @@ -0,0 +1,72 @@ +# Conditional refinement: cap interior length independently of d + +This is a separate follow-up to the uncapped proof audit. Its current status is **PASS_CONDITIONAL_PHASE_FUSION_REVIEW**, following the independent capped-gauge and capped-layout audits and an integrated root review. It preserves the weaker `OUTER-ARITHMETIC.json` as history. Under the retained supplier assumptions and the reviewed interfaces, the capped construction below gives `kappa=0.001672922349588871`. The only new analytic change is the anchored O(aB) local gauge; all altered operation counts are derived explicitly here. No upstream selected certificate is changed. + +Let the original phase-cell scale remain `d=p^epsilon`, and cap dyadic interior lengths at `B=2^floor(b log2 p)`, where `1/20. The supplier support proof needs each individual K-neighborhood to fit its original axis and total moved support o(p); it does **not** require all g controls to fit within one original axis. The sufficient auxiliary inequality `beta<1-epsilon` in the uncapped ledger must therefore be removed in this follow-up. + +The changed principal exponents, normalized by V, are + +| Component | exponent | +|---|---:| +| main transform | `1-epsilon*q` | +| fused Gaussian circuits | `epsilon-q*beta` | +| boundary algebra | `epsilon-b+(omega-1)/2` | +| sparse union movement | `epsilon+beta+1/2-b` | +| grouped scans | `epsilon-beta` | + +For the boundary row, one axis has O(V W/B) boundary records. A band solve or coupling product on W simultaneous right-hand sides costs O(W^(omega-2)) per boundary record, so all d axes cost `O(V (d/B) W^(omega-1) polylog p)`. The new factor d/B must not be omitted. For the sparse union, its fraction within a group is O(gW/B polylog p); g sequential axis operations and d/g groups give `O(V (dg/B) W polylog p)`. These are precisely the third and fourth changed rows. + +All other costs in `OUTER-AUDIT.md` remain paid. Global routing has saving a_eff; packed polynomial products have saving epsilon. Even a conservative full active-group key surcharge adds only `beta+1-epsilon` to the union exponent. Sparse core-face corrections have exponent at most `epsilon+beta-3`, using the former p² per-record allowance on their O(gA/L) copied volume. + +With the rational bound omega<45/16, put c=(3-omega_upper)/2=3/32 and take a positive boundary slack s_boundary=10^-5. Balancing the two dominant rows while leaving that boundary slack gives + +``` +epsilon=(1+q+q*(c-s_boundary-r-zeta))/(1+q)^2, +b=epsilon*(1+q)-c+s_boundary, +beta=1-b-r-zeta, +kappa_limit=epsilon*q. +``` + +Using the same q, r and zeta as the uncapped candidate yields approximately + +``` +epsilon = 0.9984838711085557536432677624420451614... +b = 0.9064167934681446247757331487619919695... +beta = 0.0935821965318553752242668512380080305... +kappa_limit = 0.0016729223595888711324653863199468081... +kappa = 0.001672922349588871 +``` + +The boundary saving equals `kappa_limit+10^-5`. The union movement exponent is below 0.686, and even its optional full-key surcharge leaves it below 0.781. The per-record phase metadata exponent `beta+2(1-epsilon)` is below 0.097, so calculating all g membership predicates remains cheaper than scanning one p-bit record. Core-face repair costs remain negligible at L=p^8. + +The common compiled H calls are still made globally across complete coarse-label ranges. Each selected bit needs only its own K=128G neighborhood to fit its original axis. Since `1-epsilon>0`, this holds eventually even though gK is larger than one axis width. The total support is O(g polylog p)=o(p), because beta<1. All support-router reservoirs, product row stock and exact endpoint restoration are those of the independently audited supplier. + +At the displayed parameters, `b>1/2`, `epsilon>b`, `beta>0`, `epsilon>beta`, `1-b-r-beta=zeta>0`, and `1-epsilon-r>0`. These respectively pay the wide-band separation, cap below original cell scale, growing group payload, number of groups, compiled local-gauge precision, and global gamma/rounding range. The raw global rounded weights may still have O(ad) logarithmic range during offline generation. The **data** only sees exact anchored ratios of range O(aB); approximating those ratios to the retained precision introduces `exp(o(p))` error amplification, already paid by the strict group guard. + +The assembled group endpoints are the same full normalized Haug inverse or Saug/4 operators as in the uncapped proof. Changing the interior partition changes only their factorization. It does not change their completed contraction bounds, intended zero-hole semantics, principal-core locality, or the final `2^D` scale compensation. Consequently the uncapped precision and error argument applies with the improved group gauge budget. + +`check_capped_outer.py` computes 44 exact positive slacks, every displayed cost margin, and five rejecting controls. The full-cell-gauge control is especially relevant: it fails, so the stronger result depends on the independently reviewed anchored-gauge proof. The local review retains the original supplier and all-size conditional assumptions. diff --git a/research/capped-phase-fusion/CAPPED-OUTER-ARITHMETIC.json b/research/capped-phase-fusion/CAPPED-OUTER-ARITHMETIC.json new file mode 100644 index 0000000000..56c871f4f3 --- /dev/null +++ b/research/capped-phase-fusion/CAPPED-OUTER-ARITHMETIC.json @@ -0,0 +1,388 @@ +{ + "status": "PASS_CONDITIONAL_PHASE_FUSION_REVIEW", + "scope": "Local conditional transfer and exact arithmetic; inherited supplier and all-size interfaces remain assumptions. 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signoff + +The earlier sparse-union layout proof extends to dyadic interior cap +`B=p^b infinity`. Individual guarded neighborhoods +fit the original long axis; the collection of all g controls does not have +to fit in a single axis. This is the same support proof already audited. + +Microedge correction is unchanged. The underlying rounded operator and its +physical bandwidth do not change when a factorization receives more cuts. +Only the principal restriction and Schur decomposition are repartitioned. +The retained Neumann/principal-window comparison still applies to that +same operator. With L=p^8 and halo A=p^3, the additional output fraction is +O(g p^-5), now about p^-4.9064. Its sparse gather, true-period correction and +old sequential local solvers are still negligible. Capping does not turn +this into a dense non-dyadic tensor halo. + +The pinned independent arithmetic checks give approximately + +``` +epsilon = .9984838711085558 +b = .9064167934681446 +beta = .0935821965318554 + +per-group union movement power = -.2192524004044339 +total union movement power = .6856492741722665 +total group scan power = .9049016745767004 +membership descriptor power = .0966144543147439. +``` + +`check_capped_layout.py` independently checks finite capped partitions, +dyadic block sizes, separator distances, worst-pattern elimination fill, +and these exact rational layout inequalities against the pinned candidate +JSON. No layout blocker was found. + +This signoff is specifically for carving, sparse support, actual stream +movement, descriptors, and inherited microedge placement. It does not +independently prove the newly anchored broad/narrow Gaussian formulas or +their final precision bound. Those remain the algebraic/analytic cap audit. diff --git a/research/capped-phase-fusion/FINAL-PACKAGE-REVIEW.json b/research/capped-phase-fusion/FINAL-PACKAGE-REVIEW.json new file mode 100644 index 0000000000..66a4af697e --- /dev/null +++ b/research/capped-phase-fusion/FINAL-PACKAGE-REVIEW.json @@ -0,0 +1,18 @@ +{ + "status": "PASS_CONDITIONAL_SCOPE_AND_PORTABLE_REPLAY_REVIEW", + "scope": "Reviewed final README, proof scope, attribution and verifier; independently reproduced 21 exact checks, 46 canonical native JSON receipts and all 58 file bindings against the completed run. Two false-acceptance controls reject. No independent human or formal theorem review is claimed.", + "conditions": [ + "Use latest support verifier and native capsule before generating final package manifest.", + "Optional matrix corollary retains omega_C<=9/4 as an explicit extra premise.", + "All inherited all-size and fixed-tape hypotheses remain." + ], + "reviewed_sha256": { + "verify.py": "93ef0f46e132485da6194587199b8f039e25b1b9f099f93bbdaeac5c0d66de7f", + "NATIVE-RECEIPT-CAPSULE.json": "a04d6bba3d15c11a219536b66a23012797484ae106ec1952ecd7b76f077ddb01", + "README.md": "61b371fe72353cd1d80cf2a1df5cf4e10df18ca78d2265acf43d263222895103", + "phase_fusion_note.tex": "b5957f38c98a68f2584a42cedf91d753180dab438129fdf749b441074f00d4e1", + "TASK.json": "494271a60db34e6651bfe164a0516c685081e899475ebf8f9fabe98a6a8014b5", + "VERIFY.json": "cb50aa523a0863a457181ca815e69a1b3b4d91ed1006a79263f4a8f18457c669", + "SOURCES.json": "14ce6fdc7c83e18ccf4fee19ff51062726c960ca44462a45af60dd686167d456" + } +} diff --git a/research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md b/research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md new file mode 100644 index 0000000000..0f7f80b3f7 --- /dev/null +++ b/research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md @@ -0,0 +1,109 @@ +# Broad forward Gaussian: finite-block and completed-matrix conditioning + +This note supports the phase-assembly branch. It distinguishes the broad forward map S from the narrow T used in N=C T D. Its conclusions concern conditioning and local circuit guards; the final simultaneous tensor layout remains that branch's obligation. + +Let the narrow parameter be `a=pi alpha^2=Theta(log p)`, `rho=t/s=1+theta`, and `lambda=1/rho`. The forward map is + +`S_(k,j)=alpha^-1 exp[-b0(j-lambda*k)^2]`, `b0=pi/alpha^2`. + +Equivalently it uses the small resampling parameter + +`b=b0 lambda^2=pi/(alpha^2 rho^2)`. + +## 1. The bilateral bound works at every positive parameter + +The finite and bilateral LDU arguments in BILATERAL-LDU.md use a>=1 only for their final numerical simplification of the product c. For any b>0, define + +`c_b=prod_(m>=1)(1-exp(-2b rho m))`. + +The same proof gives, for the periodic normalized matrix and for every finite contiguous principal block with q-gaps at least 1 and |beta_i|<=1/2, + +`||N_b^-1|| <= c_b^-4 exp(b/2)/(1-exp(-b rho theta))^2`. + +There is an explicit useful product bound. Expand the logarithm and sum nonnegative terms: + +`log(c_b^-1)=sum_(k>=1) 1/[k(exp(2b rho k)-1)]` + +` <=(1/(2b rho)) sum_(k>=1)1/k^2=pi^2/(12b rho)`. + +At `b=pi/(alpha^2 rho^2)`, this implies + +`c_b^-4 <= exp(pi alpha^2 rho/3)=exp(a rho/3)`. + +Since b*rho*theta=pi*theta/(alpha^2 rho)<1 for alpha>=2 and theta<=1, its reciprocal gap is at most `2alpha^2 rho/(pi theta)`. Thus every such normalized block inverse has norm polynomial in p when alpha^2=O(log p) and theta^-1 is polynomial in p. + +## 2. Actual target-grid phase-cell block + +Embed source j at target coordinate i=q_j. On an original phase cell q_j=j+c, every selected target coordinate i in that cell corresponds to j=i-c. A square interior of the forward target-grid matrix is therefore + +`K_(k,i)=alpha^-1 exp[-b0(i-lambda*k-c)^2]`. + +Here both i and k run over one contiguous selected-target interval inside that cell. Put + +`q'_k=k+rho*c`, `beta'_k=rho*k-q'_k=theta*k-rho*c`. + +At a selected coordinate k=q_j, beta'_k is exactly the original beta_j, hence lies in [-1/2,1/2). Also q' has consecutive gap 1. Integrality of q' was never needed in the finite LDU proof; monotone separation and the beta bound suffice. Consequently + +`K=alpha^-1 N_b diag(exp(-b*(beta'_i)^2))`, + +and + +`||K^-1|| <= alpha exp(3b/4) c_b^-4/(1-exp(-b rho theta))^2` + +` =O(alpha^5 rho^2 theta^-2 exp(pi alpha^2 rho/3))=poly(p)`. + +This controls the **completed** interior inverse. It avoids multiplying the large individual diagonal-gauge norms to estimate K inverse. Target holes should be retained as boundary vertices, so every interior coordinate has the asserted original beta bound. + +## 3. Square completion of the global forward map + +Add one identity column e_h for every target hole h, and feed zero into these extra columns. This makes an invertible square t-by-t matrix H_aug without changing the desired forward result. Permute selected q_j rows/columns first and holes second. In that order, + +`H_aug = [ alpha^-1 N_b D_b^-1 0 ; R I ]`, + +where `D_b=diag(exp(b beta_j^2))` and R consists of broad Gaussian rows at holes. A Gaussian lattice-sum bound gives `||R||=O(1)` after the alpha^-1 factor. Thus + +`H_aug^-1=[ alpha D_b N_b^-1 0 ; -R alpha D_b N_b^-1 I ]` + +has polynomial row norm. The matrix itself also has O(1) row norm. Gaussian truncation and sufficiently accurate structure-preserving coefficient rounding perturb both it and all required interior blocks by exponentially small amounts, preserving these polynomial bounds. + +If A is the block-diagonal interior matrix and S is the completed boundary Schur complement, then S inverse is the boundary principal block of H_aug inverse; hence its norm is polynomial. Also + +`||S||<=||H_BB||+||H_BI|| ||A^-1|| ||H_IB||=poly(p)`. + +This is enough for the completed Schur map. To justify a particular sequential block-LU implementation, its prefix factor norms still need to be checked. The natural physical-prefix elimination sets consist of one contiguous selected-source prefix plus disjoint future interiors, with identity hole columns. That geometry offers the same uniform contiguous-block inverse bound; it should be made explicit in the assembly proof. + +## 4. Exact local Toeplitz split and guard range + +Choose an anchor m inside the target cell, and write x=i-m, y=k-m, + +`delta=(1-lambda)m-c`, with |delta|<=1/2. + +Then the exact factorization is + +`K_(k,i)=Drow(y) G(y,x) Dcol(x)`, + +`G(y,x)=exp[-b0 lambda(x-y)^2]`, + +`Dcol(x)=exp[-b0((1-lambda)x^2+2delta*x)]`, + +`Drow(y)=alpha^-1 exp[b0 lambda(1-lambda)y^2+2b0 delta lambda*y-b0 delta^2]`. + +Expansion gives exactly `-b0(delta+x-lambda*y)^2`; no approximate recentering is involved. On a phase cell of O(d) coordinates, the two diagonal logarithmic ranges are O(d/alpha^2). These are smaller than the O(alpha^2 d) narrow inverse gauges already paid by the proposed group size. + +## 5. SPD Toeplitz conditioning for the shallow circuit + +Let `gamma=b0 lambda=pi/(alpha^2 rho)`. The full Gaussian Toeplitz symbol has, by the Gaussian Poisson identity, + +`f(omega)=sqrt(pi/gamma) sum_(k in Z) exp[-(omega+2pi*k)^2/(4gamma)]`. + +For -pi<=omega<=pi, + +`sqrt(pi/gamma) exp[-pi^2/(4gamma)] <= f(omega) <= 1+sqrt(pi/gamma)`. + +Every finite principal Gaussian Toeplitz matrix has eigenvalues between these same bounds, by integrating the squared modulus of its coefficient polynomial against f. At alpha^2=O(log p), its condition number is polynomial in p. Truncation width O(sqrt(p/gamma)) and Cp-bit rounding perturb the matrix far less than its lower eigenvalue, so the actual symmetric rounded Toeplitz Ghat remains SPD with polynomial condition number. + +The low-displacement block-LU circuit of SHALLOW-TOEPLITZ.md extends from near identity to this SPD setting: every principal Schur complement has norm at most lambda_max and inverse norm at most lambda_min^-1, and ranks grow by the same additive displacement recurrence. Shear coefficients are polynomial in p; multiplying these bounds along O(log d) recursive levels gives exp(O(log^2 p)) partial-prefix growth. This calls for O(log^2 p) internal circuit guard bits per axis, rather than claiming the near-identity theorem's O(log p) unchanged. It is still below the group budget dominated by the local Gaussian diagonals. + +Dyadic interior subdivision from LOG-A-PHASE-SCHUR-AUDIT.md applies here too, using the broader width O(sqrt(p alpha^2)). It keeps target holes and separators as boundary, adds only polylogarithmic boundary volume, and lets every interior use a dyadic shallow circuit without 2^g padding. + +No numerical final exponent follows from these conditioning statements alone. diff --git a/research/capped-phase-fusion/HALO-THEOREM.md b/research/capped-phase-fusion/HALO-THEOREM.md new file mode 100644 index 0000000000..60d59bb378 --- /dev/null +++ b/research/capped-phase-fusion/HALO-THEOREM.md @@ -0,0 +1,271 @@ +# Fast tensor convolution by core blocks and sparse face corrections + +This gives a constructive replacement for g dense halo scans. It applies to +a **translation-invariant separable compact stencil**, with optional +diagonal conjugations whose coefficients and tile gauges satisfy the +precision and sequential-access conditions below. It does not claim that +the separate Gaussian product-tree factorization has this form. + +Let g consecutive active axes have dyadic lengths S_i, with +`h=max_i log2 S_i`. Earlier coordinates are outer prefix fibers. Later +coordinates are a complete payload R, so they need no keys or transposition. +Let V denote the total bit volume, including B-bit coefficient records and +all spectators. Assume B=Theta(p), `g h=O(p)`, and the active group grows +superpolynomially. A fixed number of tapes may hold O(V) bits. + +The operation is + +``` +(Tf)(x) = sum_{z in [-L,L]^g} [product_i k_i(z_i)] f(x-z mod S), +``` + +where `L=poly(p)`. A fixed or polylogarithmic number of these operations and +diagonal scans is allowed. Kernel coefficients, coefficient products and +the required intermediates have O(p)-bit representations under the supplied +normalization contract. The retained grouped convolution engine on a +binary field of width k has cost `O(V k^(1-q) polylog p)`. Polynomial-length +one-axis convolution also has its unconditional quasilinear bit algorithm; +it does not invoke the improved bound being proved. + +Choose a dyadic tile side N dividing every S_i, with `N>4L` and `gL/N=o(1)`. +Write `b=log2 N`, `k=gb`, and `K0=O(g(h+b)+log g)=O(gh)` for a correction key. +Require `K0^2=O(B)` or retain the explicit extra metadata term below. + +The complete cost is + +``` +O(V [ (gb)^(1-a) polylog p + + (gb)^(1-q) polylog p + + polylog p + + (gL/N) (K0 + g polylog p) ]) ++ polynomial(p) setup, +``` + +plus `O((gL/N)V K0^2/B)` if metadata arithmetic is not absorbed by records. +All scans, duplication, sparse sorts, local transposes and final output +order are included. In particular, there is no gV layout term. + +## 1. One support-limited gather + +Within the active group, gather its g low b-bit fields into a contiguous +suffix using the support-at-most-2gb extension proved in +`SUPPORT-ROUTER.md`. The remaining active coarse bits remain in their +possibly scrambled order. Every coarse tile is now one contiguous block +of N^g complete R-payloads. No other active coarse bit is put into a new +stable order, and no spectator coordinate is gathered. + +The initial and inverse gather together cost +`O(V (gb)^(1-a) polylog p)`. The explicit permutation maps each physical +coarse tile name back to its logical vector; that small map is used only +for tile setup and sparse correction metadata. + +## 2. Interior outputs from ordinary cyclic convolution + +Flatten a tile by `phi(x)=sum_i x_i N^(g-1-i)` into Q=N^g slots. Put the +product kernel at offsets `psi(z)=sum_i z_i N^(g-1-i) mod Q` in an ordinary +one-dimensional cyclic convolution kernel. These offsets are distinct: +reduce a putative collision modulo N, use `2L= p^eta`, `g -> infinity`; alternatively a supplied sequential dense +kernel generator is an explicit acceptable input. + +If every tile coordinate satisfies `L<=x_ij: [0,N). +``` + +These 2g rectangles are disjoint and contain every discarded output once. +Their required unwrapped input rectangles have intervals + +``` +ij: [-L,N+L). +``` + +Coordinates outside `[0,N)` are taken from the correct neighboring coarse +tile, including periodic wrap in each actual axis. All input dimensions +are less than 2N; no assumption of independent full 2N padding is used. + +The total copied input volume is exactly + +``` +M = 6L V/N^g * sum_{j=0}^{g-1} N^j (N+2L)^(g-1-j) + = 3V [ (1+2L/N)^g - 1 ] + = O((gL/N)V). +``` + +The corrected output volume is +`V [1-(1-2L/N)^g] = O((gL/N)V)`. The formulas count corner duplication +exactly. An individual corner may have many destinations; its total traffic +is already included in M. No bound by worst-case 2^g multiplicity is charged +to all records. + +## 4. One full scan generates all correction copies + +For each original core tile, recover its logical coarse coordinates once. +Scan its local coordinate tuple with a binary odometer. Maintain flags for +membership in the four fixed edge intervals `[0,L)`, `[0,2L)`, +`[N-2L,N)` and `[N-L,N)`. Only coordinates changed by the odometer update +their flags. The number of coordinate updates over a whole tile is +`Q(1+1/N+...)<2Q`. Comparisons cost O(log N), and counters/head resets use +the retained amortized stream-counter convention. Maintain a count of +active near-boundary flags. A nonexceptional record takes no scan through +the g flags. + +For an exceptional record, enumerate candidate first-boundary axes from +the flag tape and all valid neighboring-tile choices. The exact choices +are the ones implemented in `sparse_copies` in `check_halo.py`: earlier +axes stay in the original tile; the selected face axis has its own or one +adjacent tile; each later axis has its own tile plus an adjacent tile only +when within L of that edge. Every emitted record carries + +``` +(j, side, destination coarse tile, unwrapped local coordinates). +``` + +These are K0=O(gh) bits. Precompute the plus/minus-one forms of each logical +coarse coordinate once per tile. A copy's logical neighbor is generated +by selecting those forms, then encoding the known scrambled coarse order. +All carries, periodic wraps and changed tile order are explicit. Using +ordinary binary arithmetic gives O(K0^2) safe metadata work per copy. + +The full scan costs O(V polylog p). The exceptional flag scans and address +work cost O(M K0^2/B+M) in bit-volume units when R includes the B-bit record. +Their setup per coarse tile is absorbed by its N^g payload. This is not an +unpaid nonlinear boundary-signature evaluation on every payload bit. + +## 5. Sparse radix gather and fixed-tape local processing + +Within each *earlier-coordinate prefix fiber*, stable-radix sort the copies +by the displayed key, with unwrapped local coordinates in ordinary tensor +order. Only the g active axes occur in the key; all following spectator +coordinates remain one R-block payload. A radix pass scans each complete +header and payload, splits into two bucket tapes, concatenates, and resets +the same fixed set of tapes. Since K0=O(B), each pass costs O(M); K0 passes +cost O(M K0). No simultaneous open tape per tile, face or corner is needed. + +Sorted copies form the exact input rectangles of section 3, consecutively. +Process one rectangle at a time. For each axis, expose its current local +coordinate field, apply the length-polynomial one-dimensional convolution, +and crop L positions from each end. Thus its length changes from its input +interval to the corresponding output interval. Other axes remain unchanged +until their own pass. This yields the tensor product of the restricted +one-dimensional maps, hence the exact desired face outputs. + +Every current axis has length at most N+2L=poly(p). Its mixed-radix transpose +is the retained short-field construction: pad **only that field** to its +next power of two temporarily, move its O(log p) bits using O(log p) linear +block interchanges, remove the temporary padding, and use the correct new +field length on return. The temporary factor below two is removed before +the next axis. A rectangle therefore costs its volume times +`O(g polylog p)` for movement and unconditional polynomial-length arithmetic. +Summing over rectangles costs `O(M g polylog p)`. Signed/complex dyadic +convolution can use sign splitting and ordinary Kronecker packing with +O(p+log N)=O(p) guard bits, followed by the unconditional quasilinear integer +multiplier; no improved supplier is used on this sparse path. + +After cropping, attach each output's physical coarse-tile and flat inner +address. Their keys are distinct because the face output partition is +disjoint. A second sparse radix sort orders them as holes in the retained +core-output stream. One merge scan replaces exactly those holes. The inverse +support-limited gather restores the original axis-major stream, including +the initially displaced coarse coordinates. All metadata is discarded. + +## 6. Parameters for the proposed partial fusion + +Use `d=p^epsilon`, `h=Theta(p/d)`, `g=p^(1-epsilon-zeta)` with fixed positive +zeta and epsilon>5/6. Suppose `L=O(p polylog p)` and take the dyadic side +`N=2^ceil((3/2)log2 p)`. The actual S_i eventually exceed every polynomial, +so N divides them and each periodic window is shorter than S_i. + +Then `K0=O(p^(2(1-epsilon)-zeta) polylog p)` and `K0^2=o(p)`. The two sparse +cost multipliers satisfy + +``` +(gL/N) K0 = O(p^(5/2-3epsilon-2zeta) polylog p) = o(1), +(gL/N) g = O(p^(3/2-2epsilon-2zeta) polylog p) = o(1). +``` + +All halo work is therefore O(V polylog p), below the desired +`O(V g^(1-a) polylog p)` routing term. The core convolution is +`O(V g^(1-q) polylog p)` since log N=Theta(log p). + +The bounded-gauge condition cited in `PARTIAL-FUSION.md` remains payable: +with `a0=Theta(log d)` and theta=Theta(1/d), its variation is + +``` +g [a0 theta N + a0/theta] + = O(p^(5/2-2epsilon-zeta) polylog p + + p^(1-zeta) polylog p) + = o(p) +``` + +for epsilon>5/6 and fixed zeta>0. Thus the larger polynomial tile does not +consume the O(p)-bit guard budget. This checks a layout/precision inequality; +it does not establish the missing Gaussian convolution factorization. + +## Diagonal and factorization boundary + +The theorem also applies to a fixed or polylogarithmic composition of these +translation-invariant convolutions and diagonal factors if every diagonal +scan/factor page has a proved sequential O(V polylog p) implementation and +all normalized tile-core and tile-halo values use O(p) bits. A tile-dependent +common scalar gauge may multiply every input to that tile and divide its +outputs; boundary copies must use the **destination** tile's gauge. No +enormous diagonal is formed and later canceled numerically by assertion. + +A concrete sufficient diagonal implementation is available for +`D(x)=product_i d_i(x_i)`. Supply each one-axis factor page in coordinate +order, with all normalized prefix products bounded to O(p) bits. Traverse +the rectangle by nested coordinate loops and retain the g successive prefix +products on work tape. Multiply by the innermost factor once per record; +outer prefix updates occur once per complete inner row, then once per +complete inner rectangle, and so on. With side lengths at least two their +number is less than twice the record count. Singleton axes are folded into +one constant per rectangle. Rewinding a factor page costs its length and is +charged to the corresponding completed row. Ordinary O(p)-bit multiplication +therefore gives O(V polylog p) scan cost with O(gp) temporary bits. This +argument uses bounded normalized **prefix products**; bounds on the final +product alone do not suffice. + +It does **not** cover an unbounded collection of different short kernels +inside one tile, a product-tree level with factor-two padding on all g axes, +or an unspecified transform-dependent sparse correction. Those require a +new representation theorem. In particular, the Gaussian_fast/product-tree +algorithm is not silently substituted for the translation-invariant kernel +assumed here. + +## Finite evidence + +`check_halo.py` verifies 5,632 exact outputs in three two/three-axis cyclic +examples, with asymmetric signed kernels, nontrivial diagonal conjugations, +periodic coarse carries, scrambled coarse order, unique sparse gather/scatter +keys and all local sequential crops. Pure flattening gives 3,040 wrong +boundary outputs across these examples; all are repaired. The exact copied +volumes equal `3V[(1+2L/N)^g-1]` in every case. These are constructive finite +controls; the preceding scan, metadata and volume arguments give the +conditional asymptotic claim. diff --git a/research/capped-phase-fusion/INDEPENDENT-FINAL-REVIEW.json b/research/capped-phase-fusion/INDEPENDENT-FINAL-REVIEW.json new file mode 100644 index 0000000000..121a4ebf87 --- /dev/null +++ b/research/capped-phase-fusion/INDEPENDENT-FINAL-REVIEW.json @@ -0,0 +1,24 @@ +{ + "status": "NO_BLOCKING_GAP_FOUND_CONDITIONAL", + "unconditional_integer_multiplication_theorem": false, + "reviewed_current_phase_note_sha256": "08ecb1876695beee25484158e14cfd57c3d48f551a8a41d5abc998d71d674555", + "same_assumption_kappa": "0.001672922349588871", + "additional_matrix_premise_kappa": "0.001673360105405778", + "fixed_nonblocking_finding": "Generic SPD prefix bound required absolute scale M and 1/m rather than condition number alone; current note corrected.", + "remaining_nonblocking_evidence_limit": "Union checker is an algebraic dictionary control, not a tape implementation or two-window reuse simulation.", + "reports": { + "core_review.md": "13f2df4d5b240f9b89ca0ae44bba198c935532098ff4c1161631fcbd86b3e8b1", + "gaussian_layout_review.md": "ebd7188970f55b5979b1b1a13f874a31ed0ee6c82bee329e7ad0b2a38c42c430", + "layout_subreview.md": "637e4698a9fb02978ebc4032d8939ce0db9528e89e908c94d63b9f0d2a41f64a", + "union_subreview.md": "738b35a04d6871cd73b14b4165ae69a1feb3a103eb46e064a64a22c4c7426856" + }, + "premises_not_independently_certified": [ + "finite bit/complex suppliers", + "exact common-frame/mutable-source realization", + "whole-call row hosting", + "semantic guard C1=1", + "retained prime packing and fixed-tape interfaces", + "paid arbitrary-width equal-chunk interchange", + "additional omega_C<=9/4 premise" + ] +} diff --git a/research/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md b/research/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md new file mode 100644 index 0000000000..3bcf490a8c --- /dev/null +++ b/research/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md @@ -0,0 +1,140 @@ +# Audit: logarithmic Gaussian width retains the phase-cell Schur solver + +This audit checks the analytic hypotheses of the retained rad20 phase-cell solver at `a=pi alpha^2=Theta(log p)`, `theta=Theta(1/d)`, even though `a theta` tends to zero. It also audits the shallow Toeplitz circuit algebra and identifies the actual matrix that must be precomputed. It does not certify the final tensor layout or multiplication exponent. + +Read inputs: `references/semantic-bulk/rad20/reports/review-bulk-resampling.md`, `review-phase-cell-inverse.md`, `downstream-phase-cell-inverse.md`, and `work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md`. + +## 1. The sharp elementary diagonal-dominance estimate + +Let `rho=1+theta`, `beta_i=rho*i-q_i`, and let the Gaussian row coefficient at integer offset h be + +`n_i,h=exp(-a[(rho*h+beta_i)^2-beta_(i+h)^2])`. + +The h=0 coefficient is 1. Set `g_i=q_i-q_(i-1)` and `g_(i+1)=q_(i+1)-q_i`, each in {1,2}. For the nearest coefficients, + +`n_i,-1=exp(-a*g_i*(2rho-2beta_i-g_i))`, + +`n_i,+1=exp(-a*g_(i+1)*(2rho+2beta_i-g_(i+1)))`. + +If g_i=1, the first exponent is exactly `1+2theta-2beta_i`; if g_i=2, that gap forces `beta_i=1/2-theta`, so the upper exponent is at least `1+2theta+2beta_i`. Consequently + +`n_i,-1+n_i,+1` + +` <=exp(-a(1+2theta-2beta_i))+exp(-a(1+2theta+2beta_i))` + +` <=exp(-2a theta)+exp(-2a(1+theta))`. + +The last step is convexity in beta on [-1/2,1/2]. The two nearest directions cannot both be charged as exp(-2a theta). + +For |h|>=2, + +`(rho*h+beta_i)^2-beta_(i+h)^2 >= h^2-|h|`, + +so their entire two-sided tail is below `4 exp(-2a)` for a>=1. Therefore + +`||N-I||_infinity <= exp(-2a theta)+5 exp(-2a)`. + +This includes nonzero periodic diagonal aliases in the tail; no alias is silently omitted from this bound. + +For p>=101, theta>=1/(4p), and `a>=4 log(16p)`, + +`1-exp(-2a theta) >= 1-exp(-1/(2p)) >=1/(4p)`, + +`5exp(-2a)<=5/(16p)^8<1/(8p)`. + +Hence N has row diagonal-dominance margin at least `1/(8p)` in the following robust sense: after deleting its tiny nonzero diagonal aliases, its diagonal is 1 and its off-diagonal row sum is at most `1-1/(8p)`. In particular the fixed-width inverse theorem is not needed to supply the retained strict row dominance. + +Choose the legal integer alpha by + +`alpha=ceil(sqrt((4/pi)log(16p)))`. + +Then a has the required lower bound and is O(log p). The old sufficient condition `u>=ceil(log2(8p))` need not be reused: the direct estimate above replaces that prerequisite. + +## 2. Actual rounded H, interior matrices, and guard range + +Use exactly the retained structure: + +- `w=ceil(sqrt(16p/u))+2`, u=alpha^2=a/pi; +- cell weights `a_j=exp(-a(1/4-beta_j^2)/theta)`; +- one globally fixed downward dyadic Toeplitz kernel `Ghat_delta` for |delta|<=w, zero outside, with Ghat_0=1; +- positive downward dyadic weights ahat_j; +- within a physical cell, define H_jk **exactly** as `ahat_j Ghat_(j-k)/ahat_k`; +- across cells, use the retained downward Gaussian entries; set H_jj=1. + +Thus the actual interior block being inverted is + +`A_cell=diag(ahat) Ghat_cell diag(ahat)^-1`. + +It is not the unrounded Gaussian matrix, and its retained truncated Ghat is not silently replaced by the infinite Gaussian kernel. + +The weight range is + +`0<=-log a_j<=a/(4theta)=O(a d)` + +under theta between 1/(4d) and 1/(2d-1). The old eventual condition u*d<=p holds whenever d=p^epsilon, epsilon<1, and u=O(log p). It bounds the old rational perturbation calculation by the same `2^(O(p)) eta` allowance. The stronger range `O(a d)` should be used for grouped tensor precision, rather than charging O(p) independently for every axis. + +Choose the coefficient precision high enough that `||H-N||<1/(16p)`, including the Gaussian truncation, omitted diagonal aliases and weight-ratio rounding. The retained bound `p*2^(16p+7)*eta+2^(-46p)` at eta=2^(-32768p) is more than sufficient. Then H has diagonal 1, row gap at least `1/(16p)`, row norm below 2, and inverse norm at most 16p. All clipped principal windows inherit the same row gap. The physical period cut must remain a cell boundary, as in the retained bulk review. + +## 3. Schur complement bounds without an a*theta>1 assumption + +A scalar elimination step on a strictly row-diagonally-dominant matrix cannot decrease its minimum absolute row gap. For pivot k, + +`gap_i(new) >= gap_i(old)+(|a_ik|/|a_kk|) gap_k(old)`. + +The full absolute row sum also cannot increase, since the added update row has off-pivot sum less than |a_kk|. Induction applies regardless of the signs of filled entries. Since the diagonal begins real positive and the correction magnitude is below it, the surviving diagonal remains positive. + +Consequently the exact Schur complement + +`S=H_BB-H_BI H_II^-1 H_IB` + +has row gap at least `1/(16p)`, row norm below 2, positive diagonal, and + +`||S^-1||_infinity<=16p`. + +Principal interior inverses also have norm at most 16p. Rounding S by a row perturbation below 1/(32p) preserves gap at least 1/(32p) and inverse norm at most 32p. A row has O(w) nonzeros; the retained P-grid readily meets this target. These are polynomial bounds on arbitrary signed inputs, with no factor equal to the number of cells. + +The boundary geometry is unchanged: keep first/last w vertices of each physical cell, yielding O(s*w/d) boundary vertices and circular half-bandwidth at most 2w. Schur fill only joins the two end groups of one cell, while original cross-cell entries join neighboring end groups. Actual inverse and LU coefficient tables must be precomputed for this H/S, not imported from the old parameter values. + +Thus the retained banded/cyclic boundary solver's gap-dependent proof remains applicable with fixed constants enlarged. The proposed batched boundary arithmetic still needs its own data-layout and Strassen error audit, but there is no analytic conditioning obstruction at logarithmic a. + +## 4. Apply the shallow circuit directly to Ghat inverse + +Because a is eventually larger than 4pi, the symmetric Toeplitz matrix Ghat_cell satisfies + +`max(||Ghat_cell-I||_1,||Ghat_cell-I||_infinity)<1/8`. + +Downward rounding only decreases this off-diagonal sum. The shallow full-Toeplitz theorem therefore applies directly to Ghat_cell. Reverse the resulting invertible circuit and invert its gates to obtain Ghat_cell inverse. This avoids the two-branch Gohberg-Semencul difference entirely. The interior inverse is: + +1. multiply by diag(ahat)^-1; +2. apply the reversed shallow circuit for Ghat_cell; +3. multiply by diag(ahat). + +Its only superpolynomial individual scale comes from the **local** weight range O(a d). Completed A_cell inverse has polynomial norm by the strict row gap. Across g axes, the uncompleted local weight contribution is at most exp(O(g*a*d)); choosing `g<=p/(d polylog p)` with sufficient logarithmic slack keeps the total data precision O(p). The shallow circuit contributes polynomial-in-d amplification per axis, hence only O(g log d) additional logarithmic range. This is a guard calculation, not a complete simultaneous-layer routing proof. + +## 5. Independent shallow-Toeplitz algebra audit and a length caveat + +The parity-split triangular identity, block Schur displacement formulas, Stein-generator telescoping identity, and bounded-prefix induction in SHALLOW-TOEPLITZ.md check algebraically. In particular the parity children run in parallel, avoiding the earlier erroneous sequential-depth recurrence. The full block-LU recursion also runs its two children in parallel, with O(log N) displacement rank after O(log N) levels. + +That document currently states its circuit theorem for dyadic N. Actual cell interiors are not generally dyadic. Padding every tensor axis to its next power of two would be invalid. A plausible exact extension has been communicated to its author: split an arbitrary N into balanced sizes differing by at most one; a near-square rectangular Toeplitz dirty shear is a square Toeplitz shear plus one extra row/column rank-one shear. The latter can be implemented by a binary tree of unitary 2-coordinate reductions and its inverse, without clean ancilla. The rectangular Stein identity telescopes as before. This extension needs its own written proof/controls before it is used as an established interface. + +No new final kappa is asserted by this audit. + +## 6. Dyadic interior subdivision resolves the length caveat + +No arbitrary-length shallow FFT theorem is actually needed. Refine the interior/boundary partition inside each original physical phase cell of length L as follows. Keep its first and last w vertices as boundary. Starting with the remaining interval of length R=L-2w: + +- while R>4w, choose the largest power of two B<=R-w; +- mark the next B vertices as one interior, the following w vertices as boundary, and replace R by R-B-w; +- once R<=4w, mark all remaining vertices as boundary. + +Every interior now has exactly dyadic length. If R>4w, then `B>(R-w)/2`, so the next remainder is less than R/2. There are `O(1+log(L/w))` interiors and the total boundary count in the cell is `O(w(1+log(L/w)))`. The final boundary group, including the initially reserved last w vertices, has size at most 5w; all inserted separators have size w. + +The width-w matrix H cannot connect two different new interiors: their separating boundary gap has w vertices, so their nearest physical coordinates differ by w+1. Each new interior is still a principal block of the SAME globally fixed Toeplitz-similar cell matrix. Its weights retain the original cell gauge; there is no new rescaling or rerounding at a subdivision boundary. + +Eliminating one dyadic interior fills only between its immediately neighboring boundary groups. Each such group has O(w) vertices. Original H edges also connect only the same group or neighboring groups; an interior has length greater than w in the loop above. Thus, in inherited physical boundary order, the new Schur complement has half-bandwidth O(w), not O(w log d). Only its number of rows grows by O(log d): + +`n_boundary=O((s/d) w log d)`. + +Clipped endpoint cells of size O(w) may be declared wholly boundary, adding only O(w) rows; preserve the original physical period cut. This is the same endpoint convention as the retained bulk review. + +The boundary solve and coupling work therefore gain only polylogarithmic factors. All dyadic interior blocks use the established SHALLOW-TOEPLITZ theorem directly, without padding any tensor axis and without requiring a non-dyadic Fourier circuit. 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"-0.25103009636737138762328042591872403463400943697970053427204949195" + } + } + } +} diff --git a/research/capped-phase-fusion/MATRIX-COROLLARY-REVIEW.json b/research/capped-phase-fusion/MATRIX-COROLLARY-REVIEW.json new file mode 100644 index 0000000000..fea12fbe37 --- /dev/null +++ b/research/capped-phase-fusion/MATRIX-COROLLARY-REVIEW.json @@ -0,0 +1,164 @@ +{ + "status": "PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW", + "created_utc": "2026-10-14T02:11:15.272021+00:00", + "kappa": "836680052702889/500000000000000000", + "kappa_decimal": "0.001673360105405778", + "upstream_selected_unchanged": true, + "base_transfer_preserved": "CAPPED-OUTER-ARITHMETIC.json and REVIEW-RECEIPT.json", + "additional_hypothesis": "The supplied Matrix Multiplication in Nine Fourths theorem gives omega_C<=9/4; apply the independently audited effective algebraic transfer at tau=19/8.", + "limitations": [ + "All finite supplier, common-frame, whole-call, semantic precision, fixed-tape and prime-packing assumptions of the first transfer remain.", + "No generated competitive finite matrix formula or practical runtime claim is made; the fixed formula search is effective but may have enormous constant cost.", + "This is a local conditional corollary, not an upstream selected certificate or an unconditional integer-multiplication theorem." + ], + "reviews": [ + { + "reviewer": "root agent", + "result": "Read matrix corollary and independent layout signoff; reproduced exact parameters/ledger and matrix construction controls; approved integrated conditional corollary." + }, + { + "reviewer": "matrix_boundary agent", + "result": "Read all 1022 supplied source lines and audited effective algebraic coefficient, fixed-tape, precision and rank-exponent transfer." + }, + { + "reviewer": "global_macro agent", + "result": "Independent new-parameter no-sort union, metadata, q-ordering and microhalo layout signoff." + }, + { + "reviewer": "phase_assembly agent", + "result": "Read both complete matrix transfer notes and independent layout signoff; constructed and replayed 47-slack ledger and six rejecting controls." + }, + { + "reviewer": "independent_fusion_review agent and independent children", + "result": "No blocking gap found under stated assumptions after full compiler, support, Gaussian, layout and tape-cost review. Generic SPD scale wording corrected without changing parameters. See independent FINAL-REVIEW.json." + } + ], + "arithmetic": { + "positive_slacks": 47, + "rejecting_controls": 6, + "minimum_saving": "0.0016733601154057784538170479517203762842018888767028492264912297200" + }, + "files": [ + { + "bundle_filename": "phase_fusion_note.tex", + "source_path": "work/phase_assembly_ultra/phase_fusion_note.tex", + "sha256": "08ecb1876695beee25484158e14cfd57c3d48f551a8a41d5abc998d71d674555", + "bytes": 33757 + }, + { + "bundle_filename": "MATRIX-COROLLARY.md", + "source_path": "work/phase_assembly_ultra/MATRIX-COROLLARY.md", + "sha256": "2d7fb1088e58bb86e574277733e082f135362215a69add71d9090213d32c9221", + "bytes": 5787 + }, + { + "bundle_filename": "check_matrix_corollary.py", + "source_path": "work/phase_assembly_ultra/check_matrix_corollary.py", + "sha256": "30db2d8e0f40fcc5c91e25f74cdc54e46aa4a0353c725345f9bcf5744e0b617f", + "bytes": 6692 + }, + { + "bundle_filename": "MATRIX-COROLLARY-ARITHMETIC.json", + "source_path": "work/phase_assembly_ultra/MATRIX-COROLLARY-ARITHMETIC.json", + "sha256": "c6efc4d2f6d17159cefebf5c2dd6099a6327fe0911c5dd87623cad96c7d6c56a", + "bytes": 16783 + }, + { + "bundle_filename": "PROOF-AND-COVERAGE.md", + "source_path": "work/matrix_boundary_ultra/PROOF-AND-COVERAGE.md", + "sha256": "db7509d754bb7fc3381419e0fed2674f22a9d57d34e8420fd5afa3c688fde9aa", + "bytes": 12130 + }, + { + "bundle_filename": "ALGEBRAIC-STABILITY-INDEPENDENT.md", + "source_path": "work/matrix_boundary_ultra/ALGEBRAIC-STABILITY-INDEPENDENT.md", + "sha256": "7873e7d9a99fdc9de3ee224b9925d3e277e9773d7f5f8c2d0f1c19b0e52bc87f", + "bytes": 11743 + }, + { + "bundle_filename": "MATRIX-SOURCE-COVERAGE.json", + "source_path": "work/matrix_boundary_ultra/SOURCE-COVERAGE.json", + "sha256": "fdc6849f97999a6cf99cdaa50591d24f78d6fb0bc580315f205452e5906b35cc", + "bytes": 2839 + }, + { + "bundle_filename": "MATRIX-SOURCE-CONSTRUCTIONS.json", + "source_path": "work/matrix_boundary_ultra/SOURCE-CONSTRUCTIONS.json", + "sha256": "114c1d8fb70592455a80c73d8a1bc6bfc9aca4e7395a420b6331d5474b4d1917", + "bytes": 21018 + }, + { + "bundle_filename": "MATRIX-TRANSFER-MANIFEST.json", + "source_path": "work/matrix_boundary_ultra/MANIFEST.json", + "sha256": "35d98d7579a16dae36530b4e5f4d0b8f925a069daa3f6fe1ea5276916167e0c2", + "bytes": 1159 + }, + { + "bundle_filename": "MATRIX-LAYOUT-SIGNOFF.md", + "source_path": "work/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md", + "sha256": "4c8c62f723232e842dc3bbdcc4defcc8ff1aeb86ba9f694661899dc6783b13c4", + "bytes": 3081 + }, + { + "bundle_filename": "MATRIX-LAYOUT-CONTROLS.json", + "source_path": "work/phase_layout_audit_ultra/MATRIX-LAYOUT-CONTROLS.json", + "sha256": "f8bd8729cb8438188e90fb9149f598b825461c97b4ed290584ca4367f812a48a", + "bytes": 1446 + }, + { + "bundle_filename": "PHASE-LAYOUT-AUDIT.md", + "source_path": "work/phase_layout_audit_ultra/AUDIT.md", + "sha256": "54aafc7afd09fe1a5312e69fad09c87422690543ab75c6ae35f6cbcc26b3eb8a", + "bytes": 11774 + }, + { + "bundle_filename": "CAPPED-SIGNOFF.md", + "source_path": "work/phase_layout_audit_ultra/CAPPED-SIGNOFF.md", + "sha256": "70e9175ebde200f8b016236adc88c4a1c4858945e7c3e5c453a8b6a5f79a263a", + "bytes": 4414 + }, + { + "bundle_filename": "CAPPED-GAUGE-AUDIT.md", + "source_path": "work/gaussian_fast_ultra/CAPPED-GAUGE-AUDIT.md", + "sha256": "1030a34922709ab8d3434a4c57a4a58934ae268c237fb598cb415697da411c9f", + "bytes": 9176 + }, + { + "bundle_filename": "LOG-A-PHASE-SCHUR-AUDIT.md", + "source_path": "work/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md", + "sha256": "e05d6b8596c80b6e8676569a0ff464a29fa704c3304143f58914f47cdf38b272", + "bytes": 10642 + }, + { + "bundle_filename": "FORWARD-SMALL-A-AUDIT.md", + "source_path": "work/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md", + "sha256": "e37aaa30b15671a1dff00676b14572213efaaac99df202dbba5e09f57af487c0", + "bytes": 6689 + }, + { + "bundle_filename": "SHALLOW-TOEPLITZ.md", + "source_path": "work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md", + "sha256": "b384b93f3fb4a8c97da2e617ebc14d55cebb0a739070e7f8dbb66f63e24e9310", + "bytes": 15480 + }, + { + "bundle_filename": "TENSOR-COMPILER.md", + "source_path": "work/inplace_algebra_ultra/TENSOR-COMPILER.md", + "sha256": "4d3b8466598a864a234b978e25976648d4e761840aad9153dd18d73d3dfe4d15", + "bytes": 13430 + }, + { + "bundle_filename": "SUPPLIER-SUPPORT-AUDIT.md", + "source_path": "work/inplace_algebra_ultra/SUPPLIER-SUPPORT-AUDIT.md", + "sha256": "bd5b06eca043c5c871227264f6a954473c55cd16ec85bb420ff538fb8ec1d8d6", + "bytes": 16557 + }, + { + "bundle_filename": "SUPPORT-ROUTER.md", + "source_path": "work/halo_layout_ultra/SUPPORT-ROUTER.md", + "sha256": "6ddb322e11883c1177ba6cd1a77d23a1dac827d7290b32eaf642457b1d762a54", + "bytes": 7755 + } + ], + "post_review_correction": "Generic SPD prefix bound records both M and 1/m, not condition number alone; actual Gaussian bounds already satisfy it." +} diff --git a/research/capped-phase-fusion/MATRIX-COROLLARY.md b/research/capped-phase-fusion/MATRIX-COROLLARY.md new file mode 100644 index 0000000000..f08bb57f9c --- /dev/null +++ b/research/capped-phase-fusion/MATRIX-COROLLARY.md @@ -0,0 +1,72 @@ +# Additional corollary using the supplied matrix-exponent theorem + +Status: **PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW**. The independent matrix-transfer and new-parameter layout audits passed, followed by the integrated root review and exact ledger replay. The first, Strassen-based result `kappa=0.001672922349588871` remains separately reviewed and unchanged. Neither result changes the upstream selected certificate. + +This corollary has an **additional hypothesis**: the supplied *Matrix Multiplication in Nine Fourths* theorem gives `omega_C<=9/4`. Its complete-source audit is `PROOF-AND-COVERAGE.md`; the effective approximate-bit transfer is `ALGEBRAIC-STABILITY-INDEPENDENT.md`. The result here retains every supplier and all-size assumption of the first phase-fusion theorem. + +## Effective matrix operation with the required error interface + +Fix `tau=19/8>9/4`. The matrix exponent theorem supplies a fixed block dimension m and rank r with `r0`. All local coefficient pages remain polynomial in p, while `L^g` remains superpolynomial. Microedge correction has exponent `epsilon+beta-3<-1.75`, including the generous old scalar arithmetic bound. Sparse microhalo key passes remain explicitly paid inside that negligible budget. + +All forward q-image group embeddings occur in native axis-major order before the one global low-bit gather. All compression q-image extractions occur after the inverse global gather. The entire Gaussian calculation keeps full dyadic target coordinate ranges, so every globally invoked H stage has complete coarse labels. This schedule needs no per-group gathering of the whole long-axis fields; the q-image scans have the already-paid Vd/g cost. + +The **optional full-group-key sort surcharge on the sparse union is omitted** because it is absent from the actual algorithm. The reviewed union routine extracts the complete rectangle `U_i=E_i x [L]^(g-1)`, transposes only its one polynomial-size field, computes its axis operator, and merges it into U in the original order. It does not radix-sort U. Reintroducing that sort is a rejecting control in the new ledger; it would destroy this parameter choice. This omission relies on the existing positive layout proof, not on the matrix theorem. + +`check_matrix_corollary.py` verifies 47 exact positive slacks and six rejecting controls and writes `MATRIX-COROLLARY-ARITHMETIC.json`. The rejected variants are endpoint equality, zero group guard, zero union slack, the old Vd row, the old full-cell data gauge, and adding a full-key sort to U. The first theorem's records are not overwritten. diff --git a/research/capped-phase-fusion/MATRIX-LAYOUT-CONTROLS.json b/research/capped-phase-fusion/MATRIX-LAYOUT-CONTROLS.json new file mode 100644 index 0000000000..aee3270196 --- /dev/null +++ b/research/capped-phase-fusion/MATRIX-LAYOUT-CONTROLS.json @@ -0,0 +1,30 @@ +{ + "status": "PASS_MATRIX_COROLLARY_NO_SORT_LAYOUT", + "source_sha256": "b0d4648e16df74fc0a3644baa8a82cbb82842c0aa21abe915a28c0c4c949182a", + "exponents": { + "union_fraction": { + "rational": "-3430918384747396741111767383/8006701850307232414091700000000", + "decimal": "-0.000428505825356260251146382720850643577327042013532977151967005" + }, + "membership": { + "rational": "269791328968194081006854064986174254715733926/1069348903949477650676785089185287127357866963", + "decimal": "0.252294950667420906279768139101314143625086172719573255573065" + }, + "total_union": { + "rational": "320264641396534864583914893734048954038319544983689789/320804671184843295203035526755586138207360088900000000", + "decimal": "0.998316639884594221546182952048279623715798111123297150773509" + }, + "group_scans": { + "rational": "2402698356122892965636680000000000000000000000/3208046711848432952030355267555861382073600889", + "decimal": "0.748959903622628611922902526129555589081788674143596616501459" + }, + "microhalo_fraction": { + "rational": "-380335532915361620704585/80067018503072324140917", + "decimal": "-4.75021475791267813012557319136042532178866352100676648857598" + }, + "rejected_union_full_key": { + "rational": "3334408963974161672450196077539/2668900616769077471363900000000", + "decimal": "1.24935673626196560962328042591872403463400943697970053427205" + } + } +} diff --git a/research/capped-phase-fusion/MATRIX-LAYOUT-SIGNOFF.md b/research/capped-phase-fusion/MATRIX-LAYOUT-SIGNOFF.md new file mode 100644 index 0000000000..d683e4d495 --- /dev/null +++ b/research/capped-phase-fusion/MATRIX-LAYOUT-SIGNOFF.md @@ -0,0 +1,64 @@ +# Matrix-corollary layout: positive signoff + +The no-sort union layout in `AUDIT.md` applies unchanged to +`work/phase_assembly_ultra/MATRIX-COROLLARY.md`. Its algebraic conditions are +closed axis supports E_i and a complete microbox; neither its proof nor its +implementation requires gkappa`. The minimum excess is greater than `10^-11`. Consequently every fixed logarithmic factor and each use of the unconditional quasi-linear short arithmetic kernel can be absorbed in a single positive fraction of that excess. Floors, rounded alpha, and dyadic L alter constants or logarithmic factors. All claims are eventual in p; one fixed finite prefix of inputs can use the retained elementary fallback. This statement becomes an admitted theorem only when the named implementation hypotheses receive the intended full review. + +## Parameter convention + +Let `a_bit=0.001733427039589844234749` and `b=0.001675462676808103522925` be the retained lower bounds certified by the selected PR1094 replay. For applying the older compact recurrence, lower the usable bit saving to `a_eff=(b+q)/2`; this keeps its historical convention `a_eff0` with `(1-t_stop)b>a_eff` and `q<(1-t_stop)b`. + +For each visited coarse block and group, all required local coefficient pages can instead be regenerated in polynomial time in p: there are g local windows of size at most L=Theta(p^8), their rational origins have O(p/d) bits, and the published local coefficient/Schur/circuit constructors are polynomial in that size and requested precision. Recover local phases and anchor gauges from these origins; do not form a raw uncentered global q² exponential on the data. The page contains O(p)-bit online coefficients. Its polynomial setup and sequential loading are amortized by the complete `L^g` payload, which eventually dominates every fixed power of p. This gives an explicit access method even when the coarse labels are permuted; no random global coefficient-table lookup is assumed. + +Global chirps, twists, positive diagonal normalizations and exact scale factors are separable products. The prefix-product scan supplies their products in O(V polylog p), with completed scalar range `O(d alpha^2)=o(p)`. Packed polynomial products retain their saving epsilon. The finite supplier, exact common-label frame, whole-call hosting, semantic depth bound with C1=1, phase precision, prime packing and fixed-tape input/output assumptions are inherited unchanged. They are still conditional interfaces, as in the selected result. + +## Cost ledger, with conservative wide bandwidth + +Let `h_w=1/2` and `h_K=beta+1-epsilon`. Normalize all costs by V. Fixed logarithmic factors are absorbed by strict gaps. + +| Component | power of p | reason | +|---|---:|---| +| main transform | `1-epsilon*q` | deferred reservations | +| constant global coordinate routing | `1-a_eff` | paid support/full router | +| dense Gaussian groups | `epsilon-q*beta` | `(d/g) g^(1-q)` | +| grouped bit routing | `epsilon-a_eff*beta` | same grouping, bit supplier | +| one/fixed group scans | `epsilon-beta` | d/g groups | +| boundary algebra | `(omega-1)*h_w` | batched w-column Strassen | +| sparse union, basic movement | `beta+h_w` | V g w overall | +| sparse union, conservative key surcharge | `beta+h_w+h_K` | full active-group keys | +| polynomial products | `1-epsilon` | retained short polynomial products | +| core-face corrections | `epsilon+beta+3-8+2` | old local maps on sparse copies, with a loose p² allowance | + +Use the rational upper bound `omega<45/16`, checked by `7^16<2^45`. The target balances the first and third rows exactly before a fixed 10^-11 absorption backoff. + +This ledger is accepted only after its circuit tensorizer, grouped sparse route, target-grid core/halo adapter, forward small-a inverse bound, and deferred-reservation interfaces have written proofs. 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has a constructive +implementation with its stated `V g^2 W polylog(p)/d` per-group movement +budget. It does not require sorting the full tensor or retaining d full +payload scans. The conclusion is conditional on the separately supplied +dense local tensor compiler, rounded phase-cell factorization, and precision +bounds. This audit does not prove those algebraic/analytic inputs. + +The maximum physical bandwidth **W**, including the broad forward map, must +be used in the dyadic carving and support sets below. The narrower inverse +bandwidth alone is insufficient for the broad map. The phase-assembly +branch confirms this convention. + +## Exact operator order and the union that is actually invariant + +Write H in interior/boundary order as `[A B; C D]` and let +`S=D-C A^-1 B`. The displayed identity in the parent note is correct. In +execution order its inverse can be scheduled as + +``` +diag(A^-1,I) +[I 0; -C I] +diag(I,S^-1) +diag(A,I) +[I -B; 0 I] +diag(A^-1,I). +``` + +The middle two diagonal factors commute. Thus there are three dense +interior stages and two sparse windows: the C and S^-1 stages share one +window; B uses the second window. For H itself, reverse and invert this +sequence: the same original B,C and S occur, with signs reversed and dense +A/A^-1 interchanged. No dense A^-1 B or C A^-1 shear is smuggled into the +sparse boundary cost. + +For each active axis i, let J_i be its designated boundary coordinates. +Let E_i contain J_i and every interior coordinate incident to an original +nonzero entry of B_i or C_i. Original physical bandwidth W implies that E_i +lies within O(W) of the carved endpoints. Include physical-period cuts, +artificial phase cuts and clipped microbox endpoints. Dyadic carving gives + +``` +|E_i| <= C (L/d+1) W log d, +mu := max_i |E_i|/L = O(W log d/d + W log d/L) +``` + +inside a length-L microbox. All sparse factors have the block form +`T_i = I_(E_i^c) direct_sum T_i|E_i`. In particular S_i^-1 can be dense on +J_i: J_i is already included in E_i. Let + +``` +U = {x in [L]^g : some x_i belongs to E_i}. +``` + +Then `|U|/L^g <= g mu`. Every sparse tensor-axis operation preserves U and +its complement. If x_i is changed by a nonzero off-diagonal entry, both +its old and new coordinates belong to E_i; otherwise it is unchanged. +This proves closure for an arbitrarily long sequence of these sparse +operations, not just one application. + +Dense A or A^-1 is **not** covered. It can move a near-boundary interior +coordinate into the deep interior. Scatter U before a dense stage and +regather afterwards. `check_union.py` includes an exact example in which +dense A^-1 sends union-supported data outside U. The actual schedule above +therefore uses two union gathers, not one retained union for all six stages. + +## Complete rectangles inside the union: no missing-zero assumption + +For one sparse axis i define + +``` +U_i = {x : x_i in E_i} = E_i x [L]^(g-1). +``` + +This is a **complete rectangle**, and it is wholly present in U regardless +of which other coordinates are near a boundary. All nonidentity data needed +by T_i is in U_i. A line with every other coordinate outside its own E_j +still has every E_i coordinate present. A line with another boundary +coordinate may contain more data, but those extra E_i^c entries are identity +coordinates and need no boundary computation. + +Keep a copy of U in its original microbox lexicographic order, with a short +coordinate/membership header. For each axis i: + +1. Scan U once and copy records with x_i in E_i to an auxiliary stream. + The copied stream is lexicographic E_i x [L]^(g-1). +2. Rename the increasing E_i coordinates by their ordinal 0,...,|E_i|-1. + This is a stream counter; no rank/select query is needed. +3. Expose that one field at the line suffix using the retained small-field + transpose. Its length is at most L=poly(p), so only O(log p) binary + block moves are needed. Pad only this field to its next power of two + temporarily, at a factor below two, and remove that padding before the + next axis. The other fields may be arbitrary spectators. +4. Apply the original B/C or boundary S/S^-1 factor on its complete E_i + line, retaining identity coordinates E_i\J_i when appropriate. +5. Undo the transpose at the actual field length. Merge the ordered E_i + stream back into a fresh copy of U, replacing exactly those entries. + +No array is completed by invented zero coordinates. The finite checker +verifies the extracted rectangle cardinality and the full tensor result. + +The extraction/merge costs O(|U|) per axis. Transposition costs +O(|U_i| polylog p). Consequently one sparse window costs + +``` +O(V [g^2 mu + g mu polylog p]) + = O(V g^2 W polylog p/d) + O(V g^2 W polylog p/L). +``` + +There is no factor `K=g log s` from a computed-key sort in this part of the +algorithm. A more general radix implementation would still fit with extra +small powers of g, but the complete-rectangle construction avoids it. + +## Union extraction and piece catalogues + +Only one full microbox scan is needed to form U, and one to scatter it. +For every complete p-bit coefficient record, compute its g membership flags +from its coordinate tuple and OR them. The flags and coordinate header have +O(g log L) bits. A nonzero number of flags sends the whole record once to U; +it is not duplicated for each active axis. + +This is a charged descriptor calculation. For actual source periods of +bitlength h=Theta(p/d), q_j inversion and phase-cell endpoint calculations +use O(h^2 polylog p) elementary bit operations per axis. Within a phase +cell, the greedy dyadic carve is found by at most O(log d) halvings and +integer comparisons. Thus a conservative membership cost is +`O(g h^2 polylog p)` per record, which is o(p) for +`g=p^(1-epsilon-r-zeta)` and epsilon>2/3. At the proposed epsilon this has +power about 0.005. It is absorbed by reading the p-bit record once. +Alternatively the original remainder phase generator and an odometer give +the same membership flags in sequential coordinate order. + +Factor pages may be regenerated once per complete microbox/group from its +actual coarse origins, using the supplied polynomial-time coefficient +generation. The complete inner microbox and untouched spectator payload +are superpolynomial in p and amortize this polynomial setup. This avoids +random access to a global catalogue in the scrambled coarse order created +by a support-limited gather. Online row/factor pages are then visited in +the ordered extracted E_i line. All page rewinds are within that line or +RHS batch and are included in the corresponding boundary arithmetic. + +## Batched right-hand sides do not add a large transpose + +All other inner coordinates and following tensor axes are independent +right-hand sides for an axis operation. Their count R is superpolynomial. +Batch them in a dyadic width between W and 2W; the final incomplete batch +can be padded once at constant cost. To put the batch index outside the +boundary-row field, move only that polynomial-width row field. This costs +O(log p) times the boundary volume, not a transpose of the long spectator +field. Each W-by-W factor page is scanned once per RHS batch and reused in +the block product. Reverse the small-field layout after the solve. + +Thus the assumed Strassen `W^omega` per W RHS gives the quoted total +`O(V W^(omega-1) polylog p)` boundary arithmetic. This audit checks the +layout and factor reuse; stability and the no-pivot factor theorem remain +the phase-assembly branch's responsibility. + +## q_j embeddings, selections and source order need one group scan + +For a group, each coordinate map `j -> q_j=floor(tj/s+1/2)` is increasing +and injective. Its tensor product preserves lexicographic order on its +image. Enumerate the target-group rectangle once. At target coordinate k, +set + +``` +j=max(0,ceil(s(2k-1)/(2t))). +``` + +The coordinate belongs to the image iff `j9/4`. + +If the surrounding proof allows polylog p factors, the bandwidth's polylogs remain polylogs; no positive power of p is introduced by coefficient approximation or error amplification. The algorithm provides absolute accuracy, which is what local residual bounds use. It makes no relative-error promise for nearly canceled output entries. + +The recursion can also be scheduled with sequential tapes. At one fixed-base node, its fixed number of input block combinations and output block sums require a fixed number of complete scans and fixed block permutations of `O(n^2)` records. Store child inputs consecutively, process children depth-first, and stream their returned blocks into accumulators. Copying a pending-child region or repositioning a finite set of tapes costs another `O(n^2)`-record pass at that node, already covered by the recurrence's addition term. Addresses and recursion descriptions add at most logarithmic factors. If any implementation retains an additional `polylog W` factor, choose the rank scheme first with exponent strictly between 9/4 and the stated target tau, leaving enough fixed exponent slack to absorb it. No algebraic unit-cost random-access assumption is needed for the target power tau. + +## 4. Boundary substitution and the outer ledger + +The existing capped boundary algebra uses W simultaneous right-hand sides. A square product costs `O(W^tau M(p))`; dividing by W² processed entries gives `O(W^(tau-2) M(p))` per entry. One axis has `O(V W/B)` boundary entries, and there are d axes. Its total is consequently + +`O(V (d/B) W^(tau-1) [M(p)/p] polylog p)` + +when V is the total bit volume, as in the phase-assembly notes. There are V/Theta(p) scalar entries, so the factor M(p)/p is the per-bit arithmetic charge. Equivalently, if V counts scalar entries instead, the formula has M(p) without the division. The retained quasi-linear short multiplier has M(p)/p=polylog p. + +This retains the essential d/B factor. With `d=p^epsilon`, `B=p^b`, and `W=p^(1/2) polylog p`, its normalized exponent is + +`epsilon-b+(tau-1)/2`. + +Taking `tau=19/8` gives `(tau-1)/2=11/16` and boundary saving `5/16+b-epsilon`. This is already more than enough to put the basic sparse-union cost near the binding row at b near 3/4. Pushing tau down to the unattained endpoint 9/4 is unnecessary. + +The changed matrix algorithm has the same input/output interface as the former Strassen product. It does not change the original Gaussian matrix, boundary support, anchored gauge, square-block inverse choice, or number of right-hand sides. The surrounding local-residual proof must still supply polynomial bounds on local exact inverses and partial right-hand sides. The new product's error is `poly(p)2^-P`, so that proof applies with a fixed larger guard constant. + +The optional full-coordinate-key surcharge in the old arithmetic ledger would become binding if mechanically retained at b near 3/4. It is not part of the actual rectangle extraction/one-field transpose/ordered-merge algorithm, as confirmed by the separate phase-layout audit. Dropping it is a distinct layout fact, not a consequence of matrix multiplication. It must remain explicitly justified in the final outer ledger. All retained supplier assumptions remain in force. + +No final improved outer exponent is claimed in this note; that requires the whole ledger and the independent group-layout/gauge checks. + +## 5. Exact controls run in this audit + +`check_source_constructions.py` uses only Python's exact rational arithmetic and writes `SOURCE-CONSTRUCTIONS.json`: + +* Finite Fourier/separation support for M=1,...,12: the modular relation has no wraparound and zero weight is exactly g=h,u=v. +* Determinant change of basis for a=1,...,8 and b=2,...,8, for every first-input monomial: quotient and kernel diagonal slices are exactly the two claimed neighboring convolution tensors; the upper-right block vanishes. +* Three-sector identity for a,h=1,...,14: all weights are nonnegative and each of the three retained branches is exactly the claimed tensor after its permitted identification. +* Exact rational diagonal product lower bound through a=300. + +All controls passed. They detect several relevant implementation mistakes but are not a finite substitute for the universal arguments above or for the theorem's nonconstructive rank-existence step. + +## 6. Deliberate limitations + +There is no generated competitive finite matrix scheme, no numeric claim about its constants, no use of a rank formula at exponent exactly 9/4, and no fast algorithm for finding the first successful algebraic tuple. The algorithm remains computable and uniform because the finite search halts under the proved existence statement and its cost is independent of the growing input. The outer conclusion remains conditional on exactly the previously retained supplier/computational assumptions. diff --git a/research/capped-phase-fusion/PUBLICATION-CHECKS.json b/research/capped-phase-fusion/PUBLICATION-CHECKS.json new file mode 100644 index 0000000000..befac93552 --- /dev/null +++ b/research/capped-phase-fusion/PUBLICATION-CHECKS.json @@ -0,0 +1,226 @@ +{ + "status": "PASS_PORTABLE_EXACT_PHASE_FUSION_PACKAGE", + "scope": "Exact finite controls and arithmetic plus retained native receipt binding. This does not replay the entire native dependency or discharge all-size hypotheses. The matrix corollary retains an additional external premise.", + "package_files": 180, + "headline_kappa": "1677160151714381/1000000000000000000", + "checks": [ + { + "script": "frontier_fusion/check_frontier_fusion.py", + "artifact": "frontier_fusion/FRONTIER-FUSION-ARITHMETIC.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": "frontier_fusion/independent_substitution_check.py", + "artifact": "frontier_fusion/INDEPENDENT-SUBSTITUTION-RECEIPT.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": "evidence/inplace_algebra_ultra/check_exact_algebra.py", + "artifact": "evidence/inplace_algebra_ultra/EXACT-ALGEBRA.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": "evidence/inplace_algebra_ultra/check_tensor_compiler.py", + "artifact": "evidence/inplace_algebra_ultra/TENSOR-COMPILER-CONTROLS.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": "evidence/inplace_algebra_ultra/check_supplier_capacity.py", + "artifact": "evidence/inplace_algebra_ultra/SUPPLIER-CAPACITY-CONTROLS.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": "evidence/gaussian_fast_ultra/check_lower.py", + "artifact": "evidence/gaussian_fast_ultra/LOWER-CONTROLS.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": "evidence/gaussian_fast_ultra/check_bilateral.py", + "artifact": "evidence/gaussian_fast_ultra/BILATERAL-CONTROLS.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": "evidence/gaussian_fast_ultra/check_upper_data.py", + "artifact": "evidence/gaussian_fast_ultra/UPPER-DATA-CONTROLS.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": "evidence/gaussian_fast_ultra/check_upper_merge.py", + "artifact": "evidence/gaussian_fast_ultra/UPPER-MERGE-CONTROLS.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": "evidence/gaussian_fast_ultra/check_capped_gauges.py", + "artifact": "evidence/gaussian_fast_ultra/CAPPED-GAUGE-CONTROLS.json", + "status": "PASS_EXACT_REPLAY" + }, + { + "script": 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"36f8c00c200cc375512ad0e881e91b42daba10c962fce37465040cb717b5cc60" + } + }, + "environment": { + "python": "3.14.7", + "platform": "macOS-15.7.7-arm64-arm-64bit-Mach-O", + "dependencies": { + "numpy": "2.5.4", + "scipy": "1.18.1", + "sympy": "1.14.0" + } + }, + "native_replay_mode": "Fresh native providers and retained parent; no reuse/cache flags. Complete source verification was run before publication.", + "capsule_generator_replay": "Regenerated canonical capsule byte-identical to committed capsule using explicit --native-source and --native-work paths.", + "registered_python_312": { + "python": "3.12.15", + "sympy": "1.14.0", + "status": "PASS_REGISTERED_RESEARCH_PACKAGE", + "command": "python3.12 scripts/research_package.py --package research/capped-phase-fusion", + "exact_control_replays": 21 + }, + "native_build_requirements": "C++17 compiler and Boost headers; successful run used /opt/homebrew/include for header lookup." +} diff --git a/research/capped-phase-fusion/PUBLICATION-SOURCES.json b/research/capped-phase-fusion/PUBLICATION-SOURCES.json new file mode 100644 index 0000000000..3f760bee8a --- /dev/null +++ b/research/capped-phase-fusion/PUBLICATION-SOURCES.json @@ -0,0 +1,12 @@ +{ + "research_base": "e6f9716621f4e01fc190617934cdfaf072bd976f", + "publication_base": "a5d4f08ac102d1a6da301d89379c75285eca680e", + "supplier": { + "repository": "https://github.com/CrocSwap/integer-mult-bounds", + "pr": 1661, + "commit": "f852af5e178ce035cd46181a29d9330bcbcaa206", + "directory": "research/gmx-t74-p566-ep400", + "manifest_sha256": "d90d49eaa68e48c6992a07972e736ef4b9cd80274361cbbc1ec980d43f29ad7f" + }, + "scope": "Source snapshots and exact recipe for complete cold replay; historical native receipt hashes are external provenance, not all bundled artifacts." +} diff --git a/research/capped-phase-fusion/README.md b/research/capped-phase-fusion/README.md new file mode 100644 index 0000000000..75daa3cfe9 --- /dev/null +++ b/research/capped-phase-fusion/README.md @@ -0,0 +1,169 @@ +# Capped phase-cell fusion: conditional final kappa = 0.001677160151714381 + +This package gives the conditional final integer-multiplication bound + +\[ +T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad +\kappa=\frac{1677160151714381}{10^{18}}=0.001677160151714381. +\] + +The contribution changes the **outer Gaussian resampling reduction**. It uses +shallow in-place Toeplitz circuits, a common Hadamard schedule for multiple +axes, recentered and capped Gaussian cells, and explicitly charged sparse +boundary operations. The finite supplier is the unchanged, pinned +[PR1661](https://github.com/CrocSwap/integer-mult-bounds/pull/1661) T74 construction. +Its complete native verification was run afresh, including both new provider +words and the retained parent. The final exponent is a conditional theorem +from the written transfer argument; finite checks alone do not prove it. + +Read [the proof](phase_fusion_note.tex), especially its T74 substitution section, +and the [independent substitution review](frontier_fusion/INDEPENDENT-SUBSTITUTION-REVIEW.md). +The proof's earlier numerical specializations remain valid historical cases. +“Independent” in this package means separate agent implementations/reviews, +not an independent human referee or a proof-assistant verification. + +## Exact paid inputs and result + +| Quantity | Value | +|---|---:| +| Retained bit saving c | 0.001733427039589844234749 | +| Chosen complex saving b | 0.00167971337478 | +| Effective tensor saving q | b - 10^-10 | +| Effective bit saving a | (q+b)/2 | +| Native dimension m | 73 | +| Native live width W | 66304 | +| Final kappa | 0.001677160151714381 | + +The complete native profile is +`{1:1559766, 2:337668, 3:278058, 4:84912, 5:54612, 8:5328, 48:22748}`. +All decimals above are exact rationals. Exact epsilon, xi, beta, every outer +cost power and all 45 strict inequalities are in +[FRONTIER-FUSION-ARITHMETIC.json](frontier_fusion/FRONTIER-FUSION-ARITHMETIC.json). +Five deliberate bad variants reject. The native moment has an outward rational +upper bound below one; a larger test endpoint is rejected. + +For the same pinned supplier, PR1661's original outer reduction gives +0.0016768966690212993077507..., so the new outer transfer improves its final +exponent by about 2.6348269308e-7. This is an exponent comparison, not a measured +runtime improvement or a claim of global priority. The selected-result record +is not changed by this contribution. + +The supplementary branch gives 0.001677600129305161 **only with the additional +assumption omega_C <= 9/4**, used through a fixed algebraic algorithm of exponent +19/8. That extra premise is not needed for the headline Strassen-based result. +The supplied 9/4 paper, its audit and historical local locator are recorded in +PROOF-AND-COVERAGE.md and MATRIX-SOURCE-COVERAGE.json; it is treated as an explicit +additional premise, not as a theorem certified by this package. + +## Scope and remaining assumptions + +Retained assumptions include the all-size compiler, exact common-label and +mutable-source realization, whole-call hosting, the semantic precision guard +with C1=1, fixed-tape/prime-packing interfaces, paid arbitrary-width equal-chunk +interchange, exact recovery, and the original analytic transfer. The construction +does not prove an unconditional multiplication theorem, a complete Lean theorem, +or a practical finite cutoff. The written proofs address the new outer steps; +finite algebra and layout checks support, but do not replace, those arguments. + +Original assignment: improve the bound using broad mathematical resources. +For repository indexing this is a changed scope of `uniform-transfer-audit`: +it constructs and audits a replacement outer transfer rather than only checking +one inherited implication. [TASK.json](TASK.json) records the scope and next steps. + +## Reproduce + +Python 3.12, with SymPy 1.14.0, is the registered environment. The exact checks +also ran locally with the Python version recorded in PUBLICATION-CHECKS.json. +The package verifier checks its manifest and source pins, works in a temporary +copy, and compares the regenerated mathematical receipts. It does not modify +this directory. From the repository root: + +```sh +python3 -m pip install sympy==1.14.0 +python3 -B research/capped-phase-fusion/verify.py +python3 scripts/research_package.py --package research/capped-phase-fusion --check-only +``` + +The first command after installation is the bounded outer-proof evidence replay, +not a cold native-supplier replay. The native run already completed with all 12 +top-level stages passing, 319680 owners, prefix 1441 charged at 2048, physical row +coefficient 19545 below 20161, and rejected controls: 114 admission, 12 bit, +15 controller, 8 independent owner, 6 geometry. The retained compact RESULT +and [native binding](frontier_fusion/NATIVE-ADMISSION-BINDING.json) record it. + +To reproduce the full native dependency, obtain the exact source commit +`f852af5e178ce035cd46181a29d9330bcbcaa206` from CrocSwap/integer-mult-bounds, +install its documented dependencies (NumPy, SciPy, SymPy, a C++17 compiler and +Boost headers), and run its unmodified command without cache/reuse flags: + +```sh +python3 -B /absolute/pr1661/research/gmx-t74-p566-ep400/verify.py \ + --work /absolute/new-native-work --audit-shards 4 +python3 -B research/capped-phase-fusion/verify.py \ + --native-source /absolute/pr1661/research/gmx-t74-p566-ep400 \ + --native-work /absolute/new-native-work +``` + +The second command binds a completed external native run to this package's +source and mathematical result. It compares regenerated proof-relevant content; +fresh timings and log bytes need not match the historical run. The full 121 MB +assembly and vendor tree are not duplicated here. The receipt capsule adds canonical hashes for 46 JSON receipts. The 58 hashes +in the historical native binding identify external run artifacts; they are not all bundled files. +The original bind_native_admission.py is retained as historical provenance; +use the new top-level verifier for portable binding. + +PACKAGE-MANIFEST.json is the authoritative publication manifest. MANIFEST.json, +MANIFEST-UPDATE.json and earlier review receipts are explicitly historical +snapshots. The copied upstream CLAIM.json is immutable and contains older pending +language; the later completed native RESULT records this contribution's replay. +Hosted CI and independent human review are separate from the local checks. + +## Sources, overlap and credit + +Prepared by **Vlado Vojdanovski with substantial OpenAI Codex assistance**, +including separate GPT-6.1 Sol Ultra research and audit agents. New material is +Apache-2.0. The original manuscript and source-specific notices retain their +licenses. This is not an official OpenAI release or endorsement. + +The main research snapshot was `e6f9716621f4e01fc190617934cdfaf072bd976f`; +the publication base is recorded in SOURCES.json. Exact T74 source pins are in +PUBLICATION-SOURCES.json and the source manifest. Credit **Rohan Arun** for +PR1319's T74 exact fill and PR1297's four cold kernels; **mdumitrean** for PR1625's +pivot-566 helper-donor kernel; **Jeremy Kothe** for the endpoint and outer-witness +lineage; **Midhat / midhatn** for PR1661's endpoint-grid refinement; **philippeVV** +for the fine-Mercator repair; **Jeremy Diviney, Thomas DiFiore, Hao Yan, eumemic, +icekylinx / Zhihao Chen, DreamingOfClouds**, and the credited predecessor chain +for the retained construction, recurrence, bit supplier and verification. +The [unaltered T74 NOTICE](target002_upstream/pr1661/research/gmx-t74-p566-ep400/NOTICE) +is included. Douglas Colkitt maintains the community repository; the original +OpenAI #109 framework and the Harvey–van der Hoeven reduction remain inputs. + +The in-place algebra builds on Giorgi–Grenet–Roche, +[*Fast In-place Algorithms for Polynomial Operations*](https://arxiv.org/abs/2002.10304), +and Dumas–Grenet, [*Fast in-place accumulation*](https://arxiv.org/abs/2302.13600). +SHALLOW-TOEPLITZ.md distinguishes their results from the new parity-split, +low-displacement and prefix-stability arguments. The Gaussian factorization +also relates to Chafik Boukhalfa's +[PR489](https://github.com/CrocSwap/integer-mult-bounds/pull/489), which already +gives a branch-free local Gaussian inverse but leaves joint evaluation, +routing and boundary composition open. The present package supplies a different +in-place tensor compiler and capped boundary transfer for those outer costs. +[PR1141](https://github.com/CrocSwap/integer-mult-bounds/pull/1141) concerns a +reversible quadratic phase stream and is related background, not a dependency. +The overlap check does not establish novelty beyond these identified sources. + +## Open next questions + +1. Can a new finite supplier exceed the present complex rate by a substantial + amount? Even a lossless outer transfer cannot reach 0.002 from b<0.00168. + First step: emit a complete side circuit for a lower-cost decoder and measure + its actual rank profile, helper storage and restoration. The missing piece is + the physical circuit, not an idealized central-rank estimate. +2. Can the capped Gaussian transfer be independently checked or formalized with + its complete fixed-tape schedule? First step: independently reconstruct the + sparse-union and support-local routing lemmas. Existing finite tests cover + algebraic identities, not an end-to-end all-size tape implementation. +3. Can the same outer transfer consume a future reviewed supplier without + changing its proof? First step: rebind the endpoint, prefix and completed-call + restoration contract, then replay the exact ledger. The supplier must satisfy + the full functional interface; a numerical endpoint alone is insufficient. diff --git a/research/capped-phase-fusion/REVIEW-RECEIPT.json b/research/capped-phase-fusion/REVIEW-RECEIPT.json new file mode 100644 index 0000000000..2521d02ada --- /dev/null +++ b/research/capped-phase-fusion/REVIEW-RECEIPT.json @@ -0,0 +1,53 @@ +{ + "status": "PASS_CONDITIONAL_PHASE_FUSION_REVIEW", + "created_utc": "2026-10-14T02:06:22.293042+00:00", + "kappa": "1672922349588871/1000000000000000000", + "kappa_decimal": "0.001672922349588871", + "upstream_selected_unchanged": true, + "scope": "Local all-size conditional transfer, retaining the selected finite suppliers and common-frame, whole-call, precision, fixed-tape and prime-packing hypotheses.", + "reviews": [ + { + "reviewer": "root agent", + "result": "Read standalone TeX, integrated notes and exact ledger; cap, supplier and layout audits passed; authorized local conditional status." + }, + { + "reviewer": "gaussian_fast agent", + "result": "Independent capped-gauge and finite-block audit passed; exact inherited-weight reanchoring and broad derivatives checked." + }, + { + "reviewer": "global_macro agent", + "result": "Independent capped carving, sparse-union, q-image and microedge layout audit passed." + }, + { + "reviewer": "inplace_algebra agent", + "result": "Global selected-H support and capacity audit passed; all g controls need not fit one original axis." + }, + { + "reviewer": "phase_assembly agent", + "result": "Read named proofs and relevant retained source contracts; reconstructed complete cost ledger and reviewed capped interfaces." + } + ], + "phase_assembly_replays": [ + "check_capped_outer.py: 44 positive exact slacks and 5 rejecting controls", + "check_forward_cell.py: 350 cases, 5184 factorization and 16496 completion entries", + "check_union.py: exact five-factor/tensor/complete-rectangle checks, with dense escape negative control", + "check_q_embedding.py: product order and augmented inverse checks" + ], + "files": { + "work/phase_assembly_ultra/phase_fusion_note.tex": "da2bf119be3ed1e44473c380e43643d16bdc93015c80380fc59da33d84e951de", + "work/phase_assembly_ultra/CAPPED-INTERIORS.md": "983c5cab2d2398f2192a8aec07daecb655f2de136e12a37f75672a14d27172ba", + "work/phase_assembly_ultra/check_capped_outer.py": "b4555c77de7a3ed4abc708a8d93363ef7c9a914be1091f80ab3ddea83f71230a", + "work/phase_assembly_ultra/CAPPED-OUTER-ARITHMETIC.json": "22caac9ec2ecf4d5743b7eb946b633911fbac55a714399556e87e2eae0457dec", + "work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md": "b384b93f3fb4a8c97da2e617ebc14d55cebb0a739070e7f8dbb66f63e24e9310", + "work/inplace_algebra_ultra/TENSOR-COMPILER.md": "4d3b8466598a864a234b978e25976648d4e761840aad9153dd18d73d3dfe4d15", + "work/inplace_algebra_ultra/SUPPLIER-SUPPORT-AUDIT.md": "bd5b06eca043c5c871227264f6a954473c55cd16ec85bb420ff538fb8ec1d8d6", + "work/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md": "e05d6b8596c80b6e8676569a0ff464a29fa704c3304143f58914f47cdf38b272", + "work/gaussian_fast_ultra/BILATERAL-LDU.md": "0d65e66d227464dda1a19f781ae0a0be671cbfd347c41e37ec954f3d629d8a77", + "work/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md": "e37aaa30b15671a1dff00676b14572213efaaac99df202dbba5e09f57af487c0", + "work/gaussian_fast_ultra/CAPPED-GAUGE-AUDIT.md": "1030a34922709ab8d3434a4c57a4a58934ae268c237fb598cb415697da411c9f", + "work/phase_layout_audit_ultra/AUDIT.md": "54aafc7afd09fe1a5312e69fad09c87422690543ab75c6ae35f6cbcc26b3eb8a", + "work/phase_layout_audit_ultra/CAPPED-SIGNOFF.md": "70e9175ebde200f8b016236adc88c4a1c4858945e7c3e5c453a8b6a5f79a263a", + "work/halo_layout_ultra/SUPPORT-ROUTER.md": "d4d2f5bd0a8eb51d9ebd28e4d1a38d3c69bdbf97471a705b3aa199c184364c52", + "work/halo_layout_ultra/HALO-THEOREM.md": "cb9e1da38400b658f9209e465b4e5896fadd17dc66afa862f38bae8d8c1f2058" + } +} diff --git a/research/capped-phase-fusion/ROOT-EXACT-REPLAY.json b/research/capped-phase-fusion/ROOT-EXACT-REPLAY.json new file mode 100644 index 0000000000..4f80ff6ecc --- /dev/null +++ b/research/capped-phase-fusion/ROOT-EXACT-REPLAY.json @@ -0,0 +1,39 @@ +{ + "status": "PASS_ROOT_EXACT_BASELINE_AND_CAPPED_MARGIN_REPLAY", + "selected_commit": "e6f9716621f4e01fc190617934cdfaf072bd976f", + "baseline_complex_moment_upper_minus_one": "-2785787033843904352780990891572049605608726093667665717994302747547920313/2135987035920910082395021706169552114602704522356652769947041607822219725780640550022962086936576", + "q_below_seven_step_supplier": 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Mathematical transfer depends on the accompanying proved circuit/layout/precision lemmas and inherited repository assumptions." +} diff --git a/research/capped-phase-fusion/ROOT-MATRIX-EXACT-REPLAY.json b/research/capped-phase-fusion/ROOT-MATRIX-EXACT-REPLAY.json new file mode 100644 index 0000000000..bd509b8377 --- /dev/null +++ b/research/capped-phase-fusion/ROOT-MATRIX-EXACT-REPLAY.json @@ -0,0 +1,21 @@ +{ + "status": "PASS_ROOT_MATRIX_CONDITIONAL_LEDGER", + "additional_assumption": "supplied complex matrix multiplication exponent <=9/4, used with strict 19/8 algorithm exponent", + "kappa": "836680052702889/500000000000000000", + "gain_over_selected": "34995416443/50000000000000000", + "epsilon": "3204021080669390695403468340153200000000000000/3208046711848432952030355267555861382073600889", + "cap": "6006737803538363603786961767383/8006701850307232414091700000000", + "group": "19999559600000000000000/80067018503072324140917", + "slacks": { + "main": "16040234287175308991291365140577147466485957516731679/1604023355924216476015177633777930691036800444500000000000000000", + "Gaussian": "16040234287175308991291365140577147466485957516731679/1604023355924216476015177633777930691036800444500000000000000000", + "global routing": "84100856093020917/40000000000000000000000", + "grouped routing": "36073302400837752235461073644407147466485957516731679/1604023355924216476015177633777930691036800444500000000000000000", + "scans": "399990069170827316553169929437873914639085240389379902516731679/1604023355924216476015177633777930691036800444500000000000000000", + "matrix boundary": "33307674429019250694161274836955713292695810709060319172243893/534674451974738825338392544592643563678933481500000000000000000", + "union": "16040249599476451935460767629144447487515470930957516731679/1604023355924216476015177633777930691036800444500000000000000000", + "pointwise products": "1599326431642752671058066465736543223602284795889379902516731679/1604023355924216476015177633777930691036800444500000000000000000", + "microfaces": "1470245804132326338277304830206155329249828857963126634172243893/534674451974738825338392544592643563678933481500000000000000000" + }, + "sparse_fraction_slack": "3430918384747396741111767383/8006701850307232414091700000000" +} diff --git a/research/capped-phase-fusion/SHALLOW-TOEPLITZ.md b/research/capped-phase-fusion/SHALLOW-TOEPLITZ.md new file mode 100644 index 0000000000..bc4349b2da --- /dev/null +++ b/research/capped-phase-fusion/SHALLOW-TOEPLITZ.md @@ -0,0 +1,197 @@ +# Shallow, stable, dimension-preserving Toeplitz circuits + +Status: a positive algebraic circuit construction for **dyadic local tiles**. It gives the missing full-Toeplitz primitive in the normalized Sylvester merge. It does not reconcile the anchor coordinates with the original Gaussian gauges; that remains a separate obligation. The construction uses no data ancilla, no zero-initialized coordinate and no fan-out. Fixed coefficient generation is allowed polynomial time in the local tile size and precision. + +## 1. Result + +Let N be a power of two, and let H be an N by N real or complex Toeplitz matrix with + + max(||H-I||_1, ||H-I||_infinity) <= delta < 1/4. + +There is an N-wire linear circuit applying H, with O(log^4 N) rounds of disjoint invertible 2 by 2 gates and diagonal gates, O(N log^4 N) gates, and polynomial-in-N bounds for all coefficients and all partial circuit operator norms and inverse operator norms. Constants depend only on delta. + +For any requested output accuracy 2^-p, rounding the coefficients to p+O(log N) bits gives operator error at most 2^-p; arithmetic at that precision also gives uniform forward error 2^-p on bounded inputs, after allowing the same O(log N) guard bits. A rational approximation to the input Toeplitz coefficients and all required gate coefficients can be generated using poly(N,p) bit operations. This coefficient preprocessing may use polynomial extra memory; the **data circuit** still has exactly N live coordinates. + +The same result applies to upper or lower triangular Toeplitz matrices with max(||T-I||_1,||T-I||_infinity)<=eta for **any fixed eta<1**, with only O(log^2 N) rounds and constants depending on eta. In particular eta=1/2 is permitted for the generator-normalization step below, even though the full-Toeplitz statement uses delta<1/4. Descendants need bounded norm and inverse, not continued distance below 1/4. This independent triangular construction is important: a contiguous bottom-recursion / shear / top-recursion has linear depth even though its work is quasi-linear. + +## 2. Dirty Toeplitz shear is a shallow circuit + +For disjoint m-coordinate vectors x and y, implement + + (x,y) -> (x,y+Tx), + +where T is Toeplitz, in O(log m) rounds when m is dyadic. It is an invertible square map on the existing 2m wires. + +Write the first column and row of T as c_0,...,c_(m-1) and c_0,r_1,...,r_(m-1). Set + + g_0=h_0=c_0/2, + g_j=(c_j+r_(m-j))/2, + h_j=(c_j-r_(m-j))/2, 1 <= j < m. + +Then T=C(g)+C_-(h), where C is a circulant and C_- is a negacyclic circulant. For a unitary m-point FFT F, C(g)=F^* diag(DFT(g)) F. A circulant shear therefore consists of F on x and y in parallel, the disjoint pairwise shears y_j+=DFT(g)_j x_j, and F^* on x and y. Every FFT stage consists of disjoint unitary 2 by 2 butterflies; bit reversal is a disjoint swap stage. + +For gamma=exp(pi i/m), D=diag(1,gamma,...,gamma^(m-1)), + + C_-(h)=D^-1 C(Dh) D. + +Use diagonal twists before and after the same circulant-shear circuit. The two completed shears commute. Source x is restored after each. No circulant inverse is needed, so zero Fourier multipliers cause no problem. Every gate is invertible even when T itself is singular. + +All intermediate norms are polynomial in m and the l1 norm of the Toeplitz symbol. In the 2-norm, FFT and twist stages are unitary, and the spectral diagonal has norm at most the symbol l1 norm. + +## 3. Independent triangular primitive: parity splitting + +Let T_N(f) be lower triangular Toeplitz multiplication by f modulo z^N. Write + + f(z)=e(z^2)+z o(z^2). + +Reorder the existing coordinates into even and odd powers, N=2m. Then + + T_N(f) = [ E ZO ; O E ], + +where E=T_m(e), O=T_m(o), and Z=T_m(z). All these m by m triangular Toeplitz matrices commute. Set + + k=o/e mod z^m, + s=e-z o^2/e mod z^m. + +The exact factorization is + + [ E ZO ; O E ] + = [ I 0 ; T_m(k) I ] + diag(T_m(e),T_m(s)) + [ I T_m(zk) ; 0 I ]. + +Thus apply the right Toeplitz shear, the two child triangular circuits **in parallel**, and the left Toeplitz shear. The recurrence for rounds is + + D_tri(N) <= D_tri(N/2)+O(log N), + +giving D_tri(N)=O(log^2 N). The two recursions act on disjoint coordinates. Inverses are obtained by reversing the gate sequence and inverting every gate, at the same depth. + +Stability is not proved by multiplying coefficient l1 bounds afresh at each depth. Each child is a principal submatrix or principal Schur complement of the original triangular matrix, after a coordinate permutation. If the initial T_N(f) differs from I in both 1- and infinity-norm by at most eta<1, every descendant M satisfies + + ||M-I|| <= eta+eta^2/(1-eta)=eta/(1-eta) + +in both norms, and ||M^-1||_2 <= 1/(1-eta). This follows directly by expressing the descendant as a Schur complement of a principal block of the original matrix; nested elimination and restriction preserve that description. Hence all local diagonal blocks and local Schur multipliers have constant 2-norms. Their triangular Toeplitz coefficient l1 norms are at most sqrt(N) times these constant bounds. The dirty shear circuits from section 2 therefore remain polynomially bounded. + +## 4. Full Toeplitz recursion: displacement rank grows additively + +Use the Sylvester shift displacement + + Delta_m(M)=Z_m M-M Z_m. + +A Toeplitz H has rank Delta_N(H)<=2. At a node of size 2m, partition + + M=[ A B ; C D ], + Z_(2m)=[ Z 0 ; E Z ], E=e_0 e_(m-1)^T, + K=A^-1 B, L=C A^-1, S=D-C A^-1 B. + +The usual block factorization is + + M=[ I 0 ; L I ] diag(A,S) [ I K ; 0 I ]. + +Again A and S run in parallel. To bound ranks, let U=[-C A^-1,I] and V=[-A^-1 B;I]. Then M V=[0;S], U M=[0,S], and direct block multiplication gives + + Delta_m(S)=U Delta_(2m)(M) V-S E A^-1 B. + +Therefore rank Delta(S)<=rank Delta(M)+1. Also, the upper-left block of Delta_(2m)(M) is Delta_m(A)-B E, so rank Delta(A)<=rank Delta(M)+1. At depth d, both children thus have displacement rank at most 2+d. + +The top-right block of Delta(M) equals Delta(B), while its bottom-left equals E A+Delta(C)-D E. Consequently + + rank Delta(B)<=r, + rank Delta(C)<=r+2, + rank Delta(K)<=rank Delta(A)+rank Delta(B)<=2r+1, + rank Delta(L)<=rank Delta(C)+rank Delta(A)<=2r+3. + +All shear matrices have displacement rank O(log N). + +Every recursive M is a principal Schur complement of the original H. Since H=I+R and both ||R||_1 and ||R||_infinity are at most delta, + + max(||M-I||_1,||M-I||_infinity) <= delta/(1-delta), + ||M^-1||_2 <= 1/(1-delta). + +The inverse assertion also follows because M^-1 is a principal submatrix of the inverse of a principal submatrix of H. Thus A,S,K,L are uniformly bounded in 2-norm independently of recursion depth. + +## 5. Bounded-generator shear compiler + +For an m by m matrix T define its Stein displacement + + E_T=T-Z T Z^T=-Delta_m(T) Z^T+T e_0 e_0^T. + +Its rank is at most rank Delta(T)+1. Choose an orthonormal basis g_1,...,g_r for the column space of E_T, and put h_j=E_T^* g_j. With conjugates interpreted appropriately, + + E_T=sum_j g_j h_j^*, + T=sum_j L(g_j) U(conj(h_j)). + +Here L(g) is lower triangular Toeplitz with first column g, and U(h) is upper triangular Toeplitz with first row h. The second identity is the finite telescoping sum T=sum_(k=0)^(m-1) Z^k E_T (Z^T)^k. All g_j have 2-norm 1, and all h_j have 2-norm at most ||E_T||_2=O(1). Their coefficient l1 norms are O(sqrt(m)). + +A zero or tiny first coefficient of h is harmless. Choose an integer gamma >= 2||h||_1 (a fixed conservative gamma=64N suffices under the stated bounds), and set + + P=I+U(h)/gamma. + +P and P^-1 have constant norm, and P is upper triangular Toeplitz. On the same two existing vectors, one generator term is implemented by + + x <- P x, + y <- y+gamma L(g) x, + x <- P^-1 x, + y <- y-gamma L(g) x. + +The result is y<-y+L(g)U(h)x and x is restored. Source transforms use section 3; the other two operations use section 2. Each term costs O(log^2 m) rounds. There are O(log N) terms, hence O(log^3 N) rounds for a full low-displacement-rank shear. + +Although the partial generator sum can be larger than its final T, its norm is at most the sum of O(log N) products of O(sqrt N)-norm factors, hence O(N log N). Within a term the source norm remains constant times its entry norm and the target norm is polynomially bounded. Terms are completed as shears with one common source, so their partial sums add; one must not charge a product of polynomial factors for every generator. + +The full recurrence is therefore + + D_full(N) <= D_full(N/2)+O(log^3 N)=O(log^4 N). + +## 6. Global prefixes, rounding and preprocessing + +At each block-LU node the completed right shear, completed middle block, and completed left shear have constant norm and constant inverse norm. At an intermediate time there is at most one unfinished generator compiler or one unfinished triangular compiler along each recursive path; disjoint paths form a direct sum. If B(N) bounds a full circuit prefix and inverse prefix, this gives B(N)<=max(C B(N/2),N^C), hence B(N)=N^O(1). The same argument applies to suffixes and to the triangular compiler. + +There are N polylog(N) arithmetic gates. Perturbation of one bounded gate by epsilon contributes at most N^C epsilon to the whole operator, by the prefix/suffix bounds. The sum over all gates remains N^O(1) epsilon. Therefore p+O(log N) coefficient/arithmetic bits suffice; there is no N or N^2 precision term on the data. + +The coefficients are fixed before seeing the data. One concrete polynomial-time generation route is: + +1. Approximate the original Toeplitz symbol by dyadic rationals at p+O(log N) bits, preserving Toeplitz structure. The operator perturbation is at most N times the entry perturbation. +2. Compute every recursive Schur matrix as a principal Schur complement of this original rational matrix. Its entries are ratios of minors of that original matrix, so numerator and denominator bitlengths are polynomial in N and p. Avoid an unnecessarily expanded recursive rational expression. +3. For each Stein displacement, use exact rational elimination to select independent columns and rational, unnormalized Gram-Schmidt to obtain orthogonal vectors v_j. Then g_j=v_j/sqrt(v_j^*v_j) and h_j=E_T^*v_j/sqrt(v_j^*v_j). The resulting generators have the bounded norms proved above. Nonzero rational residual norms are at least 2^-poly(N,p), so poly(N,p) offline precision suffices for their normalization. Equivalently a high-accuracy SVD may be used. +4. Compute the finite series divisions and FFT roots to p+O(log N) final bits. All final data-gate coefficients are polynomially bounded. Offline intermediate rational numerators may be longer; they never become data wires. + +This separates polynomial-time coefficient setup from bounded-precision data application. Applying an algebraic-RAM theorem directly to arbitrarily large integers would not provide this separation. + +## 7. Normalized Sylvester merge + +For two b-point children, N=2b, the normalized map supplied by the radical-reduction branch is + + S(B,A)=f(z)B(z)+z^b g(z^-1)A(z), + f(z)=product_lower(1+(x/zeta)z), + g(z^-1)=product_upper(1+(zeta/x)z^-1). + +Let h=g(z^-1)/f(z), K_ij=h_(i-b-j), H_ij=h_(i-j), 0<=i,j0. Every principal submatrix has spectrum in [mu,M]. Every principal Schur complement S also does: S^-1 is a principal block of the inverse of a principal submatrix, whose spectrum is in [1/M,1/mu]. Thus every recursive A,S has norm at most M and inverse norm at most 1/mu. The shears K,L have norm at most kappa=M/mu. The displacement-rank formulas are unchanged. + +Use orthonormal g generators and h of norm O(kappa), and choose gamma>=2||h||_1. The triangular source transform P is still near I, so the depth O(log^4 N) is unchanged. Generator-stage amplitudes and coefficients are poly(N,kappa). Across the O(log N) completed ancestor factors, the bound becomes + + log(max prefix/inverse-prefix norm) + = O(log N * log(2+M+1/mu) + log N). + +If M and 1/mu are exp(O(u)), this is O((u+1)log N). The corresponding data precision is p+O((u+1)log N), and g-axis tensorization adds O(g(u+1)log N) guard bits. This is a usable broad-Gaussian extension once a separate spectral argument supplies the stated eigenvalue interval. Reversing the circuit applies H^-1; the inverse itself need not be Toeplitz. diff --git a/research/capped-phase-fusion/SOURCES.json b/research/capped-phase-fusion/SOURCES.json new file mode 100644 index 0000000000..a5d007aae1 --- /dev/null +++ b/research/capped-phase-fusion/SOURCES.json @@ -0,0 +1,18 @@ +{ + "schema_version": 1, + "base_commit": "a5d4f08ac102d1a6da301d89379c75285eca680e", + "files": [ + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11", + "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32" + }, + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1", + "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075" + } + ] +} diff --git a/research/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md b/research/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md new file mode 100644 index 0000000000..72404c42d7 --- /dev/null +++ b/research/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md @@ -0,0 +1,139 @@ +# Independent audit: support-local simultaneous-H supplier + +Conclusion: the retained compact phase supplier can be used at width g with cost + + O(V g^(1-q) polylog(p)) + +without the previously charged additive V p^(1-a), **provided the call is made globally on the complete dyadic tensor, not separately on one fixed coarse-label microbox**. This is conditional on the same accepted finite complex circuit and arbitrary-width equal-chunk interchange as the retained supplier, and on the support-router transfer proved in `work/halo_layout_ultra/SUPPORT-ROUTER.md`. The new support-router's unequal-spacing algebra and its use here have been checked below. + +The full address still has O(p) bits. That quantity remains in descriptor arithmetic, keys and error precision; it no longer determines the width of any newly required fast coordinate interchange. + +## 1. What the actual phase frames touch + +The primary source is `work/integer-mult-current/upstream/build/sections/05-layers.tex`. + +At lines 86–139, a residual basis is a matrix M in GL(m,F_2), extended **inside F_2^m**. At width e=mf it is applied separately to the column + + (x_(1,j),...,x_(m,j)), 0<=j T as prescribed, + T\S -> S\T bijectively, + identity outside S union T + +is a coordinate permutation supported on at most 2gK positions. Its inverse restores the original coarse-coordinate interpretation. The target can be chosen before the payload and a reserved untouched suffix. No stable reordering of every coarse field is required. + +Apply the H supplier to this full global stream, whose complete coarse-coordinate ranges remain present. A fixed coarse-label microbox alone does **not** contain enough complete K-bit fields: its high coarse bits are fixed. Borrowing those fixed labels inside such a microbox would violate the supplier's complete-field hypothesis. This audit approves the global call only. + +The tensor compiler's H schedule is independent of all coarse labels. Those labels affect only diagonal coefficients between complete H calls. The diagonal scan can still be organized by coarse-label microboxes after undoing the global exposure permutation. Coarse-dependent coefficients therefore do not require separate supplier invocations. + +## 3. Deferred reservoirs and the row prefix + +At a phase with h unprocessed selected bits, reserve whole K-bit chunks for two front hG fields, one back hG field, and a leading row prefix. With K=128G, the field reservation is at most 3h/128+O(1) chunks. Row preparation needs O(log h) further chunks, including the product stock for nested bit and complex calls. For h above a fixed polylog(p) threshold the total R(h) is at most h/16. The remaining bounded/polylogarithmic h can be processed individually. + +Every such chunk is one of the current selected chunks. The phase applies C only to the other h-R(h) selected bits and identity to the reserved chunks; the latter are deferred to the next phase. + +This requires one explicit modification to the original exposition: **do not first apply C to the reserved row/front/back selected bits**. That preliminary work in the old all-axis construction only ensured those selected bits were eventually covered. It is not needed to create rows or complete temporary fields. Source `compact-control-layout.tex` lines 30–62 proves that the active invocation is the required C tensor separately on every row and identity on all reserved/spectator coordinates, for arbitrary incoming payloads. Hence dirty, unprocessed reserved selected bits are valid. + +The leading q0 complete chunks have cardinality 2^(q0 K). Choose q0 from the actual product row stock Q_rows=W_complex^D_complex W_bit^D_bit. This preserves the accepted nesting contract; replacing the product by a maximum would be incorrect. Pad the leading row count to a multiple of Q_rows once for this phase, by adding whole rows. The factor is below two. All descendants inherit the same complete within-row fields and the same row stock; no new independent bit coordinates are allocated. + +A completed active invocation is identity on the row label and every temporary coordinate. Each padded zero row is therefore zero at completion and can be deleted. Delete padding and restore the exposure layout **before the next deferred phase**. Consequently row padding does not multiply by 2 at every phase or tensor axis. + +The next deferred phase gathers only its h'=R(h) remaining neighborhoods. A gather/restore can again be extended with support at most 2h'K; it may move previously processed coordinates temporarily but fixes them after the inverse placement. Their values need not be recomputed. There is no need to move all p coordinates back into a stable global axis order. + +## 4. The logarithmic K choice is legal after tightening the statement + +The original compact-control proposition states the sufficient asymptotic condition K/log p -> infinity. K=128G does not satisfy that literal condition. The source proof, however, uses it only at lines 149–154 to obtain + + K >= G+4 ceil(log2 p)+10, + delta_bad <= 5/(128p^3). + +K=128G satisfies the displayed finite inequality directly. The earlier/later-source guard proof, complete-field layout, row stock, scalar gates and semantic guard have no other use of K/log p -> infinity. Thus the transfer must cite the tightened finite guard hypothesis; it must not claim that the new K satisfies the old stronger asymptotic wording. + +With this change, all compact interchanges retain width O(h log p), and the per-node movement overhead is O(V_node h^(1-a) polylog(p)). Here h is the current selected width, not the ambient number d of original Gaussian axes. The supplier's stopping threshold and branching analysis likewise use its own top width g; global p remains only in logarithmic guards and O(p)-bit descriptors. + +## 5. Check of the support-router extension + +The original arbitrary-source router colors active targets by their current location modulo K. Targets in one color have disjoint K-bit guards. The new router stores only the r active control digits, H=rG, even when the target spacings are unequal multiples of K. + +The four-update proof depends solely on disjointness of the guard intervals: on a good address the first y update adds 2zt within a guard, the t update adds its old low bit e without carry, and the second y update adds z(1-2(t+e)). Their net is z(1-2e), changing exactly the selected low bit. The final t update restores t. Nothing uses equality between successive target gaps. A complete F therefore fixes every nontarget source bit, even inside another guard. F; load u from the sources; F; unload implements the desired simultaneous XOR for sources outside the target set. + +All rotations remain globally bijective, because their controls precede the target after the prescribed back-field shuttle. Their current-control inverse is used on bad addresses. The ideal selected-bit XOR preserves the guard/control bad predicate, so the old sparse repair applies unchanged. Its O(p)-bit keys and polynomial per-record arithmetic remain explicit; they are not replaced by a unit-cost sort. + +For a permutation moving k positions, factor it into two involutions and three XOR stages per matching. Three disjoint 3kG-bit reservoirs can be chosen when n>=9kG; the three matching cohorts avoid the respective reservoirs. Each target residue class then uses width at most kG in each control field, rather than ceil(n/K)G. Original equal-width interchanges place and restore the fields. This gives O(V k^(1-a) polylog p), plus descriptor setup. + +When records are short, choose only ell_s=ceil(log2 p)^2 untouched binary bits, not a whole long original axis. Here is explicit noncircular suffix bookkeeping. For a requested permutation pi with moved support at most k, if n>2(k+1)ell_s, a gap in its support contains a consecutive ell_s-bit field S disjoint from the actual last ell_s bits. Use one original equal-ell_s interchange A to move S to the actual suffix. Route the conjugate A pi A^-1 on the n-ell_s prefix only, then undo A. The conjugate fixes that suffix pointwise because pi fixed S. Choose all control reservoirs inside the prefix, after this interchange, and recompute positions and residue classes after their placements. Thus no control operation touches the grouped suffix. Grouping its 2^ell_s complete values into a record preserves the data volume and gives a superpolynomial record length. If n<=2(k+1)ell_s, the retained full-span bound is already O(V(k+1)^(1-a) polylog p), so no suffix grouping is needed for this support bound. The placement costs O(V polylog p), not V ell^(1-a) for the original long axis length ell. + +For the calls here, k=O(hK)=O(h log p), and n/(k log p) tends to infinity because g=p^beta with beta<1 while n is of order p in the retained full tensor. If this inequality fails in a small regime, n=O(k polylog p), so the retained full-span bound is already O(V k^(1-a) polylog p). The support theorem's own temporary reservoirs may have O(kG) bits; this is a polylogarithmic enlargement of support, already priced in the same bound. + +## 6. Complete charge table + +| Operation | Active coordinate/data support | Charge | +| --- | --- | --- | +| Initial and inverse neighborhood exposure | At most 2gK moved bit positions | O(V g^(1-a) polylog p) | +| Deferred phase exposure at width h | At most 2hK positions | O(V h^(1-a) polylog p) | +| Fixed binary residual adapter | h selected bits; compact controls O(hG) | O(V_node h^(1-a) polylog p) | +| Control-field shuttle | At most O(hG)-bit chunks | Original O(V_node(hG)^(1-a)) | +| Modular rotations and exceptional repair | Full scan; O(p)-bit descriptors/keys | O(V_node) after record grouping | +| Residual child concatenation | Existing contiguous first-r slots | Descriptor change only | +| Source/sink/inverse phases | Selected-bit masks and coefficients | O(V_node) | +| Scalar role gates, parking and restoration | Fixed role streams at aligned addresses | O(V_node) | +| Whole-row padding and deletion | One current-phase row prefix | O(V), factor below two, removed before next phase | + +No row in this table requires an interchange of width p. + +Let tau=1-a. The retained stopped branching argument gives O(V h^(1-q) polylog p) for each phase, with q 2(k+1)L. +``` + +Indeed, after excluding the actual suffix, the at most k moved positions +cut the remaining interval into at most k+1 gaps. If every gap had length +less than L, that prefix would have length at most (k+1)L-1. The displayed +inequality contradicts this bound. + +Let A be the old disjoint equal-L swap exchanging S with the actual suffix. +Execute A, route the conjugated permutation A pi A^{-1}, then execute +A^{-1}. Because pi fixes S pointwise, its conjugate fixes the actual suffix +pointwise. It still moves at most k coordinates. All endpoints are now in +the n-L prefix. Choose the three dirty reservoirs in that prefix and run +all compact rotations, original control shuttles, elementary CNOTs, and +bad-hole repair there. Recompute endpoint names after A and after the +reservoir placement. In the application, n-L>=9kG holds with a wide +asymptotic margin. + +The suffix is a complete binary field with arbitrary contents. Reinterpreting +its 2^L values as part of a record gives record length R0*2^L, without +creating, padding, or deleting data. This length exceeds every fixed +polynomial in p. No operation whose cost relies on that record length acts +on its component coordinates. Both instances of A use the already approved +primitive and cost O(V L^(1-a))=O(V polylog p); they do not invoke the new +router. This removes the prospective circularity. + +For the general router outside this sparse branch, the retained +first-L/second-L/actual-last-L candidate convention and matching cohorts +also work. Arbitrary candidates should not be substituted blindly: a +candidate can partially overlap the actual suffix, whereas the old swap +requires equal disjoint fields. After each candidate placement, the same +n-L prefix exclusion is mandatory. If the entire binary span is only +polylogarithmic, the old elementary construction suffices. If +n=O((k+1)polylog p), the old full-span construction gives the desired +support bound up to polylogarithms; its own short-record preparation must +still use the old suffix construction. + +For a full H supplier call, reserve the suffix outside its selected field +and keep it fixed throughout the completed call. If necessary, perform +the same old L-bit placement before exposure, update the logical names of +the neighborhoods, and restore it after inverse exposure. The source +neighborhoods need not remain physically consecutive before exposure: +the exposure permutation explicitly routes their named bits. The target +complete chunks are placed before the reserved suffix. + +## 3. Whole-row padding and the next diagonal scan + +The leading row prefix and the front/back controls are assembled from +complete K-bit neighborhoods in the globally complete binary tensor. +Their bits are not assumed zero or semantically independent of coefficient +values. Completeness means every coordinate value is represented in the +stream. Multiplication by previous coarse-dependent diagonal coefficients +does not remove any address or change this property. + +Deferring the selected bits lying in reserved chunks is valid: the accepted +active invocation is the desired tensor on the other selected bits and +identity on every reserved coordinate, for arbitrary incoming coefficients. +The preliminary individual transforms in the old layout were used to +finish those selected bits, not to create complete row or control ranges. +After deferral, those transforms are performed in later phases instead. + +The row count is rounded once to a multiple of the actual product + +``` +Q_rows = W_complex^D_complex * W_bit^D_bit. +``` + +The padded row label is allowed to be a nonbinary preceding fiber count. +Every within-row field remains a full binary field. The supplier's +interchanges and controls act within those complete fields; the global +exposure router is not applied to the padded row label. Nested bit adapters +consume and restore the bit stock before later complex splits, exactly as +in the retained product-stock argument. + +At the completed invocation boundary, the operator is identity on the row +label. A padded zero row therefore returns to zero even though temporary +role computations may have combined or moved its contents. Delete those +rows before inverse exposure or the next deferred phase. This bounds each +phase's volume by twice its original volume, not by a factor exponential in +the number of phases. + +All H calls in the compiler have the same selected-bit schedule for every +coarse label; label dependence resides in the intervening diagonal tables. +Run one H call on the complete global tensor, complete every deferred +phase, remove padding, and restore the original exposed-coordinate order. +Only then run the next diagonal scan. The microbox layout used for that +scan is the restored layout in which the local coordinates are already +contiguous (or the separately paid low-bit gather layout). No padded row, +partial H result, or temporarily relocated coarse label is presented to +the diagonal-table lookup. Thus scanning different coarse-label tables +does not require separate H invocations or borrowing fixed labels inside +a microbox. + +## 4. The actual parameter inequalities + +The largest exposure moves k=O(gK)=O(p^.093 log p) binary coordinates. +With G=Theta(log p), its three dirty reservoirs require + +``` +9kG = O(p^.093 log^2 p) = o(p). +``` + +The suffix occupies only L=Theta(log^2 p) positions. In particular, + +``` +n/(kG) = Omega(p^.907/log^2 p) -> infinity, +n/((k+1)L) = Omega(p^.907/log^3 p) -> infinity. +``` + +There is ample room both for the old spectator placement and for all +subsequent prefix-only reservoirs. These inequalities concern the full +binary address, not any one original axis. + +The only per-axis inequality is + +``` +K + O(log p) < ell_axis = Theta(p^.002), +``` + +which holds eventually. The original K/log p -> infinity wording must be +replaced, as already noted in `SUPPLIER-SUPPORT-AUDIT.md`, by the finite +guard inequality K>=G+4ceil(log2 p)+10. K=128G satisfies it. + +The supplier's front/back controls are made by concatenating reserved +complete neighborhoods. Their chunk count is + +``` +ceil(2hG/K) + ceil(hG/K) <= 3h/128 + 2. +``` + +The leading row stock adds O(log h) complete chunks. Above a fixed +polylogarithmic cutoff this is at most h/16, and smaller h is handled +individually. No single original axis stores hG control bits. Indeed +gG/ell_axis=Theta(p^.091 log p) diverges; the construction works precisely +because it does not demand this false per-axis fit. + +Consequently the reviewed mechanisms support the proposed +O(V g^(1-q) polylog p) global-call cost under the retained supplier +assumptions. The suffix exclusion and completed-call boundary should be +kept explicit in any final assembly. They are necessary bookkeeping, +not a new exponent restriction or an unproved sorting primitive. diff --git a/research/capped-phase-fusion/SUPPORT-ROUTER.md b/research/capped-phase-fusion/SUPPORT-ROUTER.md new file mode 100644 index 0000000000..4f39fa9eaf --- /dev/null +++ b/research/capped-phase-fusion/SUPPORT-ROUTER.md @@ -0,0 +1,157 @@ +# Coordinate routing measured by moved support + +This is a conditional extension of the retained compact arbitrary-source +router. It uses the same equal-width chunk interchange with cost +`O(V H^(1-a))`, original fixed-tape modular rotations, counters and sparse +radix repair. It changes the packed-control length from the full address span +to the number of active target positions. No computed-key sort of the full +input is assumed. + +Let a supplied permutation of `n=O(p)` binary address coordinates move at +most `k` positions. The stream is a complete binary rectangle, with arbitrary +preceding fibers and following payload. In the regime used below, +`n/(k log p) -> infinity`; short remaining cases can use the retained +full-span router, since `n=O(k polylog p)` there. The conditional cost is + +``` +O(V (k+1)^(1-a) polylog p + poly(p)). +``` + +All other coordinate positions return to their original values. This is a +support bound, not a bound on how far a moved coordinate travels. + +## Unequally spaced target lemma + +Put `B=2^G`, `G=4 ceil(log2 p)+6`, and `K=64G`. Take target positions +`j_0<...2(k+1)ell`, a gap outside the moved support contains a contiguous ell-bit +field S disjoint from the actual last ell bits. Use an original equal-ell +interchange A to move S to that actual suffix. Route the conjugate +`A pi A^-1` on the n-ell prefix only, then undo A. The conjugate fixes the +suffix pointwise because pi fixed S. Choose every dirty control reservoir +inside the prefix after this interchange, then recompute positions and +residue colors. Thus no control operation touches the enlarged record. +Its 2^ell complete values absorb the polynomial descriptor work; no data +is padded or duplicated. If `n<=2(k+1)ell`, the retained full-span bound +already gives `O(V (k+1)^(1-a) polylog p)`. The placement itself costs only +O(V polylog p). This explicit suffix-conjugation bookkeeping was +independently supplied in `work/inplace_algebra_ultra/SUPPLIER-SUPPORT-AUDIT.md`. + +## The gather uses only S union T + +Let S list the k selected low-coordinate bits in the desired tensor order, +and T list k consecutive destination slots. Send S bijectively to T. Send +the remaining domain slots `T\S` bijectively to the remaining output slots +`S\T`, and fix everything outside `S union T`. This explicitly extends the +requested gather to a coordinate permutation supported on at most 2k bits. + +The other coarse-coordinate bits can be scrambled by this extension. Their +order is retained and described by the inverse permutation; **no stable +compaction of all coarse fields is performed**. A coarse neighbor address +can be decoded, incremented in its logical axis, then encoded in this +physical coarse order when constructing a sparse correction key. + +## Evidence and boundary + +`check_sparse_shear.py` checks all 32,768 addresses for targets 0 and 6 with +guard length 5, including 512 good addresses, inverse composition and every +bad repair. It also checks 4,000 arbitrary/constructed-good addresses with +unequal gaps 9 and 11 and sources inside guard fields. `check_halo.py` +checks three explicit support-at-most-2k gather permutations. + +These controls support the algebra; the cost argument above is the +all-size conditional transfer. It retains the original bit-supplier, +equal-width interchange, fixed-tape and common-label hypotheses. It does +not upgrade an arbitrary full computed-key sort to a fast router. + +Sources: the retained reports `compact-arbitrary-source-routing.md` and +`review-arbitrary-routing.md` under +`work/integer-mult-current/references/semantic-bulk/rad20/reports/`. diff --git a/research/capped-phase-fusion/TASK.json b/research/capped-phase-fusion/TASK.json new file mode 100644 index 0000000000..62b7e778be --- /dev/null +++ b/research/capped-phase-fusion/TASK.json @@ -0,0 +1,72 @@ +{ + "schema_version": 1, + "task_id": "uniform-transfer-audit", + "priority_revision": 3, + "base_commit": "a5d4f08ac102d1a6da301d89379c75285eca680e", + "original_card": { + "id": "uniform-transfer-audit", + "question": "Can one exact all-size/compiler, precision, hosting or restoration implication be proved or falsified?", + "starting_points": [ + "certificates/selected-result.json", + "docs/audit.md" + ], + "methods": [ + "Prove one missing implication with explicit quantified hypotheses.", + "Find an exact counterexample at the claimed interface boundary." + ], + "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.", + "first_step": { + "command": [ + "python3", + "-B", + "scripts/research_baseline.py" + ], + "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY", + "inspect": "Read the selected review scope and choose one exact all-size or precision implication.", + "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay." + }, + "first_experiment": "Write its quantified inputs and conclusion; find a missing hypothesis or construct a minimal exact adversarial instance.", + "bottleneck_link": "Finite supplier improvements are useful only under the retained transfer obligations.", + "allowed_variations": [ + "Choose one tape, precision, hosting or restoration lemma." + ], + "scope_change_triggers": [ + "Replacing a quantified obligation with another finite smoke test." + ], + "overlap_terms": [ + "uniform transfer", + "all-size", + "precision" + ] + }, + "scope_status": "changed", + "actual_scope": "Construct and audit capped phase-cell fusion and substitute the pinned fully replayed T74 supplier to obtain final conditional kappa=0.001677160151714381.", + "scope_change_reason": "The user explicitly requested stronger bounds using broad resources; the resulting work replaces several outer transfer steps rather than auditing a single inherited implication.", + "bottleneck_link": "The outer reduction repeatedly scans Gaussian axes; in-place tensor compilation and capped sparse boundaries reduce that cost. The unchanged complex supplier still caps the possible final saving.", + "budget_minutes": 180, + "stop_rule": "Publish the verified conditional transfer with exact source bindings and bounded replay; retain larger-supplier research as open work.", + "outcome": "completed", + "evidence": "phase_fusion_note.tex; frontier_fusion/FRONTIER-FUSION-ARITHMETIC.json; frontier_fusion/NATIVE-ADMISSION-BINDING.json; PUBLICATION-CHECKS.json", + "author": "Vlado Vojdanovski", + "ai_assistance": "Substantial OpenAI Codex assistance; separate GPT-6.1 Sol Ultra derivation and audit agents. No independent human or proof-assistant review claimed.", + "open_questions": [ + { + "question": "Can a new finite supplier materially exceed the present complex rate?", + "why_it_matters": "The current complex endpoint is below .00168 and cannot yield final .002.", + "first_step": "Emit and price a full native side circuit for a lower-cost central decoder.", + "blocked_by": "Actual routing, helper and restoration costs." + }, + { + "question": "Can the full capped transfer be independently formalized?", + "why_it_matters": "Finite algebra checks do not establish all-size tape complexity.", + "first_step": "Reconstruct the sparse-union and support-local routing proofs.", + "blocked_by": "Formal all-size schedule and inherited compiler interfaces." + }, + { + "question": "Can future suppliers be substituted without changing the outer proof?", + "why_it_matters": "The outer gain can compose with later finite improvements.", + "first_step": "Bind endpoint, prefix and completed-call restoration contracts and rerun the ledger.", + "blocked_by": "A complete supplier satisfying the same functional contract." + } + ] +} diff --git a/research/capped-phase-fusion/TENSOR-COMPILER.md b/research/capped-phase-fusion/TENSOR-COMPILER.md new file mode 100644 index 0000000000..19efb9cb1c --- /dev/null +++ b/research/capped-phase-fusion/TENSOR-COMPILER.md @@ -0,0 +1,160 @@ +# Tensorizing local circuits without dimension padding + +Status: an algebraic compiler and scan-cost lemma. It reduces a tensor product of different bounded-depth local circuits to a bounded sequence of simultaneous binary Hadamard calls and diagonal scans on the **same** data volume. The retained supplier must support the stated Hadamard call on selected local bits and restore the agreed endpoint record order. No new fast-transform supplier is asserted here. + +## 1. Interface and conclusion + +Let N=2^m and let g axes each have N local coordinates. A coarse-label block contains V=N^g data records in lexicographic local-coordinate order. On axis a let C_a be an N-wire linear circuit of depth D; every layer consists of disjoint invertible two-coordinate gates and one-coordinate diagonals. The permitted gates are bounded shears, swaps, FFT butterflies, and diagonals. Circuits, pair matchings, coefficients and depths can differ by axis and by coarse label. Their tables are fixed independently of data values. + +Write H=[[1,1],[1,-1]]. A simultaneous Hadamard call means the tensor product of one H acting on one selected bit in each axis, with all remaining local and coarse coordinates as spectators. It acts on existing 2^g-coordinate fibers inside the N^g records; it allocates no new coordinate. The bit selected in an axis may be prescribed by the compiler. + +After padding shorter circuits by identity **operations**, the tensor product C_1 tensor ... tensor C_g is a sequence of + + at most (8m+2)D simultaneous H calls + +and O(Dm) tensor-diagonal scans. A tensor diagonal has the form + + d(i_1,...,i_g;c)=product_(a=1)^g d_a(i_a;c), + +where c is the fixed coarse label. Each diagonal scan uses O(1) amortized coefficient multiplications and O(1) amortized coefficient-table record movements per data record, after loading gN local values. The number of arithmetic-data coordinates remains exactly N^g throughout. + +If all local gate coefficients and inverses are bounded by B and all original prefix and inverse-prefix norms by Lambda, the compiler's local prefix bounds are Lambda times poly(B), up to absolute constants. For polynomial B and Lambda this is N^O(1). The tensor-prefix logarithmic bound is O(g log N). More generally depth D=polylog(p), B=poly(p), N=poly(p) gives an O(g polylog(p)) guard even without a stronger prefix theorem. + +An outer similarity whose logarithmic range is Q_a on axis a adds only sum_a Q_a. Therefore the retained phase-cell case Q_a=O(a d) uses + + p + O(g a d + g polylog(p)) + +working bits. The condition g a d+g polylog(p)=O(p) is sufficient for O(p)-bit records. + +## 2. Fixed Hadamard words, including zero shears + +All formulas below are matrix-product order; execution reads right to left. Let U(t)=[[1,t],[0,1]]. Direct multiplication gives + + H diag(2,1) H diag(-3,1) H = [[-8,-10],[0,-6]], + diag(-1/8,-1/6) H diag(2,1) H diag(-3,1) H = U(5/4). + +For q nonzero, conjugate by A_q=diag(4q/5,1): + + U(q)=A_q diag(-1/8,-1/6) H diag(2,1) H diag(-3,1) H A_q^-1. + +This has three H factors. To include t=0 with uniformly invertible diagonals, choose a known B>=|t| and c=2B+1. Then + + U(t)=U(t+c) U(-c). + +Both parameters have magnitude at least B+1 and at most 3B+1. This is a **six-H word for every t**, with all scalar coefficients and inverses bounded by poly(B). There is no division by t and no case split at t=0. Transpose the word to obtain the same six-H count for lower shears. + +There are two bounded identity words: + + I = (1/2) H H, + I = diag((1-i)/4,(1+i)/4) H diag(1,i) H diag(1,i) H. + +The second identity uses three H factors. Its middle product before the leading diagonal equals diag(2+2i,2-2i). Complex coefficients are permitted by the retained complex H interface. + +Consequently every gate type can be given **exactly six H factors**, separated by diagonal matrices: + +- Upper/lower shear: the six-H word above. +- Diagonal, including I: append three two-H identity words. +- Swap J: use J=(1/2)H diag(1,-1)H, then append two two-H identity words. +- FFT butterfly H diag(1,w), optionally scaled by 1/sqrt(2): append a two-H identity and a three-H identity. Here |w|=1. + +Combining neighboring diagonal matrices puts every word in one common alternating shape with six H factors and seven diagonals. Thus different gate types, zero parameters and inactive pairs can coexist in one simultaneous schedule without branches or padded data coordinates. + +## 3. Every matching can be routed to adjacent pairs + +For one local layer, assign its oriented pairs (i,j) to positions (2k,2k+1). Unpaired coordinates are assigned arbitrarily to the remaining positions, in pairs carrying diagonal or identity gates. This defines a permutation pi on exactly N coordinates. The layer equals + + P_pi^-1 (direct sum of adjacent-pair gates) P_pi. + +We use a Beneš network of 2m-1 switch layers. At each layer all switches join indices differing in one fixed bit; the bit sequence is + + 0,1,...,m-1,m-2,...,1,0. + +Here is a constructive proof of rearrangeability. Form a bipartite multigraph whose left vertices are input pairs {2j,2j+1} and whose right vertices are output pairs {2k,2k+1}. Input i supplies an edge to output-pair floor(pi(i)/2). Every vertex has degree two, so each component is an even cycle (parallel edges are allowed). Alternately color each cycle's edges 0 and 1. The first switch sends an input edge into the subnetwork of its color; the last switch sends that color to the required output parity. Each color class induces a permutation on N/2 pair indices. Recursively route both color subnetworks in parallel. This gives exactly the displayed bit sequence and uses no storage coordinates. + +The switch is either I or J. Both are obtained from the common two-H word + + S_epsilon=(1/2) H diag(1,epsilon) H, + epsilon=+1 for I, epsilon=-1 for J. + +Thus all axis-dependent switch controls are diagonal entries. A physical switch on bit k simply uses a simultaneous H call selecting bit k in each axis. No interstage wire shuffle is needed. If a supplier accepts only a contiguous selected-bit field, the existing paid bit router must gather that field and undo the gather; that is a supplier-interface cost, not an uncharged permutation. + +Each forward or inverse Beneš route uses 2(2m-1) H calls. The adjacent gates use six. Hence one original matching layer uses + + 2[2(2m-1)]+6 = 8m+2 + +H calls. The schedule is identical on every axis and every coarse-label block; only diagonal tables change. + +## 4. Tensor fusion is literal multiplication, not a sum expansion + +Let the compiled axis-a circuit be D_(a,L) H_(a,L) ... H_(a,1) D_(a,0), with the same number and arrangement of H factors. Kronecker multiplication gives + + tensor_a C_a + = (tensor_a D_(a,L)) (tensor_a H_(a,L)) ... + (tensor_a H_(a,1)) (tensor_a D_(a,0)). + +Every tensor of diagonals is a single diagonal scan. Every tensor of binary H factors is one simultaneous H call. This introduces neither a sum of 2^g branches nor a 2^g padding factor. The 2^g-dimensional Hadamard fibers are subsets of the N^g existing coordinates. + +An axis with no work at a particular original depth receives an identity layer, routed and compiled with the same fixed shape. Its intermediate H operations are still present, but the complete local word is I. Coarse-label variation is equally harmless: for each fixed label the identity above holds separately, and the whole data operator is a direct sum over labels. + +## 5. Uniform intermediate bounds and precision + +A completed switch route is a permutation and has norm one. Each partial switch word has absolute constant norm and inverse norm. Between original local circuit boundaries, at most one original gate word is partially executed. Its length is constant, and its scalar coefficients and inverses are poly(B); all different pairs act as a direct sum. Therefore + + max_prefix ||C_compiled,prefix||, + max_prefix ||C_compiled,prefix^-1|| <= poly(B) Lambda. + +The length O(D log N) does not require multiplying this polynomial bound afresh at every stage: completed original prefixes already have bound Lambda; completed route and padding words have their exact intended endpoints. The corresponding suffix bound is also polynomial when the original endpoint and inverse-prefix bounds are. + +For g tensor factors, the operator norms multiply, so their logarithms add. A microbox coarse-label direct sum takes the maximum over labels, not a product over labels. Hence polynomial local bounds give exp(O(g log N)); using only the rough original bound Lambda<=(2B)^D gives exp(O(gD log B)). Adding the original cell similarity contributes exp(sum Q_a). + +The number of compiled stages is polylog(p). Allocate each rounding or supplied-transform error below the final tolerance divided by this stage count and the prefix/suffix amplification. This costs O(g polylog(p)+sum Q_a) guard bits. Supplier-internal precision charges still must be those already guaranteed for the simultaneous H call; this compiler does not erase them. + +## 6. Product diagonals with constant amortized scan cost + +Fix one diagonal stage and one coarse label. Load the gN coefficients d_a(i) on a coefficient tape in axis order. Use base-N lexicographic enumeration with axis 0 fastest. Store suffix products + + P_j=product_(a=j)^(g-1) d_a(i_a), 1<=j=1) k/N^k = O(V), + +plus O(g) initialization. No division by d_a(i) is used, so this method also permits zero diagonal entries when an external application permits them. Every cached product is recomputed from current factors, never updated by a long chain of multiplicative ratios, so numerical drift does not depend on the number of records scanned. + +The same accounting works on a fixed number of tapes. Keep the coefficient-tape head normally in axis-0's N-record table. Rewinding that table costs O(N) once per N records. A depth-k carry visits the needed tables up to axis k and returns, moving O(kN) coefficient records. Summing over carries gives + + O(V sum_(k>=1) kN/N^k)=O(V) + +record moves. The suffix-product tape is accessed near its front except on carries, with the same geometric bound. Counter/address arithmetic costs only polylog(p) per record. This does not require g work tapes or a random-access lookup every record. + +The table loading cost is O(gN) records per coarse block and stage. For N>=2 it is O(N^g), including small g, and polynomial-time table construction is negligible per N^g block in the retained asymptotic regime g=p^beta tending to infinity. If tables are supplied rather than constructed locally, no additional assumption is needed. Both generation and loading costs must be charged once per visited coarse block, not silently once for the entire input. + +At O(p) working precision, O(1) amortized scalar multiplications per record and O(1) record movements cost O(V_bits polylog(p)) under the retained pointwise-arithmetic kernel, where V_bits counts bit volume. There is no g V_bits coefficient-generation term. + +## 7. Exact controls + +`check_tensor_compiler.py` constructs the network controls and all fixed words. `TENSOR-COMPILER-CONTROLS.json` records: + +- every permutation for N=2,4,8, including all 40,320 permutations at N=8; +- 100 random permutations each at N=16,32,64; +- exact symbolic checks of upper/lower zero and nonzero shears, diagonal, Swap and FFT words; +- an exact two-axis N=4 replay with different axis matchings and gate types: 16 data coordinates, all 256 input-output matrix entries, 18 simultaneous H calls, no dimension expansion. + +The finite controls support the constructive proofs; the cycle coloring, word identities, Kronecker identity and geometric scan sums establish the asymptotic statements. + +## 8. Cost substitution into the retained supplier + +Suppose a simultaneous H call on g selected bits, across V_bits bits of data and spectators, has paid cost O(V_bits g^(1-q) polylog(p)). Suppose also that any required bit-field gather/restore has cost O(V_bits g^(1-a_bit) polylog(p)). For D=polylog(p), N=poly(p), the compiler uses only polylog(p) such calls, and its diagonal scans cost O(V_bits polylog(p)). Thus the grouped local circuit costs + + O(V_bits [g^(1-q)+g^(1-a_bit)+1] polylog(p)). + +When a_bit>q and 00 for value in row['slacks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + controls={} + tests=[('endpoint_equality',dict(q=Q('0.001675462676808103522925'))), + ('zero_group_guard',dict(zeta=Q(0))), + ('zero_boundary_slack',dict(sigma_boundary=Q(0))), + ('old_per_axis_scan',dict(old_vd=True)), + ('unrecentered_full_cell_gauge',dict(full_cell_gauge=True))] + for name,kwargs in tests: + bad=calculate(**kwargs) + failed={k:v for k,v in bad['slacks'].items() if v<=0} + assert failed,name + controls[name]=failed + output=dict(status='PASS_CONDITIONAL_PHASE_FUSION_REVIEW', + scope='Local conditional transfer and exact arithmetic; inherited supplier and all-size interfaces remain assumptions. No upstream selected certificate is changed.', + review=dict(integrated_review='Root reviewed the complete standalone note, integrated proof chain, exact ledger, and independent cap/supplier/layout audits.', + independent_cap_gauge='CAPPED-GAUGE-AUDIT.md', + independent_cap_layout='CAPPED-SIGNOFF.md', + supplier_support='SUPPLIER-SUPPORT-AUDIT.md'), + **row,controls=controls) + (ROOT/'CAPPED-OUTER-ARITHMETIC.json').write_text(json.dumps(encode(output),indent=2)+'\n') + print(output['status']) + for name in ['epsilon','capped_block_power','beta_group','kappa']: + print(name,decimal(row['parameters'][name])) + print('minimum margin',decimal(min(row['margins'].values()))) + print('gain over selected',decimal(row['slacks']['improvement'])) + print('positive strict slacks',len(row['slacks'])) + print('controls',', '.join(controls)) diff --git a/research/capped-phase-fusion/check_matrix_corollary.py b/research/capped-phase-fusion/check_matrix_corollary.py new file mode 100644 index 0000000000..3f543803d7 --- /dev/null +++ b/research/capped-phase-fusion/check_matrix_corollary.py @@ -0,0 +1,143 @@ +#!/usr/bin/env python3 +"""Exact corollary ledger, additionally assuming the supplied 9/4 theorem.""" +from fractions import Fraction as Q +from decimal import Decimal,localcontext +from pathlib import Path +import json + +ROOT=Path(__file__).resolve().parent + +def decimal(x): + with localcontext() as ctx: + ctx.prec=65 + return str(Decimal(x.numerator)/Decimal(x.denominator)) + +def calculate(q=None,zeta=None,union_slack=None,old_vd=False, + full_cell_gauge=False,add_full_key_sort=False): + bit=Q('0.001733427039589844234749') + complex_=Q('0.001675462676808103522925') + q=complex_-Q(1,10**10) if q is None else q + a=(q+complex_)/2 + r=Q(1,10**6) + zeta=Q(1,10**8) if zeta is None else zeta + union_slack=Q(1,10**5) if union_slack is None else union_slack + tau_matrix=Q(19,8) + c=(3-tau_matrix)/2 + stop=Q(1,10**9) + cap=(Q(3,2)+q-(1+q)*(r+zeta)+union_slack)/(2+q) + eps=(1+q*(1-cap-r-zeta))/(1+q) + beta=1-cap-r-zeta + kappa=Q('0.001673360105405778') + selected=Q('0.001672660197076918') + first=Q('0.001672922349588871') + tau_bit,sigma=1-a,1-complex_ + internal=tau_bit+(1-stop)*max(sigma-tau_bit,Q(0)) + leaf=sigma+stop*(1-sigma) + lam=1-(q+a)/2 + key=beta+1-eps + margins=dict( + main_transform=eps*q, + global_coordinate_router=a, + fused_gaussian=1-eps+q*beta, + group_bit_router=1-eps+a*beta, + group_scans=1-eps+beta, + boundary_algebraic_matrix=c+cap-eps, + sparse_union_movement=1-eps-beta-Q(1,2)+cap, + polynomial_products=eps, + microbox_face_corrections=1-(eps+beta+3-8+2), + ) + if old_vd:margins['unreplaced_per_axis_scan']=1-eps + if add_full_key_sort: + margins['unnecessary_union_full_key_sort']=margins['sparse_union_movement']-key + slacks=dict( + actual_bit_above_effective=bit-a, + complex_above_effective=complex_-a, + complex_below_one_over_32=Q(1,32)-complex_, + q_positive=q, + complex_endpoint=complex_-q, + q_below_effective_bit=a-q, + stop_positive=stop, + stop_below_one=1-stop, + leaf_saving_above_effective_bit=(1-stop)*complex_-a, + q_below_internal=1-internal-q, + q_below_leaf=1-leaf-q, + lambda_above_tau=lam-tau_bit, + lambda_above_sigma=lam-sigma, + lambda_above_internal=lam-internal, + lambda_prime_above_lambda=1-q-lam, + lambda_prime_above_leaf=1-q-leaf, + epsilon_positive=eps, + epsilon_below_one=1-eps, + capped_block_above_band=cap-Q(1,2), + capped_block_below_cell=eps-cap, + group_positive=beta, + group_below_dimension=eps-beta, + support_router_span=1-beta, + group_gauge_power_slack=1-cap-r-beta, + actual_logarithmic_group_guard=1-cap-beta, + global_gamma_power_slack=1-eps-r, + actual_logarithmic_gamma_guard=1-eps, + alpha_logarithm_absorption=r, + group_key_fits_record=1-key, + membership_arithmetic_fits_record=1-beta-2*(1-eps), + sparse_union_fraction_decays=cap-beta-Q(1,2), + microbox_halo_volume=8-3-beta, + union_positive_slack=union_slack, + matrix_exponent_above_source_bound=tau_matrix-Q(9,4), + matrix_exponent_below_five_halves=Q(5,2)-tau_matrix, + short_record_fallback=eps-a, + improvement_over_selected=kappa-selected, + improvement_over_first_reviewed_transfer=kappa-first, + ) + if full_cell_gauge: + slacks['unrecentered_group_gauge']=1-eps-r-beta + slacks.update({name+'_above_kappa':value-kappa for name,value in margins.items()}) + assert margins['main_transform']==margins['fused_gaussian'] + assert margins['sparse_union_movement']==margins['main_transform']+union_slack + return dict(parameters=dict(a_bit=bit,a_eff=a,complex_saving=complex_,q=q, + r_spare=r,zeta=zeta,sigma_union=union_slack,stop=stop, + epsilon=eps,capped_block_power=cap,beta_group=beta, + matrix_exponent=tau_matrix,matrix_source_bound=Q(9,4), + c_boundary=c,kappa=kappa,selected=selected,first_reviewed_transfer=first, + lambda_=lam,lambda_prime=1-q,tau=tau_bit,sigma=sigma, + microbox_power=Q(8),halo_power=Q(3)),margins=margins,slacks=slacks) + +def encode(o): + if isinstance(o,Q):return dict(rational=str(o),decimal=decimal(o)) + if isinstance(o,dict):return {k:encode(v) for k,v in o.items()} + return o + +if __name__=='__main__': + row=calculate() + assert all(value>0 for value in row['slacks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + controls={} + tests=[('endpoint_equality',dict(q=Q('0.001675462676808103522925'))), + ('zero_group_guard',dict(zeta=Q(0))), + ('zero_union_slack',dict(union_slack=Q(0))), + ('old_per_axis_scan',dict(old_vd=True)), + ('unrecentered_full_cell_gauge',dict(full_cell_gauge=True)), + ('full_key_sort_on_union',dict(add_full_key_sort=True))] + for name,kwargs in tests: + bad=calculate(**kwargs) + failed={k:v for k,v in bad['slacks'].items() if v<=0} + assert failed,name + controls[name]=failed + output=dict(status='PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW', + scope='Additional matrix-conditioned corollary. Assumes supplied omega_C<=9/4 theorem and the effective algebraic stability transfer, as well as all first-transfer assumptions.', + review=dict(integrated='Root read the matrix corollary and independent layout signoff, reproduced the exact parameters and ledger, and replayed matrix source construction controls.', + matrix_source='PROOF-AND-COVERAGE.md', + effective_stability='ALGEBRAIC-STABILITY-INDEPENDENT.md', + independent_layout='MATRIX-LAYOUT-SIGNOFF.md'), + omitted_row=dict(name='optional full-coordinate-key union surcharge', + reason='The actual U_i complete-rectangle extraction, one-field transpose, and ordered merge do not sort union records. Sparse microhalo keys remain paid in their separate cost.'), + **row,controls=controls) + (ROOT/'MATRIX-COROLLARY-ARITHMETIC.json').write_text(json.dumps(encode(output),indent=2)+'\n') + print(output['status']) + for name in ['epsilon','capped_block_power','beta_group','kappa']: + print(name,decimal(row['parameters'][name])) + print('minimum margin',decimal(min(row['margins'].values()))) + print('gain over selected',decimal(row['slacks']['improvement_over_selected'])) + print('gain over first transfer',decimal(row['slacks']['improvement_over_first_reviewed_transfer'])) + print('positive strict slacks',len(row['slacks'])) + print('controls',', '.join(controls)) diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/BILATERAL-CONTROLS.json b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/BILATERAL-CONTROLS.json new file mode 100644 index 0000000000..ed4b620d8a --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/BILATERAL-CONTROLS.json @@ -0,0 +1,7 @@ +{ + "status": "all exact assertions passed", + "arithmetic": "fractions.Fraction", + "negative_and_zero_origin_instances": 45, + "finite_LDU_and_shift_entries": 750, + "scope": "Finite-origin algebra, exact origin-shift correction, upper inverse formula and simultaneous periodic shifts; bilateral limiting argument is analytic." +} diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/BILATERAL-LDU.md b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/BILATERAL-LDU.md new file mode 100644 index 0000000000..5cad6ee19f --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/BILATERAL-LDU.md @@ -0,0 +1,131 @@ +# Bilateral Gaussian LDU and a fixed-width periodic inverse bound + +Independent audit of the parent's infinite-LDU observation. This is an analytic norm theorem. It does **not** assert that the bilateral factors already have a fully priced fast implementation. + +## Statement + +Let `1=1`, define on Z + +`G_ij=exp(-a[(rho*j-q_i)^2-beta_j^2])`. + +Let `r=exp(2a rho)` and `c=prod_(m>=1)(1-r^-m)>0`. Then G has an exact factorization on bounded bilateral sequences + +`G=L C U`, + +where L is unit lower triangular, U is unit upper triangular, C is diagonal, and all factors and their inverses are bounded. Moreover all six factors are invariant under simultaneous index translation by s. Their periodizations therefore factor the periodic Gaussian matrix N, and + +`||N^-1||_infinity <= c^-4 exp(a/2)/(1-exp(-a rho theta))^2`. + +In particular, at the fixed choice `a=1`, + +`||N^-1||_infinity < 16 theta^-2`. + +Any fixed a>=1, including a=4pi if an integer alpha>=2 is retained, gives a constant times theta^-2. No lower bound proportional to log(1/theta) on a is needed for this norm theorem. + +## 1. Finite-origin factorization and the shift correction + +Write `x_i=r^q_i`, `d_i=exp(-a q_i^2)`, and `e_i=exp(-a rho^2 i^2+a beta_i^2)`. Then `G_ij=d_i x_i^j e_j`, including negative indices, and `d_i x_i^i e_i=1`. + +For a finite square principal interval `[l,r0]`, the Vandermonde powers must begin at zero. The correct shifted row diagonal is **d_i x_i^l**, and the power is `j-l`. This removes a possible erroneous origin-dependent diagonal from the upper factor. + +Put + +`C_i(l)=prod_(l<=k=j`, + +`U^l_ij=(e_j/e_i) h_(j-i)(x_l,...,x_i)`, `j>=i`. + +Here h is the complete symmetric polynomial, h_0=1. Upper and lower inverses are + +`(U^l)^-1_ij=(-1)^(j-i)(e_j/e_i) e_(j-i)(x_l,...,x_(j-1))`, + +`(L^l)^-1_ij=(-1)^(i-j) exp(E_ij) C_i(l)/(C_j(l) D_ij)`, + +where + +`D_ij=prod_(j=j`, + +`|(U^l)^-1_ij|<=c^-1 exp(a/4) exp(-a rho theta(j-i))`, `j>=i`. + +Both use only `-1/2<=beta_k<1/2`. Thus their row norms are at most the corresponding constants divided by `1-exp(-a rho theta)`, uniformly in l and the finite size. + +The row sum of G is uniformly finite, since + +`G_i,i+h <= exp(a/4) exp(-a(rho*|h|-1/2)^2)`. + +The same positive Gaussian bound controls all finite-origin L and U factors. + +## 3. Letting the left origin tend to minus infinity + +For fixed i,j, C_i(l) converges to + +`C_i=prod_(k(i+s,j+s)`. Every displayed factor has the same invariance: ratios `x_k/x_i` are preserved, and the Gaussian/prefix expressions shift together. C_i is s-periodic. + +For an invariant matrix A with bounded absolute row sums, set + +`Pi(A)_ij=sum_(n in Z) A_(i,j+ns)`, `0<=i,j=e^2>7, and the elementary product bound gives `c>=1-sum_(m>=1)r^-m>5/6`. Also `1-exp(-rho theta)>=theta/2` for 04W, choose + +`B=min(Bcap, 2^floor(log2(R-W)))`, + +retain the next B vertices as an interior, then the next W vertices as boundary, and decrement R by B+W. Declare the final R<=4W vertices boundary. + +There are `O(L/Bcap+log Bcap)` interiors: while the cap is active, each step removes at least Bcap vertices; after it ceases to be active, the remaining length halves at each step. Every interior length is dyadic and at most Bcap. Every interior produced by the loop is longer than W; two successive interiors are separated by at least W retained coordinates. + +Total boundary count per original cell is + +`O(W(L/Bcap+log Bcap))`. + +Since original cell lengths are Theta(d), total boundary density is + +`O(W/Bcap + W log Bcap/d)=O((W/Bcap) polylog p)` + +when Bcap<=d. The same selected-input and target-output physical period cuts must be preserved. A clipped endpoint cell of O(W) length is wholly boundary. + +Interior elimination only fills between the adjacent boundary groups of one interior. Each group has O(W) vertices: inserted groups have W; an end remainder has at most 5W after joining the reserved end. Hence scalar Schur half-bandwidth remains O(W), independently of the cap and the number of pieces. No retained matrix coefficient is changed by this new partition. + +## 2. Narrow Gaussian: exact recentering of the existing rounded weights + +In one original phase cell, beta_i=theta*i-c and + +`log a_i=-a(1/4-beta_i^2)/theta`. + +For any two coordinates i,j in that same cell, + +`log(a_i/a_j)=a(i-j)(beta_i+beta_j)`. + +Since both beta values lie in [-1/2,1/2), + +`|log(a_i/a_j)|<=a|i-j|`. + +Thus a capped interior of at most Bcap vertices has ideal weight range at most exp(a Bcap), even when the absolute full-cell gauge has range exp(O(ad)). + +The actual matrix uses already-rounded positive weights ahat_i. Suppose, as ensured by the retained high-precision setup, + +`(1-epsilon) a_i <= ahat_i <= a_i`, with epsilon<=1/2. + +Then for an anchor m in the capped interior, + +`(1-epsilon) exp(-a Bcap) <= ahat_i/ahat_m` + +` <= exp(a Bcap)/(1-epsilon)`. + +Replacing ahat_i by `ahat_i/ahat_m` in both diagonal factors of + +`A=diag(ahat) Ghat diag(ahat)^-1` + +is **exact**. It changes neither A nor the fixed globally rounded H. These ratios must be formed in coefficient setup; the data circuit must not apply the full-cell diagonal and hope to cancel it later. One may anchor at the maximum ahat in the piece to make the direct diagonal contractive and its inverse at most `2 exp(a Bcap)`. + +The old setup provides absolute error below `4*2^-P` and ideal minimum at least exp(-a/(4theta)). It suffices that + +`4*2^-P exp(a/(4theta)) <= epsilon`. + +In the retained family a=Theta(log p), theta^-1=O(d), d=p^epsilon0 with fixed epsilon0<1, so ad=o(p). The existing P=Cp setup precision easily gives epsilon<=2^-cp. The **ad cost occurs in offline coefficient accuracy**, not in the runtime norm of the recentered diagonal. It is not multiplied by the number g of simultaneously executed axes. More explicitly, if the raw relative error is delta0<=4*exp(a/(4theta))*2^-P, then an anchored ratio has relative error at most 2delta0/(1-delta0). A g-fold product of anchored diagonals has error bounded by a polynomial factor times g*delta0*exp(O(g*a*Bcap)). Thus the needed precision is P>=p+O(ad+g*a*Bcap+log g+polylog p), rather than p+O(g*a*d). Both added power terms are o(p) in the proposed parameter family. + +## 3. Broad forward Gaussian: local gauges + +Use the target-grid notation from FORWARD-SMALL-A-AUDIT.md. With lambda=1/rho, b0=pi/alpha^2, original phase c, and an anchor m in the capped interior, + +`delta=(1-lambda)m-c`, `x=i-m`, `y=k-m`. + +The exact split is + +`Dcol(i)=exp[-b0((1-lambda)x^2+2delta*x)]`, + +`Drow(k)=alpha^-1 exp[b0 lambda(1-lambda)y^2+2b0 delta lambda*y-b0 delta^2]`, + +with a fixed Toeplitz kernel `exp[-b0 lambda(i-k)^2]`. + +Throughout the original selected-target phase cell, `beta'_i=theta*i-rho*c` lies in [-1/2,1/2). Therefore + +`d(log Dcol)/di = -2 b0 lambda beta'_i`, + +`d(log Drow)/dk = 2 b0 lambda^2 beta'_k`. + +Their absolute slopes are at most b0 and b0 respectively. Both diagonal logarithmic ranges on the capped block are at most `b0 Bcap`. At the anchor, Dcol=1 and Drow=alpha^-1 exp(-b0 delta^2), so their absolute values also have logarithmic size `O(b0 Bcap+log alpha)`. + +If a globally rounded broad block was already defined using a different anchor, use exact constant ratios to recenter its two diagonals: divide the column diagonal by its value at m and multiply the row diagonal by the same value. Ideal changes of anchor differ only by reciprocal constants, since the derivatives above are independent of m. Uniform relative coefficient error changes the range by only O(1), just as in the narrow case. This preserves the already-defined rounded block exactly. Alternatively the broad block can be initially defined with the capped anchor, using one globally specified rounding construction. + +Holes remain boundary. The completed narrow and broad interior inverse bounds and the completed global/Schur norm bounds proved earlier do not depend on the cap: every new interior is still a contiguous principal block of the same original phase matrix. + +## 4. Group guard and circuit count + +Each capped narrow diagonal contributes at most `a Bcap+O(1)` to the logarithmic norm; the broad contribution `O(Bcap/alpha^2+log alpha)` is smaller. Shallow Toeplitz circuit amplification contributes polylog(p) per axis. The number of capped blocks per line does not multiply these bounds: they are disjoint direct-sum blocks and their circuits run in parallel, padding completed smaller circuits with identity gates. + +Consequently g simultaneously executed axes require + +`O(g a Bcap + g polylog p)` + +extra logarithmic precision. With `Bcap<=p^b`, `g<=p^beta`, a=Theta(log p), and beta+b=1-r-zeta with r,zeta>0, + +`g a Bcap=O(p^(1-r-zeta) log p)=o(p)`. + +In particular no remaining full-cell term g*a*d may be charged after exact recentering. Offline accuracy still needs to dominate ad=o(p), as above. + +## 5. Schur solves and whole-group ordering + +For the narrow matrix, strict row dominance is inherited by any interior elimination, so all completed Schur and inverse norms remain polynomial under the cap. + +For the broad matrix, completed Schur/inverse norms follow from the square identity-hole completion and polynomial principal-interior bounds in FORWARD-SMALL-A-AUDIT.md. To obtain uniform norms for a banded online factorization, eliminate **whole boundary groups** of W to O(W) vertices, rather than appealing without proof to scalar prefixes inside one group. + +After all interiors and a prefix of whole boundary groups are eliminated, the eliminated set is a direct sum of one physical contiguous prefix and disjoint future interiors: each remaining full group contains at least W uneliminated coordinates, blocking every width-W connection across it. The same holds after adding the next whole group. A physical prefix reduces, under selected/hole ordering, to the small-b normalized Gaussian block plus identity hole columns, so it has polynomial inverse norm. Each future interior also has polynomial inverse norm. Thus every required eliminated-set inverse is polynomial, uniformly in the number of pieces. + +At the final group on a full periodic line, use the already-established full periodic H inverse bound instead of a nonperiodic-prefix bound. For a clipped principal window the complement-gap condition excludes wrap edges, so the prefix statement applies directly. Each block pivot inverse is a principal block of the inverse after adding that group. Block Schur entries are bounded by `H+H H_EE^-1 H`, hence polynomial. There are only O(1) neighboring boundary groups in a band row. The resulting block-L and block-U norms are polynomial; their full inverses are polynomial as well, from `L^-1=U S^-1` and `U^-1=S^-1 L`. A scalar expansion inside a group is unnecessary: precompute its dense O(W)-square inverse and apply the block updates to batches by Strassen. + +This whole-group order prevents the subtle gap in which a partially consumed separator of fewer than W vertices permits a connection to the next interior. It preserves the claimed batched boundary exponent and is unaffected by capped subdivision. + +## Scope + +The capped local-gauge change is valid. The listed exact controls check carving, analytic slope identities and preservation of the already-rounded narrow matrix. The final recurrence arithmetic and tensor routing remain separate reviewed components. diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/CAPPED-GAUGE-CONTROLS.json b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/CAPPED-GAUGE-CONTROLS.json new file mode 100644 index 0000000000..6ea5c59eab --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/CAPPED-GAUGE-CONTROLS.json @@ -0,0 +1,11 @@ +{ + "status": "all exact assertions passed", + "arithmetic": "integers and fractions.Fraction", + "checks": { + "partitions": 27216, + "narrow_slope_pairs": 6132, + "rounded_ratio_and_exact_reanchoring_pairs": 6132, + "broad_slope_pairs": 14754 + }, + "scope": "Capped dyadic carving, narrow exact log-slope, positive perturbed global weights, exact local reanchoring, and broad gauge log-slope. Final tensor schedule/ledger excluded." +} diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/CYCLIC-SCAN-CONTROLS.json b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/CYCLIC-SCAN-CONTROLS.json new file mode 100644 index 0000000000..eddd860c96 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/CYCLIC-SCAN-CONTROLS.json @@ -0,0 +1,5 @@ +{ + "status": "all exact assertions passed", + "cyclic_signed_scan_systems": 203, + "scope": "Two-pass cyclic affine recurrence against dense exact inversion; bilateral coefficient estimates are analytic." +} diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/DYADIC-INTERIOR-CONTROLS.json b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/DYADIC-INTERIOR-CONTROLS.json new file mode 100644 index 0000000000..e42eac6b93 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/DYADIC-INTERIOR-CONTROLS.json @@ -0,0 +1,6 @@ +{ + "status": "all exact assertions passed", + "cell_partitions": 80000, + "reachable_boundary_records": 6413386, + "scope": "Dyadic lengths, exact coverage, w-wide separation, logarithmic block count, and O(w) Schur boundary span." +} diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-CELL-CONTROLS.json b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-CELL-CONTROLS.json new file mode 100644 index 0000000000..c74f144886 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-CELL-CONTROLS.json @@ -0,0 +1,7 @@ +{ + "status": "all exact assertions passed", + "phase_cell_cases": 350, + "local_two_factorization_entries": 5184, + "selected_completion_entries": 16496, + "scope": "Exact target-grid coordinate conversion, local Toeplitz/Gaussian-normalized factorizations and selected block of square hole completion. Norm estimates are analytic." +} diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md new file mode 100644 index 0000000000..0f7f80b3f7 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md @@ -0,0 +1,109 @@ +# Broad forward Gaussian: finite-block and completed-matrix conditioning + +This note supports the phase-assembly branch. It distinguishes the broad forward map S from the narrow T used in N=C T D. Its conclusions concern conditioning and local circuit guards; the final simultaneous tensor layout remains that branch's obligation. + +Let the narrow parameter be `a=pi alpha^2=Theta(log p)`, `rho=t/s=1+theta`, and `lambda=1/rho`. The forward map is + +`S_(k,j)=alpha^-1 exp[-b0(j-lambda*k)^2]`, `b0=pi/alpha^2`. + +Equivalently it uses the small resampling parameter + +`b=b0 lambda^2=pi/(alpha^2 rho^2)`. + +## 1. The bilateral bound works at every positive parameter + +The finite and bilateral LDU arguments in BILATERAL-LDU.md use a>=1 only for their final numerical simplification of the product c. For any b>0, define + +`c_b=prod_(m>=1)(1-exp(-2b rho m))`. + +The same proof gives, for the periodic normalized matrix and for every finite contiguous principal block with q-gaps at least 1 and |beta_i|<=1/2, + +`||N_b^-1|| <= c_b^-4 exp(b/2)/(1-exp(-b rho theta))^2`. + +There is an explicit useful product bound. Expand the logarithm and sum nonnegative terms: + +`log(c_b^-1)=sum_(k>=1) 1/[k(exp(2b rho k)-1)]` + +` <=(1/(2b rho)) sum_(k>=1)1/k^2=pi^2/(12b rho)`. + +At `b=pi/(alpha^2 rho^2)`, this implies + +`c_b^-4 <= exp(pi alpha^2 rho/3)=exp(a rho/3)`. + +Since b*rho*theta=pi*theta/(alpha^2 rho)<1 for alpha>=2 and theta<=1, its reciprocal gap is at most `2alpha^2 rho/(pi theta)`. Thus every such normalized block inverse has norm polynomial in p when alpha^2=O(log p) and theta^-1 is polynomial in p. + +## 2. Actual target-grid phase-cell block + +Embed source j at target coordinate i=q_j. On an original phase cell q_j=j+c, every selected target coordinate i in that cell corresponds to j=i-c. A square interior of the forward target-grid matrix is therefore + +`K_(k,i)=alpha^-1 exp[-b0(i-lambda*k-c)^2]`. + +Here both i and k run over one contiguous selected-target interval inside that cell. Put + +`q'_k=k+rho*c`, `beta'_k=rho*k-q'_k=theta*k-rho*c`. + +At a selected coordinate k=q_j, beta'_k is exactly the original beta_j, hence lies in [-1/2,1/2). Also q' has consecutive gap 1. Integrality of q' was never needed in the finite LDU proof; monotone separation and the beta bound suffice. Consequently + +`K=alpha^-1 N_b diag(exp(-b*(beta'_i)^2))`, + +and + +`||K^-1|| <= alpha exp(3b/4) c_b^-4/(1-exp(-b rho theta))^2` + +` =O(alpha^5 rho^2 theta^-2 exp(pi alpha^2 rho/3))=poly(p)`. + +This controls the **completed** interior inverse. It avoids multiplying the large individual diagonal-gauge norms to estimate K inverse. Target holes should be retained as boundary vertices, so every interior coordinate has the asserted original beta bound. + +## 3. Square completion of the global forward map + +Add one identity column e_h for every target hole h, and feed zero into these extra columns. This makes an invertible square t-by-t matrix H_aug without changing the desired forward result. Permute selected q_j rows/columns first and holes second. In that order, + +`H_aug = [ alpha^-1 N_b D_b^-1 0 ; R I ]`, + +where `D_b=diag(exp(b beta_j^2))` and R consists of broad Gaussian rows at holes. A Gaussian lattice-sum bound gives `||R||=O(1)` after the alpha^-1 factor. Thus + +`H_aug^-1=[ alpha D_b N_b^-1 0 ; -R alpha D_b N_b^-1 I ]` + +has polynomial row norm. The matrix itself also has O(1) row norm. Gaussian truncation and sufficiently accurate structure-preserving coefficient rounding perturb both it and all required interior blocks by exponentially small amounts, preserving these polynomial bounds. + +If A is the block-diagonal interior matrix and S is the completed boundary Schur complement, then S inverse is the boundary principal block of H_aug inverse; hence its norm is polynomial. Also + +`||S||<=||H_BB||+||H_BI|| ||A^-1|| ||H_IB||=poly(p)`. + +This is enough for the completed Schur map. To justify a particular sequential block-LU implementation, its prefix factor norms still need to be checked. The natural physical-prefix elimination sets consist of one contiguous selected-source prefix plus disjoint future interiors, with identity hole columns. That geometry offers the same uniform contiguous-block inverse bound; it should be made explicit in the assembly proof. + +## 4. Exact local Toeplitz split and guard range + +Choose an anchor m inside the target cell, and write x=i-m, y=k-m, + +`delta=(1-lambda)m-c`, with |delta|<=1/2. + +Then the exact factorization is + +`K_(k,i)=Drow(y) G(y,x) Dcol(x)`, + +`G(y,x)=exp[-b0 lambda(x-y)^2]`, + +`Dcol(x)=exp[-b0((1-lambda)x^2+2delta*x)]`, + +`Drow(y)=alpha^-1 exp[b0 lambda(1-lambda)y^2+2b0 delta lambda*y-b0 delta^2]`. + +Expansion gives exactly `-b0(delta+x-lambda*y)^2`; no approximate recentering is involved. On a phase cell of O(d) coordinates, the two diagonal logarithmic ranges are O(d/alpha^2). These are smaller than the O(alpha^2 d) narrow inverse gauges already paid by the proposed group size. + +## 5. SPD Toeplitz conditioning for the shallow circuit + +Let `gamma=b0 lambda=pi/(alpha^2 rho)`. The full Gaussian Toeplitz symbol has, by the Gaussian Poisson identity, + +`f(omega)=sqrt(pi/gamma) sum_(k in Z) exp[-(omega+2pi*k)^2/(4gamma)]`. + +For -pi<=omega<=pi, + +`sqrt(pi/gamma) exp[-pi^2/(4gamma)] <= f(omega) <= 1+sqrt(pi/gamma)`. + +Every finite principal Gaussian Toeplitz matrix has eigenvalues between these same bounds, by integrating the squared modulus of its coefficient polynomial against f. At alpha^2=O(log p), its condition number is polynomial in p. Truncation width O(sqrt(p/gamma)) and Cp-bit rounding perturb the matrix far less than its lower eigenvalue, so the actual symmetric rounded Toeplitz Ghat remains SPD with polynomial condition number. + +The low-displacement block-LU circuit of SHALLOW-TOEPLITZ.md extends from near identity to this SPD setting: every principal Schur complement has norm at most lambda_max and inverse norm at most lambda_min^-1, and ranks grow by the same additive displacement recurrence. Shear coefficients are polynomial in p; multiplying these bounds along O(log d) recursive levels gives exp(O(log^2 p)) partial-prefix growth. This calls for O(log^2 p) internal circuit guard bits per axis, rather than claiming the near-identity theorem's O(log p) unchanged. It is still below the group budget dominated by the local Gaussian diagonals. + +Dyadic interior subdivision from LOG-A-PHASE-SCHUR-AUDIT.md applies here too, using the broader width O(sqrt(p alpha^2)). It keeps target holes and separators as boundary, adds only polylogarithmic boundary volume, and lets every interior use a dyadic shallow circuit without 2^g padding. + +No numerical final exponent follows from these conditioning statements alone. diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md new file mode 100644 index 0000000000..3bcf490a8c --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md @@ -0,0 +1,140 @@ +# Audit: logarithmic Gaussian width retains the phase-cell Schur solver + +This audit checks the analytic hypotheses of the retained rad20 phase-cell solver at `a=pi alpha^2=Theta(log p)`, `theta=Theta(1/d)`, even though `a theta` tends to zero. It also audits the shallow Toeplitz circuit algebra and identifies the actual matrix that must be precomputed. It does not certify the final tensor layout or multiplication exponent. + +Read inputs: `references/semantic-bulk/rad20/reports/review-bulk-resampling.md`, `review-phase-cell-inverse.md`, `downstream-phase-cell-inverse.md`, and `work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md`. + +## 1. The sharp elementary diagonal-dominance estimate + +Let `rho=1+theta`, `beta_i=rho*i-q_i`, and let the Gaussian row coefficient at integer offset h be + +`n_i,h=exp(-a[(rho*h+beta_i)^2-beta_(i+h)^2])`. + +The h=0 coefficient is 1. Set `g_i=q_i-q_(i-1)` and `g_(i+1)=q_(i+1)-q_i`, each in {1,2}. For the nearest coefficients, + +`n_i,-1=exp(-a*g_i*(2rho-2beta_i-g_i))`, + +`n_i,+1=exp(-a*g_(i+1)*(2rho+2beta_i-g_(i+1)))`. + +If g_i=1, the first exponent is exactly `1+2theta-2beta_i`; if g_i=2, that gap forces `beta_i=1/2-theta`, so the upper exponent is at least `1+2theta+2beta_i`. Consequently + +`n_i,-1+n_i,+1` + +` <=exp(-a(1+2theta-2beta_i))+exp(-a(1+2theta+2beta_i))` + +` <=exp(-2a theta)+exp(-2a(1+theta))`. + +The last step is convexity in beta on [-1/2,1/2]. The two nearest directions cannot both be charged as exp(-2a theta). + +For |h|>=2, + +`(rho*h+beta_i)^2-beta_(i+h)^2 >= h^2-|h|`, + +so their entire two-sided tail is below `4 exp(-2a)` for a>=1. Therefore + +`||N-I||_infinity <= exp(-2a theta)+5 exp(-2a)`. + +This includes nonzero periodic diagonal aliases in the tail; no alias is silently omitted from this bound. + +For p>=101, theta>=1/(4p), and `a>=4 log(16p)`, + +`1-exp(-2a theta) >= 1-exp(-1/(2p)) >=1/(4p)`, + +`5exp(-2a)<=5/(16p)^8<1/(8p)`. + +Hence N has row diagonal-dominance margin at least `1/(8p)` in the following robust sense: after deleting its tiny nonzero diagonal aliases, its diagonal is 1 and its off-diagonal row sum is at most `1-1/(8p)`. In particular the fixed-width inverse theorem is not needed to supply the retained strict row dominance. + +Choose the legal integer alpha by + +`alpha=ceil(sqrt((4/pi)log(16p)))`. + +Then a has the required lower bound and is O(log p). The old sufficient condition `u>=ceil(log2(8p))` need not be reused: the direct estimate above replaces that prerequisite. + +## 2. Actual rounded H, interior matrices, and guard range + +Use exactly the retained structure: + +- `w=ceil(sqrt(16p/u))+2`, u=alpha^2=a/pi; +- cell weights `a_j=exp(-a(1/4-beta_j^2)/theta)`; +- one globally fixed downward dyadic Toeplitz kernel `Ghat_delta` for |delta|<=w, zero outside, with Ghat_0=1; +- positive downward dyadic weights ahat_j; +- within a physical cell, define H_jk **exactly** as `ahat_j Ghat_(j-k)/ahat_k`; +- across cells, use the retained downward Gaussian entries; set H_jj=1. + +Thus the actual interior block being inverted is + +`A_cell=diag(ahat) Ghat_cell diag(ahat)^-1`. + +It is not the unrounded Gaussian matrix, and its retained truncated Ghat is not silently replaced by the infinite Gaussian kernel. + +The weight range is + +`0<=-log a_j<=a/(4theta)=O(a d)` + +under theta between 1/(4d) and 1/(2d-1). The old eventual condition u*d<=p holds whenever d=p^epsilon, epsilon<1, and u=O(log p). It bounds the old rational perturbation calculation by the same `2^(O(p)) eta` allowance. The stronger range `O(a d)` should be used for grouped tensor precision, rather than charging O(p) independently for every axis. + +Choose the coefficient precision high enough that `||H-N||<1/(16p)`, including the Gaussian truncation, omitted diagonal aliases and weight-ratio rounding. The retained bound `p*2^(16p+7)*eta+2^(-46p)` at eta=2^(-32768p) is more than sufficient. Then H has diagonal 1, row gap at least `1/(16p)`, row norm below 2, and inverse norm at most 16p. All clipped principal windows inherit the same row gap. The physical period cut must remain a cell boundary, as in the retained bulk review. + +## 3. Schur complement bounds without an a*theta>1 assumption + +A scalar elimination step on a strictly row-diagonally-dominant matrix cannot decrease its minimum absolute row gap. For pivot k, + +`gap_i(new) >= gap_i(old)+(|a_ik|/|a_kk|) gap_k(old)`. + +The full absolute row sum also cannot increase, since the added update row has off-pivot sum less than |a_kk|. Induction applies regardless of the signs of filled entries. Since the diagonal begins real positive and the correction magnitude is below it, the surviving diagonal remains positive. + +Consequently the exact Schur complement + +`S=H_BB-H_BI H_II^-1 H_IB` + +has row gap at least `1/(16p)`, row norm below 2, positive diagonal, and + +`||S^-1||_infinity<=16p`. + +Principal interior inverses also have norm at most 16p. Rounding S by a row perturbation below 1/(32p) preserves gap at least 1/(32p) and inverse norm at most 32p. A row has O(w) nonzeros; the retained P-grid readily meets this target. These are polynomial bounds on arbitrary signed inputs, with no factor equal to the number of cells. + +The boundary geometry is unchanged: keep first/last w vertices of each physical cell, yielding O(s*w/d) boundary vertices and circular half-bandwidth at most 2w. Schur fill only joins the two end groups of one cell, while original cross-cell entries join neighboring end groups. Actual inverse and LU coefficient tables must be precomputed for this H/S, not imported from the old parameter values. + +Thus the retained banded/cyclic boundary solver's gap-dependent proof remains applicable with fixed constants enlarged. The proposed batched boundary arithmetic still needs its own data-layout and Strassen error audit, but there is no analytic conditioning obstruction at logarithmic a. + +## 4. Apply the shallow circuit directly to Ghat inverse + +Because a is eventually larger than 4pi, the symmetric Toeplitz matrix Ghat_cell satisfies + +`max(||Ghat_cell-I||_1,||Ghat_cell-I||_infinity)<1/8`. + +Downward rounding only decreases this off-diagonal sum. The shallow full-Toeplitz theorem therefore applies directly to Ghat_cell. Reverse the resulting invertible circuit and invert its gates to obtain Ghat_cell inverse. This avoids the two-branch Gohberg-Semencul difference entirely. The interior inverse is: + +1. multiply by diag(ahat)^-1; +2. apply the reversed shallow circuit for Ghat_cell; +3. multiply by diag(ahat). + +Its only superpolynomial individual scale comes from the **local** weight range O(a d). Completed A_cell inverse has polynomial norm by the strict row gap. Across g axes, the uncompleted local weight contribution is at most exp(O(g*a*d)); choosing `g<=p/(d polylog p)` with sufficient logarithmic slack keeps the total data precision O(p). The shallow circuit contributes polynomial-in-d amplification per axis, hence only O(g log d) additional logarithmic range. This is a guard calculation, not a complete simultaneous-layer routing proof. + +## 5. Independent shallow-Toeplitz algebra audit and a length caveat + +The parity-split triangular identity, block Schur displacement formulas, Stein-generator telescoping identity, and bounded-prefix induction in SHALLOW-TOEPLITZ.md check algebraically. In particular the parity children run in parallel, avoiding the earlier erroneous sequential-depth recurrence. The full block-LU recursion also runs its two children in parallel, with O(log N) displacement rank after O(log N) levels. + +That document currently states its circuit theorem for dyadic N. Actual cell interiors are not generally dyadic. Padding every tensor axis to its next power of two would be invalid. A plausible exact extension has been communicated to its author: split an arbitrary N into balanced sizes differing by at most one; a near-square rectangular Toeplitz dirty shear is a square Toeplitz shear plus one extra row/column rank-one shear. The latter can be implemented by a binary tree of unitary 2-coordinate reductions and its inverse, without clean ancilla. The rectangular Stein identity telescopes as before. This extension needs its own written proof/controls before it is used as an established interface. + +No new final kappa is asserted by this audit. + +## 6. Dyadic interior subdivision resolves the length caveat + +No arbitrary-length shallow FFT theorem is actually needed. Refine the interior/boundary partition inside each original physical phase cell of length L as follows. Keep its first and last w vertices as boundary. Starting with the remaining interval of length R=L-2w: + +- while R>4w, choose the largest power of two B<=R-w; +- mark the next B vertices as one interior, the following w vertices as boundary, and replace R by R-B-w; +- once R<=4w, mark all remaining vertices as boundary. + +Every interior now has exactly dyadic length. If R>4w, then `B>(R-w)/2`, so the next remainder is less than R/2. There are `O(1+log(L/w))` interiors and the total boundary count in the cell is `O(w(1+log(L/w)))`. The final boundary group, including the initially reserved last w vertices, has size at most 5w; all inserted separators have size w. + +The width-w matrix H cannot connect two different new interiors: their separating boundary gap has w vertices, so their nearest physical coordinates differ by w+1. Each new interior is still a principal block of the SAME globally fixed Toeplitz-similar cell matrix. Its weights retain the original cell gauge; there is no new rescaling or rerounding at a subdivision boundary. + +Eliminating one dyadic interior fills only between its immediately neighboring boundary groups. Each such group has O(w) vertices. Original H edges also connect only the same group or neighboring groups; an interior has length greater than w in the loop above. Thus, in inherited physical boundary order, the new Schur complement has half-bandwidth O(w), not O(w log d). Only its number of rows grows by O(log d): + +`n_boundary=O((s/d) w log d)`. + +Clipped endpoint cells of size O(w) may be declared wholly boundary, adding only O(w) rows; preserve the original physical period cut. This is the same endpoint convention as the retained bulk review. + +The boundary solve and coupling work therefore gain only polylogarithmic factors. All dyadic interior blocks use the established SHALLOW-TOEPLITZ theorem directly, without padding any tensor axis and without requiring a non-dyadic Fourier circuit. This resolves the length caveat in section 5 for the intended phase-cell application; it does not prove the separate simultaneous tensor-layer routing claim. diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-SCHUR-CONTROLS.json b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-SCHUR-CONTROLS.json new file mode 100644 index 0000000000..301815a2c5 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/LOG-A-SCHUR-CONTROLS.json @@ -0,0 +1,11 @@ +{ + "status": "all exact assertions passed", + "arithmetic": "fractions.Fraction", + "checks": { + "nearest_exponent_checks": 147618, + "far_exponent_checks": 442854, + "schur_eliminations": 468, + "schur_entries": 19032 + }, + "scope": "Nearest/far Gaussian exponent inequalities and signed/asymmetric Schur row-gap preservation; the transcendental row-sum estimate and tensor transfer are analytic." +} diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-CONTROLS.json b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-CONTROLS.json new file mode 100644 index 0000000000..df9fc504c1 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-CONTROLS.json @@ -0,0 +1,13 @@ +{ + "status": "all exact assertions passed", + "arithmetic": "fractions.Fraction", + "checks": { + "instances": 63, + "inverse_entries": 2424, + "tile_entries": 4110, + "suffix_outputs": 1377, + "exponent_bounds": 2424, + "upper_entries": 2424 + }, + "scope": "Nonperiodic Newton L inverse and U inverse algebra; exact signed cross-tile data application; no runtime or tensor-fusion certificate." +} diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-FACTOR.md b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-FACTOR.md new file mode 100644 index 0000000000..35af6c7b99 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/LOWER-FACTOR.md @@ -0,0 +1,180 @@ +# A fast, stable application of the general Newton lower factor + +Status: a proved one-dimensional **nonperiodic lower-factor** algorithm and exact controls. It does not by itself implement the entire periodic inverse. The upper factor has a polynomial norm bound but its fast normalization/fusion remains to be supplied. No final multiplication exponent is claimed. + +## 1. Exact Gaussian LDU + +Let `rho=1+theta`, `q_i=floor(rho*i+1/2)`, `beta_i=rho*i-q_i`, `x_i=r^q_i`, `r=exp(2a rho)>1`, and `d_i=exp(-a q_i^2)`, for `0<=i=j`, zero otherwise. + +The ordinary monic Newton expansion of `X^j` gives exactly + +`G=L diag(C_i) U`, + +where `L=diag(d_i) T diag(d_i)^-1`, `L_ii=U_ii=1`, and + +`(U^-1)_ij=(-1)^(j-i) (e_j/e_i) e_(j-i)(x_0,...,x_(j-1))`, + +`e_j=exp(-a rho^2 j^2+a beta_j^2)`. + +The pivot calculation is exact: + +`d_i P_i(x_i) e_i = C_i`. + +Define + +`A_i=d_i x_i^i C_i/prod_(k<=i)x_k` + +` =C_i exp(-a beta_i^2-a rho^2 i+2a rho sum_(k<=i) beta_k)`. + +Divided differences give the exact inverse lower entries + +`(L^-1)_ij=(-1)^(i-j) (A_i/A_j)/D_(i,j)`, + +`D_(i,j)=prod_(j=1)(1-r^-m)`. Because `q_(i+1)-q_i>=1`, + +`c<=C_i,D_(i,j)<=1`. + +For `i>=j`, direct summation gives + +`log |A_i/A_j| <= -log c+a/4-a rho theta(i-j)`. + +Consequently + +`||L^-1||_infinity <= c^-2 exp(a/4)/(1-exp(-a rho theta))`. + +For the upper factor, compare every elementary-symmetric term with the product of the largest `j-i` variables. The sorted deficits form a partition, hence + +`1<= e_(j-i)(x_0,...,x_(j-1))/prod_(i<=k=-1/2` gives + +`||U^-1||_infinity <= c^-1 exp(a/4)/(1-exp(-a rho theta))`. + +Also `||diag(C_i)^-1||<=c^-1`. For `a>=1`, `r>=e^2`, so `c` has an absolute positive lower bound. For `a=O(log(1/theta))`, both factor norms are polynomial in `theta^-1`. These bounds apply to **arbitrary signed/complex input vectors**, not only correlated outputs of another factor. + +## 3. One stable scan plus a local correction + +Let + +`D_j^+=prod_(j=0`, + +`0<=1-H_i(x_j)<=sum_(k>i)x_j/x_k <= r^(-h-1)/(1-r^-1)`. + +The sum of discarded correction entries for `h>=W` is therefore at most + +`c^-2 exp(a/4) r^(-W-1)/[(1-r^-1)(1-exp(-a rho theta)/r)]`. + +Thus `W=O((p+a+log(1/theta))/a+1)` makes the omitted correction less than `2^-p` in operator norm. At `a=O(log(1/theta))`, `theta>=p^-O(1)`, this is `O(p/a+1)`. + +Partition the line into pages of W indices. Compute the exact correction on each diagonal page and each immediately lower adjacent page, and omit it elsewhere. Every omitted entry has `h>=W`; every retained tile has O(W) rows/columns. A diagonal page is recursively split into ordered cross tiles plus its diagonal. Only O(log W) tree levels are required. The adjacent-page rectangle can use the tile algorithm below directly. Subtracting the scan's contribution on those same tiles gives exactly the desired retained correction. + +## 4. The bounded-polynomial cross-tile algorithm + +Let every column in J precede every row in `I=[l,b]`, and suppose both sets lie in an index interval of length O(W). Put + +`z_j=x_j/x_l` (j in J), `v_k=x_l/x_k` (l=0) M_m Z^m=sum_j w_j/(1-z_j Z)=Q(Z)/P(Z)`. + +Build a balanced tree with leaf pairs `(P,Q)=(1-z_j Z,w_j)` and merge by + +`P=P_left P_right`, + +`Q=Q_left P_right+Q_right P_left`. + +Truncate to the needed degree at each node. Since `sum z_j<=1/(r-1)`, all product coefficient l1 norms are at most `exp(1/(r-1))<2`; numerator norms are at most twice the absolute input weight sum. The power-series reciprocal of P has l1 norm at most `c^-1`. Standard fast power-series inversion, or doubling the geometric series of `1-P` (whose l1 norm is <1/2 for r>=e^2), is therefore stable with polynomially bounded intermediates. Product-tree complexity is soft-O(|J|+|I|) scalar operations. + +### Fast suffix-polynomial evaluation + +The second operation is the transpose of multiplication by the suffix products. Split `[l+1,b]` at m. Right-hand output rows use the moment vector restricted to their degree range. For left rows, replace the moment vector by + +`M'_h=sum_k [Z^k]prod_(m=1; at a=1 this is `<16 theta^-2`. The independent exact negative-origin controls are BILATERAL-CONTROLS.json. +2. **Complete one-dimensional data application:** LOWER-FACTOR.md supplies the lower factor with logarithmic guard overhead. UPPER-DATA-ALGORITHM.md supplies the upper factor and a direct periodic implementation with `O(s p polylog p)` data cost, shared per-axis coefficient preprocessing, and **Cp internal precision**. This is stated in the target `s>>poly(p)` regime. Exact controls are LOWER-CONTROLS.json, UPPER-DATA-CONTROLS.json, and CYCLIC-SCAN-CONTROLS.json. +3. **Length-preserving upper merge:** UPPER-LIFTING.md proves the Newton merge and bounded exact shear algebra. UPPER-MERGE-CONTROLS.json checks its degree indexing and factorization. +4. **Still open:** a realization of the normalized Hankel/Toeplitz steps with only logarithmic internal guard bits, and a bounded-depth width-preserving tensor circuit. Neither the analytic norm theorem nor the one-dimensional time bound establishes those, and no improved final multiplication exponent is claimed. + +Replay all finite controls with Python 3: + +``` +python3 work/gaussian_fast_ultra/check_lower.py +python3 work/gaussian_fast_ultra/check_upper_merge.py +python3 work/gaussian_fast_ultra/check_bilateral.py +python3 work/gaussian_fast_ultra/check_upper_data.py +python3 work/gaussian_fast_ultra/check_cyclic_scan.py +``` + +The package uses no third-party source result beyond the elementary Newton polynomial identities stated and proved by their constructions. It retains the parent Gaussian notation and does not modify the frozen upstream manuscript. + +Focused phase-fusion audit added later: + +- `LOG-A-PHASE-SCHUR-AUDIT.md`: verifies strict row dominance at a=Theta(log p), polynomial Schur bounds, actual rounded Toeplitz interiors, and dyadic interior subdivision with only a logarithmic increase in boundary count. `LOG-A-SCHUR-CONTROLS.json` and `DYADIC-INTERIOR-CONTROLS.json` contain independent exact checks. +- `FORWARD-SMALL-A-AUDIT.md`: handles the distinct broad forward Gaussian S via b=pi/(alpha^2 rho^2), gives polynomial finite-block/completed-matrix inverse bounds, the exact target-grid chirp factorization, and the SPD Toeplitz circuit guard estimate. `FORWARD-CELL-CONTROLS.json` checks the coordinate and matrix identities. + +These are interface components for the parent's focused partial-fusion proposal. They do not independently certify its full tensor schedule or final exponent. + +- `CAPPED-GAUGE-AUDIT.md` independently validates capped dyadic interiors and exact reanchoring of the already-rounded narrow matrix, with local log range O(aB) rather than O(ad). It also validates the broad O(B/alpha^2) gauge and whole-boundary-group solve order. `CAPPED-GAUGE-CONTROLS.json` contains 27,216 partition checks and exact narrow/broad gauge controls. The capped final exponent ledger remains a separate assembly artifact. diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-ALGORITHM.md b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-ALGORITHM.md new file mode 100644 index 0000000000..f07f2c68c7 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-ALGORITHM.md @@ -0,0 +1,166 @@ +# A complete upper-inverse data algorithm with linear precision overhead + +Status: the construction below closes the one-dimensional **data application** of the nonperiodic upper inverse in `O(s p polylog p)` bit operations, after fixed per-axis coefficient preprocessing, using `C p` internal precision for an absolute constant C. It does not obtain logarithmic internal guard bits and does not prove tensor fusion. The lower-factor algorithm in LOWER-FACTOR.md does have logarithmic guard overhead. + +Let `p` be the required output precision, `a>=1`, `theta>=p^-O(1)`, and `a=O(p)`. In the target family a is fixed or logarithmic and every line length s grows faster than any fixed power of p. Preprocessing depending on the axis is shared by all its lines, as allowed by the parent reduction; it is not charged anew for every line. The coefficient words actually read by one data application have total `O(s p polylog p)` bits. + +## 1. Upper inverse is a stable suffix scan plus bandwidth O(p/a) + +Use the column diagonal `e_i=exp(-a rho^2 i^2+a beta_i^2)`. Define + +`B_j=e_j prod_(0<=k=i`, write + +`C_i,j=e_(j-i)(x_0,...,x_(j-1))/prod_(i<=k0. Put `d=m-i`, + +`gamma_i=(U^-1)_i,m`, + +`tau_j=(-1)^(j-m)(B_j/B_m)`. + +An exact Laurent-polynomial coefficient identity gives + +`(U^-1)_ij=gamma_i tau_j sum_(h>=0) (alpha_(d+h)/alpha_d) beta_j,h`. + +The signs are already in gamma and tau; the remaining coefficient sum is nonnegative. Both gamma and tau are polynomially bounded, with the same exponential-distance primary bounds as before. + +To verify the identity, write X=x_m Z and factor + +`P_j(X)=X^m prod_(k=1)(1+r^-k) <= 2/c`. + +A truncated product/sum tree computes M_0,...,M_R in `O((|J|+R) polylog p)` scalar operations. At a split, its exact rule is `S=S_left+P_left S_right`, with P a fixed product. Intermediate norms are bounded by a constant times the absolute weighted input sum, independently of tile length. No large Gaussian diagonal occurs in this step. + +## 3. Exponentially small high degrees + +The partition bound on alpha and beta gives + +`alpha_(d+h)/alpha_d <= c^-1 r^(-d*h-h(h+1)/2)`, + +`beta_j,h <= c^-1 r^(-h(h-1)/2)`. + +Thus the h-term, after summing input weights, is bounded by + +`c^-2 r^(-d*h-h^2) sum_j |tau_j u_j|`. + +Group row distances dyadically: D<=d<2D. For that group retain only + +`R=min(|J|, O(sqrt(P)), O(P/D))`, `P=(p+a+log(1/theta))/a`, + +with sufficiently large absolute constants. If |J| is smaller, the polynomial is already exact. Otherwise all omitted terms sum to less than `2^-p` after allowing the polynomial gamma/tau factors. There are O(log p) row groups. + +## 4. Fast normalized Hankel application, with Cp internal bits + +The remaining map for a row group is + +`y_d=sum_(0<=h<=R) (alpha_(d+h)/alpha_d) M_h`. + +Split the d-range into consecutive blocks of length at most R (or the whole group if shorter). For a block beginning at d0, divide all alpha coefficients used by alpha_d0. Then the numerator is one reversed ordinary convolution of that coefficient segment with M. Divide output d by alpha_d/alpha_d0. + +The decisive height bound is local, not global. If k>=0, comparing dominant products and using q-gaps in {1,2} gives + +`c r^(-2*d0*k-k(k+1)) <= alpha_(d0+k)/alpha_d0 <= c^-1`. + +Here the lower bound is needed only when the coefficient exists; every queried alpha index is <=m. If alpha vanishes because its degree exceeds m, store exact zero and never divide by it. Since d0=O(D), k=O(R), and `D R+R^2=O(P)`, every nonzero coefficient ratio and every divisor in this block lies between `exp(-O(aP))` and an absolute constant. Thus **Cp internal precision** suffices for these convolutions and the final divisions. + +The number of scalar operations for one row group is `O(D polylog p)`; its separate bounded-prefix computation costs `O((|J|+R) polylog p)`. Summing the O(log p) row groups gives `O((|I|+|J|) polylog p)` scalar operations per ordered tile. Recursing through one W-page or processing adjacent pages therefore costs `O(W p polylog p)` bits. Over all pages the data cost is + +`O(s p polylog p)`. + +The Gaussian factors gamma and tau can be evaluated from bounded dominant-product ratios rather than raw e_i or B_i. Precomputed alpha tables can similarly be stored with dominant-product gauges. Every alpha segment actually read by the data algorithm has length O(|I|+|J|) per row-group and tile level; this accounts for the coefficient tape scans in the same bound. + +A conservative preprocessing implementation may compute every required alpha table from O(poly(p)) nearby nodes separately, and can therefore cost `s poly(p)` per axis. This is a one-time axis cost, not an unpriced per-line cost. In the parent regime it is negligible compared with the full tensor data volume. A uniformly quasi-linear preprocessing theorem is not asserted here. + +## 5. Complete nonperiodic and periodic application + +Combining this upper map with the lower map and the bounded diagonal gives the complete nonperiodic inverse `G^-1=U^-1 C^-1 L^-1` in `O(s p polylog p)` data time, at Cp internal precision, with polynomial operator norms. Requested final error is achieved by a fixed additional factor in precision plus O(log p+log(1/theta)) endpoint guard bits. + +For the periodic matrix one may either implement the bilateral factors in BILATERAL-LDU.md or use a finite boundary correction. The latter needs only two nonperiodic inverse data applications: + +- Truncate nonzero Gaussian aliases to the two corner blocks of width `b=O(sqrt((p+log(1/theta))/a)+1)`, obtaining `K=G+UV` and a row-norm tail smaller than `2^-Cp`. +- U inserts O(b) boundary coordinates and V reads a bounded O(b)-by-O(b) corner matrix. +- Precompute `S^-1`, `S=I+V G^-1 U`. Its norm is polynomial because `S^-1=I-V K^-1 U` and the periodic inverse norm plus the tiny discarded tail bounds K inverse. +- Apply `x=G^-1 z`, `w=S^-1 Vx`, and return `x-G^-1 Uw`. + +The boundary data work is polynomial in p. For the target line lengths s growing faster than every fixed polynomial in p, it is negligible relative to sp. The updated matrix in the Woodbury inverse identity is K, not the base G. The periodic Gaussian tail and final correction error are bounded using the fixed-a theorem in BILATERAL-LDU.md. + +## What is still not established + +The normalized Hankel step uses Cp working precision because its local alpha divisors may be exp(-Cp). It has not been converted to a circuit whose every intermediate operator has polynomial norm. Therefore this result does not meet the stronger **logarithmic guard at every internal stage** requirement needed by the proposed full-dimensional fusion. It supplies a complete one-dimensional data algorithm and a precise remaining normalization target; it supplies no improved final kappa. + +## 6. Direct periodic implementation from the bilateral factors + +The bilateral theorem also permits the Woodbury step above to be omitted. This is cleaner in the target regime `s>>W`. + +For the bilateral lower inverse, use + +`C_i=prod_(ki)(1-x_i/x_k)`. + +The same lower formula and scan/correction split hold, with the defining products now bilateral. The adjacent A ratio is still given by the beta formula in LOWER-FACTOR.md. For the upper inverse, use elementary polynomials in the infinite prefix `x_k, k=1-exp(-a)`. Intermediate partial products and scan norms have the polynomial bounds already proved. This supplies a stable two-pass scan on the original s records. + +Retain the correction only for one-sided distances below W on the integer cover of the circle. With `s>>W`, each retained coefficient belongs to its diagonal page or the single adjacent cyclic page; virtual negative or beyond-s indices are generated by periodic q/beta formulas. The same local tile algorithms apply. Periodization cannot increase the row norm of any discarded tail, so all the previous error estimates remain valid. + +In the upper tile algorithm, alpha_d is now the coefficient of the infinite small-root prefix + +`prod_(k=d+O(q/a+1)`: a contribution involving an omitted node has partition deficit at least K-d, and summing the corresponding partition tail gives `O(r^(-(K-d)/2))` relative error (bound the tail by r to half the negative deficit times the convergent partition generating product). For the O(W)-degree tables at q=Cp, this is O(W+p/a) nodes, independent of s. Normalized table storage and data access are unchanged. + +Thus, after fixed per-axis coefficient preprocessing, the bilateral factorization itself gives the **periodic** inverse data application in `O(s p polylog p)` time with Cp working precision, in the target regime `s>>poly(p)`. It needs neither a large raw Vandermonde nor a boundary Schur system. The unresolved logarithmic-internal-guard and tensor-fusion qualifications remain exactly as stated above. diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-CONTROLS.json b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-CONTROLS.json new file mode 100644 index 0000000000..0d9410b6a8 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-DATA-CONTROLS.json @@ -0,0 +1,12 @@ +{ + "status": "all exact assertions passed", + "arithmetic": "fractions.Fraction", + "checks": { + "instances": 75, + "suffix_recurrences": 2715, + "tile_entries": 6413, + "signed_prefix_tree_and_hankel_outputs": 1833, + "alpha_ratio_bounds": 10008 + }, + "scope": "Algebra, signs, degree indexing, suffix-tail recurrence, and signed blocked upper data application. Cp-bit stability and asymptotic data cost are analytic, not certified by finite tests." +} diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-LIFTING.md b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-LIFTING.md new file mode 100644 index 0000000000..0985ad16cd --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-LIFTING.md @@ -0,0 +1,57 @@ +# The upper factor has a length-preserving Newton merge + +This is a structural advance, not yet the missing fast normalized FFT implementation. It concerns the nonperiodic factor from LOWER-FACTOR.md. + +Write `d_i=exp(-a rho^2 i^2+a beta_i^2)` in this note (this is the column diagonal called `e_i` in LOWER-FACTOR.md). The upper inverse is + +`(U^-1)_ij=(d_j/d_i) [X^i] prod_(k=m and h=j-i, the logarithm of the last Gaussian factor is at most + +`a/4-a rho^2 h^2-a rho(2rho-1)h-2a rho^2(i-m)h`. + +Thus D inverse also has row norm `O(exp(a/4))`, and the shear B D inverse has polynomial norm. These statements hold on arbitrary signed/complex records. + +## Remaining realization problem + +The algebra is length preserving and the exact blocks have bounded norm. Directly executing their diagonal-Toeplitz-diagonal presentation is not yet safe for full-dimensional tensor fusion: the raw d_i ratio can have logarithmic range `Theta(a(r-l)^2)`. A local FFT using that presentation therefore reintroduces an O(p) guard for blocks of length sqrt(p/a). + +A plain unit triangular Toeplitz map can be implemented by recursive Toeplitz shears and in-place cyclic/negacyclic Fourier transforms, as suggested by the parent investigation. To close this upper-factor route, that primitive needs a version for the **Gaussian-conjugated** Toeplitz blocks above whose intermediate norms stay polynomial, or a different bounded lifting factorization of them. The final exact block norms alone do not prove the intermediate FFT normalization claim. + +The numerator Sylvester merge in the lower-factor moment tree remains a separate lifting problem; the one-product Newton merge here avoids that two-product merge for the upper factor only. diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-MERGE-CONTROLS.json b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-MERGE-CONTROLS.json new file mode 100644 index 0000000000..7e851839f8 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/UPPER-MERGE-CONTROLS.json @@ -0,0 +1,6 @@ +{ + "status": "all exact assertions passed", + "arithmetic": "fractions.Fraction", + "normalized_node_merges_and_shear_factorizations": 286, + "scope": "Algebra and degree indexing, not bounded FFT realization or tensor fusion." +} diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_bilateral.py b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_bilateral.py new file mode 100644 index 0000000000..b2f527e951 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_bilateral.py @@ -0,0 +1,45 @@ +#!/usr/bin/env python3 +from fractions import Fraction as F +from pathlib import Path +import json +from check_lower import matmul, inv, elem + +def pp2(n):return F(2)**n +def prod(v): + z=F(1) + for x in v:z*=x + return z +def complete(xs,n): + c=[F(1)]+[F(0)]*n + for x in xs: + for k in range(1,n+1):c[k]+=x*c[k-1] + return c[n] +def case(s,t,l,n): + q=lambda i:(2*t*i+s)//(2*s) + x=lambda i:pp2(2*s*t*q(i)) + e=lambda i:pp2(-t*t*i*i+(t*i-s*q(i))**2) + G=lambda i,j:pp2(-(t*j-s*q(i))**2+(t*j-s*q(j))**2) + C=lambda i,o:prod(1-x(k)/x(i) for k in range(o,i)) + L=lambda i,j,o:G(i,j)*prod(1-x(k)/x(i) for k in range(o,j))/C(j,o) if i>=j else F(0) + U=lambda i,j,o:e(j)/e(i)*complete([x(k) for k in range(o,i+1)],j-i) if j>=i else F(0) + ids=list(range(l,l+n)); LL=[[L(i,j,l) for j in ids] for i in ids];UU=[[U(i,j,l) for j in ids] for i in ids] + CC=[[C(i,l) if i==j else F(0) for j in ids] for i in ids] + assert matmul(matmul(LL,CC),UU)==[[G(i,j) for j in ids] for i in ids] + UI=inv(UU) + for ii,i in enumerate(ids): + for jj,j in enumerate(ids): + assert L(i+s,j+s,l+s)==L(i,j,l) + assert U(i+s,j+s,l+s)==U(i,j,l) + assert G(i+s,j+s)==G(i,j) + if j>=i: + assert UI[ii][jj]==(-1)**(j-i)*e(j)/e(i)*elem([x(k) for k in range(l,j)],j-i) + return n*n + +def main(): + instances=entries=0 + for s,t in [(2,3),(3,4),(3,5),(4,5),(4,7)]: + for l in [-5,-2,0]: + for n in [3,4,5]:entries+=case(s,t,l,n);instances+=1 + out={'status':'all exact assertions passed','arithmetic':'fractions.Fraction','negative_and_zero_origin_instances':instances,'finite_LDU_and_shift_entries':entries,'scope':'Finite-origin algebra, exact origin-shift correction, upper inverse formula and simultaneous periodic shifts; bilateral limiting argument is analytic.'} + Path(__file__).with_name('BILATERAL-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) +if __name__=='__main__':main() diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_capped_gauges.py b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_capped_gauges.py new file mode 100644 index 0000000000..d6f5a14cb2 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_capped_gauges.py @@ -0,0 +1,73 @@ +#!/usr/bin/env python3 +from fractions import Fraction as F +from pathlib import Path +import json + +def p2(n):return F(2)**n + +def carve(L,W,cap): + if L<=6*W:return [('B',0,L)] + ps=[('B',0,W)];pos=W;R=L-2*W + while R>4*W: + B=min(cap,1<<((R-W).bit_length()-1)) + ps.append(('I',pos,pos+B));pos+=B + ps.append(('B',pos,pos+W));pos+=W;R-=B+W + if R:ps.append(('B',pos,pos+R));pos+=R + ps.append(('B',pos,L)) + return ps + +def main(): + c={'partitions':0,'narrow_slope_pairs':0,'rounded_ratio_and_exact_reanchoring_pairs':0,'broad_slope_pairs':0} + for W in [1,2,3,5,9,17]: + for power in range(2,11): + cap=1<=W for x,y in zip(ins,ins[1:])) + budget=(L+cap-1)//cap+cap.bit_length()+10 + assert len(ins)<=budget + assert sum(b-a for typ,a,b in ps if typ=='B')<=W*budget + c['partitions']+=1 + for s in range(3,31): + for K in range(1,min(s,6)): + rho=F(s+K,s);theta=rho-1;lam=1/rho + q=[(2*(s+K)*i+s)//(2*s) for i in range(s)] + cells={} + for i in range(s):cells.setdefault(q[i]-i,[]).append(i) + # a/log(2)=4*s*K makes every original narrow weight dyadic. + exps=[4*(K*i-s*(q[i]-i))**2-s*s for i in range(s)] + aa=[p2(e) for e in exps];eta=p2(-2*s*s-100) + ah=[x-(1+i%4)*eta for i,x in enumerate(aa)] + for phase,inds in cells.items(): + for start in range(0,len(inds),3): + block=inds[start:start+3];m=block[0] + for i in block: + for j in block: + assert abs(exps[i]-exps[j])<=4*s*K*abs(i-j) + c['narrow_slope_pairs']+=1 + assert F(1,2)*p2(-4*s*K*abs(i-j))<=ah[i]/ah[j]<=2*p2(4*s*K*abs(i-j)) + kernel=F(2,3) if i!=j else F(1) + before=ah[i]*kernel/ah[j] + after=(ah[i]/ah[m])*kernel/(ah[j]/ah[m]) + assert before==after + c['rounded_ratio_and_exact_reanchoring_pairs']+=1 + # Broad gauges on target coordinates in the original physical phase cell. + target=[q[i] for i in inds];anchor=target[len(target)//2] + delta=(1-lam)*anchor-phase + def col(k): + x=k-anchor + return -((1-lam)*x*x+2*delta*x) + def row(k): + y=k-anchor + return lam*(1-lam)*y*y+2*delta*lam*y-delta*delta + for i in target: + for j in target: + assert abs(col(i)-col(j))<=abs(i-j) + assert abs(row(i)-row(j))<=abs(i-j) + c['broad_slope_pairs']+=1 + out={'status':'all exact assertions passed','arithmetic':'integers and fractions.Fraction','checks':c,'scope':'Capped dyadic carving, narrow exact log-slope, positive perturbed global weights, exact local reanchoring, and broad gauge log-slope. Final tensor schedule/ledger excluded.'} + Path(__file__).with_name('CAPPED-GAUGE-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) +if __name__=='__main__':main() diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_cyclic_scan.py b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_cyclic_scan.py new file mode 100644 index 0000000000..7f8ebba6af --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_cyclic_scan.py @@ -0,0 +1,25 @@ +#!/usr/bin/env python3 +from fractions import Fraction as F +from pathlib import Path +import json +from check_lower import matmul,inv,eye + +def scan(lam,u): + P=F(1);Q=F(0) + for a,b in zip(lam,u):P=a*P;Q=a*Q+b + prev=Q/(1-P);out=[] + for a,b in zip(lam,u):prev=a*prev+b;out.append(prev) + return out + +count=0 +for n in range(2,31): + for offset in range(7): + lam=[F((-1)**(i+offset)*(i%3+1),5+i%4) for i in range(n)] + u=[F((-1)**i*(2*i+offset+1),i+1) for i in range(n)] + A=eye(n) + for i,a in enumerate(lam):A[i][(i-1)%n]-=a + expected=[r[0] for r in matmul(inv(A),[[x] for x in u])] + assert scan(lam,u)==expected + count+=1 +out={'status':'all exact assertions passed','cyclic_signed_scan_systems':count,'scope':'Two-pass cyclic affine recurrence against dense exact inversion; bilateral coefficient estimates are analytic.'} +Path(__file__).with_name('CYCLIC-SCAN-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_dyadic_interiors.py b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_dyadic_interiors.py new file mode 100644 index 0000000000..9a970cb94e --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_dyadic_interiors.py @@ -0,0 +1,38 @@ +#!/usr/bin/env python3 +from pathlib import Path +import json + +def parts(L,w): + if L<=2*w:return [('B',0,L)] + out=[('B',0,w)];pos=w;R=L-2*w + while R>4*w: + B=1<<((R-w).bit_length()-1) + out.append(('I',pos,pos+B));pos+=B + out.append(('B',pos,pos+w));pos+=w;R-=B+w + if R:out.append(('B',pos,pos+R));pos+=R + out.append(('B',pos,L)) + return out +cases=edges=0 +for w in range(1,41): + for L in range(1,2001): + ps=parts(L,w) + assert ps[0][1]==0 and ps[-1][2]==L + assert all(a[2]==b[1] for a,b in zip(ps,ps[1:])) + ins=[x for x in ps if x[0]=='I'] + assert all((b-a)&(b-a-1)==0 for _,a,b in ins) + assert all(y[1]-x[2]>=w for x,y in zip(ins,ins[1:])) + assert len(ins)<=2+max(0,(max(1,L//w)).bit_length()) + assert sum(b-a for t,a,b in ps if t=='B')<=w*(8+max(1,L//w).bit_length()) + # For each new interior, all boundary neighbors reachable by an original band edge + # lie in its two adjacent retained blocks. Eliminating it can fill only their union. + labels=[-1]*L;bid=0 + for typ,a,b in ps: + if typ=='B': + for i in range(a,b):labels[i]=bid;bid+=1 + for _,a,b in ins: + ns=[labels[i] for i in list(range(max(0,a-w),a))+list(range(b,min(L,b+w))) if labels[i]>=0] + if ns:assert max(ns)-min(ns)<=10*w + edges+=len(ns) + cases+=1 +out={'status':'all exact assertions passed','cell_partitions':cases,'reachable_boundary_records':edges,'scope':'Dyadic lengths, exact coverage, w-wide separation, logarithmic block count, and O(w) Schur boundary span.'} +Path(__file__).with_name('DYADIC-INTERIOR-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_forward_cell.py b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_forward_cell.py new file mode 100644 index 0000000000..b91b7a6bad --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_forward_cell.py @@ -0,0 +1,40 @@ +#!/usr/bin/env python3 +from fractions import Fraction as F +from pathlib import Path +import json + +def p2(k):return F(2)**k +cases=entries=completion=0 +for s in range(3,22): + for t in range(s+1,min(2*s,s+5)+1): + rho=F(t,s);lam=1/rho;q=[(2*t*j+s)//(2*s) for j in range(s)] + cells={} + for j,i in enumerate(q):cells.setdefault(i-j,[]).append(i) + for c,inds in cells.items(): + m=inds[len(inds)//2];delta=F((t-s)*m-t*c,t) + assert abs(delta)<=F(1,2) + for k in inds: + betak=(rho-1)*k-rho*c + assert -F(1,2)<=betak=1+2*th-2*b + assert upper>=1+2*th+2*b + counts['nearest_exponent_checks']+=2 + for h in [-4,-3,-2,2,3,4]: + assert (rho*h+b)**2-beta(i+h)**2>=h*h-abs(h) + counts['far_exponent_checks']+=1 + # Both signs and asymmetric rows: no positivity hypothesis on filled off-diagonals. + for n in range(3,15): + for seed in range(6): + A=[[F(1) if i==j else F(((-1)**(i+j+seed))*((7*i+3*j+seed)%11+1),40*n) for j in range(n)] for i in range(n)] + original_gap=min(gap(A,i) for i in range(n)); original_norm=norm(A) + while len(A)>2: + k=(seed+len(A))%len(A);idx=[i for i in range(len(A)) if i!=k] + B=[[A[i][j]-A[i][k]*A[k][j]/A[k][k] for j in idx] for i in idx] + assert min(gap(B,i) for i in range(len(B)))>=original_gap + assert norm(B)<=original_norm + assert all(B[i][i]>0 for i in range(len(B))) + counts['schur_eliminations']+=1;counts['schur_entries']+=len(B)**2 + A=B + return counts +if __name__=='__main__': + out={'status':'all exact assertions passed','arithmetic':'fractions.Fraction','checks':run(),'scope':'Nearest/far Gaussian exponent inequalities and signed/asymmetric Schur row-gap preservation; the transcendental row-sum estimate and tensor transfer are analytic.'} + Path(__file__).with_name('LOG-A-SCHUR-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_lower.py b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_lower.py new file mode 100644 index 0000000000..1260c26dd1 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_lower.py @@ -0,0 +1,92 @@ +#!/usr/bin/env python3 +"""Exact finite controls for LOWER-FACTOR.md; no floating point.""" +from fractions import Fraction as F +from math import prod +import json +from pathlib import Path + +def matmul(A,B): + return [[sum((a*b for a,b in zip(row,col)),F(0)) for col in zip(*B)] for row in A] +def eye(n): return [[F(i==j) for j in range(n)] for i in range(n)] +def inv(A): + n=len(A); B=[list(r)+list(e) for r,e in zip(A,eye(n))] + for k in range(n): + v=B[k][k]; assert v; B[k]=[x/v for x in B[k]] + for i in range(n): + if i!=k: + v=B[i][k]; B[i]=[x-v*y for x,y in zip(B[i],B[k])] + return [r[n:] for r in B] +def poly_mul(a,b): + out=[F(0)]*(len(a)+len(b)-1) + for i,x in enumerate(a): + for j,y in enumerate(b): out[i+j]+=x*y + return out +def newton_poly(xs): + out=[F(1)] + for x in xs: out=poly_mul(out,[-x,F(1)]) + return out +def elem(xs,k): return (-1)**k*newton_poly(xs)[len(xs)-k] +def suffix_pull(vs,M): + """All =i)(1-v_k Z),M>, exact D&C transpose tree.""" + if not vs: return [M[0]] + if len(vs)==1: return [M[0]-vs[0]*M[1],M[0]] + mid=len(vs)//2; left,right=vs[:mid],vs[mid:] + P=[F(1)] + for v in right: P=poly_mul(P,[F(1),-v]) + LM=[sum((P[k]*M[h+k] for k in range(len(P))),F(0)) for h in range(len(left)+1)] + return suffix_pull(left,LM)[:-1]+suffix_pull(right,M[:len(right)+1]) + +def run(): + count={'instances':0,'inverse_entries':0,'tile_entries':0,'suffix_outputs':0,'exponent_bounds':0,'upper_entries':0} + for s in range(3,12): + for t in range(s+1,2*s+1): + rho=F(t,s); q=[(2*t*i+s)//(2*s) for i in range(s)] + beta=[rho*i-q[i] for i in range(s)]; x=[F(2)**k for k in q] + # Arbitrary exact rational diagonal tests all identities independent of Gaussian values. + d=[F(3)**(-q[i]*q[i]) for i in range(s)] + C=[prod((1-x[k]/x[i] for k in range(i)),start=F(1)) for i in range(s)] + D=lambda i,j:prod((1-x[j]/x[k] for k in range(j+1,i+1)),start=F(1)) + A=[d[i]*x[i]**i*C[i]/prod(x[:i+1],start=F(1)) for i in range(s)] + L=[[F(0)]*s for _ in range(s)] + for i in range(s): + for j in range(i+1): + L[i][j]=d[i]/d[j]*prod((x[i]-x[k] for k in range(j)),start=F(1))/prod((x[j]-x[k] for k in range(j)),start=F(1)) + LI=inv(L) + for i in range(s): + for j in range(i+1): + assert LI[i][j]==(-1)**(i-j)*A[i]/A[j]/D(i,j) + count['inverse_entries']+=1 + h=i-j + exponent=-beta[i]**2+beta[j]**2-rho**2*h+2*rho*sum(beta[j+1:i+1],F(0)) + assert exponent<=F(1,4)-rho*(rho-1)*h + count['exponent_bounds']+=1 + # Newton U inverse coefficients and V=L D U, with row diagonal removed. + V=[[x[i]**j for j in range(s)] for i in range(s)] + T=[[L[i][j]*d[j]/d[i] for j in range(s)] for i in range(s)] + U=matmul(inv(T),V) + for i in range(s): + pivot=prod((x[i]-x[k] for k in range(i)),start=F(1)) + U[i]=[v/pivot for v in U[i]] + UI=inv(U) + for i in range(s): + for j in range(i,s): + assert UI[i][j]==(-1)**(j-i)*elem(x[:j],j-i) + count['upper_entries']+=1 + # Every adjacent-cut tile, arbitrary signed inputs, exact moment/suffix algorithm. + u=[F((-1)**j*(2*j+1),j+1) for j in range(s)] + for l in range(1,s): + b=min(s-1,l+3); J=list(range(max(0,l-4),l)); I=list(range(l,b+1)) + z=[x[j]/x[l] for j in J]; v=[x[l]/x[k] for k in range(l+1,b+1)] + w=[(-1)**j*A[l]/A[j]/D(b,j)*u[j] for j in J] + M=[sum((wj*zj**m for wj,zj in zip(w,z)),F(0)) for m in range(len(I))] + pulled=suffix_pull(v,M) + for i,got in zip(I,pulled): + got*=(-1)**i*A[i]/A[l] + expect=sum((LI[i][j]*u[j] for j in J),F(0)) + assert got==expect + count['suffix_outputs']+=1; count['tile_entries']+=len(J) + count['instances']+=1 + return count +if __name__=='__main__': + out={'status':'all exact assertions passed','arithmetic':'fractions.Fraction','checks':run(),'scope':'Nonperiodic Newton L inverse and U inverse algebra; exact signed cross-tile data application; no runtime or tensor-fusion certificate.'} + p=Path(__file__).with_name('LOWER-CONTROLS.json');p.write_text(json.dumps(out,indent=2)+'\n'); print(json.dumps(out,indent=2)) diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_upper_data.py b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_upper_data.py new file mode 100644 index 0000000000..ad06229d84 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_upper_data.py @@ -0,0 +1,79 @@ +#!/usr/bin/env python3 +"""Exact checks of the upper suffix recurrence, tile identity, and blocked data algorithm.""" +from fractions import Fraction as F +from pathlib import Path +import json +from check_lower import poly_mul, elem + +def prod(xs): + z=F(1) + for x in xs:z*=x + return z + +def add(a,b): + out=[F(0)]*max(len(a),len(b)) + for i,x in enumerate(a):out[i]+=x + for i,x in enumerate(b):out[i]+=x + return out + +def prefix_sum(roots,weights): + """sum weights[j] product_{k=0 + assert delta<=2*F(4)**(-(j-i+1)) + out['suffix_recurrences']+=1 + u=[F((-1)**j*(j+2),j+1) for j in range(s)] + for m in range(1,s): + l=max(0,m-4);r=min(s,m+5);I=list(range(l,m));J=list(range(m,r)) + aa=[elem([x/xs[m] for x in xs[:m]],h) for h in range(m+1)] + tau=[(-1)**(j-m)*BB[j]/BB[m] for j in J] + gamma={i:U(i,m) for i in I} + roots=[xs[m]/xs[k] for k in range(m,r-1)] + moments=prefix_sum(roots,[tau[k]*u[j] for k,j in enumerate(J)]) + R=len(moments)-1 + for i in I: + d=m-i + for j in J: + b=[elem([xs[m]/xs[k] for k in range(m,j)],h) if h<=j-m else F(0) for h in range(R+1)] + v=sum((aa[d+h]/aa[d]*b[h] for h in range(R+1) if d+h<=m),F(0)) + assert gamma[i]*tau[j-m]*v==U(i,j) + out['tile_entries']+=1 + # Blocks of length two for normalized Hankel convolution. + for d0 in range(1,m-l+1,2): + B=min(2,m-l-d0+1) + a=[aa[d0+k]/aa[d0] if d0+k<=m else F(0) for k in range(B+R)] + cv=poly_mul(a,list(reversed(moments))) + for off in range(B): + d=d0+off;i=m-d + got=gamma[i]*cv[off+R]/(aa[d]/aa[d0]) + expected=sum((U(i,j)*u[j] for j in J),F(0)) + assert got==expected + out['signed_prefix_tree_and_hankel_outputs']+=1 + for d in range(1,m+1): + for k in range(m-d+1): + rat=aa[d+k]/aa[d] + assert F(1,2)*F(4)**(-2*d*k-k*(k+1))<=rat<=2 + out['alpha_ratio_bounds']+=1 + out['instances']+=1 + return out +if __name__=='__main__': + out={'status':'all exact assertions passed','arithmetic':'fractions.Fraction','checks':run(),'scope':'Algebra, signs, degree indexing, suffix-tail recurrence, and signed blocked upper data application. Cp-bit stability and asymptotic data cost are analytic, not certified by finite tests.'} + Path(__file__).with_name('UPPER-DATA-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) diff --git a/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_upper_merge.py b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_upper_merge.py new file mode 100644 index 0000000000..3dda87e144 --- /dev/null +++ b/research/capped-phase-fusion/evidence/gaussian_fast_ultra/check_upper_merge.py @@ -0,0 +1,43 @@ +#!/usr/bin/env python3 +from check_lower import F, poly_mul, newton_poly, matmul, eye, inv +from pathlib import Path +import json + +def local(xs,ds,l,r): + n=r-l;A=[[F(0)]*n for _ in range(n)] + for j in range(l,r): + P=newton_poly(xs[l:j]) + for k,p in enumerate(P):A[k][j-l]=ds[j]*p/ds[l+k] + return A + +def main(): + count=0 + for n in range(2,13): + xs=[F(2)**(i+i//3) for i in range(n)] + ds=[F(3)**(-i*i) for i in range(n)] + for l in range(n-1): + for r in range(l+2,n+1): + m=(l+r)//2;n1=m-l;n2=r-m + P=newton_poly(xs[l:m]);M=eye(r-l) + for i in range(l,r): + for j in range(m,r): + k=i-l-(j-m) + M[i-l][j-l]=ds[j]/ds[i]*P[k] if 0<=k4L` and `gL/N=o(1)`. +Write `b=log2 N`, `k=gb`, and `K0=O(g(h+b)+log g)=O(gh)` for a correction key. +Require `K0^2=O(B)` or retain the explicit extra metadata term below. + +The complete cost is + +``` +O(V [ (gb)^(1-a) polylog p + + (gb)^(1-q) polylog p + + polylog p + + (gL/N) (K0 + g polylog p) ]) ++ polynomial(p) setup, +``` + +plus `O((gL/N)V K0^2/B)` if metadata arithmetic is not absorbed by records. +All scans, duplication, sparse sorts, local transposes and final output +order are included. In particular, there is no gV layout term. + +## 1. One support-limited gather + +Within the active group, gather its g low b-bit fields into a contiguous +suffix using the support-at-most-2gb extension proved in +`SUPPORT-ROUTER.md`. The remaining active coarse bits remain in their +possibly scrambled order. Every coarse tile is now one contiguous block +of N^g complete R-payloads. No other active coarse bit is put into a new +stable order, and no spectator coordinate is gathered. + +The initial and inverse gather together cost +`O(V (gb)^(1-a) polylog p)`. The explicit permutation maps each physical +coarse tile name back to its logical vector; that small map is used only +for tile setup and sparse correction metadata. + +## 2. Interior outputs from ordinary cyclic convolution + +Flatten a tile by `phi(x)=sum_i x_i N^(g-1-i)` into Q=N^g slots. Put the +product kernel at offsets `psi(z)=sum_i z_i N^(g-1-i) mod Q` in an ordinary +one-dimensional cyclic convolution kernel. These offsets are distinct: +reduce a putative collision modulo N, use `2L= p^eta`, `g -> infinity`; alternatively a supplied sequential dense +kernel generator is an explicit acceptable input. + +If every tile coordinate satisfies `L<=x_ij: [0,N). +``` + +These 2g rectangles are disjoint and contain every discarded output once. +Their required unwrapped input rectangles have intervals + +``` +ij: [-L,N+L). +``` + +Coordinates outside `[0,N)` are taken from the correct neighboring coarse +tile, including periodic wrap in each actual axis. All input dimensions +are less than 2N; no assumption of independent full 2N padding is used. + +The total copied input volume is exactly + +``` +M = 6L V/N^g * sum_{j=0}^{g-1} N^j (N+2L)^(g-1-j) + = 3V [ (1+2L/N)^g - 1 ] + = O((gL/N)V). +``` + +The corrected output volume is +`V [1-(1-2L/N)^g] = O((gL/N)V)`. The formulas count corner duplication +exactly. An individual corner may have many destinations; its total traffic +is already included in M. No bound by worst-case 2^g multiplicity is charged +to all records. + +## 4. One full scan generates all correction copies + +For each original core tile, recover its logical coarse coordinates once. +Scan its local coordinate tuple with a binary odometer. Maintain flags for +membership in the four fixed edge intervals `[0,L)`, `[0,2L)`, +`[N-2L,N)` and `[N-L,N)`. Only coordinates changed by the odometer update +their flags. The number of coordinate updates over a whole tile is +`Q(1+1/N+...)<2Q`. Comparisons cost O(log N), and counters/head resets use +the retained amortized stream-counter convention. Maintain a count of +active near-boundary flags. A nonexceptional record takes no scan through +the g flags. + +For an exceptional record, enumerate candidate first-boundary axes from +the flag tape and all valid neighboring-tile choices. The exact choices +are the ones implemented in `sparse_copies` in `check_halo.py`: earlier +axes stay in the original tile; the selected face axis has its own or one +adjacent tile; each later axis has its own tile plus an adjacent tile only +when within L of that edge. Every emitted record carries + +``` +(j, side, destination coarse tile, unwrapped local coordinates). +``` + +These are K0=O(gh) bits. Precompute the plus/minus-one forms of each logical +coarse coordinate once per tile. A copy's logical neighbor is generated +by selecting those forms, then encoding the known scrambled coarse order. +All carries, periodic wraps and changed tile order are explicit. Using +ordinary binary arithmetic gives O(K0^2) safe metadata work per copy. + +The full scan costs O(V polylog p). The exceptional flag scans and address +work cost O(M K0^2/B+M) in bit-volume units when R includes the B-bit record. +Their setup per coarse tile is absorbed by its N^g payload. This is not an +unpaid nonlinear boundary-signature evaluation on every payload bit. + +## 5. Sparse radix gather and fixed-tape local processing + +Within each *earlier-coordinate prefix fiber*, stable-radix sort the copies +by the displayed key, with unwrapped local coordinates in ordinary tensor +order. Only the g active axes occur in the key; all following spectator +coordinates remain one R-block payload. A radix pass scans each complete +header and payload, splits into two bucket tapes, concatenates, and resets +the same fixed set of tapes. Since K0=O(B), each pass costs O(M); K0 passes +cost O(M K0). No simultaneous open tape per tile, face or corner is needed. + +Sorted copies form the exact input rectangles of section 3, consecutively. +Process one rectangle at a time. For each axis, expose its current local +coordinate field, apply the length-polynomial one-dimensional convolution, +and crop L positions from each end. Thus its length changes from its input +interval to the corresponding output interval. Other axes remain unchanged +until their own pass. This yields the tensor product of the restricted +one-dimensional maps, hence the exact desired face outputs. + +Every current axis has length at most N+2L=poly(p). Its mixed-radix transpose +is the retained short-field construction: pad **only that field** to its +next power of two temporarily, move its O(log p) bits using O(log p) linear +block interchanges, remove the temporary padding, and use the correct new +field length on return. The temporary factor below two is removed before +the next axis. A rectangle therefore costs its volume times +`O(g polylog p)` for movement and unconditional polynomial-length arithmetic. +Summing over rectangles costs `O(M g polylog p)`. Signed/complex dyadic +convolution can use sign splitting and ordinary Kronecker packing with +O(p+log N)=O(p) guard bits, followed by the unconditional quasilinear integer +multiplier; no improved supplier is used on this sparse path. + +After cropping, attach each output's physical coarse-tile and flat inner +address. Their keys are distinct because the face output partition is +disjoint. A second sparse radix sort orders them as holes in the retained +core-output stream. One merge scan replaces exactly those holes. The inverse +support-limited gather restores the original axis-major stream, including +the initially displaced coarse coordinates. All metadata is discarded. + +## 6. Parameters for the proposed partial fusion + +Use `d=p^epsilon`, `h=Theta(p/d)`, `g=p^(1-epsilon-zeta)` with fixed positive +zeta and epsilon>5/6. Suppose `L=O(p polylog p)` and take the dyadic side +`N=2^ceil((3/2)log2 p)`. The actual S_i eventually exceed every polynomial, +so N divides them and each periodic window is shorter than S_i. + +Then `K0=O(p^(2(1-epsilon)-zeta) polylog p)` and `K0^2=o(p)`. The two sparse +cost multipliers satisfy + +``` +(gL/N) K0 = O(p^(5/2-3epsilon-2zeta) polylog p) = o(1), +(gL/N) g = O(p^(3/2-2epsilon-2zeta) polylog p) = o(1). +``` + +All halo work is therefore O(V polylog p), below the desired +`O(V g^(1-a) polylog p)` routing term. The core convolution is +`O(V g^(1-q) polylog p)` since log N=Theta(log p). + +The bounded-gauge condition cited in `PARTIAL-FUSION.md` remains payable: +with `a0=Theta(log d)` and theta=Theta(1/d), its variation is + +``` +g [a0 theta N + a0/theta] + = O(p^(5/2-2epsilon-zeta) polylog p + + p^(1-zeta) polylog p) + = o(p) +``` + +for epsilon>5/6 and fixed zeta>0. Thus the larger polynomial tile does not +consume the O(p)-bit guard budget. This checks a layout/precision inequality; +it does not establish the missing Gaussian convolution factorization. + +## Diagonal and factorization boundary + +The theorem also applies to a fixed or polylogarithmic composition of these +translation-invariant convolutions and diagonal factors if every diagonal +scan/factor page has a proved sequential O(V polylog p) implementation and +all normalized tile-core and tile-halo values use O(p) bits. A tile-dependent +common scalar gauge may multiply every input to that tile and divide its +outputs; boundary copies must use the **destination** tile's gauge. No +enormous diagonal is formed and later canceled numerically by assertion. + +A concrete sufficient diagonal implementation is available for +`D(x)=product_i d_i(x_i)`. Supply each one-axis factor page in coordinate +order, with all normalized prefix products bounded to O(p) bits. Traverse +the rectangle by nested coordinate loops and retain the g successive prefix +products on work tape. Multiply by the innermost factor once per record; +outer prefix updates occur once per complete inner row, then once per +complete inner rectangle, and so on. With side lengths at least two their +number is less than twice the record count. Singleton axes are folded into +one constant per rectangle. Rewinding a factor page costs its length and is +charged to the corresponding completed row. Ordinary O(p)-bit multiplication +therefore gives O(V polylog p) scan cost with O(gp) temporary bits. This +argument uses bounded normalized **prefix products**; bounds on the final +product alone do not suffice. + +It does **not** cover an unbounded collection of different short kernels +inside one tile, a product-tree level with factor-two padding on all g axes, +or an unspecified transform-dependent sparse correction. Those require a +new representation theorem. In particular, the Gaussian_fast/product-tree +algorithm is not silently substituted for the translation-invariant kernel +assumed here. + +## Finite evidence + +`check_halo.py` verifies 5,632 exact outputs in three two/three-axis cyclic +examples, with asymmetric signed kernels, nontrivial diagonal conjugations, +periodic coarse carries, scrambled coarse order, unique sparse gather/scatter +keys and all local sequential crops. Pure flattening gives 3,040 wrong +boundary outputs across these examples; all are repaired. The exact copied +volumes equal `3V[(1+2L/N)^g-1]` in every case. These are constructive finite +controls; the preceding scan, metadata and volume arguments give the +conditional asymptotic claim. diff --git a/research/capped-phase-fusion/evidence/halo_layout_ultra/MANIFEST.json b/research/capped-phase-fusion/evidence/halo_layout_ultra/MANIFEST.json new file mode 100644 index 0000000000..48dbb2a6bf --- /dev/null +++ b/research/capped-phase-fusion/evidence/halo_layout_ultra/MANIFEST.json @@ -0,0 +1,20 @@ +{ + "EMBEDDING-CONTROLS.json": "d0182e172f16b55ec64a5edfabb10816e416348ef2db15449f3ed4b47f3fec1f", + "HALO-CONTROLS.json": "ae370f7d7421f2f96e101a4b9d8082110507c47211220fd6ead065c2366aa828", + "HALO-THEOREM.md": "cb9e1da38400b658f9209e465b4e5896fadd17dc66afa862f38bae8d8c1f2058", + "PARAMETER-CHECK.json": "e38cdfc13f1959186d888383458c9030c4bd3c8c3915aea4743d55637d2f8d76", + "RAGGED-CONTROLS.json": "c3688f65565486b585727374abfbe478f7544aa61381bd73cc6ffa168b4221b6", + "RAGGED-PERIODS.md": "c6e4dfe7f58dac7c5a513ffb3a9ab4801ab0717efae5c1e764e570335402ff25", + "README.md": "575a3acfc18a0e05ffe04a8563318a2770ce32cd7bf3a5e6123ce2de39a1cd4c", + "RECTANGULAR-EMBEDDING.md": "e64447bff7e7717802ee049290faaf29e45db3e36a6f7fcec109184478f15f26", + "SPARSE-SHEAR-CONTROLS.json": "d3b9763f7ef6a3a5752c6c11a8a4d5f465646ad75b178a1c1b3363540fcc4871", + "SUPPORT-ROUTER.md": "6ddb322e11883c1177ba6cd1a77d23a1dac827d7290b32eaf642457b1d762a54", + "check_embedding.py": "597cb5304498456575c99d46a7fb503e04db28d5a7f57857b102b50f703b7464", + "check_halo.py": "0a3257cbb8f37b6cc9bb44019cb114ea2ae9b28738a416bff4bc712d1306e1b7", + "check_ragged.py": "54b761cfe3404d524af3a2abdf09bcea8dfb5cca6199c0c182cee310e3ca8c78", + "check_sparse_shear.py": "6e22c989e6bae92c4cd209c9336aad4359956cf370bdbd5f6b70609144598121", + "embedding.log": "d0182e172f16b55ec64a5edfabb10816e416348ef2db15449f3ed4b47f3fec1f", + "halo.log": "ae370f7d7421f2f96e101a4b9d8082110507c47211220fd6ead065c2366aa828", + "ragged.log": "c3688f65565486b585727374abfbe478f7544aa61381bd73cc6ffa168b4221b6", + "sparse-shear.log": "d3b9763f7ef6a3a5752c6c11a8a4d5f465646ad75b178a1c1b3363540fcc4871" +} diff --git a/research/capped-phase-fusion/evidence/halo_layout_ultra/PARAMETER-CHECK.json b/research/capped-phase-fusion/evidence/halo_layout_ultra/PARAMETER-CHECK.json new file mode 100644 index 0000000000..5844ead5c0 --- /dev/null +++ b/research/capped-phase-fusion/evidence/halo_layout_ultra/PARAMETER-CHECK.json @@ -0,0 +1,10 @@ +{ + "epsilon": "0.99833013302245016130167750009820221155623107175809", + "g_exponent": "0.00166985697754983869832249990179778844376892824191", + "K0_exponent": "0.00333972395509967739664499980359557688753785648382", + "halo_sort_exponent": "-0.4949904190673504839050325002946066346686932152743", + "halo_local_axes_exponent": "-0.4966602860449003226033550001964044231124621435162", + "tile_gauge_exponent": "0.50333972395509967739664499980359557688753785648381", + "K0_squared_exponent": "0.00667944791019935479328999960719115377507571296764", + "initial_gather_exponent": "0.0016669624023127060364416759538459168913455129064535" +} diff --git a/research/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-CONTROLS.json b/research/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-CONTROLS.json new file mode 100644 index 0000000000..553ea1a36a --- /dev/null +++ b/research/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-CONTROLS.json @@ -0,0 +1,36 @@ +{ + "status": "PASS_RAGGED_TRUE_PERIOD_SPARSE_HALO", + "cases": [ + { + "S": [ + 19, + 22 + ], + "N": 8, + "L": 1, + "g": 2, + "input_records": 418, + "halo_records": 732, + "full_halo_candidate_pairs": 700, + "corrected_outputs": 274, + "source_and_rectangle_gathers_identical": true, + "true_period_tensor_convolution_checked": true + }, + { + "S": [ + 17, + 18, + 20 + ], + "N": 8, + "L": 1, + "g": 3, + "input_records": 6120, + "halo_records": 18064, + "full_halo_candidate_pairs": 14352, + "corrected_outputs": 4392, + "source_and_rectangle_gathers_identical": true, + "true_period_tensor_convolution_checked": true + } + ] +} diff --git a/research/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-PERIODS.md b/research/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-PERIODS.md new file mode 100644 index 0000000000..fdfe26d197 --- /dev/null +++ b/research/capped-phase-fusion/evidence/halo_layout_ultra/RAGGED-PERIODS.md @@ -0,0 +1,89 @@ +# True periods inside a near-one dyadic container + +The primary halo theorem extends to arbitrary true periods s_i when an +already supplied dyadic rectangle t_i>=s_i has bounded total inflation +`product t_i / product s_i=O(1)`. Simultaneous zero embedding and its inverse +cost one ordered scan each by `RECTANGULAR-EMBEDDING.md`. This prerequisite +is satisfied only where the inherited near-one capacity bound applies; it +must not be replaced by independently rounding every axis to its next power. + +Take N dividing every t_i. Write `s_i=q_i N+r_i`, `0<=r_i0, one ragged core in axis i. The initial +support-limited gather uses the dyadic storage coordinates, not mixed-radix +division. It may produce invalid all-zero storage tiles; discard those. + +Apply ordinary flat cyclic convolution only to tiles whose every active +side has valid length N. Their interior outputs remain exact. Regular tile +face corrections use the earlier first-boundary-axis rectangles, but every +halo coordinate is now reduced modulo **s_i**, never t_i. + +If any side of a tile is ragged, discard its entire flat-convolution output. +For its actual side lengths r_i(tile), compute every valid output in +`product [0,r_i(tile))` using the input rectangle +`product [-L,r_i(tile)+L)`. All sides are at most N+2L. The same g successive +polynomial-length convolution/crop passes therefore produce exact outputs. +Full ragged tiles and regular face outputs form a disjoint correction set. + +Let Q_i=q_i+1_(r_i>0). The total full-halo volume of *all* valid tiles is + +``` +H_all = product_i [s_i + 2L Q_i]. +``` + +Their all-full-tile contribution is `product_i q_i(N+2L)`. The sum of the +ragged-tile halo volumes is their difference. Expanding by a first ragged +axis, or simply telescoping the product, gives + +``` +H_ragged <= H_all * sum_i (N+2L)/s_i + = O(S gN/s_min), +``` + +since `H_all/S <= exp(2L sum_i Q_i/s_i)=O(1)` when gL/N=o(1). +Regular face input volume is O(S gL/N). Thus total sparse copy volume is + +``` +M=O(V [gL/N + gN/s_min]). +``` + +The new term is smaller than V divided by every fixed polynomial in p when +`s_min=2^Theta(p/d)` and `N,g=poly(p)`. It pays all the same key sorting, +metadata and local computation factors. No new per-axis full scan appears. + +## One source-driven enumerator + +For each valid source coordinate x_i, list the valid destination tiles t +for which an unwrapped representative + +``` +y_i = x_i - tN + k s_i +``` + +lies in `[-L,r_i(t)+L)`. Only the home and neighboring core intervals can +qualify; there are at most three choices because L5/6, +all sparse halo work is O(V polylog p). The initial gather costs +`O(V g^(1-a_bit) polylog p)` and the grouped convolution costs +`O(V g^(1-q) polylog p)`. The cited bounded-gauge variation is still o(p). +Actual parameter exponents are recorded in `PARAMETER-CHECK.json`. + +Read: + +1. `SUPPORT-ROUTER.md`: arbitrary-spacing guarded shears, support-sized + reservoirs, fixed-tape repair, and the support-at-most-2k gather. +2. `HALO-THEOREM.md`: tensor identity, exact sparse halo volume, all record + movement, key lengths, local transposes, final output order and costs. +3. `RECTANGULAR-EMBEDDING.md`: simultaneous near-one zero embedding and crop + in one scan, using B=Omega(p) record width and an O(p)-bit descriptor. +4. `RAGGED-PERIODS.md`: arbitrary true periods inside that dyadic container, + using full ragged tiles as exponentially sparse additional exceptions. + +No rank/select oracle or sort of the complete tensor by a computed key is +used. Full scans preserve stream order. Only sparse correction records +are sorted, with every key bit pass charged. + +Reproduce, from the chat workspace: + +``` +python3 -B work/halo_layout_ultra/check_sparse_shear.py +python3 -B work/halo_layout_ultra/check_halo.py +python3 -B work/halo_layout_ultra/check_embedding.py +python3 -B work/halo_layout_ultra/check_ragged.py +``` + +Controls include an exhaustive 32,768-address shear permutation, 4,000 +unequal-spacing probes, 5,632 regular-grid convolution outputs and 6,538 +true-period inputs for ragged gather/correction checks. Wrong naive cyclic +flattening is explicitly detected before correction. Source/result hashes +are in `MANIFEST.json`. + +This is not a complete Gaussian resampler or a larger multiplication +exponent certificate. The kernel must be translation invariant and +separable; a fixed/polylogarithmic composition with bounded and sequentially +generated diagonal conjugations is admitted. A product tree with many +different short kernels or per-axis factor-two padding is not admitted by +renaming it a convolution. The missing Gaussian representation/normalization +theorem remains separate. The original bit/complex supplier, fixed-tape, +precision and all-size interfaces remain conditional inputs. + +Prepared with substantial OpenAI Codex assistance, continuing the retained +arbitrary-source router and bulk-resampling proofs in +`work/integer-mult-current/references/semantic-bulk/rad20/reports/`. +No frozen source or selected certificate was changed. diff --git a/research/capped-phase-fusion/evidence/halo_layout_ultra/RECTANGULAR-EMBEDDING.md b/research/capped-phase-fusion/evidence/halo_layout_ultra/RECTANGULAR-EMBEDDING.md new file mode 100644 index 0000000000..191867c72e --- /dev/null +++ b/research/capped-phase-fusion/evidence/halo_layout_ultra/RECTANGULAR-EMBEDDING.md @@ -0,0 +1,51 @@ +# One-pass near-one rectangular embedding and crop + +This supplies the dyadic-container qualification for the halo theorem. It +does not require one full payload scan per axis. + +Let input axes have arbitrary lengths `s_1,...,s_D` and let dyadic target +lengths `t_i>=s_i` be given, with total address width +`A=sum_i log2 t_i=O(p)`. The input and output use lexicographic axis-major +order, the last coordinate changing fastest. Every payload record has at +least `c p` bits, as in the retained coefficient representation. Assume +`T=product t_i <= C product s_i = C S`; C is the separately proved +near-one capacity bound, not a product of per-axis dyadic roundings. + +Enumerate the complete target rectangle in lexicographic order with its +ordinary A-bit binary counter. For each target address x, output the next +source record if `x_i infinity`; short remaining cases can use the retained +full-span router, since `n=O(k polylog p)` there. The conditional cost is + +``` +O(V (k+1)^(1-a) polylog p + poly(p)). +``` + +All other coordinate positions return to their original values. This is a +support bound, not a bound on how far a moved coordinate travels. + +## Unequally spaced target lemma + +Put `B=2^G`, `G=4 ceil(log2 p)+6`, and `K=64G`. Take target positions +`j_0<...2(k+1)ell`, a gap outside the moved support contains a contiguous ell-bit +field S disjoint from the actual last ell bits. Use an original equal-ell +interchange A to move S to that actual suffix. Route the conjugate +`A pi A^-1` on the n-ell prefix only, then undo A. The conjugate fixes the +suffix pointwise because pi fixed S. Choose every dirty control reservoir +inside the prefix after this interchange, then recompute positions and +residue colors. Thus no control operation touches the enlarged record. +Its 2^ell complete values absorb the polynomial descriptor work; no data +is padded or duplicated. If `n<=2(k+1)ell`, the retained full-span bound +already gives `O(V (k+1)^(1-a) polylog p)`. The placement itself costs only +O(V polylog p). This explicit suffix-conjugation bookkeeping was +independently supplied in `work/inplace_algebra_ultra/SUPPLIER-SUPPORT-AUDIT.md`. + +## The gather uses only S union T + +Let S list the k selected low-coordinate bits in the desired tensor order, +and T list k consecutive destination slots. Send S bijectively to T. Send +the remaining domain slots `T\S` bijectively to the remaining output slots +`S\T`, and fix everything outside `S union T`. This explicitly extends the +requested gather to a coordinate permutation supported on at most 2k bits. + +The other coarse-coordinate bits can be scrambled by this extension. Their +order is retained and described by the inverse permutation; **no stable +compaction of all coarse fields is performed**. A coarse neighbor address +can be decoded, incremented in its logical axis, then encoded in this +physical coarse order when constructing a sparse correction key. + +## Evidence and boundary + +`check_sparse_shear.py` checks all 32,768 addresses for targets 0 and 6 with +guard length 5, including 512 good addresses, inverse composition and every +bad repair. It also checks 4,000 arbitrary/constructed-good addresses with +unequal gaps 9 and 11 and sources inside guard fields. `check_halo.py` +checks three explicit support-at-most-2k gather permutations. + +These controls support the algebra; the cost argument above is the +all-size conditional transfer. It retains the original bit-supplier, +equal-width interchange, fixed-tape and common-label hypotheses. It does +not upgrade an arbitrary full computed-key sort to a fast router. + +Sources: the retained reports `compact-arbitrary-source-routing.md` and +`review-arbitrary-routing.md` under +`work/integer-mult-current/references/semantic-bulk/rad20/reports/`. diff --git a/research/capped-phase-fusion/evidence/halo_layout_ultra/check_embedding.py b/research/capped-phase-fusion/evidence/halo_layout_ultra/check_embedding.py new file mode 100644 index 0000000000..54556ee93d --- /dev/null +++ b/research/capped-phase-fusion/evidence/halo_layout_ultra/check_embedding.py @@ -0,0 +1,22 @@ +"""Exact simultaneous zero embedding/cropping in lexicographic order.""" +from itertools import product +from pathlib import Path +import json,math +HERE=Path(__file__).resolve().parent +def check(s,t): + original=list(product(*(range(n) for n in s)));it=iter(enumerate(original,1)) + embedded=[];valid=[] + for x in product(*(range(n) for n in t)): + if all(a=N-2*L] + for j in active: + for side in (0,1): + choices=[] + for i,a in enumerate(x): + if i=N-L:opts.append((1,a-N)) + else: + if a>=N-2*L:opts.append((0,a)) + if a=N-L:opts.append((1,a-N)) + choices.append(opts) + for selected in product(*choices): + dest=tuple((tile[i]+selected[i][0])%Q[i] for i in range(g)) + y=tuple(v[1] for v in selected) + result.append(((j,side,dest,y),value)) + return result + +def radix(records,key): + pairs=[(key(k),k,v) for k,v in records] + bits=max((k.bit_length() for k,_,_ in pairs),default=0) + for b in range(bits): + zero=[];one=[] + for row in pairs:(one if row[0]>>b&1 else zero).append(row) + pairs=zero+one + assert [k for k,_,_ in pairs]==sorted(k for k,_,_ in pairs) + return [(k,v) for _,k,v in pairs],bits + +def key_encoder(N,L,g,Q): + # Deliberately scramble coarse address significance. This represents the + # displaced coordinate slots in a support-limited gather. + coarse=[(i,j) for i,q in enumerate(Q) for j in range(q.bit_length()-1)] + coarse=coarse[::2]+coarse[1::2][::-1] + def enc(key): + j,side,tile,y=key + out=j*2+side + for i,b in coarse:out=(out<<1)|((tile[i]>>b)&1) + for a in y:out=(out*(N+2*L))+(a+L) + return out + return enc + +def replay(N,L,g,Q): + assert N>4*L and N&(N-1)==0 and all(q>=2 and q&(q-1)==0 for q in Q) + S=[N*q for q in Q];global_ranges=[(0,s) for s in S] + raw={x:1+sum((i+3)*a*a+(i+1)*a for i,a in enumerate(x)) for x in box(global_ranges)} + pre=lambda x:1+sum((i+1)*a for i,a in enumerate(x))%3 + post=lambda x:1+sum((i+2)*a for i,a in enumerate(x))%2 + data={x:v*pre(x) for x,v in raw.items()} + kernels=[[((i+2)*(z+L+1))*(-1 if z==0 else 1) for z in range(-L,L+1)] for i in range(g)] + shifts=list(product(range(-L,L+1),repeat=g)) + weight={z:math.prod(kernels[i][a+L] for i,a in enumerate(z)) for z in shifts} + direct={x:post(x)*sum(weight[z]*data[tuple((a-b)%s for a,b,s in zip(x,z,S))] for z in shifts) for x in data} + merged={};wrong=interior=0 + for tile in product(*(range(q) for q in Q)): + core=[data[tuple(t*N+a for t,a in zip(tile,unflat(u,N,g)))] for u in range(N**g)] + for u in range(N**g): + x=unflat(u,N,g);glob=tuple(t*N+a for t,a in zip(tile,x)) + val=post(glob)*sum(w*core[(u-sum(z[i]*N**(g-1-i) for i in range(g)))%(N**g)] for z,w in weight.items()) + wrong+=val!=direct[glob] + if all(L<=a0 + tiles=math.prod(Q) + expected_copies=tiles*2*sum((N**j)*(3*L)*(N+2*L)**(g-j-1) for j in range(g)) + assert len(copies)==expected_copies + expected_boundary=tiles*(N**g-(N-2*L)**g) + assert boundary==expected_boundary + return dict(N=N,L=L,g=g,coarse_sizes=Q,input_records=len(data),interior_outputs=interior, + corrected_boundary_outputs=boundary,gathered_halo_records=len(copies),predicted_halo_records=expected_copies, + direct_flattening_wrong_outputs=wrong,radix_key_bits=keybits, + exact_tensor_outputs_checked=len(data),diagonal_conjugations_checked=True, + scalar_coefficients='signed asymmetric integer product kernel') + +def main(): + cases=[replay(8,1,2,[2,4]),replay(8,1,3,[2,2,2]),replay(16,2,2,[2,2])] + perms=[gather_permutation([5,7,6],3),gather_permutation([4,4,4],3),gather_permutation([9,6],4)] + out=dict(status='PASS_EXACT_CORE_PLUS_SPARSE_BOUNDARY_TENSOR_CONVOLUTION',cases=cases, + support_limited_gathers=perms, + scope='Finite exact map, sparse-copy completeness, unique scatter, radix ordering and support controls; asymptotic fixed-tape costs are in the companion proof.') + (HERE/'HALO-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n');print(json.dumps(out,indent=2)) +if __name__=='__main__':main() diff --git a/research/capped-phase-fusion/evidence/halo_layout_ultra/check_ragged.py b/research/capped-phase-fusion/evidence/halo_layout_ultra/check_ragged.py new file mode 100644 index 0000000000..4665063452 --- /dev/null +++ b/research/capped-phase-fusion/evidence/halo_layout_ultra/check_ragged.py @@ -0,0 +1,69 @@ +"""Independent source-driven versus rectangle-driven ragged halo gather.""" +from itertools import product +from collections import defaultdict +from pathlib import Path +import json,math +HERE=Path(__file__).resolve().parent +def points(ranges):return product(*(range(a,b) for a,b in ranges)) +def rectangles(S,N,L): + Q=[(s+N-1)//N for s in S];g=len(S) + for tile in product(*(range(q) for q in Q)): + sizes=[min(N,s-t*N) for s,t in zip(S,tile)] + if min(sizes)>(i*G))&((1<>(j+1))&((1<<(K-1))-1))<(1<<(K-1))-2*B + for i,j in enumerate(targets)) + +def ops(targets,sources,G): + out=[] + F=[('y1',1),('t1',1),('y2',1),('t2',-1)] + return F+[('load',1)]+F+[('load',-1)] + +def run(state,targets,sources,G,n,inverse=False): + u,t,y=state;H=len(targets)*G;mask=(1<>j)&1)<<(i*G) for i,j in enumerate(targets)))&mask + elif tag=='t2':t=(t+sign*sum((((y>>j)&1)^(digit(u,i,G)&1))<<(i*G) for i,j in enumerate(targets)))&mask + else:u=(u+sign*sum(((y>>s)&1)<<(i*G) for i,s in enumerate(sources)))&mask + return u,t,y + +def ideal(state,targets,sources): + u,t,y=state + return u,t,y^sum(((y>>s)&1)<=K for a,b in zip(sorted(targets),sorted(targets)[1:])) + assert all(j+K<=n for j in targets) + count=good_count=bad_count=corrected=0;images=set() + for state in states: + count+=1;out=run(state,targets,sources,G,n) + assert run(out,targets,sources,G,n,True)==state + assert good(*out,targets,G,K)==good(*state,targets,G,K) + assert good(*ideal(state,targets,sources),targets,G,K)==good(*state,targets,G,K) + if good(*state,targets,G,K): + good_count+=1;assert out==ideal(state,targets,sources) + else: + bad_count+=1 + fixed=ideal(run(out,targets,sources,G,n,True),targets,sources) + assert fixed==ideal(state,targets,sources) + corrected+=fixed!=out + images.add(out) + return dict(targets=targets,sources=sources,G=G,K=K,n=n,states=count, + good=good_count,bad=bad_count,changed_by_repair=corrected,distinct_images=len(images)) + +def main(): + small=check_case([0,6],[4,8],1,5,11, + product(range(4),range(4),range(1<<11))) + assert small['states']==small['distinct_images']==32768 + rng=random.Random(713) + targets=[1,10,21];sources=[5,15,25];G=2;K=7;n=29 + arbitrary=[(rng.randrange(64),rng.randrange(64),rng.randrange(1< infinity`. Their aggregate and all helper reservoirs + fit the full address because beta<1. There is no necessary constraint + beta<1-epsilon; a group is allowed to be wider than one original axis. +* The width-dependent phase sums are geometric. With q=2`, the exponent is at least +`h^2-|h|`; the two-sided tail is at most `4 exp(-2a)`. Thus the claimed +`exp(-2a theta)+5 exp(-2a)` bound follows. In particular, counting both nearest +neighbors separately at the larger bound would be invalid, but the supplied +argument does not do so. + +The actual H is specified by globally fixed rounded weights and a globally +fixed truncated Toeplitz kernel inside each physical phase cell. Its diagonal +is one, with the omitted aliases included in the perturbation. The candidate's +perturbation budget is strictly below the displayed row-dominance gap. Principal +submatrices preserve this row gap. Adding identity hole rows/columns preserves +it as well, and increases the physical bandwidth by at most the q-image spacing +factor, which is bounded by two. + +For `F=H_aug-I`, `||F||_infinity<=1-delta` with +`delta>=1/(16p)`. Powers through order J agree on a core row whose distance from +the principal-window edge exceeds J times the bandwidth. The two inverse tails +are bounded by `2(1-delta)^J/delta`. With `J=C p^2`, this is exponentially small +in p. The distance requirement is `O(p^(5/2) polylog p)`, below the candidate's +`A=C_A p^3` eventually. The preserved physical period cut and the fact that +true axes exceed every polynomial in p exclude spurious short wrap edges. + +## 2. Broad Gaussian conditioning is a different argument, and it works + +Sources: candidate lines 151–201; +`work/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md` lines 13–73 and 75–105; +`work/gaussian_fast_ultra/BILATERAL-LDU.md` lines 25–119. + +For the finite LDU formulas, I checked the origin-shift correction and the +triangular inverse estimates. If `h=j-i`, the elementary-polynomial estimate +for the upper inverse reduces its exponential part to + +`b[beta_j^2-beta_i^2-rho^2 h-2 rho sum_{k=i}^{j-1} beta_k]`. + +Using `|beta_k|<=1/2` gives at most `b/4-b rho theta h`. The lower inverse uses +the corresponding expression with the other sign on the beta sum and has the +same decay. The factors `c_b^-1` and `c_b^-2` come from the explicitly bounded +products. This proves the stated uniform finite-interval bound; integrality of +the q-values is not needed, only gaps at least one and the beta bound. + +The broad target-cell identity also checks directly. With +`q'_k=k+rho c`, `beta'_k=theta k-rho c`, and +`b=b0/rho^2`, one obtains + +`K=alpha^-1 N_b diag(exp(-b beta'^2))`. + +At a selected target index, `beta'_k` equals the original source beta. Therefore +the finite LDU bound really applies to the block K used in the candidate; it is +not a bound for a different narrow matrix. The product estimate +`log(c_b^-1)<=pi^2/(12 b rho)` gives +`c_b^-4<=exp(pi alpha^2 rho/3)=p^O(1)`. The reciprocal gap is also polynomial +because `theta^-1=poly(p)` and `alpha^2=O(log p)`. + +The square identity-hole completion has block form `[A 0; R I]`. Its inverse is +`[A^-1 0; -R A^-1 I]`. Hence bounded Gaussian row sums and the selected-row +inverse bound give a polynomial full inverse bound. Dividing the completion +by four is contractive even on numerical hole errors. Its extra factor of +one half per axis is exactly compensated by the stated `2^D` shift. + +For the Toeplitz core, the Poisson symbol bound supplies a polynomial lower +eigenvalue, since its exponent is `O(alpha^2)=O(log p)`. An exponentially small +truncation/rounding perturbation preserves that bound. The extension from +these SPD cores to shallow circuits is outside this subreview, but the +Gaussian side supplies the conditioning claimed for that extension. + +## 3. Capping changes the runtime gauge, without changing H + +Sources: candidate lines 203–221 and 273–283; +`work/gaussian_fast_ultra/CAPPED-GAUGE-AUDIT.md` lines 31–107; +`work/phase_assembly_ultra/CAPPED-INTERIORS.md` lines 13–29 and 68–70. + +Within a fixed narrow phase cell, + +`log(a_i/a_j)=a(i-j)(beta_i+beta_j)`. + +Its absolute value is at most `a|i-j|`. Taking exact ratios of the already +rounded positive weights at one local anchor cancels the same common scalar +from both sides of the similarity, so the matrix is unchanged. Relative +rounding error of the original weights remains small because their minimum +is `exp(-O(ad))` and the absolute setup precision is `Cp`, with `ad=o(p)`. +Thus the local circuit sees range `O(aB)`, not the original `O(ad)`. + +For the broad split, expansion of the displayed exponents gives exactly +`-b0(delta+x-lambda y)^2`. The two logarithmic derivatives are +`-2 b0 lambda beta'_i` and `2 b0 lambda^2 beta'_k`; each is bounded in absolute +value by b0. Changing anchor changes the two ideal diagonals by reciprocal +constants. Applying the same exact reciprocal change to previously rounded +diagonals preserves their product with the Toeplitz core. Thus the broad +rounding scheme can consistently use the same global matrix and locally +recenter its capped restrictions. + +The number of pieces does not multiply the norm of their direct sum. The +group logarithmic range is consequently `O(gaB+g polylog p)`. The inherited +absolute weight accuracy adds `O(ad)` once to the precision requirement, rather +than once for every tensor axis. The candidate's condition +`beta+xi+r0=1-zeta`, with `a=O(log p)<=p^r0` eventually, makes this range o(p). +The total precision is therefore `p+O(ad+gaB+g polylog p)=O(p)` in both parameter +families. This conclusion concerns asymptotics; no reasonable finite cutoff +is supplied or required by the statement. + +## 4. Boundary factorization and rounding do not introduce a window-count factor + +Sources: candidate lines 225–240; +`work/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md` lines 78–98; +`work/integer-mult-current/references/semantic-bulk/rad20/reports/review-phase-cell-inverse.md` +lines 203–301. + +The five-factor identities check by ordinary block multiplication. The shears +use B and C from the original banded matrix. They do not use the generally +dense products `A^-1 B` or `C A^-1`. For forward expansion, F is only applied; +its inverse is not required by that formula. + +For a strictly row-diagonally-dominant matrix, one elimination step gives + +`gap_i(new)>=gap_i(old)+|a_ik/a_kk| gap_k(old)`. + +Its absolute row norm does not increase. Thus the narrow Schur matrix retains +the polynomial gap and bounded row norm. A noncyclic cut of its banded matrix +also retains row dominance. In its LU factors, U has bounded row norm and the +unit-lower factor has only O(W) entries per row, each bounded by the reciprocal +pivot gap. Therefore `||L||=poly(p)`. The identities +`L^-1=U A^-1` and `U^-1=A^-1 L` then give polynomial inverse-factor norms. +The periodic correction has rank O(W) and is controlled by the full completed +inverse, as in the retained identity `K^-1=I-V A^-1 U`. + +The exact partial right-hand sides are consequently polynomially bounded. +Rounding a completed block update creates a row residual bounded by +`poly(p) 2^-P`. Equivalently, rounding and coefficient error can be treated as +a perturbation whose norm is small enough to close the same bound on computed +values. Applying the polynomial full inverse bounds the output error by that +residual. Infinity norm takes the maximum over windows rather than summing +their count. Strassen's fixed recursion contributes polynomial W growth and +O(log W) guard bits, already covered by the precision allowance. + +## 5. Density, stream layout, and global calls + +Sources: candidate lines 207–211, 242–271, and 368–385; +`work/phase_layout_audit_ultra/AUDIT.md` lines 15–188; +`work/phase_layout_audit_ultra/CAPPED-SIGNOFF.md` lines 7–65; +`work/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md` lines 16–58; +`work/halo_layout_ultra/HALO-THEOREM.md` lines 88–196. + +The capped carve contributes O(d/B+log B) separators per physical phase cell. +Retaining W-wide separators ensures distinct interiors are uncoupled. Each +elimination only joins adjacent O(W)-sized boundary groups, so the Schur +bandwidth remains O(W). The extra clipped-endpoint term in an L-core is +O(W polylog p), negligible relative to `L W/B polylog p`. + +For the union U, every sparse factor changes its own axis only inside E_i; +therefore it preserves U. The dense A stage does not preserve U, and the +candidate explicitly scatters and regathers around it. For one sparse axis, +`U_i=E_i x [L]^(g-1)` is a complete rectangle inside U. Extracting it in order, +transposing only its polynomial-width axis field, processing, undoing the +transpose, and merging back needs no full g-axis key sort. The full union +scan is charged once per sparse axis, producing the stated g-squared factor +per group. The matrix corollary cannot afford an extra full-key union sort, +but this algorithm does not require one. + +The q-image predicate checks the exact half-open interval +`s(2k-1)<=2tj 2(k+1)L. +``` + +Indeed, after excluding the actual suffix, the at most k moved positions +cut the remaining interval into at most k+1 gaps. If every gap had length +less than L, that prefix would have length at most (k+1)L-1. The displayed +inequality contradicts this bound. + +Let A be the old disjoint equal-L swap exchanging S with the actual suffix. +Execute A, route the conjugated permutation A pi A^{-1}, then execute +A^{-1}. Because pi fixes S pointwise, its conjugate fixes the actual suffix +pointwise. It still moves at most k coordinates. All endpoints are now in +the n-L prefix. Choose the three dirty reservoirs in that prefix and run +all compact rotations, original control shuttles, elementary CNOTs, and +bad-hole repair there. Recompute endpoint names after A and after the +reservoir placement. In the application, n-L>=9kG holds with a wide +asymptotic margin. + +The suffix is a complete binary field with arbitrary contents. Reinterpreting +its 2^L values as part of a record gives record length R0*2^L, without +creating, padding, or deleting data. This length exceeds every fixed +polynomial in p. No operation whose cost relies on that record length acts +on its component coordinates. Both instances of A use the already approved +primitive and cost O(V L^(1-a))=O(V polylog p); they do not invoke the new +router. This removes the prospective circularity. + +For the general router outside this sparse branch, the retained +first-L/second-L/actual-last-L candidate convention and matching cohorts +also work. Arbitrary candidates should not be substituted blindly: a +candidate can partially overlap the actual suffix, whereas the old swap +requires equal disjoint fields. After each candidate placement, the same +n-L prefix exclusion is mandatory. If the entire binary span is only +polylogarithmic, the old elementary construction suffices. If +n=O((k+1)polylog p), the old full-span construction gives the desired +support bound up to polylogarithms; its own short-record preparation must +still use the old suffix construction. + +For a full H supplier call, reserve the suffix outside its selected field +and keep it fixed throughout the completed call. If necessary, perform +the same old L-bit placement before exposure, update the logical names of +the neighborhoods, and restore it after inverse exposure. The source +neighborhoods need not remain physically consecutive before exposure: +the exposure permutation explicitly routes their named bits. The target +complete chunks are placed before the reserved suffix. + +## 3. Whole-row padding and the next diagonal scan + +The leading row prefix and the front/back controls are assembled from +complete K-bit neighborhoods in the globally complete binary tensor. +Their bits are not assumed zero or semantically independent of coefficient +values. Completeness means every coordinate value is represented in the +stream. Multiplication by previous coarse-dependent diagonal coefficients +does not remove any address or change this property. + +Deferring the selected bits lying in reserved chunks is valid: the accepted +active invocation is the desired tensor on the other selected bits and +identity on every reserved coordinate, for arbitrary incoming coefficients. +The preliminary individual transforms in the old layout were used to +finish those selected bits, not to create complete row or control ranges. +After deferral, those transforms are performed in later phases instead. + +The row count is rounded once to a multiple of the actual product + +``` +Q_rows = W_complex^D_complex * W_bit^D_bit. +``` + +The padded row label is allowed to be a nonbinary preceding fiber count. +Every within-row field remains a full binary field. The supplier's +interchanges and controls act within those complete fields; the global +exposure router is not applied to the padded row label. Nested bit adapters +consume and restore the bit stock before later complex splits, exactly as +in the retained product-stock argument. + +At the completed invocation boundary, the operator is identity on the row +label. A padded zero row therefore returns to zero even though temporary +role computations may have combined or moved its contents. Delete those +rows before inverse exposure or the next deferred phase. This bounds each +phase's volume by twice its original volume, not by a factor exponential in +the number of phases. + +All H calls in the compiler have the same selected-bit schedule for every +coarse label; label dependence resides in the intervening diagonal tables. +Run one H call on the complete global tensor, complete every deferred +phase, remove padding, and restore the original exposed-coordinate order. +Only then run the next diagonal scan. The microbox layout used for that +scan is the restored layout in which the local coordinates are already +contiguous (or the separately paid low-bit gather layout). No padded row, +partial H result, or temporarily relocated coarse label is presented to +the diagonal-table lookup. Thus scanning different coarse-label tables +does not require separate H invocations or borrowing fixed labels inside +a microbox. + +## 4. The actual parameter inequalities + +The largest exposure moves k=O(gK)=O(p^.093 log p) binary coordinates. +With G=Theta(log p), its three dirty reservoirs require + +``` +9kG = O(p^.093 log^2 p) = o(p). +``` + +The suffix occupies only L=Theta(log^2 p) positions. In particular, + +``` +n/(kG) = Omega(p^.907/log^2 p) -> infinity, +n/((k+1)L) = Omega(p^.907/log^3 p) -> infinity. +``` + +There is ample room both for the old spectator placement and for all +subsequent prefix-only reservoirs. These inequalities concern the full +binary address, not any one original axis. + +The only per-axis inequality is + +``` +K + O(log p) < ell_axis = Theta(p^.002), +``` + +which holds eventually. The original K/log p -> infinity wording must be +replaced, as already noted in `SUPPLIER-SUPPORT-AUDIT.md`, by the finite +guard inequality K>=G+4ceil(log2 p)+10. K=128G satisfies it. + +The supplier's front/back controls are made by concatenating reserved +complete neighborhoods. Their chunk count is + +``` +ceil(2hG/K) + ceil(hG/K) <= 3h/128 + 2. +``` + +The leading row stock adds O(log h) complete chunks. Above a fixed +polylogarithmic cutoff this is at most h/16, and smaller h is handled +individually. No single original axis stores hG control bits. Indeed +gG/ell_axis=Theta(p^.091 log p) diverges; the construction works precisely +because it does not demand this false per-axis fit. + +Consequently the reviewed mechanisms support the proposed +O(V g^(1-q) polylog p) global-call cost under the retained supplier +assumptions. The suffix exclusion and completed-call boundary should be +kept explicit in any final assembly. They are necessary bookkeeping, +not a new exponent restriction or an unproved sorting primitive. diff --git a/research/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER-CONTROLS.json b/research/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER-CONTROLS.json new file mode 100644 index 0000000000..80cb2c8083 --- /dev/null +++ b/research/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER-CONTROLS.json @@ -0,0 +1,154 @@ +{ + "status": "pass", + "arithmetic": "exact symbolic rational/complex", + "routing": [ + { + "N": 2, + "permutations": 2, + "switch_depth": 1 + }, + { + "N": 4, + "permutations": 24, + "switch_depth": 3 + }, + { + "N": 8, + "permutations": 40320, + "switch_depth": 5 + }, + { + "N": 16, + "random_permutations": 100, + "switch_depth": 7 + }, + { + "N": 32, + "random_permutations": 100, + "switch_depth": 9 + }, + { + "N": 64, + "random_permutations": 100, + "switch_depth": 11 + } + ], + "fixed_H_word_checks": [ + { + "kind": "upper", + "parameter": "-10", + "H_calls": 6 + }, + { + "kind": "lower", + "parameter": "-10", + "H_calls": 6 + }, + { + "kind": "upper", + "parameter": "-3/7", + "H_calls": 6 + }, + { + "kind": "lower", + "parameter": "-3/7", + "H_calls": 6 + }, + { + "kind": "upper", + "parameter": "0", + "H_calls": 6 + }, + { + "kind": "lower", + "parameter": "0", + "H_calls": 6 + }, + { + "kind": "upper", + "parameter": "7/9", + "H_calls": 6 + }, + { + "kind": "lower", + "parameter": "7/9", + "H_calls": 6 + }, + { + "kind": "upper", + "parameter": "10", + "H_calls": 6 + }, + { + "kind": "lower", + "parameter": "10", + "H_calls": 6 + }, + { + "kind": "diag", + "H_calls": 6 + }, + { + "kind": "swap", + "H_calls": 6 + }, + { + "kind": "fft", + "H_calls": 6 + }, + { + "kind": "diag", + "H_calls": 6 + } + ], + "coefficient_odometer": [ + { + "N": 2, + "g": 1, + "records": 2, + "coefficient_products": 2, + "products_per_record": 1.0, + "table_movement_bound": 2, + "zeros_in_table": true + }, + { + "N": 2, + "g": 8, + "records": 256, + "coefficient_products": 510, + "products_per_record": 1.9921875, + "table_movement_bound": 510, + "zeros_in_table": true + }, + { + "N": 4, + "g": 5, + "records": 1024, + "coefficient_products": 1364, + "products_per_record": 1.33203125, + "table_movement_bound": 1364, + "zeros_in_table": true + }, + { + "N": 8, + "g": 4, + "records": 4096, + "coefficient_products": 4680, + "products_per_record": 1.142578125, + "table_movement_bound": 4680, + "zeros_in_table": true + } + ], + "tensor_test": { + "N": 4, + "axes": 2, + "data_coordinates": 16, + "input_output_matrix_entries": 256, + "simultaneous_H_calls": 18, + "expected_H_calls": 18, + "dimension_expansion": false, + "different_axis_matchings": true, + "different_axis_gate_kinds": true, + "local_matching_endpoints_checked": true + } +} diff --git a/research/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER.md b/research/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER.md new file mode 100644 index 0000000000..19efb9cb1c --- /dev/null +++ b/research/capped-phase-fusion/evidence/inplace_algebra_ultra/TENSOR-COMPILER.md @@ -0,0 +1,160 @@ +# Tensorizing local circuits without dimension padding + +Status: an algebraic compiler and scan-cost lemma. It reduces a tensor product of different bounded-depth local circuits to a bounded sequence of simultaneous binary Hadamard calls and diagonal scans on the **same** data volume. The retained supplier must support the stated Hadamard call on selected local bits and restore the agreed endpoint record order. No new fast-transform supplier is asserted here. + +## 1. Interface and conclusion + +Let N=2^m and let g axes each have N local coordinates. A coarse-label block contains V=N^g data records in lexicographic local-coordinate order. On axis a let C_a be an N-wire linear circuit of depth D; every layer consists of disjoint invertible two-coordinate gates and one-coordinate diagonals. The permitted gates are bounded shears, swaps, FFT butterflies, and diagonals. Circuits, pair matchings, coefficients and depths can differ by axis and by coarse label. Their tables are fixed independently of data values. + +Write H=[[1,1],[1,-1]]. A simultaneous Hadamard call means the tensor product of one H acting on one selected bit in each axis, with all remaining local and coarse coordinates as spectators. It acts on existing 2^g-coordinate fibers inside the N^g records; it allocates no new coordinate. The bit selected in an axis may be prescribed by the compiler. + +After padding shorter circuits by identity **operations**, the tensor product C_1 tensor ... tensor C_g is a sequence of + + at most (8m+2)D simultaneous H calls + +and O(Dm) tensor-diagonal scans. A tensor diagonal has the form + + d(i_1,...,i_g;c)=product_(a=1)^g d_a(i_a;c), + +where c is the fixed coarse label. Each diagonal scan uses O(1) amortized coefficient multiplications and O(1) amortized coefficient-table record movements per data record, after loading gN local values. The number of arithmetic-data coordinates remains exactly N^g throughout. + +If all local gate coefficients and inverses are bounded by B and all original prefix and inverse-prefix norms by Lambda, the compiler's local prefix bounds are Lambda times poly(B), up to absolute constants. For polynomial B and Lambda this is N^O(1). The tensor-prefix logarithmic bound is O(g log N). More generally depth D=polylog(p), B=poly(p), N=poly(p) gives an O(g polylog(p)) guard even without a stronger prefix theorem. + +An outer similarity whose logarithmic range is Q_a on axis a adds only sum_a Q_a. Therefore the retained phase-cell case Q_a=O(a d) uses + + p + O(g a d + g polylog(p)) + +working bits. The condition g a d+g polylog(p)=O(p) is sufficient for O(p)-bit records. + +## 2. Fixed Hadamard words, including zero shears + +All formulas below are matrix-product order; execution reads right to left. Let U(t)=[[1,t],[0,1]]. Direct multiplication gives + + H diag(2,1) H diag(-3,1) H = [[-8,-10],[0,-6]], + diag(-1/8,-1/6) H diag(2,1) H diag(-3,1) H = U(5/4). + +For q nonzero, conjugate by A_q=diag(4q/5,1): + + U(q)=A_q diag(-1/8,-1/6) H diag(2,1) H diag(-3,1) H A_q^-1. + +This has three H factors. To include t=0 with uniformly invertible diagonals, choose a known B>=|t| and c=2B+1. Then + + U(t)=U(t+c) U(-c). + +Both parameters have magnitude at least B+1 and at most 3B+1. This is a **six-H word for every t**, with all scalar coefficients and inverses bounded by poly(B). There is no division by t and no case split at t=0. Transpose the word to obtain the same six-H count for lower shears. + +There are two bounded identity words: + + I = (1/2) H H, + I = diag((1-i)/4,(1+i)/4) H diag(1,i) H diag(1,i) H. + +The second identity uses three H factors. Its middle product before the leading diagonal equals diag(2+2i,2-2i). Complex coefficients are permitted by the retained complex H interface. + +Consequently every gate type can be given **exactly six H factors**, separated by diagonal matrices: + +- Upper/lower shear: the six-H word above. +- Diagonal, including I: append three two-H identity words. +- Swap J: use J=(1/2)H diag(1,-1)H, then append two two-H identity words. +- FFT butterfly H diag(1,w), optionally scaled by 1/sqrt(2): append a two-H identity and a three-H identity. Here |w|=1. + +Combining neighboring diagonal matrices puts every word in one common alternating shape with six H factors and seven diagonals. Thus different gate types, zero parameters and inactive pairs can coexist in one simultaneous schedule without branches or padded data coordinates. + +## 3. Every matching can be routed to adjacent pairs + +For one local layer, assign its oriented pairs (i,j) to positions (2k,2k+1). Unpaired coordinates are assigned arbitrarily to the remaining positions, in pairs carrying diagonal or identity gates. This defines a permutation pi on exactly N coordinates. The layer equals + + P_pi^-1 (direct sum of adjacent-pair gates) P_pi. + +We use a Beneš network of 2m-1 switch layers. At each layer all switches join indices differing in one fixed bit; the bit sequence is + + 0,1,...,m-1,m-2,...,1,0. + +Here is a constructive proof of rearrangeability. Form a bipartite multigraph whose left vertices are input pairs {2j,2j+1} and whose right vertices are output pairs {2k,2k+1}. Input i supplies an edge to output-pair floor(pi(i)/2). Every vertex has degree two, so each component is an even cycle (parallel edges are allowed). Alternately color each cycle's edges 0 and 1. The first switch sends an input edge into the subnetwork of its color; the last switch sends that color to the required output parity. Each color class induces a permutation on N/2 pair indices. Recursively route both color subnetworks in parallel. This gives exactly the displayed bit sequence and uses no storage coordinates. + +The switch is either I or J. Both are obtained from the common two-H word + + S_epsilon=(1/2) H diag(1,epsilon) H, + epsilon=+1 for I, epsilon=-1 for J. + +Thus all axis-dependent switch controls are diagonal entries. A physical switch on bit k simply uses a simultaneous H call selecting bit k in each axis. No interstage wire shuffle is needed. If a supplier accepts only a contiguous selected-bit field, the existing paid bit router must gather that field and undo the gather; that is a supplier-interface cost, not an uncharged permutation. + +Each forward or inverse Beneš route uses 2(2m-1) H calls. The adjacent gates use six. Hence one original matching layer uses + + 2[2(2m-1)]+6 = 8m+2 + +H calls. The schedule is identical on every axis and every coarse-label block; only diagonal tables change. + +## 4. Tensor fusion is literal multiplication, not a sum expansion + +Let the compiled axis-a circuit be D_(a,L) H_(a,L) ... H_(a,1) D_(a,0), with the same number and arrangement of H factors. Kronecker multiplication gives + + tensor_a C_a + = (tensor_a D_(a,L)) (tensor_a H_(a,L)) ... + (tensor_a H_(a,1)) (tensor_a D_(a,0)). + +Every tensor of diagonals is a single diagonal scan. Every tensor of binary H factors is one simultaneous H call. This introduces neither a sum of 2^g branches nor a 2^g padding factor. The 2^g-dimensional Hadamard fibers are subsets of the N^g existing coordinates. + +An axis with no work at a particular original depth receives an identity layer, routed and compiled with the same fixed shape. Its intermediate H operations are still present, but the complete local word is I. Coarse-label variation is equally harmless: for each fixed label the identity above holds separately, and the whole data operator is a direct sum over labels. + +## 5. Uniform intermediate bounds and precision + +A completed switch route is a permutation and has norm one. Each partial switch word has absolute constant norm and inverse norm. Between original local circuit boundaries, at most one original gate word is partially executed. Its length is constant, and its scalar coefficients and inverses are poly(B); all different pairs act as a direct sum. Therefore + + max_prefix ||C_compiled,prefix||, + max_prefix ||C_compiled,prefix^-1|| <= poly(B) Lambda. + +The length O(D log N) does not require multiplying this polynomial bound afresh at every stage: completed original prefixes already have bound Lambda; completed route and padding words have their exact intended endpoints. The corresponding suffix bound is also polynomial when the original endpoint and inverse-prefix bounds are. + +For g tensor factors, the operator norms multiply, so their logarithms add. A microbox coarse-label direct sum takes the maximum over labels, not a product over labels. Hence polynomial local bounds give exp(O(g log N)); using only the rough original bound Lambda<=(2B)^D gives exp(O(gD log B)). Adding the original cell similarity contributes exp(sum Q_a). + +The number of compiled stages is polylog(p). Allocate each rounding or supplied-transform error below the final tolerance divided by this stage count and the prefix/suffix amplification. This costs O(g polylog(p)+sum Q_a) guard bits. Supplier-internal precision charges still must be those already guaranteed for the simultaneous H call; this compiler does not erase them. + +## 6. Product diagonals with constant amortized scan cost + +Fix one diagonal stage and one coarse label. Load the gN coefficients d_a(i) on a coefficient tape in axis order. Use base-N lexicographic enumeration with axis 0 fastest. Store suffix products + + P_j=product_(a=j)^(g-1) d_a(i_a), 1<=j=1) k/N^k = O(V), + +plus O(g) initialization. No division by d_a(i) is used, so this method also permits zero diagonal entries when an external application permits them. Every cached product is recomputed from current factors, never updated by a long chain of multiplicative ratios, so numerical drift does not depend on the number of records scanned. + +The same accounting works on a fixed number of tapes. Keep the coefficient-tape head normally in axis-0's N-record table. Rewinding that table costs O(N) once per N records. A depth-k carry visits the needed tables up to axis k and returns, moving O(kN) coefficient records. Summing over carries gives + + O(V sum_(k>=1) kN/N^k)=O(V) + +record moves. The suffix-product tape is accessed near its front except on carries, with the same geometric bound. Counter/address arithmetic costs only polylog(p) per record. This does not require g work tapes or a random-access lookup every record. + +The table loading cost is O(gN) records per coarse block and stage. For N>=2 it is O(N^g), including small g, and polynomial-time table construction is negligible per N^g block in the retained asymptotic regime g=p^beta tending to infinity. If tables are supplied rather than constructed locally, no additional assumption is needed. Both generation and loading costs must be charged once per visited coarse block, not silently once for the entire input. + +At O(p) working precision, O(1) amortized scalar multiplications per record and O(1) record movements cost O(V_bits polylog(p)) under the retained pointwise-arithmetic kernel, where V_bits counts bit volume. There is no g V_bits coefficient-generation term. + +## 7. Exact controls + +`check_tensor_compiler.py` constructs the network controls and all fixed words. `TENSOR-COMPILER-CONTROLS.json` records: + +- every permutation for N=2,4,8, including all 40,320 permutations at N=8; +- 100 random permutations each at N=16,32,64; +- exact symbolic checks of upper/lower zero and nonzero shears, diagonal, Swap and FFT words; +- an exact two-axis N=4 replay with different axis matchings and gate types: 16 data coordinates, all 256 input-output matrix entries, 18 simultaneous H calls, no dimension expansion. + +The finite controls support the constructive proofs; the cycle coloring, word identities, Kronecker identity and geometric scan sums establish the asymptotic statements. + +## 8. Cost substitution into the retained supplier + +Suppose a simultaneous H call on g selected bits, across V_bits bits of data and spectators, has paid cost O(V_bits g^(1-q) polylog(p)). Suppose also that any required bit-field gather/restore has cost O(V_bits g^(1-a_bit) polylog(p)). For D=polylog(p), N=poly(p), the compiler uses only polylog(p) such calls, and its diagonal scans cost O(V_bits polylog(p)). Thus the grouped local circuit costs + + O(V_bits [g^(1-q)+g^(1-a_bit)+1] polylog(p)). + +When a_bit>q and 0= j else row[j-i] + for j in range(n)] for i in range(n)]) + + +def lower(c): + return toep(c) + + +def invseries(f): + n = len(f) + ans = np.zeros(n, complex) + ans[0] = 1 / f[0] + for j in range(1, n): + ans[j] = -sum(f[k] * ans[j-k] for k in range(1, j+1)) / f[0] + return ans + + +def conv(a, b, n): + return np.convolve(a, b)[:n] + + +def diag(indices, values): + return [[((int(i),), np.array([[v]], complex)) for i, v in zip(indices, values)]] + + +def inverse(plan): + return [[(ids, np.linalg.inv(g)) for ids, g in layer] + for layer in reversed(plan)] + + +def parallel(a, b): + return [(a[j] if j < len(a) else []) + (b[j] if j < len(b) else []) + for j in range(max(len(a), len(b)))] + + +def fft(indices): + n = len(indices) + if n <= 1: + return [] + assert n & (n - 1) == 0 + bits = n.bit_length() - 1 + rev = [int(f"{j:0{bits}b}"[::-1], 2) for j in range(n)] + plan = [[((indices[i], indices[j]), np.array([[0, 1], [1, 0]], complex)) + for i, j in enumerate(rev) if i < j]] + size = 2 + while size <= n: + layer = [] + for start in range(0, n, size): + for j in range(size // 2): + w = np.exp(-2j * np.pi * j / size) + layer.append(((indices[start+j], indices[start+j+size//2]), + np.array([[1, w], [1, -w]], complex) / math.sqrt(2))) + plan.append(layer) + size *= 2 + return plan + + +def circ_shear(target, source, kernel): + """target += circulant(kernel) source, using these wires only.""" + p = parallel(fft(target), fft(source)) + lams = np.fft.fft(kernel) + mid = [[((source[j], target[j]), np.array([[1, 0], [lams[j], 1]], complex)) + for j in range(len(source))]] + return p + mid + inverse(p) + + +def toeplitz_shear(target, source, T): + """Cyclic + negacyclic decomposition; source and target are disjoint.""" + n = len(source) + assert T.shape == (n, n) and len(target) == n + c, r = T[:, 0], T[0, :] + g = np.zeros(n, complex) + h = np.zeros(n, complex) + g[0] = h[0] = c[0] / 2 + for j in range(1, n): + g[j] = (c[j] + r[n-j]) / 2 + h[j] = (c[j] - r[n-j]) / 2 + gamma = np.exp(1j * np.pi / n) + d = gamma ** np.arange(n) + twists = parallel(diag(target, d), diag(source, d)) + return (circ_shear(target, source, g) + twists + + circ_shear(target, source, d * h) + inverse(twists)) + + +def triangle_plan(indices, f): + n = len(indices) + STATS["triangle_nodes"] += 1 + STATS["max_triangle_coefficient"] = max( + STATS["max_triangle_coefficient"], float(np.max(np.abs(f)))) + if n == 1: + return diag(indices, f) + assert n % 2 == 0 + m = n // 2 + ev, od = indices[::2], indices[1::2] + e, o = np.asarray(f[::2]), np.asarray(f[1::2]) + k = conv(o, invseries(e), m) + zk = np.r_[0, k[:-1]] + s = e - np.r_[0, conv(o, k, m)[:-1]] + right = toeplitz_shear(ev, od, lower(zk)) + middle = parallel(triangle_plan(ev, e), triangle_plan(od, s)) + left = toeplitz_shear(od, ev, lower(k)) + return right + middle + left + + +def lowdisp_shear(target, source, T): + n = len(source) + Z = np.eye(n, k=-1) + Delta = T - Z @ T @ Z.T + U, vals, Vh = np.linalg.svd(Delta) + cutoff = 1e-11 * max(1.0, vals[0]) + rank = int(np.sum(vals > cutoff)) + STATS["max_displacement_rank"] = max(STATS["max_displacement_rank"], rank) + STATS["generator_shears"] += 1 + result = [] + reconstructed = np.zeros_like(T) + for j in range(rank): + g = U[:, j] + h = vals[j] * Vh[j, :] + STATS["max_generator_entry"] = max( + STATS["max_generator_entry"], float(np.max(np.abs(g))), + float(np.max(np.abs(h)))) + reconstructed += lower(g) @ lower(h).T + gamma = max(1.0, 2 * float(np.sum(np.abs(h)))) + p = h / gamma + p[0] += 1 + # U(h)=J L(h) J. This is a genuinely invertible source transform. + transform = triangle_plan(source[::-1], p) + result += transform + result += toeplitz_shear(target, source, gamma * lower(g)) + result += inverse(transform) + result += toeplitz_shear(target, source, -gamma * lower(g)) + assert np.max(np.abs(T - reconstructed)) < 1e-9 + return result + + +def full_plan(indices, M): + n = len(indices) + STATS["full_nodes"] += 1 + if n == 1: + return diag(indices, [M[0, 0]]) + m = n // 2 + A, B, C, D = M[:m, :m], M[:m, m:], M[m:, :m], M[m:, m:] + K = np.linalg.solve(A, B) + L = np.linalg.solve(A.T, C.T).T + S = D - C @ K + right = lowdisp_shear(indices[:m], indices[m:], K) + mid = parallel(full_plan(indices[:m], A), full_plan(indices[m:], S)) + left = lowdisp_shear(indices[m:], indices[:m], L) + return right + mid + left + + +def replay(plan, n, track=True): + out = np.eye(n, dtype=complex) + peak = 1.0 + min_det = 1.0 + for layer in plan: + used = set() + for ids, g in layer: + assert not used.intersection(ids), "non-disjoint layer" + used.update(ids) + min_det = min(min_det, float(abs(np.linalg.det(g)))) + out[list(ids), :] = g @ out[list(ids), :] + if track: + peak = max(peak, float(np.linalg.norm(out, 2))) + STATS["min_gate_determinant"] = min(STATS["min_gate_determinant"], min_det) + return out, peak + + +def main(): + rng = np.random.default_rng(20261013) + rows = [] + for n in [2, 4, 8, 16]: + f = rng.normal(size=n) / (12 * n) + f[0] += 1 + p = triangle_plan(list(range(n)), f) + actual, peak = replay(p, n) + rows.append({"kind": "triangular", "n": n, "rounds": len(p), + "gates": sum(map(len, p)), + "max_error": float(np.max(np.abs(actual-lower(f)))), + "peak_prefix_norm_2": peak}) + + c = rng.normal(size=n) / (20 * n) + r = rng.normal(size=n) / (20 * n) + c[0] += 1 + H = toep(c, r) + p = full_plan(list(range(n)), H) + actual, peak = replay(p, n, track=n <= 8) + rows.append({"kind": "full_toeplitz", "n": n, "rounds": len(p), + "gates": sum(map(len, p)), + "max_error": float(np.max(np.abs(actual-H))), + "peak_prefix_norm_2": peak if n <= 8 else None, + "input_distance_from_I_2": float(np.linalg.norm(H-np.eye(n), 2))}) + # Destructive control: dropping the second subtraction computes the wrong map. + # Independent direct parity block algebra across a range of sizes. + identities = [] + for n in [2, 4, 8, 16, 32]: + f = rng.normal(size=n) / n + f[0] += 2 + m = n // 2 + e, o = f[::2], f[1::2] + k = conv(o, invseries(e), m) + s = e - np.r_[0, conv(o, k, m)[:-1]] + E, O, K = lower(e), lower(o), lower(k) + Z = np.eye(m, k=-1) + left = np.block([[np.eye(m), np.zeros((m,m))], [K, np.eye(m)]]) + diagblock = np.block([[E, np.zeros((m,m))], [np.zeros((m,m)), lower(s)]]) + right = np.block([[np.eye(m), Z@K], [np.zeros((m,m)), np.eye(m)]]) + order = list(range(0,n,2)) + list(range(1,n,2)) + exact = lower(f)[np.ix_(order,order)] + identities.append({"n": n, "error": float(np.max(np.abs(left@diagblock@right-exact)))}) + sylvester = [] + for b in [1, 2, 4, 8]: + f = np.array([1.0]) + g = np.array([1.0]) + for j in range(b): + f = np.convolve(f, [1.0, 256.0 ** (j-b+0.5)]) + g = np.convolve(g, [1.0, 256.0 ** (-j-0.5)]) + fp = np.r_[f, np.zeros(3*b+2-len(f))] + fi = invseries(fp) + def hc(k): + return sum(g[v]*fi[k+v] for v in range(b+1) if 0 <= k+v < len(fi)) + K = np.array([[hc(i-b-j) for j in range(b)] for i in range(b)]) + H = np.array([[hc(i-j) for j in range(b)] for i in range(b)]) + S = np.zeros((2*b, 2*b), complex) + for j in range(b): + for k in range(b+1): + if j+k < 2*b: + S[j+k, j] += f[k] + if 0 <= b+j-k < 2*b: + S[b+j-k, b+j] += g[k] + inds = list(range(2*b)) + p = toeplitz_shear(inds[:b], inds[b:], K) + p += full_plan(inds[b:], H) + p += triangle_plan(inds, np.r_[f, np.zeros(2*b-len(f))]) + actual, peak = replay(p, 2*b, track=b <= 4) + sylvester.append({"b": b, "total_wires": 2*b, "rounds": len(p), + "gates": sum(map(len, p)), + "max_error": float(np.max(np.abs(actual-S))), + "H_minus_I_norm_1": float(np.linalg.norm(H-np.eye(b), 1)), + "H_minus_I_norm_infinity": float(np.linalg.norm(H-np.eye(b), np.inf)), + "peak_prefix_norm_2": peak if b <= 4 else None}) + # Removing the compensating dirty subtraction must fail. + b = 4 + gi, hi = np.array([1, .02, .01, .005]), np.array([.1, .04, .02, .01]) + pi = hi.copy() + pi[0] += 1 + target = np.block([[np.eye(b), np.zeros((b,b))], [lower(gi)@lower(hi).T, np.eye(b)]]) + pp = triangle_plan(list(range(b))[::-1], pi) + correct = pp + toeplitz_shear(list(range(b, 2*b)), list(range(b)), lower(gi)) + inverse(pp) + broken_actual, _ = replay(correct, 2*b, track=False) + correct += toeplitz_shear(list(range(b, 2*b)), list(range(b)), -lower(gi)) + correct_actual, _ = replay(correct, 2*b, track=False) + omission = {"correct_error": float(np.max(np.abs(correct_actual-target))), + "omit_second_subtraction_error": float(np.max(np.abs(broken_actual-target)))} + result = {"seed": 20261013, "note": "Floating replay supports, does not replace, algebraic proof.", + "controls": rows, "polyphase_identity": identities, + "normalized_sylvester": sylvester, "omission_control": omission, "stats": STATS} + (ROOT / "CIRCUIT-REPLAY.json").write_text(json.dumps(result, indent=2)+"\n") + print(json.dumps(result, indent=2)) + + +if __name__ == "__main__": + main() diff --git a/research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py b/research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py new file mode 100644 index 0000000000..4f0180c88f --- /dev/null +++ b/research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_exact_algebra.py @@ -0,0 +1,157 @@ +"""Exact rational checks of parity LU and displacement rank closure.""" +from fractions import Fraction as F +from pathlib import Path +import json +import random + + +def zeros(n, m=None): + return [[F(0) for _ in range(n if m is None else m)] for _ in range(n)] + + +def eye(n): + a = zeros(n) + for i in range(n): + a[i][i] = F(1) + return a + + +def mm(a,b): + return [[sum(a[i][k]*b[k][j] for k in range(len(b))) + for j in range(len(b[0]))] for i in range(len(a))] + + +def add(a,b,sgn=1): + return [[x+sgn*y for x,y in zip(r,s)] for r,s in zip(a,b)] + + +def inv(a): + n=len(a) + q=[r[:] + s for r,s in zip(a,eye(n))] + for j in range(n): + k=next(k for k in range(j,n) if q[k][j]) + q[j],q[k]=q[k],q[j] + v=q[j][j] + q[j]=[x/v for x in q[j]] + for i in range(n): + if i!=j: + v=q[i][j] + q[i]=[x-v*y for x,y in zip(q[i],q[j])] + return [r[n:] for r in q] + + +def rank(a): + a=[r[:] for r in a] + k=0 + for j in range(len(a[0])): + p=next((i for i in range(k,len(a)) if a[i][j]),None) + if p is None: + continue + a[k],a[p]=a[p],a[k] + v=a[k][j] + a[k]=[x/v for x in a[k]] + for i in range(k+1,len(a)): + v=a[i][j] + a[i]=[x-v*y for x,y in zip(a[i],a[k])] + k+=1 + if k==len(a): + break + return k + + +def block(a,b,c,d): + return [r+s for r,s in zip(a,b)] + [r+s for r,s in zip(c,d)] + + +def lo(f): + return [[f[i-j] if i>=j else F(0) for j in range(len(f))] + for i in range(len(f))] + + +def shift(n): + a=zeros(n) + for i in range(1,n): + a[i][i-1]=F(1) + return a + + +def displacement(a): + z=shift(len(a)) + return add(mm(z,a),mm(a,z),-1) + + +def series_inv(f): + g=[1/f[0]] + for i in range(1,len(f)): + g.append(-sum(f[j]*g[i-j] for j in range(1,i+1))/f[0]) + return g + + +def conv(a,b,n): + return [sum(a[j]*b[i-j] for j in range(i+1) + if j=j else r[j-i] for j in range(n)] for i in range(n)] + walk(H) + receipt={"status":"pass","arithmetic":"exact rational", + "parity_instances":parity,"schur_nodes":trace} + Path(__file__).with_name('EXACT-ALGEBRA.json').write_text(json.dumps(receipt,indent=2)+'\n') + print(json.dumps({"status":"pass","parity_instances":len(parity), + "parity_entries":sum(x['matrix_entries_checked'] for x in parity), + "schur_nodes":len(trace)},indent=2)) + + +if __name__=='__main__': + main() diff --git a/research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_supplier_capacity.py b/research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_supplier_capacity.py new file mode 100644 index 0000000000..65b20b8b68 --- /dev/null +++ b/research/capped-phase-fusion/evidence/inplace_algebra_ultra/check_supplier_capacity.py @@ -0,0 +1,47 @@ +"""Exact capacity controls in the large-g, gG >> single-axis-ell regime.""" +from pathlib import Path +from fractions import Fraction +import hashlib +import json + + +def main(): + # p itself has one million binary exponent bits. Only short descriptions + # and modest exact powers are needed; no input tensor is materialized. + logp=1_000_000 + eps=Fraction(998,1000) + beta=Fraction(93,1000) + interior=Fraction(907,1000) + d_exp=int(eps*logp) + g_exp=int(beta*logp) + ell_exp=logp-d_exp + g=1<ell + assert reserved<=g//16 + assert (g*K*G).bit_length() 2` is a fixed rational number. There is a single, uniform algorithm, with a fixed finite description, with the following properties. + +* Its recursive block multiplication uses a fixed finite bilinear formula whose coefficients are computable algebraic complex numbers. +* Given two `W` by `W` complex dyadic matrices with component magnitudes at most `2^h`, it computes their product with maximum entrywise absolute error at most `2^{-p}` in + +\[ +O\!\left(W^\tau M(p+h+\log(W+1))\right) +\] + +bit operations. Here `M(t)` is a regular integer multiplication bound: `M(t) >= t`, `M(Ct)=O_C(M(t))` for fixed `C`, and precision doubling sums to `O(M(t))`. All constants can depend on the chosen fixed block formula. +* If the input encoding has `h = O(p)`, this is + +\[ +O\!\left(W^\tau M(p+\log(W+1))\right). +\] + +If also `log W = O(p)`, it is exactly `O(W^tau M(p))`, without a suppressed logarithmic factor. In particular this holds for `W = Theta(sqrt(p))`. + +The strict exponent slack is essential to this statement: an exponent infimum does not by itself supply a finite scheme attaining that infimum. If a source proves `omega_C <= 9/4`, one may take any fixed rational `tau` with `9/4 < tau < 5/2`, for example `tau = 19/8`. + +The proof below also gives an exponent strictly below a specified threshold from only the promise that `omega_C` is below that threshold. It does not give useful numerical constants or an efficient way to discover the block formula. + +## 1. A finite exact bilinear formula exists with exponent slack + +Write `R(b)` for the exact complex tensor rank of `b` by `b` matrix multiplication. Use the usual equivalent definition + +\[ +\omega_{\mathbb C}=\inf_{b\ge2}\log_b R(b). +\] + +Thus some fixed integers `b >= 2` and `r` satisfy + +\[ +R(b)\le r= 2`, `r >= 1` satisfying `r < b^tau`; +2. finite encoded algebraic tuples of the required length; +3. exact tests of all rank equations. + +For rational `tau=a/d`, the first comparison is the integer comparison `r^d < b^a`. The search halts under the stated exponent promise, by Sections 1 and 2. Its output supplies the single fixed block formula. This search may be extremely expensive, but its cost is an absolute constant independent of `W,p,h`, so including it does not change the asymptotic bound. The same result can be phrased by fixing the first successful tuple once and for all. + +For just the promise `omega_C < 5/2`, search using `r^2 < b^5`. Upon success choose a rational `sigma` with + +\[ +\max\{2,\log_b r\}<\sigma<5/2. +\] + +The resulting algorithm has exponent at most `sigma < 5/2`. Thus one need not know a numerical gap in advance to obtain a uniform algorithm below the threshold. A practical certificate or practical constant is a separate issue. + +## 4. Each fixed algebraic coefficient has an ordinary `O(M(q))` precision algorithm + +Every chosen coefficient has a fixed minimal polynomial and a fixed isolating region. In characteristic zero its minimal-polynomial root is simple. Fix a rational starting approximation in a sufficiently small Newton convergence neighborhood. Newton iteration with successively doubled precision computes the root to absolute error `2^{-q}` with `O(M(q))` bit operations. + +The polynomial degree, coefficient heights, root separation, initial precision, and bounds on the derivative and its reciprocal are all fixed constants here. Polynomial evaluation, complex division, and the finite number of operations per Newton step cost `O(M(t))` at precision `t`; the usual precision-doubling sum is `O(M(q))`. This is not a claim uniform in algebraic degree or height. + +There are only finitely many block coefficients. Compute their dyadic approximations once at the precision needed for the current multiplication, and reuse them at all recursive nodes. This costs `O(M(q))` total, with a formula-dependent constant, rather than a logarithmic factor per scalar operation. + +## 5. Recursive arithmetic work + +Pad the matrices with zeros to size `N=b^L`, where `W <= N < bW` (and handle `W=1` directly). At each node, form the fixed `r` input linear combinations, recursively multiply the corresponding submatrices, and form the fixed output linear combinations. + +The arithmetic operation count satisfies + +\[ +T(N)\le rT(N/b)+O(N^2). +\] + +Let `rho=log_b r`. If `rho>2`, this is `O(N^rho)`; if `rho=2`, it is `O(N^2 log N)`; and if `rho<2`, it is `O(N^2)`. Every case is `O(N^tau)` because `rho2`. In particular the possible equality case `r=b^2` creates no extra factor in the advertised bound: its logarithm is absorbed by the fixed exponent slack. + +One could instead tensor a fixed formula a fixed number of times and add a zero product to arrange `b^2 < r < b^tau`, obtaining an exact recursion exponent `rho` strictly between two and `tau`. This refinement is unnecessary for the advertised `O(W^tau)` conclusion. + +## 6. Fixed-point roundoff grows only polynomially in the dimension + +Use fixed-point complex arithmetic with `q` fractional bits, round after scalar products, and use dyadic block coefficients whose absolute errors are at most + +\[ +\epsilon=2^{-q}. +\] + +Complex arithmetic introduces only absolute constant factors. Additions may be done exactly on this fixed grid; allowing an additional absolute error `O(epsilon)` after an addition also satisfies the following bounds. + +For a multiplication node with remaining depth `d`, let `a,beta >= 1` bound the maximum absolute entries of its two inputs. There is a fixed constant `K`, depending only on the block formula, such that the computed result obeys + +\[ +\|\widehat T_d(A,B)-AB\|_{\max} +\le \epsilon E_d a\,\beta, +\qquad +E_d\le K E_{d-1}+K b^{d-1},\quad E_0\le K. +\] + +Here the symbol `b` in `b^{d-1}` is the fixed block dimension; `beta` is the second input bound. + +To verify the recurrence, every local input linear combination has fixed length and fixed coefficients. Its rounded value differs from its exact value by at most `c epsilon a` or `c epsilon beta`, and its size is at most `S a` or `S beta`, for fixed `c,S`. Applying the inductive child bound therefore contributes at most + +\[ +c\epsilon E_{d-1}a\beta. +\] + +Perturbing the two child inputs in their *exact* matrix product contributes at most + +\[ +c\epsilon b^{d-1}a\beta, +\] + +because each entry of that child product sums `b^{d-1}` scalar products. Finally, the output reconstruction has fixed length and fixed coefficients. Propagation through its exact coefficients is a fixed factor; coefficient approximation and reconstruction rounding contribute at most + +\[ +c\epsilon\bigl(b^{d-1}+\epsilon E_{d-1}+1\bigr)a\beta. +\] + +For `epsilon <= 1`, all these terms have the stated recurrence after enlarging `K`. Taking `K >= 2b` gives `E_d <= C K^d` for a fixed `C`. Thus at the root + +\[ +\|\widehat T_L(A,B)-AB\|_{\max} +\le C\,2^{-q}K^L a\beta +=2^{-q}N^{O(1)}a\beta. +\] + +For root input magnitude at most `2^h`, take `a=beta=max(1,2^h)` and choose + +\[ +q=p+2\max(h,0)+\lceil L\log_2K\rceil+O(1). +\] + +The final absolute error is then at most `2^{-p}`. + +There is also no hidden exponential-in-precision magnitude growth. Along a path, each preprocessing step multiplies a magnitude envelope by a fixed `S`. A node at depth `t` therefore receives inputs bounded by `S^t a` and `S^t beta`. Its computed output is bounded by its exact product size plus the proved error: + +\[ +\bigl(b^{L-t}+\epsilon C K^{L-t}\bigr)S^{2t}a\beta. +\] + +Local scalar products and partial sums obey the same bound up to fixed factors. All intermediate magnitudes are consequently `N^{O(1)} max(1,2^{2h})`. Only `O(h+log N)` integer bits are needed. The total working bit width is therefore `O(p+h+log N)`. + +The polynomial error estimate exploits the structure of a bilinear recursion: linear preprocessing, one variable-by-variable scalar product at a leaf, and linear reconstruction. Bounding an arbitrary multiplication by the largest intermediate magnitude at every level would give an unnecessarily pessimistic estimate and is not the argument used here. + +## 7. Bit complexity and scope + +Every scalar operation uses `O(p+h+log W)` bits and costs `O(M(p+h+log W))`. Multiplying by the `O(W^tau)` arithmetic operation count proves the statement. Padding, input conversion, output rounding, and storage of the fixed coefficient approximations fit the same bound. + +The result supplies absolute accuracy. It does not promise small relative error in entries canceled nearly to zero. It applies to arbitrary fixed finite complex bilinear formulas; large fixed coefficient heights or norms worsen only the constants and the fixed multiple of `log W` in the precision requirement. + +An exponent theorem alone therefore suffices for computable, stable approximate multiplication with any fixed strictly larger exponent. It does not justify using the exact infimum exponent without slack, treating a varying family of coefficients as fixed, or claiming a concrete efficient implementation without constructing its constants. diff --git a/research/capped-phase-fusion/evidence/matrix_boundary_ultra/MANIFEST.json b/research/capped-phase-fusion/evidence/matrix_boundary_ultra/MANIFEST.json new file mode 100644 index 0000000000..5a943f0fd6 --- /dev/null +++ b/research/capped-phase-fusion/evidence/matrix_boundary_ultra/MANIFEST.json @@ -0,0 +1,35 @@ +{ + "status": "MATRIX_BOUNDARY_TRANSFER_SIGNED_OFF", + "scope": "Full 9/4 source audit plus effective fixed algebraic rank scheme and approximate p-bit transfer; no final outer ledger promotion.", + "target_exponent": "19/8", + "source_rank_exponent_bound": "9/4", + "poly_W_error_growth": true, + "computable_fixed_selection": true, + "files": [ + { + "name": "PROOF-AND-COVERAGE.md", + "bytes": 12130, + "sha256": "db7509d754bb7fc3381419e0fed2674f22a9d57d34e8420fd5afa3c688fde9aa" + }, + { + "name": "ALGEBRAIC-STABILITY-INDEPENDENT.md", + "bytes": 11743, + "sha256": "7873e7d9a99fdc9de3ee224b9925d3e277e9773d7f5f8c2d0f1c19b0e52bc87f" + }, + { + "name": "SOURCE-COVERAGE.json", + "bytes": 2839, + "sha256": "fdc6849f97999a6cf99cdaa50591d24f78d6fb0bc580315f205452e5906b35cc" + }, + { + "name": "SOURCE-CONSTRUCTIONS.json", + "bytes": 21018, + "sha256": "114c1d8fb70592455a80c73d8a1bc6bfc9aca4e7395a420b6331d5474b4d1917" + }, + { + "name": "check_source_constructions.py", + "bytes": 6388, + "sha256": "9d387098403c0355be8e93158df02bae5a435b2366bd28efdf0b127e470c8c1c" + } + ] +} diff --git a/research/capped-phase-fusion/evidence/matrix_boundary_ultra/PROOF-AND-COVERAGE.md b/research/capped-phase-fusion/evidence/matrix_boundary_ultra/PROOF-AND-COVERAGE.md new file mode 100644 index 0000000000..1a98011b1f --- /dev/null +++ b/research/capped-phase-fusion/evidence/matrix_boundary_ultra/PROOF-AND-COVERAGE.md @@ -0,0 +1,91 @@ +# Matrix boundary transfer: full-source audit and interface signoff + +Status: the supplied 9/4 source supports the matrix boundary substitution, with strict exponent slack. This is a theorem-level, computable asymptotic transfer. It is not a numerical finite-size rank certificate and it does not remove any supplier or computational-interface assumptions in the outer integer-multiplication argument. + +The independent reader for this note read all 1,022 lines of +`/Users/vlado/Downloads/math-main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/build/paper.tex` +in order, including the full character-existence appendix and all bibliography entries, and also read the README. The companion 2.258 paper is unnecessary for this transfer and is not claimed as read in this new audit. + +## 1. Source coverage + +| Source lines | Material read | Audit result | +|---|---|---| +| 1–36 | Preamble, title, abstract | Establishes complex arithmetic setting. | +| 37–141 | Introduction, theorem and proof route | Explicitly promises an asymptotic exponent, not a competitive finite size. | +| 142–203 | Tensor semiring, rank, characters, detecting lemma | Uses exact rank and mutual restriction. Rank need only be subadditive; no false additivity assumption. | +| 204–235 | Oriented dot-product exponents | Monotonic completely multiplicative functions give power laws; tensor product of the three orientations is square matrix multiplication. | +| 236–269 | Degenerations and interpolation | Fixed degeneration degree is handled before the tensor-power limit. No uniform-in-power degree assumption is made. | +| 270–349 | Finite shared-leg separation | Fourier relation is exact since the phase has absolute size below 5M. Surviving weights equal `(g-h)^2`; the retained auxiliary dot index is distinct from the direct-sum block index. | +| 350–391 | Shared-leg entropy | The factor M cancels correctly before the N-th-root limit. Fixed-frequency counts and continuity cover every probability vector. | +| 392–432 | Polynomial multiplication and symmetrized profile | Rank is exactly a+b−1 by interpolation and output flattening. All six leg permutations are needed for the later middle branch. | +| 433–523 | Determinant filtration and discrete concavity | Fixed input/output bases work for every first-input slice. Multiplication preserves the determinant submodule; the forbidden upper-right block vanishes. The residual lower-left block receives positive weight. | +| 524–604 | Three sectors and shifted tripling | Outer sectors equal C(a,h); the middle is obtained by reversing the first basis and exchanging its second/third legs. This is an identification of an individual block and does not apply incompatible maps to the shared first leg. | +| 605–651 | Diagonal growth | Concavity gives the stated two-increment recurrence; H_a grows at least as a^(1/3). | +| 652–689 | Spectral bound, rank bound, recursion | Detecting characters may depend on d. The conclusion is `nu<=9/4`; selecting a fixed rank scheme requires a strict larger exponent. Rectangular block substitution preserves product order. | +| 690–699 | Algebraic-coefficient remark | The source explicitly allows replacing a complex solution of rational rank equations by a point over Qbar. | +| 700–769 | Appendix: invariant state family and multiplicativity | The fixed-point map is continuous, not asserted affine. Freezing its scalar eigenvalue produces a compact convex invariant subset, permitting finite intersections. | +| 770–905 | Appendix: detecting state existence | Finite rational separation gives an additive catalyst. Reuse keeps one s factor; no tensor cancellation is assumed. Amplification with a second multiplication tensor gives the contradiction to k9/4`. + +If the surrounding proof allows polylog p factors, the bandwidth's polylogs remain polylogs; no positive power of p is introduced by coefficient approximation or error amplification. The algorithm provides absolute accuracy, which is what local residual bounds use. It makes no relative-error promise for nearly canceled output entries. + +The recursion can also be scheduled with sequential tapes. At one fixed-base node, its fixed number of input block combinations and output block sums require a fixed number of complete scans and fixed block permutations of `O(n^2)` records. Store child inputs consecutively, process children depth-first, and stream their returned blocks into accumulators. Copying a pending-child region or repositioning a finite set of tapes costs another `O(n^2)`-record pass at that node, already covered by the recurrence's addition term. Addresses and recursion descriptions add at most logarithmic factors. If any implementation retains an additional `polylog W` factor, choose the rank scheme first with exponent strictly between 9/4 and the stated target tau, leaving enough fixed exponent slack to absorb it. No algebraic unit-cost random-access assumption is needed for the target power tau. + +## 4. Boundary substitution and the outer ledger + +The existing capped boundary algebra uses W simultaneous right-hand sides. A square product costs `O(W^tau M(p))`; dividing by W² processed entries gives `O(W^(tau-2) M(p))` per entry. One axis has `O(V W/B)` boundary entries, and there are d axes. Its total is consequently + +`O(V (d/B) W^(tau-1) [M(p)/p] polylog p)` + +when V is the total bit volume, as in the phase-assembly notes. There are V/Theta(p) scalar entries, so the factor M(p)/p is the per-bit arithmetic charge. Equivalently, if V counts scalar entries instead, the formula has M(p) without the division. The retained quasi-linear short multiplier has M(p)/p=polylog p. + +This retains the essential d/B factor. With `d=p^epsilon`, `B=p^b`, and `W=p^(1/2) polylog p`, its normalized exponent is + +`epsilon-b+(tau-1)/2`. + +Taking `tau=19/8` gives `(tau-1)/2=11/16` and boundary saving `5/16+b-epsilon`. This is already more than enough to put the basic sparse-union cost near the binding row at b near 3/4. Pushing tau down to the unattained endpoint 9/4 is unnecessary. + +The changed matrix algorithm has the same input/output interface as the former Strassen product. It does not change the original Gaussian matrix, boundary support, anchored gauge, square-block inverse choice, or number of right-hand sides. The surrounding local-residual proof must still supply polynomial bounds on local exact inverses and partial right-hand sides. The new product's error is `poly(p)2^-P`, so that proof applies with a fixed larger guard constant. + +The optional full-coordinate-key surcharge in the old arithmetic ledger would become binding if mechanically retained at b near 3/4. It is not part of the actual rectangle extraction/one-field transpose/ordered-merge algorithm, as confirmed by the separate phase-layout audit. Dropping it is a distinct layout fact, not a consequence of matrix multiplication. It must remain explicitly justified in the final outer ledger. All retained supplier assumptions remain in force. + +No final improved outer exponent is claimed in this note; that requires the whole ledger and the independent group-layout/gauge checks. + +## 5. Exact controls run in this audit + +`check_source_constructions.py` uses only Python's exact rational arithmetic and writes `SOURCE-CONSTRUCTIONS.json`: + +* Finite Fourier/separation support for M=1,...,12: the modular relation has no wraparound and zero weight is exactly g=h,u=v. +* Determinant change of basis for a=1,...,8 and b=2,...,8, for every first-input monomial: quotient and kernel diagonal slices are exactly the two claimed neighboring convolution tensors; the upper-right block vanishes. +* Three-sector identity for a,h=1,...,14: all weights are nonnegative and each of the three retained branches is exactly the claimed tensor after its permitted identification. +* Exact rational diagonal product lower bound through a=300. + +All controls passed. They detect several relevant implementation mistakes but are not a finite substitute for the universal arguments above or for the theorem's nonconstructive rank-existence step. + +## 6. Deliberate limitations + +There is no generated competitive finite matrix scheme, no numeric claim about its constants, no use of a rank formula at exponent exactly 9/4, and no fast algorithm for finding the first successful algebraic tuple. The algorithm remains computable and uniform because the finite search halts under the proved existence statement and its cost is independent of the growing input. 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bilinear structure; unbounded sequential multiplication does not follow", + "endpoint_exponent_9/4": "reject: rank infimum only supplies fixed algorithms strictly above 9/4", + "arbitrary_complex_constants": "reject: computability requires algebraic-point replacement or explicit computable coefficients" + } +} diff --git a/research/capped-phase-fusion/evidence/matrix_boundary_ultra/SOURCE-COVERAGE.json b/research/capped-phase-fusion/evidence/matrix_boundary_ultra/SOURCE-COVERAGE.json new file mode 100644 index 0000000000..c6275bc1cb --- /dev/null +++ b/research/capped-phase-fusion/evidence/matrix_boundary_ultra/SOURCE-COVERAGE.json @@ -0,0 +1,114 @@ +{ + "status": "FULL_SOURCE_READ_AND_AUDITED", + "source": "/Users/vlado/Downloads/math-main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/build/paper.tex", + "sha256": "c396f2580a711b8b9e51ecf1265cb0e691acd745943f81bfc14ec0df209e1028", + "line_count": 1022, + "coverage": [ + { + "first_line": 1, + "last_line": 36, + "topic": "preamble and abstract", + "read": true + }, + { + "first_line": 37, + "last_line": 141, + "topic": "introduction and theorem", + "read": true + }, + { + "first_line": 142, + "last_line": 203, + "topic": "tensor semiring and rank", + "read": true + }, + { + "first_line": 204, + "last_line": 235, + "topic": "dot exponents", + "read": true + }, + { + "first_line": 236, + "last_line": 269, + "topic": "interpolation", + "read": true + }, + { + "first_line": 270, + "last_line": 349, + "topic": "finite shared-leg separation", + "read": true + }, + { + "first_line": 350, + "last_line": 391, + "topic": "entropy", + "read": true + }, + { + "first_line": 392, + "last_line": 432, + "topic": "polynomial profile", + "read": true + }, + { + "first_line": 433, + "last_line": 523, + "topic": "determinant filtration", + "read": true + }, + { + "first_line": 524, + "last_line": 604, + "topic": "tripling", + "read": true + }, + { + "first_line": 605, + "last_line": 651, + "topic": "diagonal growth", + "read": true + }, + { + "first_line": 652, + "last_line": 689, + "topic": "rank and arithmetic conclusion", + "read": true + }, + { + "first_line": 690, + "last_line": 699, + "topic": "algebraic coefficients", + "read": true + }, + { + "first_line": 700, + "last_line": 769, + "topic": "multiplicativity appendix", + "read": true + }, + { + "first_line": 770, + "last_line": 905, + "topic": "detecting-state existence appendix", + "read": true + }, + { + "first_line": 906, + "last_line": 1022, + "topic": "bibliography and ending", + "read": true + } + ], + "readme": { + "path": "/Users/vlado/Downloads/math-main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/README.md", + "sha256": "1857fc06d04a43588207ba2f726679c50a418299a5fd0313423a354680a783a7", + "read": true + }, + "companion_2_258_claimed_read": false, + "companion_required": false, + "proof_dependencies": "The required new interpolation, determinant, entropy, growth and spectral detection arguments are contained in this source. Standard background dependencies are listed in PROOF-AND-COVERAGE.md.", + "finite_rank_scheme_generated": false, + "outer_improvement_claimed_by_this_audit": false +} diff --git a/research/capped-phase-fusion/evidence/matrix_boundary_ultra/check_source_constructions.py b/research/capped-phase-fusion/evidence/matrix_boundary_ultra/check_source_constructions.py new file mode 100644 index 0000000000..c8e362e90d --- /dev/null +++ b/research/capped-phase-fusion/evidence/matrix_boundary_ultra/check_source_constructions.py @@ -0,0 +1,164 @@ +#!/usr/bin/env python3 +"""Exact finite controls for the constructions in the supplied 9/4 paper. + +These checks are controls for an end-to-end human proof audit, not substitutes +for the universal proofs or an executable fast matrix multiplication scheme. +""" +from fractions import Fraction as Q +from pathlib import Path +import hashlib +import json + +ROOT = Path(__file__).resolve().parent +SOURCE = Path('/Users/vlado/Downloads/math-main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026/build/paper.tex') + +def inverse(a): + n = len(a) + t = [[Q(v) for v in row] + [Q(i == j) for j in range(n)] + for i, row in enumerate(a)] + for j in range(n): + p = next(i for i in range(j, n) if t[i][j]) + t[j], t[p] = t[p], t[j] + z = t[j][j] + t[j] = [v/z for v in t[j]] + for i in range(n): + if i != j: + z = t[i][j] + t[i] = [v-z*w for v,w in zip(t[i],t[j])] + return [row[n:] for row in t] + +def mm(a,b): + return [[sum(x*y for x,y in zip(row,col)) for col in zip(*b)] + for row in a] + +def basis(e): + # Ambient coordinates: u^(e-i)v^i s, then u^(e-i)v^i w. + n = 2*(e+1) + cols = [] + for i in range(e+1): + v = [Q(0)]*n + v[i] = 1 + cols.append(v) + v = [Q(0)]*n + v[-1] = 1 + cols.append(v) + for j in range(e): + v = [Q(0)]*n + v[j+1] = -1 + v[e+1+j] = 1 + cols.append(v) + return [list(row) for row in zip(*cols)] + +def check_determinant(): + cases = [] + for a in range(1,9): + for b in range(2,9): + ei, eo = b-1, a+b-2 + bi, bo = basis(ei), basis(eo) + inv = inverse(bo) + for k in range(a): + mult = [[Q(0)]*(2*b) for _ in range(2*(a+b-1))] + for side in range(2): + for i in range(b): + mult[side*(a+b-1)+i+k][side*b+i] = 1 + t = mm(inv,mm(mult,bi)) + qo, qi = a+b, b+1 + for i,row in enumerate(t): + for j,val in enumerate(row): + if i < qo and j < qi: + assert val == Q(i == j+k) + elif i < qo and j >= qi: + assert val == 0 + elif i >= qo and j >= qi: + assert val == Q(i-qo == j-qi+k) + # Only the lower-left block may remain, with weight +1. + cases.append({'a':a,'b':b,'slices_checked':a}) + return cases + +def check_fourier(): + cases = [] + for m in range(1,13): + terms = surviving = zeros = 0 + for h in range(1,m+1): + for g in range(1,m+1): + for u in range(1,m+1): + for v in range(1,m+1): + terms += 1 + phase = u-v+2*(g-h) + assert abs(phase) < 5*m + if phase % (5*m) == 0: + surviving += 1 + assert phase == 0 + weight = g*g+h*u-h*h-h*v + assert weight == (g-h)**2 >= 0 + if weight == 0: + zeros += 1 + assert g == h and u == v + assert zeros == m*m + cases.append({'M':m,'potential_terms':terms, + 'fourier_survivors':surviving,'weight_zero_terms':zeros}) + return cases + +def check_tripling(): + cases = [] + for a in range(1,15): + for h in range(1,15): + B=3*h+a-1 + sectors=[set(),set(),set()] + for x in range(a): + for y in range(B): + z=x+y + sy=0 if y= 0 + if weight==0: + assert sy==sz + if sy==0: + t=(x,y,z) + elif sy==2: + t=(x,y-(2*h+a-1),z-(2*h+a-1)) + else: + # Swap second/third legs; reverse first-leg basis. + t=(a-1-x,z-(h+a-1),y-h) + assert t[2]==t[0]+t[1] + sectors[sy].add(t) + expected={(x,y,x+y) for x in range(a) for y in range(h)} + assert all(s==expected for s in sectors) + cases.append({'a':a,'h':h,'terms_in_each_branch':a*h}) + return cases + +def check_growth(): + h=Q(1) + for a in range(1,301): + if a>1: h *= 1+Q(1,3*(a-1)) + assert h**3 >= a + assert Q(3*a-1,2) >= a + return {'maximum_a':300,'exact_product_inequalities':'PASS'} + +if __name__ == '__main__': + raw=SOURCE.read_bytes() + result={ + 'status':'PASS_EXACT_SOURCE_CONTROLS', + 'scope':'Finite controls accompany a full universal proof audit; no rank algorithm synthesized.', + 'source':str(SOURCE), + 'source_sha256':hashlib.sha256(raw).hexdigest(), + 'source_lines':len(raw.splitlines()), + 'fourier':check_fourier(), + 'determinant_filtration':check_determinant(), + 'tripling':check_tripling(), + 'growth':check_growth(), + 'negative_controls':{ + 'drop_fourier_phase': 'reject: weights need not be nonnegative, e.g. M=2,h=2,g=1,u=1,v=2 has weight -5', + 'keep_only_one_outer_branch':'reject: factor 3 requires both outer branches and the leg-swapped middle branch', + 'arbitrary_arithmetic_circuit':'reject: stability proof requires recursive bilinear structure; unbounded sequential multiplication does not follow', + 'endpoint_exponent_9/4':'reject: rank infimum only supplies fixed algorithms strictly above 9/4', + 'arbitrary_complex_constants':'reject: computability requires algebraic-point replacement or explicit computable coefficients', + }, + } + (ROOT/'SOURCE-CONSTRUCTIONS.json').write_text(json.dumps(result,indent=2)+'\n') + print(result['status']) + print('source lines',result['source_lines']) + print('fourier M cases',len(result['fourier'])) + print('determinant cases',len(result['determinant_filtration'])) + print('tripling cases',len(result['tripling'])) diff --git a/research/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-INTERIORS.md b/research/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-INTERIORS.md new file mode 100644 index 0000000000..62c3fa27b6 --- /dev/null +++ b/research/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-INTERIORS.md @@ -0,0 +1,72 @@ +# Conditional refinement: cap interior length independently of d + +This is a separate follow-up to the uncapped proof audit. Its current status is **PASS_CONDITIONAL_PHASE_FUSION_REVIEW**, following the independent capped-gauge and capped-layout audits and an integrated root review. It preserves the weaker `OUTER-ARITHMETIC.json` as history. Under the retained supplier assumptions and the reviewed interfaces, the capped construction below gives `kappa=0.001672922349588871`. The only new analytic change is the anchored O(aB) local gauge; all altered operation counts are derived explicitly here. No upstream selected certificate is changed. + +Let the original phase-cell scale remain `d=p^epsilon`, and cap dyadic interior lengths at `B=2^floor(b log2 p)`, where `1/20. The supplier support proof needs each individual K-neighborhood to fit its original axis and total moved support o(p); it does **not** require all g controls to fit within one original axis. The sufficient auxiliary inequality `beta<1-epsilon` in the uncapped ledger must therefore be removed in this follow-up. + +The changed principal exponents, normalized by V, are + +| Component | exponent | +|---|---:| +| main transform | `1-epsilon*q` | +| fused Gaussian circuits | `epsilon-q*beta` | +| boundary algebra | `epsilon-b+(omega-1)/2` | +| sparse union movement | `epsilon+beta+1/2-b` | +| grouped scans | `epsilon-beta` | + +For the boundary row, one axis has O(V W/B) boundary records. A band solve or coupling product on W simultaneous right-hand sides costs O(W^(omega-2)) per boundary record, so all d axes cost `O(V (d/B) W^(omega-1) polylog p)`. The new factor d/B must not be omitted. For the sparse union, its fraction within a group is O(gW/B polylog p); g sequential axis operations and d/g groups give `O(V (dg/B) W polylog p)`. These are precisely the third and fourth changed rows. + +All other costs in `OUTER-AUDIT.md` remain paid. Global routing has saving a_eff; packed polynomial products have saving epsilon. Even a conservative full active-group key surcharge adds only `beta+1-epsilon` to the union exponent. Sparse core-face corrections have exponent at most `epsilon+beta-3`, using the former p² per-record allowance on their O(gA/L) copied volume. + +With the rational bound omega<45/16, put c=(3-omega_upper)/2=3/32 and take a positive boundary slack s_boundary=10^-5. Balancing the two dominant rows while leaving that boundary slack gives + +``` +epsilon=(1+q+q*(c-s_boundary-r-zeta))/(1+q)^2, +b=epsilon*(1+q)-c+s_boundary, +beta=1-b-r-zeta, +kappa_limit=epsilon*q. +``` + +Using the same q, r and zeta as the uncapped candidate yields approximately + +``` +epsilon = 0.9984838711085557536432677624420451614... +b = 0.9064167934681446247757331487619919695... +beta = 0.0935821965318553752242668512380080305... +kappa_limit = 0.0016729223595888711324653863199468081... +kappa = 0.001672922349588871 +``` + +The boundary saving equals `kappa_limit+10^-5`. The union movement exponent is below 0.686, and even its optional full-key surcharge leaves it below 0.781. The per-record phase metadata exponent `beta+2(1-epsilon)` is below 0.097, so calculating all g membership predicates remains cheaper than scanning one p-bit record. Core-face repair costs remain negligible at L=p^8. + +The common compiled H calls are still made globally across complete coarse-label ranges. Each selected bit needs only its own K=128G neighborhood to fit its original axis. Since `1-epsilon>0`, this holds eventually even though gK is larger than one axis width. The total support is O(g polylog p)=o(p), because beta<1. All support-router reservoirs, product row stock and exact endpoint restoration are those of the independently audited supplier. + +At the displayed parameters, `b>1/2`, `epsilon>b`, `beta>0`, `epsilon>beta`, `1-b-r-beta=zeta>0`, and `1-epsilon-r>0`. These respectively pay the wide-band separation, cap below original cell scale, growing group payload, number of groups, compiled local-gauge precision, and global gamma/rounding range. The raw global rounded weights may still have O(ad) logarithmic range during offline generation. The **data** only sees exact anchored ratios of range O(aB); approximating those ratios to the retained precision introduces `exp(o(p))` error amplification, already paid by the strict group guard. + +The assembled group endpoints are the same full normalized Haug inverse or Saug/4 operators as in the uncapped proof. Changing the interior partition changes only their factorization. It does not change their completed contraction bounds, intended zero-hole semantics, principal-core locality, or the final `2^D` scale compensation. Consequently the uncapped precision and error argument applies with the improved group gauge budget. + +`check_capped_outer.py` computes 44 exact positive slacks, every displayed cost margin, and five rejecting controls. The full-cell-gauge control is especially relevant: it fails, so the stronger result depends on the independently reviewed anchored-gauge proof. The local review retains the original supplier and all-size conditional assumptions. diff --git a/research/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-OUTER-ARITHMETIC.json b/research/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-OUTER-ARITHMETIC.json new file mode 100644 index 0000000000..56c871f4f3 --- /dev/null +++ b/research/capped-phase-fusion/evidence/phase_assembly_ultra/CAPPED-OUTER-ARITHMETIC.json @@ -0,0 +1,388 @@ +{ + "status": "PASS_CONDITIONAL_PHASE_FUSION_REVIEW", + "scope": "Local conditional transfer and exact arithmetic; inherited supplier and all-size interfaces remain assumptions. 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supplier, common-frame, whole-call, semantic precision, fixed-tape and prime-packing assumptions of the first transfer remain.", + "No generated competitive finite matrix formula or practical runtime claim is made; the fixed formula search is effective but may have enormous constant cost.", + "This is a local conditional corollary, not an upstream selected certificate or an unconditional integer-multiplication theorem." + ], + "reviews": [ + { + "reviewer": "root agent", + "result": "Read matrix corollary and independent layout signoff; reproduced exact parameters/ledger and matrix construction controls; approved integrated conditional corollary." + }, + { + "reviewer": "matrix_boundary agent", + "result": "Read all 1022 supplied source lines and audited effective algebraic coefficient, fixed-tape, precision and rank-exponent transfer." + }, + { + "reviewer": "global_macro agent", + "result": "Independent new-parameter no-sort union, metadata, q-ordering and microhalo layout signoff." + }, + { + "reviewer": "phase_assembly agent", + "result": "Read both complete matrix transfer notes and independent layout signoff; constructed and replayed 47-slack ledger and six rejecting controls." + }, + { + "reviewer": "independent_fusion_review agent and independent children", + "result": "No blocking gap found under stated assumptions after full compiler, support, Gaussian, layout and tape-cost review. Generic SPD scale wording corrected without changing parameters. See independent FINAL-REVIEW.json." + } + ], + "arithmetic": { + "positive_slacks": 47, + "rejecting_controls": 6, + "minimum_saving": "0.0016733601154057784538170479517203762842018888767028492264912297200" + }, + "files": [ + { + "bundle_filename": "phase_fusion_note.tex", + "source_path": "work/phase_assembly_ultra/phase_fusion_note.tex", + "sha256": "08ecb1876695beee25484158e14cfd57c3d48f551a8a41d5abc998d71d674555", + "bytes": 33757 + }, + { + "bundle_filename": "MATRIX-COROLLARY.md", + "source_path": "work/phase_assembly_ultra/MATRIX-COROLLARY.md", + "sha256": "2d7fb1088e58bb86e574277733e082f135362215a69add71d9090213d32c9221", + "bytes": 5787 + }, + { + "bundle_filename": "check_matrix_corollary.py", + "source_path": "work/phase_assembly_ultra/check_matrix_corollary.py", + "sha256": "30db2d8e0f40fcc5c91e25f74cdc54e46aa4a0353c725345f9bcf5744e0b617f", + "bytes": 6692 + }, + { + "bundle_filename": "MATRIX-COROLLARY-ARITHMETIC.json", + "source_path": "work/phase_assembly_ultra/MATRIX-COROLLARY-ARITHMETIC.json", + "sha256": "c6efc4d2f6d17159cefebf5c2dd6099a6327fe0911c5dd87623cad96c7d6c56a", + "bytes": 16783 + }, + { + "bundle_filename": "PROOF-AND-COVERAGE.md", + "source_path": "work/matrix_boundary_ultra/PROOF-AND-COVERAGE.md", + "sha256": "db7509d754bb7fc3381419e0fed2674f22a9d57d34e8420fd5afa3c688fde9aa", + "bytes": 12130 + }, + { + "bundle_filename": "ALGEBRAIC-STABILITY-INDEPENDENT.md", + "source_path": "work/matrix_boundary_ultra/ALGEBRAIC-STABILITY-INDEPENDENT.md", + "sha256": "7873e7d9a99fdc9de3ee224b9925d3e277e9773d7f5f8c2d0f1c19b0e52bc87f", + "bytes": 11743 + }, + { + "bundle_filename": "MATRIX-SOURCE-COVERAGE.json", + "source_path": "work/matrix_boundary_ultra/SOURCE-COVERAGE.json", + "sha256": "fdc6849f97999a6cf99cdaa50591d24f78d6fb0bc580315f205452e5906b35cc", + "bytes": 2839 + }, + { + "bundle_filename": "MATRIX-SOURCE-CONSTRUCTIONS.json", + "source_path": "work/matrix_boundary_ultra/SOURCE-CONSTRUCTIONS.json", + "sha256": "114c1d8fb70592455a80c73d8a1bc6bfc9aca4e7395a420b6331d5474b4d1917", + "bytes": 21018 + }, + { + "bundle_filename": "MATRIX-TRANSFER-MANIFEST.json", + "source_path": "work/matrix_boundary_ultra/MANIFEST.json", + "sha256": "35d98d7579a16dae36530b4e5f4d0b8f925a069daa3f6fe1ea5276916167e0c2", + "bytes": 1159 + }, + { + "bundle_filename": "MATRIX-LAYOUT-SIGNOFF.md", + "source_path": "work/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md", + "sha256": "4c8c62f723232e842dc3bbdcc4defcc8ff1aeb86ba9f694661899dc6783b13c4", + "bytes": 3081 + }, + { + "bundle_filename": "MATRIX-LAYOUT-CONTROLS.json", + "source_path": "work/phase_layout_audit_ultra/MATRIX-LAYOUT-CONTROLS.json", + "sha256": "f8bd8729cb8438188e90fb9149f598b825461c97b4ed290584ca4367f812a48a", + "bytes": 1446 + }, + { + "bundle_filename": "PHASE-LAYOUT-AUDIT.md", + "source_path": "work/phase_layout_audit_ultra/AUDIT.md", + "sha256": "54aafc7afd09fe1a5312e69fad09c87422690543ab75c6ae35f6cbcc26b3eb8a", + "bytes": 11774 + }, + { + "bundle_filename": "CAPPED-SIGNOFF.md", + "source_path": "work/phase_layout_audit_ultra/CAPPED-SIGNOFF.md", + "sha256": "70e9175ebde200f8b016236adc88c4a1c4858945e7c3e5c453a8b6a5f79a263a", + "bytes": 4414 + }, + { + "bundle_filename": "CAPPED-GAUGE-AUDIT.md", + "source_path": "work/gaussian_fast_ultra/CAPPED-GAUGE-AUDIT.md", + "sha256": "1030a34922709ab8d3434a4c57a4a58934ae268c237fb598cb415697da411c9f", + "bytes": 9176 + }, + { + "bundle_filename": "LOG-A-PHASE-SCHUR-AUDIT.md", + "source_path": "work/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md", + "sha256": "e05d6b8596c80b6e8676569a0ff464a29fa704c3304143f58914f47cdf38b272", + "bytes": 10642 + }, + { + "bundle_filename": "FORWARD-SMALL-A-AUDIT.md", + "source_path": "work/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md", + "sha256": "e37aaa30b15671a1dff00676b14572213efaaac99df202dbba5e09f57af487c0", + "bytes": 6689 + }, + { + "bundle_filename": "SHALLOW-TOEPLITZ.md", + "source_path": "work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md", + "sha256": "b384b93f3fb4a8c97da2e617ebc14d55cebb0a739070e7f8dbb66f63e24e9310", + "bytes": 15480 + }, + { + "bundle_filename": "TENSOR-COMPILER.md", + "source_path": "work/inplace_algebra_ultra/TENSOR-COMPILER.md", + "sha256": "4d3b8466598a864a234b978e25976648d4e761840aad9153dd18d73d3dfe4d15", + "bytes": 13430 + }, + { + "bundle_filename": "SUPPLIER-SUPPORT-AUDIT.md", + "source_path": "work/inplace_algebra_ultra/SUPPLIER-SUPPORT-AUDIT.md", + "sha256": "bd5b06eca043c5c871227264f6a954473c55cd16ec85bb420ff538fb8ec1d8d6", + "bytes": 16557 + }, + { + "bundle_filename": "SUPPORT-ROUTER.md", + "source_path": "work/halo_layout_ultra/SUPPORT-ROUTER.md", + "sha256": "6ddb322e11883c1177ba6cd1a77d23a1dac827d7290b32eaf642457b1d762a54", + "bytes": 7755 + } + ], + "post_review_correction": "Generic SPD prefix bound records both M and 1/m, not condition number alone; actual Gaussian bounds already satisfy it." +} diff --git a/research/capped-phase-fusion/evidence/phase_assembly_ultra/MATRIX-COROLLARY.md b/research/capped-phase-fusion/evidence/phase_assembly_ultra/MATRIX-COROLLARY.md new file mode 100644 index 0000000000..f08bb57f9c --- /dev/null +++ b/research/capped-phase-fusion/evidence/phase_assembly_ultra/MATRIX-COROLLARY.md @@ -0,0 +1,72 @@ +# Additional corollary using the supplied matrix-exponent theorem + +Status: **PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW**. The independent matrix-transfer and new-parameter layout audits passed, followed by the integrated root review and exact ledger replay. The first, Strassen-based result `kappa=0.001672922349588871` remains separately reviewed and unchanged. Neither result changes the upstream selected certificate. + +This corollary has an **additional hypothesis**: the supplied *Matrix Multiplication in Nine Fourths* theorem gives `omega_C<=9/4`. Its complete-source audit is `PROOF-AND-COVERAGE.md`; the effective approximate-bit transfer is `ALGEBRAIC-STABILITY-INDEPENDENT.md`. The result here retains every supplier and all-size assumption of the first phase-fusion theorem. + +## Effective matrix operation with the required error interface + +Fix `tau=19/8>9/4`. The matrix exponent theorem supplies a fixed block dimension m and rank r with `r0`. All local coefficient pages remain polynomial in p, while `L^g` remains superpolynomial. Microedge correction has exponent `epsilon+beta-3<-1.75`, including the generous old scalar arithmetic bound. Sparse microhalo key passes remain explicitly paid inside that negligible budget. + +All forward q-image group embeddings occur in native axis-major order before the one global low-bit gather. All compression q-image extractions occur after the inverse global gather. The entire Gaussian calculation keeps full dyadic target coordinate ranges, so every globally invoked H stage has complete coarse labels. This schedule needs no per-group gathering of the whole long-axis fields; the q-image scans have the already-paid Vd/g cost. + +The **optional full-group-key sort surcharge on the sparse union is omitted** because it is absent from the actual algorithm. The reviewed union routine extracts the complete rectangle `U_i=E_i x [L]^(g-1)`, transposes only its one polynomial-size field, computes its axis operator, and merges it into U in the original order. It does not radix-sort U. Reintroducing that sort is a rejecting control in the new ledger; it would destroy this parameter choice. This omission relies on the existing positive layout proof, not on the matrix theorem. + +`check_matrix_corollary.py` verifies 47 exact positive slacks and six rejecting controls and writes `MATRIX-COROLLARY-ARITHMETIC.json`. The rejected variants are endpoint equality, zero group guard, zero union slack, the old Vd row, the old full-cell data gauge, and adding a full-key sort to U. 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"lambda_prime_above_lambda": { + "rational": "0", + "decimal": "0" + }, + "lambda_prime_above_leaf": { + "rational": "-67018507072324140917/40000000000000000000000000000000", + "decimal": "-1.675462676808103522925E-12" + } + } + }, + "zero_group_guard": { + "rejected_slacks": { + "group_guard": { + "rational": "0", + "decimal": "0" + } + } + }, + "old_per_axis_scan": { + "rejected_slacks": { + "unreplaced_per_axis_scan_above_kappa": { + "rational": "-18703131735503606020965614630299327/6689006167690774713639000000000000000000", + "decimal": "-0.0000027961002377069763404147924773107404647435514691532087293295859953" + } + } + } + }, + "theorem_obligations": [ + "retained finite suppliers and exact common-frame/mutable-source compiler", + "deferred reservations and paid global synthetic layout", + "shallow Toeplitz plus SPD extension and tensor compiler", + "small-a forward conditioning and exact target-grid square completion", + "grouped sparse-union route and batched boundary arithmetic", + "dyadic core with sparse microbox-face repair and monotone q-image scans", + "scalar product-diagonal scan and retained all-size semantic/prime/error transfer" + ] +} diff --git a/research/capped-phase-fusion/evidence/phase_assembly_ultra/OUTER-AUDIT.md b/research/capped-phase-fusion/evidence/phase_assembly_ultra/OUTER-AUDIT.md new file mode 100644 index 0000000000..974b8db67f --- /dev/null +++ b/research/capped-phase-fusion/evidence/phase_assembly_ultra/OUTER-AUDIT.md @@ -0,0 +1,93 @@ +# Phase-cell fusion: outer assembly audit + +Status: assembly audit in progress. The exact arithmetic in this directory is a **conditional target**, not a replacement of the selected certificate. The remaining proof interfaces are recorded explicitly below. + +## Conditional transfer statement + +Assume the same finite bit/complex suppliers, common-label arithmetic compiler, semantic guard and all-size fixed-tape interfaces as the selected conditional PR1094 result. Also assume the new deferred-reservation, support-limited H supplier, shallow-circuit tensorization, and grouped sparse/core-halo lemmas below have their stated uniform implementations. Then the retained integer-multiplication reduction runs in + +`O(n (log n)^(1-kappa))`, `kappa=0.001672664764353193`. + +Here p=Theta(log n) and the live scalar bit volume V=Theta(n). Global one-axis preprocessing is polynomial in the original axis sizes and p, hence `n^o(1)`. Table construction or page loading repeated per coarse block is charged to its superpolynomial `L^g` payload by the tensor/scan lemmas; it is included in the displayed V costs, rather than called one global preprocessing operation. Every cost row below is `O(V p^e polylog p)` with `1-e>kappa`. The minimum excess is greater than `10^-11`. Consequently every fixed logarithmic factor and each use of the unconditional quasi-linear short arithmetic kernel can be absorbed in a single positive fraction of that excess. Floors, rounded alpha, and dyadic L alter constants or logarithmic factors. All claims are eventual in p; one fixed finite prefix of inputs can use the retained elementary fallback. This statement becomes an admitted theorem only when the named implementation hypotheses receive the intended full review. + +## Parameter convention + +Let `a_bit=0.001733427039589844234749` and `b=0.001675462676808103522925` be the retained lower bounds certified by the selected PR1094 replay. For applying the older compact recurrence, lower the usable bit saving to `a_eff=(b+q)/2`; this keeps its historical convention `a_eff0` with `(1-t_stop)b>a_eff` and `q<(1-t_stop)b`. + +For each visited coarse block and group, all required local coefficient pages can instead be regenerated in polynomial time in p: there are g local windows of size at most L=Theta(p^8), their rational origins have O(p/d) bits, and the published local coefficient/Schur/circuit constructors are polynomial in that size and requested precision. Recover local phases and anchor gauges from these origins; do not form a raw uncentered global q² exponential on the data. The page contains O(p)-bit online coefficients. Its polynomial setup and sequential loading are amortized by the complete `L^g` payload, which eventually dominates every fixed power of p. This gives an explicit access method even when the coarse labels are permuted; no random global coefficient-table lookup is assumed. + +Global chirps, twists, positive diagonal normalizations and exact scale factors are separable products. The prefix-product scan supplies their products in O(V polylog p), with completed scalar range `O(d alpha^2)=o(p)`. Packed polynomial products retain their saving epsilon. The finite supplier, exact common-label frame, whole-call hosting, semantic depth bound with C1=1, phase precision, prime packing and fixed-tape input/output assumptions are inherited unchanged. They are still conditional interfaces, as in the selected result. + +## Cost ledger, with conservative wide bandwidth + +Let `h_w=1/2` and `h_K=beta+1-epsilon`. Normalize all costs by V. Fixed logarithmic factors are absorbed by strict gaps. + +| Component | power of p | reason | +|---|---:|---| +| main transform | `1-epsilon*q` | deferred reservations | +| constant global coordinate routing | `1-a_eff` | paid support/full router | +| dense Gaussian groups | `epsilon-q*beta` | `(d/g) g^(1-q)` | +| grouped bit routing | `epsilon-a_eff*beta` | same grouping, bit supplier | +| one/fixed group scans | `epsilon-beta` | d/g groups | +| boundary algebra | `(omega-1)*h_w` | batched w-column Strassen | +| sparse union, basic movement | `beta+h_w` | V g w overall | +| sparse union, conservative key surcharge | `beta+h_w+h_K` | full active-group keys | +| polynomial products | `1-epsilon` | retained short polynomial products | +| core-face corrections | `epsilon+beta+3-8+2` | old local maps on sparse copies, with a loose p² allowance | + +Use the rational upper bound `omega<45/16`, checked by `7^16<2^45`. The target balances the first and third rows exactly before a fixed 10^-11 absorption backoff. + +This ledger is accepted only after its circuit tensorizer, grouped sparse route, target-grid core/halo adapter, forward small-a inverse bound, and deferred-reservation interfaces have written proofs. Exact positive inequalities alone do not discharge those interfaces. diff --git a/research/capped-phase-fusion/evidence/phase_assembly_ultra/README.md b/research/capped-phase-fusion/evidence/phase_assembly_ultra/README.md new file mode 100644 index 0000000000..c9bd9bfd6f --- /dev/null +++ b/research/capped-phase-fusion/evidence/phase_assembly_ultra/README.md @@ -0,0 +1,15 @@ +# Reviewed conditional phase-cell fusion + +With the explicit additional supplied matrix-exponent premise `omega_C<=9/4`, the reviewed local corollary gives **`kappa=0.001673360105405778`**, a gain of `6.9990832886e-7` over the selected conditional exponent. Its status is `PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW`. + +The separate Strassen-based transfer gives `kappa=0.001672922349588871` with status `PASS_CONDITIONAL_PHASE_FUSION_REVIEW`, without that additional matrix premise. Both retain the original supplier and all-size computational assumptions. The upstream selected certificate is unchanged. + +- `phase_fusion_note.tex` is the standalone proof note. +- `CAPPED-INTERIORS.md` records the local gauge and boundary-count change. +- `check_capped_outer.py` recomputes 44 positive rational slacks and five rejecting controls; `CAPPED-OUTER-ARITHMETIC.json` records the reviewed local status. +- `MATRIX-COROLLARY.md` and `check_matrix_corollary.py` give the additional corollary; `MATRIX-COROLLARY-ARITHMETIC.json` contains 47 positive slacks and six rejecting controls. `MATRIX-COROLLARY-REVIEW.json` records its separate review and source hashes. +- `OUTER-AUDIT.md`, `check_outer.py`, and `OUTER-ARITHMETIC.json` preserve the weaker uncapped arithmetic as history. + +Run `python3 -B work/phase_assembly_ultra/check_capped_outer.py` and `python3 -B work/phase_assembly_ultra/check_matrix_corollary.py` from the workspace root. + +The old `Vd` row and the unrecentered full-cell gauge both reject this result. The improvement uses the reviewed Gaussian circuits, support-limited compact supplier, anchored capped gauges, and grouped sparse/halo scans. Tightening the old arithmetic alone cannot produce it. diff --git a/research/capped-phase-fusion/evidence/phase_assembly_ultra/REVIEW-RECEIPT.json b/research/capped-phase-fusion/evidence/phase_assembly_ultra/REVIEW-RECEIPT.json new file mode 100644 index 0000000000..2521d02ada --- /dev/null +++ b/research/capped-phase-fusion/evidence/phase_assembly_ultra/REVIEW-RECEIPT.json @@ -0,0 +1,53 @@ +{ + "status": "PASS_CONDITIONAL_PHASE_FUSION_REVIEW", + "created_utc": "2026-10-14T02:06:22.293042+00:00", + "kappa": "1672922349588871/1000000000000000000", + "kappa_decimal": "0.001672922349588871", + "upstream_selected_unchanged": true, + "scope": "Local all-size conditional transfer, retaining the selected finite suppliers and common-frame, whole-call, precision, fixed-tape and prime-packing hypotheses.", + "reviews": [ + { + "reviewer": "root agent", + "result": "Read standalone TeX, integrated notes and exact ledger; cap, supplier and layout audits passed; authorized local conditional status." + }, + { + "reviewer": "gaussian_fast agent", + "result": "Independent capped-gauge and finite-block audit passed; exact inherited-weight reanchoring and broad derivatives checked." + }, + { + "reviewer": "global_macro agent", + "result": "Independent capped carving, sparse-union, q-image and microedge layout audit passed." + }, + { + "reviewer": "inplace_algebra agent", + "result": "Global selected-H support and capacity audit passed; all g controls need not fit one original axis." + }, + { + "reviewer": "phase_assembly agent", + "result": "Read named proofs and relevant retained source contracts; reconstructed complete cost ledger and reviewed capped interfaces." + } + ], + "phase_assembly_replays": [ + "check_capped_outer.py: 44 positive exact slacks and 5 rejecting controls", + "check_forward_cell.py: 350 cases, 5184 factorization and 16496 completion entries", + "check_union.py: exact five-factor/tensor/complete-rectangle checks, with dense escape negative control", + "check_q_embedding.py: product order and augmented inverse checks" + ], + "files": { + "work/phase_assembly_ultra/phase_fusion_note.tex": "da2bf119be3ed1e44473c380e43643d16bdc93015c80380fc59da33d84e951de", + "work/phase_assembly_ultra/CAPPED-INTERIORS.md": "983c5cab2d2398f2192a8aec07daecb655f2de136e12a37f75672a14d27172ba", + "work/phase_assembly_ultra/check_capped_outer.py": "b4555c77de7a3ed4abc708a8d93363ef7c9a914be1091f80ab3ddea83f71230a", + "work/phase_assembly_ultra/CAPPED-OUTER-ARITHMETIC.json": "22caac9ec2ecf4d5743b7eb946b633911fbac55a714399556e87e2eae0457dec", + "work/inplace_algebra_ultra/SHALLOW-TOEPLITZ.md": "b384b93f3fb4a8c97da2e617ebc14d55cebb0a739070e7f8dbb66f63e24e9310", + "work/inplace_algebra_ultra/TENSOR-COMPILER.md": "4d3b8466598a864a234b978e25976648d4e761840aad9153dd18d73d3dfe4d15", + "work/inplace_algebra_ultra/SUPPLIER-SUPPORT-AUDIT.md": "bd5b06eca043c5c871227264f6a954473c55cd16ec85bb420ff538fb8ec1d8d6", + "work/gaussian_fast_ultra/LOG-A-PHASE-SCHUR-AUDIT.md": "e05d6b8596c80b6e8676569a0ff464a29fa704c3304143f58914f47cdf38b272", + "work/gaussian_fast_ultra/BILATERAL-LDU.md": "0d65e66d227464dda1a19f781ae0a0be671cbfd347c41e37ec954f3d629d8a77", + "work/gaussian_fast_ultra/FORWARD-SMALL-A-AUDIT.md": "e37aaa30b15671a1dff00676b14572213efaaac99df202dbba5e09f57af487c0", + "work/gaussian_fast_ultra/CAPPED-GAUGE-AUDIT.md": "1030a34922709ab8d3434a4c57a4a58934ae268c237fb598cb415697da411c9f", + "work/phase_layout_audit_ultra/AUDIT.md": "54aafc7afd09fe1a5312e69fad09c87422690543ab75c6ae35f6cbcc26b3eb8a", + "work/phase_layout_audit_ultra/CAPPED-SIGNOFF.md": "70e9175ebde200f8b016236adc88c4a1c4858945e7c3e5c453a8b6a5f79a263a", + "work/halo_layout_ultra/SUPPORT-ROUTER.md": "d4d2f5bd0a8eb51d9ebd28e4d1a38d3c69bdbf97471a705b3aa199c184364c52", + "work/halo_layout_ultra/HALO-THEOREM.md": "cb9e1da38400b658f9209e465b4e5896fadd17dc66afa862f38bae8d8c1f2058" + } +} diff --git a/research/capped-phase-fusion/evidence/phase_assembly_ultra/check_capped_outer.py b/research/capped-phase-fusion/evidence/phase_assembly_ultra/check_capped_outer.py new file mode 100644 index 0000000000..83223e0a2b --- /dev/null +++ b/research/capped-phase-fusion/evidence/phase_assembly_ultra/check_capped_outer.py @@ -0,0 +1,131 @@ +#!/usr/bin/env python3 +"""Independent rational ledger for capped phase interiors; proof review separate.""" +from fractions import Fraction as Q +from decimal import Decimal, localcontext +from pathlib import Path +import json + +ROOT=Path(__file__).resolve().parent + +def decimal(x): + with localcontext() as ctx: + ctx.prec=65 + return str(Decimal(x.numerator)/Decimal(x.denominator)) + +def calculate(q=None,zeta=None,sigma_boundary=None,old_vd=False,full_cell_gauge=False): + bit=Q('0.001733427039589844234749') + complex_=Q('0.001675462676808103522925') + q=complex_-Q(1,10**10) if q is None else q + a=(q+complex_)/2 + r=Q(1,10**6) + zeta=Q(1,10**8) if zeta is None else zeta + sigma_boundary=Q(1,10**5) if sigma_boundary is None else sigma_boundary + c=Q(3,32) + stop=Q(1,10**9) + eps=(1+q+q*(c-sigma_boundary-r-zeta))/(1+q)**2 + b=eps*(1+q)-c+sigma_boundary + beta=1-b-r-zeta + kappa=Q('0.001672922349588871') + selected=Q('0.001672660197076918') + tau,s=1-a,1-complex_ + internal=tau+(1-stop)*max(s-tau,Q(0)) + leaf=s+stop*(1-s) + lam=1-(q+a)/2 + key=beta+1-eps + margins=dict( + main_transform=eps*q, + global_coordinate_router=a, + fused_gaussian=1-eps+q*beta, + group_bit_router=1-eps+a*beta, + group_scans=1-eps+beta, + boundary_batched_algebra=c+b-eps, + sparse_union_movement=1-eps-beta-Q(1,2)+b, + sparse_union_full_key=1-eps-beta-Q(1,2)+b-key, + polynomial_products=eps, + microbox_face_corrections=1-(eps+beta+3-8+2), + ) + if old_vd: margins['unreplaced_per_axis_scan']=1-eps + slacks=dict( + actual_bit_above_effective=bit-a, + complex_above_effective=complex_-a, + complex_below_one_over_32=Q(1,32)-complex_, + q_positive=q, + complex_endpoint=complex_-q, + q_below_effective_bit=a-q, + stop_positive=stop, + stop_below_one=1-stop, + leaf_saving_above_effective_bit=(1-stop)*complex_-a, + q_below_internal=1-internal-q, + q_below_leaf=1-leaf-q, + lambda_above_tau=lam-tau, + lambda_above_sigma=lam-s, + lambda_above_internal=lam-internal, + lambda_prime_above_lambda=1-q-lam, + lambda_prime_above_leaf=1-q-leaf, + epsilon_positive=eps, + epsilon_below_one=1-eps, + capped_block_above_band=b-Q(1,2), + capped_block_below_cell=eps-b, + group_positive=beta, + group_below_dimension=eps-beta, + support_router_span=1-beta, + group_gauge_power_slack=1-b-r-beta, + actual_logarithmic_group_guard=1-b-beta, + global_gamma_power_slack=1-eps-r, + actual_logarithmic_gamma_guard=1-eps, + alpha_logarithm_absorption=r, + group_key_fits_record=1-key, + membership_arithmetic_fits_record=1-beta-2*(1-eps), + microbox_halo_volume=8-3-beta, + boundary_positive_slack=sigma_boundary, + short_record_fallback=eps-a, + improvement=kappa-selected, + ) + if full_cell_gauge: + slacks['unrecentered_group_gauge']=1-eps-r-beta + slacks.update({name+'_above_kappa':value-kappa for name,value in margins.items()}) + assert margins['main_transform']==margins['fused_gaussian'] + assert margins['boundary_batched_algebra']==margins['main_transform']+sigma_boundary + return dict(parameters=dict(a_bit=bit,a_eff=a,complex_saving=complex_,q=q, + r_spare=r,zeta=zeta,sigma_boundary=sigma_boundary,stop=stop, + epsilon=eps,capped_block_power=b,beta_group=beta, + omega_upper=Q(45,16),c_boundary=c,kappa=kappa,selected=selected, + lambda_=lam,lambda_prime=1-q,tau=tau,sigma=s, + microbox_power=Q(8),halo_power=Q(3)),margins=margins,slacks=slacks) + +def encode(o): + if isinstance(o,Q):return dict(rational=str(o),decimal=decimal(o)) + if isinstance(o,dict):return {k:encode(v) for k,v in o.items()} + return o + +if __name__=='__main__': + row=calculate() + assert 7**16<2**45 + assert all(value>0 for value in row['slacks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + controls={} + tests=[('endpoint_equality',dict(q=Q('0.001675462676808103522925'))), + ('zero_group_guard',dict(zeta=Q(0))), + ('zero_boundary_slack',dict(sigma_boundary=Q(0))), + ('old_per_axis_scan',dict(old_vd=True)), + ('unrecentered_full_cell_gauge',dict(full_cell_gauge=True))] + for name,kwargs in tests: + bad=calculate(**kwargs) + failed={k:v for k,v in bad['slacks'].items() if v<=0} + assert failed,name + controls[name]=failed + output=dict(status='PASS_CONDITIONAL_PHASE_FUSION_REVIEW', + scope='Local conditional transfer and exact arithmetic; inherited supplier and all-size interfaces remain assumptions. No upstream selected certificate is changed.', + review=dict(integrated_review='Root reviewed the complete standalone note, integrated proof chain, exact ledger, and independent cap/supplier/layout audits.', + independent_cap_gauge='CAPPED-GAUGE-AUDIT.md', + independent_cap_layout='CAPPED-SIGNOFF.md', + supplier_support='SUPPLIER-SUPPORT-AUDIT.md'), + **row,controls=controls) + (ROOT/'CAPPED-OUTER-ARITHMETIC.json').write_text(json.dumps(encode(output),indent=2)+'\n') + print(output['status']) + for name in ['epsilon','capped_block_power','beta_group','kappa']: + print(name,decimal(row['parameters'][name])) + print('minimum margin',decimal(min(row['margins'].values()))) + print('gain over selected',decimal(row['slacks']['improvement'])) + print('positive strict slacks',len(row['slacks'])) + print('controls',', '.join(controls)) diff --git a/research/capped-phase-fusion/evidence/phase_assembly_ultra/check_matrix_corollary.py b/research/capped-phase-fusion/evidence/phase_assembly_ultra/check_matrix_corollary.py new file mode 100644 index 0000000000..3f543803d7 --- /dev/null +++ b/research/capped-phase-fusion/evidence/phase_assembly_ultra/check_matrix_corollary.py @@ -0,0 +1,143 @@ +#!/usr/bin/env python3 +"""Exact corollary ledger, additionally assuming the supplied 9/4 theorem.""" +from fractions import Fraction as Q +from decimal import Decimal,localcontext +from pathlib import Path +import json + +ROOT=Path(__file__).resolve().parent + +def decimal(x): + with localcontext() as ctx: + ctx.prec=65 + return str(Decimal(x.numerator)/Decimal(x.denominator)) + +def calculate(q=None,zeta=None,union_slack=None,old_vd=False, + full_cell_gauge=False,add_full_key_sort=False): + bit=Q('0.001733427039589844234749') + complex_=Q('0.001675462676808103522925') + q=complex_-Q(1,10**10) if q is None else q + a=(q+complex_)/2 + r=Q(1,10**6) + zeta=Q(1,10**8) if zeta is None else zeta + union_slack=Q(1,10**5) if union_slack is None else union_slack + tau_matrix=Q(19,8) + c=(3-tau_matrix)/2 + stop=Q(1,10**9) + cap=(Q(3,2)+q-(1+q)*(r+zeta)+union_slack)/(2+q) + eps=(1+q*(1-cap-r-zeta))/(1+q) + beta=1-cap-r-zeta + kappa=Q('0.001673360105405778') + selected=Q('0.001672660197076918') + first=Q('0.001672922349588871') + tau_bit,sigma=1-a,1-complex_ + internal=tau_bit+(1-stop)*max(sigma-tau_bit,Q(0)) + leaf=sigma+stop*(1-sigma) + lam=1-(q+a)/2 + key=beta+1-eps + margins=dict( + main_transform=eps*q, + global_coordinate_router=a, + fused_gaussian=1-eps+q*beta, + group_bit_router=1-eps+a*beta, + group_scans=1-eps+beta, + boundary_algebraic_matrix=c+cap-eps, + sparse_union_movement=1-eps-beta-Q(1,2)+cap, + polynomial_products=eps, + microbox_face_corrections=1-(eps+beta+3-8+2), + ) + if old_vd:margins['unreplaced_per_axis_scan']=1-eps + if add_full_key_sort: + margins['unnecessary_union_full_key_sort']=margins['sparse_union_movement']-key + slacks=dict( + actual_bit_above_effective=bit-a, + complex_above_effective=complex_-a, + complex_below_one_over_32=Q(1,32)-complex_, + q_positive=q, + complex_endpoint=complex_-q, + q_below_effective_bit=a-q, + stop_positive=stop, + stop_below_one=1-stop, + leaf_saving_above_effective_bit=(1-stop)*complex_-a, + q_below_internal=1-internal-q, + q_below_leaf=1-leaf-q, + lambda_above_tau=lam-tau_bit, + lambda_above_sigma=lam-sigma, + lambda_above_internal=lam-internal, + lambda_prime_above_lambda=1-q-lam, + lambda_prime_above_leaf=1-q-leaf, + epsilon_positive=eps, + epsilon_below_one=1-eps, + capped_block_above_band=cap-Q(1,2), + capped_block_below_cell=eps-cap, + group_positive=beta, + group_below_dimension=eps-beta, + support_router_span=1-beta, + group_gauge_power_slack=1-cap-r-beta, + actual_logarithmic_group_guard=1-cap-beta, + global_gamma_power_slack=1-eps-r, + actual_logarithmic_gamma_guard=1-eps, + alpha_logarithm_absorption=r, + group_key_fits_record=1-key, + membership_arithmetic_fits_record=1-beta-2*(1-eps), + sparse_union_fraction_decays=cap-beta-Q(1,2), + microbox_halo_volume=8-3-beta, + union_positive_slack=union_slack, + matrix_exponent_above_source_bound=tau_matrix-Q(9,4), + matrix_exponent_below_five_halves=Q(5,2)-tau_matrix, + short_record_fallback=eps-a, + improvement_over_selected=kappa-selected, + improvement_over_first_reviewed_transfer=kappa-first, + ) + if full_cell_gauge: + slacks['unrecentered_group_gauge']=1-eps-r-beta + slacks.update({name+'_above_kappa':value-kappa for name,value in margins.items()}) + assert margins['main_transform']==margins['fused_gaussian'] + assert margins['sparse_union_movement']==margins['main_transform']+union_slack + return dict(parameters=dict(a_bit=bit,a_eff=a,complex_saving=complex_,q=q, + r_spare=r,zeta=zeta,sigma_union=union_slack,stop=stop, + epsilon=eps,capped_block_power=cap,beta_group=beta, + matrix_exponent=tau_matrix,matrix_source_bound=Q(9,4), + c_boundary=c,kappa=kappa,selected=selected,first_reviewed_transfer=first, + lambda_=lam,lambda_prime=1-q,tau=tau_bit,sigma=sigma, + microbox_power=Q(8),halo_power=Q(3)),margins=margins,slacks=slacks) + +def encode(o): + if isinstance(o,Q):return dict(rational=str(o),decimal=decimal(o)) + if isinstance(o,dict):return {k:encode(v) for k,v in o.items()} + return o + +if __name__=='__main__': + row=calculate() + assert all(value>0 for value in row['slacks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + controls={} + tests=[('endpoint_equality',dict(q=Q('0.001675462676808103522925'))), + ('zero_group_guard',dict(zeta=Q(0))), + ('zero_union_slack',dict(union_slack=Q(0))), + ('old_per_axis_scan',dict(old_vd=True)), + ('unrecentered_full_cell_gauge',dict(full_cell_gauge=True)), + ('full_key_sort_on_union',dict(add_full_key_sort=True))] + for name,kwargs in tests: + bad=calculate(**kwargs) + failed={k:v for k,v in bad['slacks'].items() if v<=0} + assert failed,name + controls[name]=failed + output=dict(status='PASS_CONDITIONAL_MATRIX_PHASE_FUSION_REVIEW', + scope='Additional matrix-conditioned corollary. Assumes supplied omega_C<=9/4 theorem and the effective algebraic stability transfer, as well as all first-transfer assumptions.', + review=dict(integrated='Root read the matrix corollary and independent layout signoff, reproduced the exact parameters and ledger, and replayed matrix source construction controls.', + matrix_source='PROOF-AND-COVERAGE.md', + effective_stability='ALGEBRAIC-STABILITY-INDEPENDENT.md', + independent_layout='MATRIX-LAYOUT-SIGNOFF.md'), + omitted_row=dict(name='optional full-coordinate-key union surcharge', + reason='The actual U_i complete-rectangle extraction, one-field transpose, and ordered merge do not sort union records. Sparse microhalo keys remain paid in their separate cost.'), + **row,controls=controls) + (ROOT/'MATRIX-COROLLARY-ARITHMETIC.json').write_text(json.dumps(encode(output),indent=2)+'\n') + print(output['status']) + for name in ['epsilon','capped_block_power','beta_group','kappa']: + print(name,decimal(row['parameters'][name])) + print('minimum margin',decimal(min(row['margins'].values()))) + print('gain over selected',decimal(row['slacks']['improvement_over_selected'])) + print('gain over first transfer',decimal(row['slacks']['improvement_over_first_reviewed_transfer'])) + print('positive strict slacks',len(row['slacks'])) + print('controls',', '.join(controls)) diff --git a/research/capped-phase-fusion/evidence/phase_assembly_ultra/check_outer.py b/research/capped-phase-fusion/evidence/phase_assembly_ultra/check_outer.py new file mode 100644 index 0000000000..2bd3f6eb68 --- /dev/null +++ b/research/capped-phase-fusion/evidence/phase_assembly_ultra/check_outer.py @@ -0,0 +1,131 @@ +#!/usr/bin/env python3 +"""Exact arithmetic only. This does not automatically certify proof obligations.""" +from fractions import Fraction as Q +from decimal import Decimal, localcontext +from pathlib import Path +import json + +ROOT = Path(__file__).resolve().parent + +def dec(x): + with localcontext() as ctx: + ctx.prec = 65 + return str(Decimal(x.numerator) / Decimal(x.denominator)) + +def compute(q=None, zeta=None, old_vd=False): + b = Q('0.001675462676808103522925') + a_bit = Q('0.001733427039589844234749') + q = b-Q(1,10**10) if q is None else q + r = Q(1,10**6) + zeta = Q(1,10**8) if zeta is None else zeta + a_eff = (q+b)/2 + t_stop = Q(1,10**9) + tau, sigma = 1-a_eff, 1-b + lam_prime = 1-q + lam = 1-(q+a_eff)/2 + internal = tau + (1-t_stop)*max(sigma-tau,Q(0)) + leaf = sigma+t_stop*(1-sigma) + eps = (1+q-q*(r+zeta))/(1+2*q) + beta = 1-eps-r-zeta + kappa = Q('0.001672664764353193') + selected = Q('0.001672660197076918') + omega_upper = Q(45,16) + # alpha^2=O(log p), so w=O(sqrt(p) polylog(p)). + wide = Q(1,2) + key = beta+1-eps + margins = dict( + main_transform=eps*q, + global_coordinate_router=a_eff, + fused_gaussian=1-eps+q*beta, + group_bit_router=1-eps+a_eff*beta, + group_scans=1-eps+beta, + boundary_batched_algebra=1-(omega_upper-1)*wide, + sparse_union_movement=1-beta-wide, + sparse_union_full_key=1-beta-wide-key, + polynomial_products=eps, + # gA/L copies, g sequential passes, each p^2 conservatively. + microbox_face_corrections=1-(eps+beta+3-8+2), + ) + if old_vd: + margins['unreplaced_per_axis_scan'] = 1-eps + slacks = dict( + actual_bit_above_effective=a_bit-a_eff, + complex_above_effective=b-a_eff, + complex_below_one_over_32=Q(1,32)-b, + q_positive=q, + complex_endpoint=b-q, + q_below_effective_bit=a_eff-q, + stop_positive=t_stop, + stop_below_one=1-t_stop, + leaf_saving_above_effective_bit=(1-t_stop)*b-a_eff, + q_below_internal=1-internal-q, + q_below_leaf=1-leaf-q, + lambda_above_tau=lam-tau, + lambda_above_sigma=lam-sigma, + lambda_above_internal=lam-internal, + lambda_prime_above_lambda=lam_prime-lam, + lambda_prime_above_leaf=lam_prime-leaf, + lambda_prime_below_one=q, + epsilon_positive=eps, + epsilon_below_one=1-eps, + group_positive=beta, + group_below_dimension=eps-beta, + group_controls_below_one_axis_width=1-eps-beta, + support_router_span=1-beta, + group_guard=1-eps-r-beta, + alpha_logarithm_absorption=r, + local_gauge_sublinear=1-eps-r, + core_above_band=eps-wide, + record_suffix=1-eps, + prime_interval_packing=1-eps, + semantic_dimension_guard=1-eps, + microbox_larger_than_halo=Q(8-3)-beta, + short_record_fallback=eps-a_eff, + improvement=kappa-selected, + ) + slacks.update({name+'_above_kappa':val-kappa for name,val in margins.items()}) + assert eps*q == 1-eps+q*beta + return dict(parameters=dict(a_bit=a_bit,a_eff=a_eff,b=b,q=q,r=r,zeta=zeta, + t_stop=t_stop,tau=tau,sigma=sigma,lambda_=lam,lambda_prime=lam_prime, + epsilon=eps,beta_group=beta,kappa=kappa,selected=selected, + omega_upper=omega_upper,microbox_power=Q(8),halo_power=Q(3)), + margins=margins,slacks=slacks) + +def convert(o): + if isinstance(o,Q): return {'rational':str(o),'decimal':dec(o)} + if isinstance(o,dict): return {k:convert(v) for k,v in o.items()} + return o + +if __name__ == '__main__': + row=compute() + assert 7**16 < 2**45 + assert all(x>0 for x in row['slacks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + controls={} + for name,kw in [('complex_endpoint_equality',{'q':Q('0.001675462676808103522925')}), + ('zero_group_guard',{'zeta':Q(0)}), + ('old_per_axis_scan',{'old_vd':True})]: + bad=compute(**kw) + failed={n:v for n,v in bad['slacks'].items() if v<=0} + assert failed, name + controls[name]={'rejected_slacks':failed} + result=dict(status='PASS_EXACT_ARITHMETIC_PENDING_FULL_PROOF_REVIEW', + scope='Exact outer arithmetic; no automatic theorem promotion.', + **row, controls=controls, + theorem_obligations=[ + 'retained finite suppliers and exact common-frame/mutable-source compiler', + 'deferred reservations and paid global synthetic layout', + 'shallow Toeplitz plus SPD extension and tensor compiler', + 'small-a forward conditioning and exact target-grid square completion', + 'grouped sparse-union route and batched boundary arithmetic', + 'dyadic core with sparse microbox-face repair and monotone q-image scans', + 'scalar product-diagonal scan and retained all-size semantic/prime/error transfer', + ]) + path=ROOT/'OUTER-ARITHMETIC.json' + path.write_text(json.dumps(convert(result),indent=2)+'\n') + print(result['status']) + print('kappa',dec(row['parameters']['kappa'])) + print('minimum margin',dec(min(row['margins'].values()))) + print('gain',dec(row['slacks']['improvement'])) + print('positive exact slacks',len(row['slacks'])) + print('negative controls',', '.join(controls)) diff --git a/research/capped-phase-fusion/evidence/phase_layout_audit_ultra/AUDIT.md b/research/capped-phase-fusion/evidence/phase_layout_audit_ultra/AUDIT.md new file mode 100644 index 0000000000..bbeeee7431 --- /dev/null +++ b/research/capped-phase-fusion/evidence/phase_layout_audit_ultra/AUDIT.md @@ -0,0 +1,242 @@ +# Grouped phase-cell layout audit + +The sparse-union layout in `REFINED-PHASE-FUSION.md` has a constructive +implementation with its stated `V g^2 W polylog(p)/d` per-group movement +budget. It does not require sorting the full tensor or retaining d full +payload scans. The conclusion is conditional on the separately supplied +dense local tensor compiler, rounded phase-cell factorization, and precision +bounds. This audit does not prove those algebraic/analytic inputs. + +The maximum physical bandwidth **W**, including the broad forward map, must +be used in the dyadic carving and support sets below. The narrower inverse +bandwidth alone is insufficient for the broad map. The phase-assembly +branch confirms this convention. + +## Exact operator order and the union that is actually invariant + +Write H in interior/boundary order as `[A B; C D]` and let +`S=D-C A^-1 B`. The displayed identity in the parent note is correct. In +execution order its inverse can be scheduled as + +``` +diag(A^-1,I) +[I 0; -C I] +diag(I,S^-1) +diag(A,I) +[I -B; 0 I] +diag(A^-1,I). +``` + +The middle two diagonal factors commute. Thus there are three dense +interior stages and two sparse windows: the C and S^-1 stages share one +window; B uses the second window. For H itself, reverse and invert this +sequence: the same original B,C and S occur, with signs reversed and dense +A/A^-1 interchanged. No dense A^-1 B or C A^-1 shear is smuggled into the +sparse boundary cost. + +For each active axis i, let J_i be its designated boundary coordinates. +Let E_i contain J_i and every interior coordinate incident to an original +nonzero entry of B_i or C_i. Original physical bandwidth W implies that E_i +lies within O(W) of the carved endpoints. Include physical-period cuts, +artificial phase cuts and clipped microbox endpoints. Dyadic carving gives + +``` +|E_i| <= C (L/d+1) W log d, +mu := max_i |E_i|/L = O(W log d/d + W log d/L) +``` + +inside a length-L microbox. All sparse factors have the block form +`T_i = I_(E_i^c) direct_sum T_i|E_i`. In particular S_i^-1 can be dense on +J_i: J_i is already included in E_i. Let + +``` +U = {x in [L]^g : some x_i belongs to E_i}. +``` + +Then `|U|/L^g <= g mu`. Every sparse tensor-axis operation preserves U and +its complement. If x_i is changed by a nonzero off-diagonal entry, both +its old and new coordinates belong to E_i; otherwise it is unchanged. +This proves closure for an arbitrarily long sequence of these sparse +operations, not just one application. + +Dense A or A^-1 is **not** covered. It can move a near-boundary interior +coordinate into the deep interior. Scatter U before a dense stage and +regather afterwards. `check_union.py` includes an exact example in which +dense A^-1 sends union-supported data outside U. The actual schedule above +therefore uses two union gathers, not one retained union for all six stages. + +## Complete rectangles inside the union: no missing-zero assumption + +For one sparse axis i define + +``` +U_i = {x : x_i in E_i} = E_i x [L]^(g-1). +``` + +This is a **complete rectangle**, and it is wholly present in U regardless +of which other coordinates are near a boundary. All nonidentity data needed +by T_i is in U_i. A line with every other coordinate outside its own E_j +still has every E_i coordinate present. A line with another boundary +coordinate may contain more data, but those extra E_i^c entries are identity +coordinates and need no boundary computation. + +Keep a copy of U in its original microbox lexicographic order, with a short +coordinate/membership header. For each axis i: + +1. Scan U once and copy records with x_i in E_i to an auxiliary stream. + The copied stream is lexicographic E_i x [L]^(g-1). +2. Rename the increasing E_i coordinates by their ordinal 0,...,|E_i|-1. + This is a stream counter; no rank/select query is needed. +3. Expose that one field at the line suffix using the retained small-field + transpose. Its length is at most L=poly(p), so only O(log p) binary + block moves are needed. Pad only this field to its next power of two + temporarily, at a factor below two, and remove that padding before the + next axis. The other fields may be arbitrary spectators. +4. Apply the original B/C or boundary S/S^-1 factor on its complete E_i + line, retaining identity coordinates E_i\J_i when appropriate. +5. Undo the transpose at the actual field length. Merge the ordered E_i + stream back into a fresh copy of U, replacing exactly those entries. + +No array is completed by invented zero coordinates. The finite checker +verifies the extracted rectangle cardinality and the full tensor result. + +The extraction/merge costs O(|U|) per axis. Transposition costs +O(|U_i| polylog p). Consequently one sparse window costs + +``` +O(V [g^2 mu + g mu polylog p]) + = O(V g^2 W polylog p/d) + O(V g^2 W polylog p/L). +``` + +There is no factor `K=g log s` from a computed-key sort in this part of the +algorithm. A more general radix implementation would still fit with extra +small powers of g, but the complete-rectangle construction avoids it. + +## Union extraction and piece catalogues + +Only one full microbox scan is needed to form U, and one to scatter it. +For every complete p-bit coefficient record, compute its g membership flags +from its coordinate tuple and OR them. The flags and coordinate header have +O(g log L) bits. A nonzero number of flags sends the whole record once to U; +it is not duplicated for each active axis. + +This is a charged descriptor calculation. For actual source periods of +bitlength h=Theta(p/d), q_j inversion and phase-cell endpoint calculations +use O(h^2 polylog p) elementary bit operations per axis. Within a phase +cell, the greedy dyadic carve is found by at most O(log d) halvings and +integer comparisons. Thus a conservative membership cost is +`O(g h^2 polylog p)` per record, which is o(p) for +`g=p^(1-epsilon-r-zeta)` and epsilon>2/3. At the proposed epsilon this has +power about 0.005. It is absorbed by reading the p-bit record once. +Alternatively the original remainder phase generator and an odometer give +the same membership flags in sequential coordinate order. + +Factor pages may be regenerated once per complete microbox/group from its +actual coarse origins, using the supplied polynomial-time coefficient +generation. The complete inner microbox and untouched spectator payload +are superpolynomial in p and amortize this polynomial setup. This avoids +random access to a global catalogue in the scrambled coarse order created +by a support-limited gather. Online row/factor pages are then visited in +the ordered extracted E_i line. All page rewinds are within that line or +RHS batch and are included in the corresponding boundary arithmetic. + +## Batched right-hand sides do not add a large transpose + +All other inner coordinates and following tensor axes are independent +right-hand sides for an axis operation. Their count R is superpolynomial. +Batch them in a dyadic width between W and 2W; the final incomplete batch +can be padded once at constant cost. To put the batch index outside the +boundary-row field, move only that polynomial-width row field. This costs +O(log p) times the boundary volume, not a transpose of the long spectator +field. Each W-by-W factor page is scanned once per RHS batch and reused in +the block product. Reverse the small-field layout after the solve. + +Thus the assumed Strassen `W^omega` per W RHS gives the quoted total +`O(V W^(omega-1) polylog p)` boundary arithmetic. This audit checks the +layout and factor reuse; stability and the no-pivot factor theorem remain +the phase-assembly branch's responsibility. + +## q_j embeddings, selections and source order need one group scan + +For a group, each coordinate map `j -> q_j=floor(tj/s+1/2)` is increasing +and injective. Its tensor product preserves lexicographic order on its +image. Enumerate the target-group rectangle once. At target coordinate k, +set + +``` +j=max(0,ceil(s(2k-1)/(2t))). +``` + +The coordinate belongs to the image iff `j infinity`. Individual guarded neighborhoods +fit the original long axis; the collection of all g controls does not have +to fit in a single axis. This is the same support proof already audited. + +Microedge correction is unchanged. The underlying rounded operator and its +physical bandwidth do not change when a factorization receives more cuts. +Only the principal restriction and Schur decomposition are repartitioned. +The retained Neumann/principal-window comparison still applies to that +same operator. With L=p^8 and halo A=p^3, the additional output fraction is +O(g p^-5), now about p^-4.9064. Its sparse gather, true-period correction and +old sequential local solvers are still negligible. Capping does not turn +this into a dense non-dyadic tensor halo. + +The pinned independent arithmetic checks give approximately + +``` +epsilon = .9984838711085558 +b = .9064167934681446 +beta = .0935821965318554 + +per-group union movement power = -.2192524004044339 +total union movement power = .6856492741722665 +total group scan power = .9049016745767004 +membership descriptor power = .0966144543147439. +``` + +`check_capped_layout.py` independently checks finite capped partitions, +dyadic block sizes, separator distances, worst-pattern elimination fill, +and these exact rational layout inequalities against the pinned candidate +JSON. No layout blocker was found. + +This signoff is specifically for carving, sparse support, actual stream +movement, descriptors, and inherited microedge placement. It does not +independently prove the newly anchored broad/narrow Gaussian formulas or +their final precision bound. Those remain the algebraic/analytic cap audit. diff --git a/research/capped-phase-fusion/evidence/phase_layout_audit_ultra/MANIFEST.json b/research/capped-phase-fusion/evidence/phase_layout_audit_ultra/MANIFEST.json new file mode 100644 index 0000000000..1fd05118b7 --- /dev/null +++ b/research/capped-phase-fusion/evidence/phase_layout_audit_ultra/MANIFEST.json @@ -0,0 +1,16 @@ +{ + "AUDIT.md": "54aafc7afd09fe1a5312e69fad09c87422690543ab75c6ae35f6cbcc26b3eb8a", + "CAPPED-LAYOUT-CONTROLS.json": "6c1342c436fb487a5aa39cf563d2a2e2fddd94d3e441a6f4889cb1e2b73a478c", + "CAPPED-SIGNOFF.md": "70e9175ebde200f8b016236adc88c4a1c4858945e7c3e5c453a8b6a5f79a263a", + "MATRIX-LAYOUT-CONTROLS.json": "f8bd8729cb8438188e90fb9149f598b825461c97b4ed290584ca4367f812a48a", + "MATRIX-LAYOUT-SIGNOFF.md": "4c8c62f723232e842dc3bbdcc4defcc8ff1aeb86ba9f694661899dc6783b13c4", + "PARAMETERS.json": "9f72cfffa34844764ca65a6b4431cc5c779141751df746019db9d3ad8f1dee92", + "Q-EMBEDDING-CONTROLS.json": "24ead3534482397c00c23ad3dbbd4e930e02290e00c97765353d8783c6575c7d", + "README.md": "0ec721f76dfa943e808afd3c2cc39ee69f1a6e5f4f77ba3c7c1373ad60db25af", + "UNION-CONTROLS.json": "6c5262144215cc603bbc3bee93a9a6d9d2318d0177a59115fe560762998498f9", + "check_capped_layout.py": "267d93ab5114ec160f88a6e65fd34c9f0696a686c36ba2dd194d6b70fa32a6bd", + "check_q_embedding.py": "ac858a14bbaf37e0e412a7ada282cd1586632f52b8a979733e775f50130969b6", + "check_union.py": "74608380d72567edcd4470c8d226b15848f9d3a4e5d77d652bdd8963480ab9ce", + "q-embedding.log": "24ead3534482397c00c23ad3dbbd4e930e02290e00c97765353d8783c6575c7d", + "union.log": "6c5262144215cc603bbc3bee93a9a6d9d2318d0177a59115fe560762998498f9" +} diff --git a/research/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-CONTROLS.json b/research/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-CONTROLS.json new file mode 100644 index 0000000000..aee3270196 --- /dev/null +++ b/research/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-CONTROLS.json @@ -0,0 +1,30 @@ +{ + "status": "PASS_MATRIX_COROLLARY_NO_SORT_LAYOUT", + "source_sha256": "b0d4648e16df74fc0a3644baa8a82cbb82842c0aa21abe915a28c0c4c949182a", + "exponents": { + "union_fraction": { + "rational": "-3430918384747396741111767383/8006701850307232414091700000000", + "decimal": "-0.000428505825356260251146382720850643577327042013532977151967005" + }, + "membership": { + "rational": "269791328968194081006854064986174254715733926/1069348903949477650676785089185287127357866963", + "decimal": "0.252294950667420906279768139101314143625086172719573255573065" + }, + "total_union": { + "rational": "320264641396534864583914893734048954038319544983689789/320804671184843295203035526755586138207360088900000000", + "decimal": "0.998316639884594221546182952048279623715798111123297150773509" + }, + "group_scans": { + "rational": "2402698356122892965636680000000000000000000000/3208046711848432952030355267555861382073600889", + "decimal": "0.748959903622628611922902526129555589081788674143596616501459" + }, + "microhalo_fraction": { + "rational": "-380335532915361620704585/80067018503072324140917", + "decimal": "-4.75021475791267813012557319136042532178866352100676648857598" + }, + "rejected_union_full_key": { + "rational": "3334408963974161672450196077539/2668900616769077471363900000000", + "decimal": "1.24935673626196560962328042591872403463400943697970053427205" + } + } +} diff --git a/research/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md b/research/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md new file mode 100644 index 0000000000..d683e4d495 --- /dev/null +++ b/research/capped-phase-fusion/evidence/phase_layout_audit_ultra/MATRIX-LAYOUT-SIGNOFF.md @@ -0,0 +1,64 @@ +# Matrix-corollary layout: positive signoff + +The no-sort union layout in `AUDIT.md` applies unchanged to +`work/phase_assembly_ultra/MATRIX-COROLLARY.md`. Its algebraic conditions are +closed axis supports E_i and a complete microbox; neither its proof nor its +implementation requires g2*W and B&(B-1)==0 and n>=2*W + J=set(range(W))|set(range(n-W,n));blocks=[];c=W;stop=n-W + while stop-c>=B+W: + blocks.append(list(range(c,c+B)));J.update(range(c+B,c+B+W));c+=B+W + while stop-c>4*W: + rem=stop-c;k=1<<((rem-2*W).bit_length()-1) + assert k<=B + J.update(range(c,c+W));blocks.append(list(range(c+W,c+W+k))) + J.update(range(c+W+k,c+2*W+k));c+=2*W+k + J.update(range(c,stop)) + assert sorted(J|{j for b in blocks for j in b})==list(range(n)) + assert sum(map(len,blocks))+len(J)==n + assert all(len(b)<=B and len(b)&(len(b)-1)==0 for b in blocks) + assert all(b[0]-a[-1]>W for a,b in zip(blocks,blocks[1:])) + assert len(J)<=10*W*((n+B-1)//B+B.bit_length()+1) + return blocks,J + +def fill_bound(n,blocks,J,W): + adj=[set(j for j in range(n) if j!=i and abs(i-j)<=W) for i in range(n)] + for block in blocks: + for v in block: + nbr=list(adj[v]) + for x in nbr: + adj[x].update(y for y in nbr if y!=x);adj[x].discard(v) + adj[v].clear() + order={v:i for i,v in enumerate(sorted(J))} + bandwidth=max((abs(order[i]-order[j]) for i in J for j in adj[i]),default=0) + assert bandwidth<=2*W + return bandwidth + +def main(): + total=0;examples=[] + for W in (1,2,3,4): + for B in (8,16,32,64): + if B<=2*W:continue + for n in range(2*W,4*B+1): + blocks,J=carve(n,B,W);total+=1 + n=4*B+7;blocks,J=carve(n,B,W);bw=fill_bound(n,blocks,J,W) + examples.append(dict(W=W,B=B,n=n,blocks=len(blocks),boundary=len(J),schur_boundary_bandwidth=bw)) + raw=SOURCE.read_bytes();source=json.loads(raw);p={k:Q(v['rational']) for k,v in source['parameters'].items()} + e,b,g=p['epsilon'],p['capped_block_power'],p['beta_group'];w=Q(1,2);h=1-e + exponents=dict(per_group_union_movement=2*g+w-b,total_union_movement=e+g+w-b, + total_group_scans=e-g,metadata_membership=g+2*h,microbox_key=g, + global_sparse_key=g+h,guarded_support_span=g, + sparse_microhalo_volume=g+3-8, + hypothetical_fullkey_union=e+2*g+h+w-b) + assert Q(1,2)1, theta=rho-1, q_i=floor(rho i+1/2), beta_i=rho i-q_i for all integer i. Define + +``` +x_i=exp(2a rho q_i), e_i=exp(-a rho^2 i^2+a beta_i^2), +G_ij=exp(-a q_i^2) x_i^j e_j, +r=exp(2a rho), c=product_(m>=1)(1-r^-m)>0. +``` + +The sequence x_i increases at least by a factor r at every step and tends geometrically to zero toward minus infinity. + +## Finite-origin identities + +For an integer origin l, Newton interpolation of columns j>=l gives, on every finite principal interval starting at l, + +``` +C_i(l)=product_(l<=k=j, +U_ij(l)=(e_j/e_i) h_(j-i)(x_l,...,x_i), j>=i, +G=L(l) diag(C_i(l)) U(l). +``` + +Here h denotes complete homogeneous symmetric polynomials, and both triangular factors have diagonal one. The powers arising from moving the monomial origin from zero to l cancel exactly: the pivot is C_i(l), not a residual power of x_i. + +Since C_i(l)>=c and the normalized complete-homogeneous coefficient is at most c^-1, + +``` +|L_ij(l)| <= c^-1 G_ij, +|U_ij(l)| <= c^-1 G_ij. +``` + +All limits as l tends to minus infinity exist, including fixed-degree h over the infinite summable past. The displayed Gaussian domination permits passing the factorization to the bilateral limit. + +## Inverse factors + +For i>=j and h=i-j, + +``` +(L(l)^-1)_ij=(-1)^h C_i(l)/C_j(l) + * exp(a[-beta_i^2+beta_j^2-rho^2 h+2rho sum_(j=i and h=j-i, + +``` +(U(l)^-1)_ij=(-1)^h (e_j/e_i) e_h(x_l,...,x_(j-1)). +``` + +The latter e_h is elementary symmetric, distinct from the scalar normalization e_j. Its dominant product consists of x_i,...,x_(j-1), so its normalized value is at most c^-1. + +Using |beta_i|<=1/2 in the two exact exponents gives uniform estimates + +``` +|(L(l)^-1)_ij| <= c^-2 exp(a/4-a rho theta h), +|(U(l)^-1)_ij| <= c^-1 exp(a/4-a rho theta h). +``` + +These pass to the bilateral limits. Products of triangular factors and their inverse factors have only finitely many terms for each supported matrix entry, so both inverse identities survive directly. All four factors define bounded absolutely row-summable operators. + +## Periodization + +Every limiting factor is simultaneously s-periodic in its row and column indices. For a bounded absolutely row-summable periodic array A, define + +``` +Pi(A)_ij=sum_(n in Z) A_(i,j+ns), 0<=i,j=1**, this is O(theta^-2), with a constant depending only on a. Thus a fixed admissible integer Gaussian width alpha suffices for the norm and perturbation estimates. No logarithmic growth of alpha is needed for conditioning. + +The theorem changes an analytic parameter constraint. Applying these factors quickly on the tape remains a separate constructive obligation. diff --git a/research/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md b/research/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md new file mode 100644 index 0000000000..59d9128d75 --- /dev/null +++ b/research/capped-phase-fusion/evidence/radical_reduction_ultra/HOLES-AND-Q-LDU.md @@ -0,0 +1,211 @@ +# Independent checks of Gaussian interpolation and holes completion + +Status: exact algebraic identities and analytic bounds below; no complete new inverse implementation, and no final kappa claim. This continues the parent investigation after the first outer-reduction package. + +## 1. Local scaled Vandermonde identity + +Write `a=pi*alpha^2`, `rho=t/s`, `q_i=floor(rho*i+1/2)`, and `beta_j=rho*j-q_j`. On any nonwrapping index interval, + +``` +N_ij = exp[-a*(rho*j-q_i)^2 + a*beta_j^2] + = exp[-a*q_i^2] * (exp[2a*rho*q_i])^j + * exp[-a*rho^2*j^2+a*beta_j^2]. +``` + +Translations of both index origins preserve a diagonally scaled Vandermonde representation. If the local span is L, centered nodes have logarithms O(aL), and the product-tree polynomials, Lagrange denominators, and outer diagonals have logarithmic height O(aL^2+L). Thus L=O(sqrt(p/a)) is compatible with Cp-bit working precision. Distinct nodes are explicitly separated, so the inverse of a *single supplied principal block* can be computed by multipoint interpolation with that precision and O(L p polylog p) bit cost. + +This does not justify independent short blocks. The current inverse has strong directed coupling near the end of a phase cell. Its exact q-LDU below shows why a halo of width merely sqrt(p/a) can lose much more than 2^-p. + +## 2. Exact Gaussian Toeplitz LDU in a no-gap phase cell + +Within a cell `q_i=i+c`, put `theta=rho-1`, + +``` +q=exp(-a*rho), Q=q^2, +P_j=product_(k=1..j) (1-Q^k), P_0=1, +D_i=exp(a*theta*i^2-2a*c*i), +G_ij=q^((i-j)^2). +``` + +Then `N=diag(D_i)*G*diag(D_i)^-1`. With the Gaussian binomial + +``` +{i choose j}_Q = P_i/(P_j*P_(i-j)), +``` + +the exact lower factor and its inverse are + +``` +L_ij = q^((i-j)^2) * {i choose j}_Q, i>=j, +Linv_ij = (-q)^(i-j) * {i choose j}_Q, i>=j, +G = L * diag(P_j) * L^T. +``` + +These follow from the ordinary Newton factorization of the Vandermonde matrix at nodes Q^-i; eliminating its successive monic Newton polynomials gives the displayed Gaussian binomial factors and pivot P_j. The inverse follows from the finite q-binomial theorem, or multiplying the two triangular factors and grouping equal powers of Q. `check_q_ldu.py` verifies both identities exactly at 30 rational q/size choices (1,155 matrix entries), including diagonal conjugations. + +Each triangular application is a diagonally scaled Toeplitz convolution: + +``` +L = diag(P_i) T_(q^(h^2)/P_h) diag(P_i)^-1, +Linv = diag(P_i) T_((-q)^h/P_h) diag(P_i)^-1. +``` + +The N factors add only the D_i conjugations. Their inverse lower entries have exponent + +``` +a*((2*beta_i-rho)*h - theta*h^2), h=i-j. +``` + +Since beta_i<=1/2, + +``` +abs((D Linv D^-1)_ij) + <= C exp[-a*theta*h*(h+1)], +C = product_(k>=1)(1-Q^k)^-1. +``` + +The upper inverse obeys the same bound using beta_i>=-1/2. For a>=pi and rho>1, C<2 by a geometric bound on the logarithm of the infinite product. Hence both inverse triangular factors have row norm O(1+(a theta)^-1/2). The P_j diagonal is bounded between 1/C and 1. This avoids using cancellation of exponentially large factors to bound the *completed* inverse. Over one cell, the explicit D_i range costs O(a/theta) guard bits, so the retained condition alpha^2/theta=O(p) still suffices for a straightforward implementation. + +This supplies a clean alternative to the retained Gohberg–Semencul interior solve. The latter already has linear convolution cost; a new exponent requires treating the gaps between cells or their boundary system. + +## 3. Exact complementary hole positions + +Let 11, x_k=r^k, and V_t=(x_k^j)_(0<=k,j=k, and 0 otherwise, +(V_holes^T)_(j,h) = x_h^j. +``` + +The coefficient of X^(t-1-j) in P(X)/(X-x_h) equals +`sum_(k=0..j) p_(t-k) x_h^(j-k)`, by synthetic division from the leading end. Lagrange interpolation gives the identity. + +Given values only on selected nodes, set the H missing values initially to zero and do one full interpolation. Solve A for the missing values that cancel the H degree-tail coefficients, then interpolate with the completed values. The resulting polynomial has degree=1, + +``` +abs(B_ij) <= exp[-a*rho*(rho'*k^2-k)]. +``` + +Indeed the relevant distance is at least rho'k-1/2, and the subtractive delta_j^2 is at most 1/4. Thus off-diagonal row sums are bounded by a convergent geometric Gaussian tail, with first exponent a rho(rho'-1). For small theta, rho' is Theta(1/theta), so this exponent is Theta(a/theta). To obtain error 2^-p a direct truncated solve only needs complementary width + +``` +O(1+sqrt(p/(a*rho*rho'))) = O(1+sqrt(p theta/a)). +``` + +This rigorously confirms the conditioning and narrow-band part of the parent's holes-completion proposal. It is not a cost proof for the complete original inverse: the Toeplitz T and full geometric interpolation must first be implemented with their enormous diagonal scalings canceled symbolically. + +## 6. A useful normalized erasure-coefficient bound + +Let hole exponents h_0<...=g, let R=r^g>1, and e_k be the kth elementary symmetric polynomial in r^h. Let M_k be the product of the k largest nodes. Then + +``` +1 <= e_k/M_k <= product_(j>=1)(1-R^-j)^-1. +``` + +For any selected k-index subset, compare its indices with the k largest indices. Its nonnegative deficits form a partition, and its product divided by M_k is at most R to the negative partition size. Summing over partitions proves the bound. The contribution from deficit >=m is at most + +``` +R^(-m/2) * product_(j>=1)(1-R^(-j/2))^-1. +``` + +For R>=4 the latter product is below 4, so retaining deficit less than `m >= (2p+4)/log2 R` suffices for relative error below 2^-p. Thus normalized coefficients have bounded height even when unnormalized coefficients contain Theta(a theta t^2) exponent bits. Only O(p theta/a) partition energy near the subset boundary is relevant. + +Likewise, for a nonhole i, pull `product_h max(r^i,r^h)` out of |E(r^i)|. The residual is `product_h(1-r^-|i-h|)`, which lies between `product_(k>=1)(1-r^-k)^2` and 1. It is uniformly bounded away from zero for fixed r>1. These are positive products and sums; the bounds do not depend on fortuitous numerical cancellation. + +The unresolved algorithmic question is performing the needed polynomial multiplication/division while storing these *degree-dependent* scales, rather than one common scale. Feeding the raw coefficients to an ordinary FFT requires Theta(a theta t^2) bits; declaring that cost absent is invalid. The positive partition estimates give a concrete starting point for a banded or locally normalized implementation. + +## 7. Correct boundary of the result + +- Exact q-LDU, exact hole indexing, exact completion, complementary conditioning, and the normalized erasure bounds above are established algebraically. +- `Q-LDU-CONTROLS.json` and `HOLES-CONTROLS.json` contain finite independent checks. +- No stable all-size global holes solver is implemented here. Its fully paid precision/tape cost remains open. +- No final multiplication exponent follows from these identities alone. + +## 8. Weighted full-geometric interpolation: a stable local implementation + +This is a further positive lemma for one component of holes completion. It does **not** complete the Toeplitz/complement adapters. + +Let beta=H/t=theta/rho, let h_l be the actual holes above, and let h*_l=(l+1/2)/beta be the ideal equally spaced holes. For x in [0,t], define + +``` +R(x) = 2a*rho * sum_l (x-h_l)_+ - a*theta*x^2. +``` + +This is the logarithm of the dominant erasure evaluation after canceling the full-geometric Gaussian input diagonal, up to a constant independent of x. That constant can be kept symbolically and canceled at the final adapter. + +For ideal holes, put m=floor(beta*x+1/2). Then + +``` +sum_l (x-h*_l)_+ = m*x - m^2/(2beta), +R*(x) = -(a*rho/beta)*(m-beta*x)^2, +-a*rho/(4beta) <= R*(x) <= 0. +``` + +Since |h_l-h*_l|<=1/2, compare increments on [x,x+v]. Holes below both endpoints give the same increment v under both locations, and holes above both give zero. Only O(beta*v+1) holes whose ideal or actual locations meet the interval can contribute; each contributes an error at most 1/2. Consequently, for 0<=x<=x+v<=t, + +``` +|[R-R*](x+v)-[R-R*](x)| <= a*theta*v + O(a*rho), +|R(x+v)-R(x)| <= a*theta*v + K, +K <= 4a/theta (01. It is a norm lemma, not a complete fast inverse algorithm. + +Let s>=5, s=1. The exact periodic normalized Gaussian matrix is + +``` +N_(i,j mod s) = sum_(k congruent j-i mod s) + exp[-a*((beta_i+rho*k)^2-beta_(i+k)^2)]. +``` + +Its principal k=0 term is one. Let L be its k=-1 weighted cyclic shift and U its k=+1 shift. Thus + +``` +L_i = exp[-a*((beta_i-rho)^2-beta_(i-1)^2)], +U_i = exp[-a*((beta_i+rho)^2-beta_(i+1)^2)]. +``` + +All other k terms form R. The aliases k=±s and beyond remain in R; they are not discarded. + +For h consecutive lower factors the exponent telescopes to + +``` +beta_i^2-beta_(i-h)^2 - 2rho sum_(j=0..h-1) beta_(i-j) + h rho^2 + >= h rho theta - 1/4. +``` + +The upper factors obey the same bound. Therefore both alternating Neumann inverses exist, including over arbitrarily many cyclic wraps, and + +``` +||(I+L)^-1||, ||(I+U)^-1|| + <= B = exp(a/4)/(1-exp(-a rho theta)) + <= 2 exp(a/4)/theta. +``` + +The final inequality uses a>=1 and 0=2, + +``` +(beta_i+rho*k)^2-beta_(i+k)^2 + >= (rho*|k|-1/2)^2-1/4 + >= k^2-|k|. +``` + +Consequently + +``` +||R|| <= 2 exp(-2a)/(1-exp(-4a)) < 4 exp(-2a). +``` + +Write `A=(I+L)(I+U)`. Then N=A+(R-LU), and + +``` +||A^-1(R-LU)|| <= 5 B^2 exp(-2a) + <= 20 theta^-2 exp(-3a/2). +``` + +If `a>=4 ln(16/theta)`, this is bounded above by +`20 theta^4/16^6 < 1/2`. Hence + +``` +||N^-1|| <= 2 B^2 <= 8 exp(a/2)/theta^2. +``` + +Choose the integer `alpha=ceil(sqrt(4 ln(16/theta)/pi))`. With +`a0=4 ln(16/theta)`, the elementary ceiling inequality gives +`a<=2a0+2pi`; therefore + +``` +||N^-1|| <= 8 exp(pi) 16^4 theta^-6. +``` + +This is an explicit polynomial bound. For theta=Theta(1/d), alpha^2=O(log d) and the total original tensor scale has logarithm O(d log d)=o(p) for every fixed d=p^epsilon, epsilon<1. The period and aliases introduce no exception. + +## Exact controls + +`check_nearest_neighbor.py` checks 69 admissible finite (s,t) pairs and 93,976 product identities/bounds, including all skip gaps and two complete cyclic wraps. It uses exact rational exponents and no floating point. The exponential conclusions follow from monotonicity and the displayed convergent geometric sums. The output is `NEAREST-NEIGHBOR-CONTROLS.json`. + +## Remaining algorithmic obligation + +This lemma establishes that logarithmic alpha squared is analytically permitted. The old inverse implementation still incurs too much work at that alpha. A fast stable inverse, such as a fully normalized holes-completion algorithm, must still supply its actual movement, rounding and operation count. Erasing the inverse cost from the outer ledger would not follow from this norm proof alone. diff --git a/research/capped-phase-fusion/evidence/radical_reduction_ultra/PARTIAL-FUSION.md b/research/capped-phase-fusion/evidence/radical_reduction_ultra/PARTIAL-FUSION.md new file mode 100644 index 0000000000..876f44a23e --- /dev/null +++ b/research/capped-phase-fusion/evidence/radical_reduction_ultra/PARTIAL-FUSION.md @@ -0,0 +1,60 @@ +# Partial tensor fusion: a strictly better prospective outer balance + +This is a prospective cost theorem, not an implemented resampler. It changes the full-fusion target to a smaller interface whose precision is payable. + +Suppose the normalized holes construction is completed so that a g-axis Gaussian group is a fixed number of tensor geometric/Toeplitz convolutions, diagonal factors, and corrections supported on the hole rows, and suppose its halo gather/scatter has the same asymptotic cost. Let d=p^epsilon and a=Theta(log d), theta=Theta(1/d). + +The exact normalizations developed in HOLES-AND-Q-LDU.md have local logarithmic range O(a/theta)=O(a d) per axis. Therefore g axes can be fused with O(p) guard bits if + +``` +g*a*d = O(p). +``` + +Taking `g=p^(1-epsilon-zeta)` for fixed zeta>0 pays the logarithm and leaves strict slack. There are d/g groups. A group convolution using the retained supplier with saving q costs, after division by full volume, + +``` +g^(1-q) polylog p. +``` + +Thus all Gaussian groups cost + +``` +(d/g)*g^(1-q) = d/g^q + = p^[epsilon-q*(1-epsilon-zeta)]. +``` + +Its saving is `(1+q)*(1-epsilon)-q*zeta`. The main synthetic transform, using deferred reservations, retains saving epsilon*q. Ignoring the arbitrarily small fixed zeta and strict absorption gaps, balancing yields + +``` +epsilon=(1+q)/(1+2q), +kappa=q*(1+q)/(1+2q). +``` + +This exceeds the old `q/(1+q)` by exactly + +``` +q^3 / ((1+q)*(1+2q)). +``` + +That is a new outer balance, rather than tighter rational margins in the old family. At the selected complex endpoint the difference is about 4.7e-9. For example take q equal to the selected complex saving minus 10^-10, zeta=10^-8, the displayed epsilon, and kappa ten^-11 below the smaller main/Gaussian margin. The resulting exact arithmetic target is approximately + +``` +0.00167266475042434534254271472, +``` + +about 4.55335e-9 above the selected 0.001672660197076918. This is only a prospective target until both missing interfaces below are established. + +## Other paid terms + +The bit routing saving is larger than q at the selected pair. Moving a group of O(g log p) small coordinate bits across spectator fields costs O(V g^(1-a_bit) polylog p) per group, hence O(V d/g^a_bit polylog p), below the convolution cost since a_bit>q. A superpolynomial untouched spectator suffix remains available while processing a group, so the compact supplier's descriptor work can still be paid; pointwise small-polynomial products require a separately paid small-field transpose but only logarithmic-width fields. + +The g-axis scalar normalization costs O(g a), and the stronger local weights cost O(g a d); both fit O(p) by construction. For local convolution window L=O(p/a+d), take a dyadic microtile N≫gL. The total halo volume is bounded by `(1+O(L/N))^g=O(1)`. The gauge increment bound implies total logarithmic variation O(g[a theta N+a/theta]); with N=Theta(gL), the extra condition is g^2<=O(d). It holds at the proposed epsilon near one, since 2(1-epsilon)q`, and requires `dK=o(p)`. Thus even if all Gaussian work disappeared, retaining that reservation argument would still force `epsilon<1/(1+q)`. It is insufficient to erase just the Gaussian row from a ledger. + +The selected outer logic was read in `research/complex-pr615-ceiling/vendor/multicold/engines/outer.py`; its seven margins and 47 slacks match the current 1094 assembly. The underlying detailed interfaces are in upstream sections 02, 04, 05, 06, 07, 08, 09, 10 and the cited `references/semantic-bulk/rad20/reports/` transfer notes. In particular, the current bulk Gaussian proof already uses local principal windows and short fields; proposing that same improvement again would not change the selected result. + +## 2. Deferred reservations: removing the power-width reservation condition + +This is an all-size **conditional transfer argument**, assuming the retained complete rectangular compact gadget, exact common-frame circuit, arbitrary coordinate router, and linear semantic guard. It does not require a new finite supplier identity. + +Let `G=4 ceil(log2 p)+6`, and choose `K=128G`. At a layer on `h` still-unprocessed selected bits, reserve enough complete K-bit chunks to provide the two front H-bit fields, one back H-bit field, and row bits, where `H=hG`. The number of reserved chunks is + +``` +R(h) = ceil(3hG/K) + O(log h). +``` + +For `h` above a fixed polynomial in `log p`, `R(h) <= h/16`. Select those chunks from the still-unprocessed axes. Apply the retained compact circuit to the other `h-R(h)` bits, borrowing all three dirty fields from the reserved subset. At the completed-call boundary it is exactly `C` on the active bits and identity on every reserved or spectator bit, for arbitrary initial coefficients and arbitrary dirty addresses. Therefore all reserved bits retain their original complete ranges and can be processed next. + +Repeat on the reserved subset. The processed subsets are disjoint. Stop once `h=polylog(p)` and apply those remaining kernels individually. This is **not** a recursive call on a truncated record or a clean-scratch claim: the entire original stream is still present, with the same full polynomial suffix, and every previously processed coordinate is an untouched spectator. Completed kernels on disjoint selected bits commute. All row padding is removed before moving to the next phase, so temporary factor-two padding never multiplies across phases. + +To establish the time bound, retain the original `m,W,s` recurrence and stop its internal recursion at `d^beta`. With `sigma=1-b`, any `lambda>max(1-a,sigma)` gives, with logarithmic factors suppressed, + +``` +F(h) <= O(h^lambda + h^sigma d^(beta*(1-sigma)) + 1). +``` + +This follows because a packed local basis change has width `O(hG)` and hence cost `O(V h^(1-a) polylog p)`. The term formerly written `(h d^c)^(1-a)` is now `h^(1-a) polylog p`. All compact-gadget exceptional fractions remain polynomially small with K=128G; the retained repair proof uses the current call's O(p)-bit address and is not replaced by a global exception count. + +Since `h_j <= d/16^j` above the polylogarithmic tail, + +``` +sum_j F(h_j) + = O((d^lambda + d^(sigma+beta*(1-sigma))) polylog p). +``` + +The extra physical rearrangements of complete existing fields cost +`O(V p^(1-a) polylog p)` by the retained arbitrary coordinate router, over O(log d) phases. Its polynomial descriptor work is paid by the unchanged superpolynomial polynomial-record suffix, at every internal call. This is additive, not free routing. + +Consequently, for every fixed `q < min(a,(1-beta)*b)`, the whole layer has + +``` +O(V [d^(1-q) + p^(1-a)] polylog p). +``` + +The common guard remains O(d): each completed phase changes semantic numerator/denominator bounds by O(h_j), and `sum h_j=O(d)`. Only the currently active phase contributes its internal excess. There is still exactly one truncation after the entire normalized layer. No power K=d^c and no condition c>q remains. K=o(ell) now follows from `ell=p^(1-epsilon)` for every fixed epsilon<1. + +The extra top round on the long axes can use the same layer after a paid coordinate rearrangement collecting its selected bits and K-bit complete neighborhoods. The common rounds use a single paid global layout router, rather than `p/K` separate K-swaps. Thus the transformed version of the synthetic cost is + +``` +O(V [p^(1-a) + ell*d^(1-q)] polylog p). +``` + +This is the strongest positive transfer established by the present note. It removes the geometric source of `q/(1+q)` in the compact compiler. It does not remove the independent O(Vd) Gaussian and scalar costs. + +### Checks + +`check_constructions.py` checks 1,000 exact reservation schedules and 5,460 entries of actual Gaussian-rational `C` tensor matrices. Every axis is processed once; the phasewise output equals the independently calculated closed tensor matrix. Omitting the deferred subset is a retained failing control. These checks support the written argument; they are not an all-size compiler implementation. + +## 3. Batched right-hand sides for the Gaussian boundary inverse + +This refinement applies to the precomputed noncyclic banded L/U factors in the retained principal-window solver. It uses the many spectator lines carrying the same axis matrix. + +For a triangular matrix with scalar bandwidth w, group consecutive rows into blocks of a power-of-two width `B in [w,2w)`. There is only one preceding block in a lower solve (one following block for an upper solve). Precompute each diagonal-block inverse. For B simultaneous right-hand sides, the update is + +``` +X_j = L_jj^-1 * (Y_j - L_j,j-1 * X_j-1). +``` + +Two B-by-B products implement each update. The seven-product identity gives `O(B^omega)` scalar operations, `omega=log_2(7)`. For z boundary rows and R>=B right-hand sides, splitting the columns into B-column batches gives + +``` +O(z R B^(omega-2)) +``` + +scalar work. Incomplete edge blocks and the last RHS batch are zero-padded only for the current product; this is a constant temporary volume increase, not a tensor-axis padding factor. The matrices may vary from row block to row block. No Toeplitz assumption is made. + +The original boundary has `z=O(M w/d + w)` rows in a length-M local window. Summed over axes and lines, the old regular contribution `V w^2` becomes + +``` +O(V w^(omega-1) polylog p). +``` + +The artificial-end contribution is handled by a polynomial-size microbox L; it is +`O(V d w^(omega-1)/L polylog p)`. The O(Vd polylog p) local field motion and interior-convolution cost is still paid and is still present. + +### Tape layout + +Expose the same axis and factor page used by the retained solver. Gather B adjacent RHS indices with a B-valued local field, moving that O(log p)-bit field across the current row field by the retained small-field transposition. It costs O(V log p) per axis. This charge belongs to the existing d-pass term. Read factor pages and RHS row blocks sequentially. Keep the immediately preceding solved B-by-B block on a fixed work tape. Products use a depth-first seven-product recursion, parking/restoring the other finitely many matrices at each node; scanning and quadrant rearrangement is O(B^2) per level and is dominated by B^omega. There is no random factor lookup and no separate tape for every RHS. + +### Precision + +Keep the retained O(p)-bit rounded factor entries and compute each B-by-B product exactly as a dyadic matrix product before one component rounding. An individual seven-product recursion has depth log B. Additions grow integers by only O(log B) bits at every product boundary, so exact products and their sums have O(p+log B)=O(p) bits. A block inverse is precomputed at the retained high precision, with its row-norm perturbation charged explicitly. + +Round after a completed block update, not between scalar products. The residual of the completed triangular system is bounded rowwise by `poly(p)*2^-P*(1+||X||)`, with P the retained large constant multiple of p. The retained inverse-norm bounds then give a uniform global solution error after absorption, independent of the number of row blocks. This is the same residual closure used by the original banded solver; it must be instantiated with new block-factor rounding constants in any adoption. An unproved claim of numerical stability of fast floating-point multiplication is unnecessary. + +For `u=p^r`, `w^2=O(p^(1-r))`. The revised regular boundary exponent is + +``` +((omega-1)/2)*(1-r), +``` + +whose saving at r=0 is `(3-log_2 7)/2 > 0.096`. Thus an implemented version could remove the old requirement r>kappa while keeping `epsilon+r<1`. It cannot remove the separate O(Vd) scan term by itself. + +### Exact checks and remaining check + +The script independently compares seven-product multiplication against classical multiplication, then compares 60 varied-size blocked triangular solves against exact Gaussian-elimination inverses (1,976 output entries). A dropped-predecessor-block control fails. This establishes finite algebraic orientation, padding and dependencies. A complete adaptation of the original numerical residual constants is still required before adopting the changed analytic interface. + +## 4. A concrete route toward tensor Gaussian fusion + +This section is **conjectural construction work**, not an established algorithm. The missing step is stated so it can be attacked. + +A possible way to avoid d separate Gaussian passes is to compute the tensor Gaussian on a microbox through one tensor convolution. For `rho=s/t`, a real Gaussian row has the exact scalar identity + +``` +exp[-a(j-rho*k)^2] + = exp[-a(1-rho)j^2] + exp[+a*rho(1-rho)k^2] + exp[-a*rho(j-k)^2]. +``` + +Integer translations of j and k add only linear factors, so each local rectangle is a diagonal, Toeplitz convolution, diagonal. Across axes, the diagonal factors fuse into one per-record multiplication. The central convolution is a tensor cyclic convolution after sufficient halos. + +With the current theta≈1/d, a local Gaussian halo has radius +`A=O(alpha*sqrt(p))`. To keep product halo volume constant, the core needs length `L=Omega(dA)`. But the fused diagonal logarithmic range is then +`O(d*theta*L^2/alpha^2)=O(d^2 p)`, not O(p). Thus the most obvious tensor-chirp fusion fails its precision ledger. This explicit failure should prevent a false positive. + +One changed hypothesis makes that favorable: choose source primes much closer to t, with `theta=Theta(d^-3)`. Then at `L=Theta(dA)`, that logarithmic range is O(p), and the product halo overhead is still constant. Such prime lengths are available eventually for every fixed epsilon<1: the explicit prime-counting error from Fiori–Kadiri–Swidinsky, Corollary 22, is `O(x sqrt(log x) exp(-c sqrt(log x)))`; on an interval of width `x/d^3`, with `log x=Theta(p^(1-epsilon))`, its error is o(x/(d^3 log x)). The interval contains far more than d primes, so distinct choices and `S/T=1-o(1)` follow. Trial division setup remains `n^o(1)`. This is an asymptotic existence argument with an enormous cutoff, not a practical prime-search claim. + +However two serious costs remain unproved: + +1. A power-of-two convolution window N has a useful core N-2A. Reinterpreting an axis by these variable non-dyadic cores is not the known binary-coordinate router. Padding every N-2A field to the next power of two costs 2^d. Performing the known small-field reshape one axis at a time costs Vd. A new simultaneous variable-core gather/scatter is needed, with full payload movement charged. +2. The inverse Gaussian correction is not itself one convolution. With theta=d^-3, the current phase-cell similarity weights range like `exp(O(u/theta))`, too large for its old O(p)-bit proof. The generic banded solve can avoid those enormous weights, but the present batched bound still has a d factor outside its favorable structured regime. The tensor inverse needs a different stable factorization or a sparse-boundary construction with a fully paid ledger. + +Therefore “choose closer primes” plus “use a multidimensional FFT” is not a proved bypass. It is an explicit construction program with two specified lemmas. + +## 5. Exact optimistic target and fully paid comparison + +Assume a new tensor-resampling interface gives + +``` +O(V [p^(1-a) + d^(1-q) + w^(omega-1)] polylog p) +``` + +for the complete expansion and inverse, including variable-core layout, coefficient factors, stable inverse, all restoration and only O(p)-bit records. Combine it with Section 2. The complete normalized saving is at least + +``` +min(a, epsilon*q, 1-epsilon*(1-q), + 1-((omega-1)/2)*(1-r), epsilon), +``` + +with strict `epsilon+r<1`, `q 0`, the complex endpoint exceeds q by `0.000000462676808103522925`, and the gamma gap is `0.000005`. The target exceeds the selected kappa by `0.000002319802923082` (about 0.1386894%). Letting epsilon approach one and q approach b would recover the complex saving itself, up to arbitrary strict slack. + +**Fully paid ledger today:** without the unproved tensor-resampling interface, the unchanged Vd term remains. It imposes `kappa<1-epsilon`, so maximizing the proven/reused terms still gives at most `b/(1+b)`. Sections 2 and 3 alone do not improve final kappa. Under the displayed sample, the paid Gaussian margin is only `0.00001`, so that sample FAILS the current interface. It is deliberately not emitted as a passing outer certificate. + +## 6. Reproduction and evidence + +Run from the parent workspace: + +``` +python3 work/radical_reduction_ultra/check_constructions.py +python3 work/radical_reduction_ultra/check_ledger.py +``` + +`CONTROLS.json` contains exact finite controls. `LEDGER.json` contains the optimistic row and the correctly rejected fully paid row. Neither imports a producer or mutates the source checkout. + +## Open next questions + +- Instantiate Section 2 against the exact current compact field-layout code and reject malformed reservoir choices. This is a bounded integration task; it could remove a real interface restriction even before a new final exponent is obtained. +- Carry the block residual argument in Section 3 through the retained P=32768p constants and factor-page format. The algebraic and layout reductions are explicit; numerical certification is the remaining bounded task. +- For the fusion program, first prove or disprove a simultaneous variable-core gather/scatter with O(V p^(1-a) polylog p) cost and no 2^d padding. This isolates the data-layout bottleneck before more analytic effort. +- Seek a stable tensor inverse for near-identity Gaussian resampling with theta≈d^-3 that avoids both enormous diagonal similarities and d sequential full scans. A finite small-d result is not enough; require the exact operator, record width and mass recurrence. + +## Primary sources and attribution + +- CrocSwap/integer-mult-bounds pinned upstream and transfer reports, in particular 02-streams, 04-swap, 05-layers, 06-transforms, 07-resampling, 08-assembly, 09-exact-arithmetic, 10-transposition; reviewed contributions by Douglas Colkitt and the cited independent contributors are retained unchanged. +- Harvey and van der Hoeven's original Gaussian resampling and multiplication interfaces are inherited through those source proofs. +- V. Strassen, *Gaussian Elimination is Not Optimal*, Numerische Mathematik 13 (1969), 354–356, DOI `10.1007/BF02165411`. The seven-product identity is reproduced and independently checked here; no modern matrix multiplication exponent is needed. +- A. Fiori, H. Kadiri, J. Swidinsky, *Sharper bounds for the error term in the prime number theorem*, arXiv:2206.12557v2, Corollary 22: https://arxiv.org/pdf/2206.12557 . Used only for the clearly marked close-prime construction program. + +## Later constructive developments + +The positive interpolation-tree and global two-tail results are in `STABLE-POSITIVE-TREE.md`, with exact controls in `POSITIVE-TREE-CONTROLS.json`. They provide bounded gauges for every signed-data intermediate and show that all high-level rank-one contributions collapse to two global running sums. Its Section 6 gives a normalized Sylvester factorization into a dirty Toeplitz shear, one well-conditioned full Toeplitz map, and one triangular Toeplitz map; `SYLVESTER-CONTROLS.json` checks that identity. + +The stronger fixed-width Gaussian norm theorem is independently audited in `FIXED-A-PERIODIC-NORM-AUDIT.md`, with `FIXED-A-LDU-CONTROLS.json`. It replaces the earlier sufficient logarithmic-width condition: for any fixed admissible a>=1, the periodic inverse has norm O(theta^-2). This is an analytic parameter improvement, not a completed fast inverse. + +The outer partial-fusion candidate in `PARTIAL-FUSION.md` remains prospective. Neither the stable, shallow in-place algebra for plain Toeplitz matrices nor bounded final Gaussian conditioning alone proves that the curved Gaussian coefficient gauges can be processed with only O(p) total guard bits when many axes are fused. That interface, and the actual halo movement cost, must be resolved before claiming a new kappa. diff --git a/research/capped-phase-fusion/evidence/radical_reduction_ultra/REFINED-PHASE-FUSION.md b/research/capped-phase-fusion/evidence/radical_reduction_ultra/REFINED-PHASE-FUSION.md new file mode 100644 index 0000000000..51920b3b76 --- /dev/null +++ b/research/capped-phase-fusion/evidence/radical_reduction_ultra/REFINED-PHASE-FUSION.md @@ -0,0 +1,125 @@ +# A narrower positive route: fuse retained phase-cell interiors + +Status: a complete parameter ledger for a concrete conditional transfer. It avoids the unresolved normalized-erasure adapter. It still requires the new shallow in-place structured-matrix circuit to be integrated with the retained fast tensor supplier and a grouped sparse boundary gather. No new selected certificate is asserted. + +## Construction + +Keep the already reviewed phase-cell Schur decomposition, with + +``` +d=p^epsilon, u=alpha^2=Theta(p^r), w=Theta(p^((1-r)/2)), +0(1-r)/2. +``` + +The positive r may be arbitrarily small; it eventually dominates the old logarithmic lower bound on u. Thus this route can retain the established diagonal-dominance, rounded H, local inverse, and Schur residual proofs. The stronger fixed-a theorem is not needed here. + +Within each no-gap phase cell, the interior matrix is exactly + +``` +A=diag(a_j) G diag(a_j)^-1, +G_ij=G_(i-j), log(max a_j/min a_j)=O(u d). +``` + +The retained G is a symmetric truncated Gaussian Toeplitz matrix with `||G-I||<1/8`; hence G and G^-1 are uniformly conditioned. Apply its inverse by the new shallow in-place low-displacement circuit, rather than summing the two Gohberg-Semencul branches. The circuit's Toeplitz coefficients and Schur generators are precomputed. This is the algebraic connection to the new in-place work; the old source proof did not provide such a tensor-fusible circuit. + +Fuse + +``` +g=p^beta, beta=1-epsilon-r-zeta, +``` + +axes at a time, with fixed zeta>0. The total diagonal similarity guard is +`O(g u d)=O(p^(1-zeta))=o(p)`. Polylogarithmic circuit depth and bounded polynomial factor growth add only `O(g polylog p)=o(p)` bits. Every completed group has polynomial norm; error is allocated at the completed-group boundaries, preserving the retained O(p)-bit record model. + +### Dyadic interiors without exponential padding + +Inside a phase cell, repeatedly choose the largest power of two K such that `K+2w` fits the remaining interval, while its remaining length exceeds 4w. Make a piece of length K+2w, designate w endpoints on each side as boundary, and use its K remaining indices as the interior. The leftover interval of length at most 4w is all boundary. + +The remaining length at least halves on each step. Thus there are O(log d) pieces per original cell, every interior has dyadic length, and the total boundary size grows by only O(log d). No padding by a factor two per tensor axis is used. Schur fill still joins only the two w-sized endpoint groups of a piece, so its scalar bandwidth remains O(w). Distinct dyadic size classes require only O(log p) circuit schedules; inactive axes use identity stages. + +### Boundary arithmetic + +Let R denote the number of spectator right-hand sides, which eventually exceeds every polynomial in p. Batch them in groups of w. Every dense w-by-w block multiplication in the banded forward/backward solve, the border correction, and the boundary/interior couplings then costs O(w^omega) scalar products for w right-hand sides, omega=log_2(7)<2.8081. The retained no-pivot factors are precomputed; Strassen creates only polynomial intermediate growth in w, hence O(log p) extra guard bits. Use the existing global residual argument to bound the completed solve error. + +There are O(s*w*log(d)/d) boundary rows per axis. Consequently the total boundary arithmetic over all d axes is + +``` +O(V * w^(omega-1) * polylog(p)), +``` + +where V counts p-bit data volume. This includes H_BI and H_IB couplings; doing those couplings scalarly would incorrectly reintroduce V*w^2. + +### Five stages separate fused interiors from sparse corrections + +For the interior/boundary order write H=[A B; C D] and S=D-C A^-1 B. The exact identity + +``` +H^-1 = diag(A^-1,I) [I -B;0 I] diag(A,S^-1) + [I 0;-C I] diag(A^-1,I) +``` + +is verified by multiplication. Thus the only full-interior stages are A and A^-1. The two shears use original banded B and C, whose support is within O(w) of each artificial cell boundary. The middle factor splits into the commuting interior A and boundary S^-1. + +For a g-axis group, collect the union U of the O(w log d)-wide endpoint slabs. Its volume is O(V*g*w*log(d)/d). All sparse shear and boundary stages preserve this union; outside it they are identity. Process the g axis operators sequentially on U, without g scans of the full V data. Even a deliberately loose movement cost is + +``` +O(V*g^2*w*polylog(p)/d) per group, +O(V*g*w*polylog(p)) over all d/g groups. +``` + +Add one full scan per group to gather/scatter U, for O(V*d/g). This is below the proposed fused interior cost at the displayed parameters. Dense boundary arithmetic is still batched as above and costs O(V*w^(omega-1)*polylog p); scanning inactive entries in U is included in the movement bound. + +The union-slab gather/scatter and the evaluation of its piece catalogues must be implemented in the retained macro-routing model. The address fields have polynomial length and g*polylog(p)=o(p), but that observation alone is not a tape routing proof. Importantly, the cost budget now tolerates a sequential factor g on the **sparse union**, so the required route is weaker than a fully fused boundary solver. + +## Paid exponents + +Let q be strictly below the supplied complex saving and below the bit saving. The proposed dominant costs, normalized by V, are + +| Component | Power of p | Saving | +|---|---:|---:| +| Main transform with deferred reservations | 1-epsilon*q | epsilon*q | +| Fused Gaussian interior / convolution groups | epsilon-q*beta | 1-epsilon+q*beta | +| Group bit routing | epsilon-a_bit*beta | 1-epsilon+a_bit*beta | +| Sparse group scans | epsilon-beta | 1-epsilon+beta | +| Boundary algebra | (omega-1)*(1-r)/2 | 1-(omega-1)*(1-r)/2 | +| Sequential operations on sparse union | beta+(1-r)/2 | (1+r)/2-beta | + +Every entry has an explicit cost source above. The similarity/normalization condition is `epsilon+r+beta<1`. The old full-axis scan Vd is not part of the claimed implementation; until its replacement is proved, this remains conditional. + +Ignoring arbitrarily small zeta and strict absorption margins, balancing the first two rows gives + +``` +epsilon=(1+q-q*r)/(1+2q), +beta=(q-(1+q)*r)/(1+2q), +kappa=q*(1+q-q*r)/(1+2q). +``` + +This is feasible for `0 0. +``` + +Thus positive r is compatible with a strictly larger exponent, while retaining the old phase-cell analytic hypotheses. + +One explicit arithmetic target uses + +``` +q=0.001675462576808103522925, +r=0.000001, +zeta=0.00000001, +epsilon=(1+q-q*(r+zeta))/(1+2q), +beta=1-epsilon-r-zeta, +kappa=epsilon*q-0.00000000001. +``` + +It yields + +``` +epsilon = 0.9983301313358845139763404147924773... +beta = 0.0016688586641154860236595852075227... +kappa = 0.0016726647643531934852000135266363... +``` + +The proposed kappa exceeds the selected 0.001672660197076918 by about 4.5672763e-9. Its boundary saving is greater than 0.09595, its sparse-scan saving is greater than 0.0033387, its sparse-union saving exceeds 0.49833, and the bit saving is strictly larger than q. This target is deliberately **not** promoted: the new circuit/supplier integration and grouped sparse tape route must first be proved. diff --git a/research/capped-phase-fusion/evidence/radical_reduction_ultra/STABLE-POSITIVE-TREE.md b/research/capped-phase-fusion/evidence/radical_reduction_ultra/STABLE-POSITIVE-TREE.md new file mode 100644 index 0000000000..687c668a9d --- /dev/null +++ b/research/capped-phase-fusion/evidence/radical_reduction_ultra/STABLE-POSITIVE-TREE.md @@ -0,0 +1,187 @@ +# Positive interpolation trees and two global inverse tails + +Status: exact algebra and an error bound for a compressed inverse representation. This note does not yet claim that the local tree operators admit the pure convolution factorization required for tensor fusion. + +## 1. Setup and signs + +Let `x_i=r^{q_i}`, `r=exp(2 a rho)>1`, and `q_i=floor(rho i+1/2)` for `0<=iu) x_j, +W_I(k)=d_(u-k)/M_I, 0<=k<|I|. +``` + +The equality `u-k=s-1-(k+c_I)` is the connection to the final reversed degree. + +Suppose `||y||_infinity<=1` and `||N0^-1||_infinity<=B`. Replace all leaves by their absolute values and call the resulting polynomial `R_I^+`. Completing the interval by the single outside monomial that chooses every node above I and no node below I gives + +``` +M_I * [z^k] R_I^+ <= [z^(k+c_I)] R_all^+ <= B d_(u-k). +``` + +Consequently + +``` +|[z^k] R_I| <= [z^k] R_I^+ <= B W_I(k). +``` + +This holds for every node of any contiguous binary interpolation tree, for arbitrary signed inputs. It does not require roots to remain correlated after rounding. In particular, it avoids the independently-rounded-erasure-quotient failure in ERASURE-ROUNDING-CONTROL.json. + +The merge identity is + +``` +R_(A union B)=F_B R_A + F_A R_B, +``` + +where A is the lower child and B the upper child. All fixed polynomial coefficients in this identity are positive. Each intermediate contribution, after multiplication by the chosen outside monomial, is a sub-sum of the final absolute Lagrange sum. Therefore bounded intermediate amplitude follows from the completed inverse bound, including in the presence of cancellation between input leaves. + +This is a height bound, not by itself a fast multiplication theorem. The gauges are degree dependent. Ordinary convolution of their unscaled coefficients can still require excessive precision. + +## 3. Partition bound for arbitrary selected nodes + +For any increasing nodes satisfying `x_(i+1)/x_i>=r`, let `e_k` be the kth elementary symmetric polynomial, and `M_k` the product of the k largest nodes. Comparing a selected k-subset to the k largest indices gives a partition of nonnegative index deficits. Hence + +``` +1 <= e_k/M_k <= C_r, +C_r=product_(h>=1)(1-r^-h)^-1. +``` + +For `r>=4`, `C_r<2`. This applies to every interval and to intervals with a node deleted. It controls the normalized fixed polynomial factors used by the tree. + +## 4. Two globally compatible far fields + +Write `E_k=e_k(x_0,...,x_(s-1))`, with E_0=1. Let `k=s-1-j`. + +If `i<=j-m`, then + +``` +0 <= 1-e_k(X without x_i)/E_k + = x_i e_(k-1)(X without x_i)/E_k + <= C_r x_i/x_(j+1) + <= C_r r^(-m-1). +``` + +The endpoint k=0 is exact and is read separately. If `i>=j+m`, then + +``` +e_k(X without x_i) + = E_(k+1)/x_i - e_(k+1)(X without x_i)/x_i, +``` + +and the second term relative to `E_(k+1)/x_i` is at most + +``` +C_r x_(j-1)/x_i <= C_r r^(-m-1). +``` + +The endpoint j=0 is again exact. Thus, after allowing a harmless factor `1/(1-C_r r^(-m-1))` to convert the relative approximation to a bound by the true positive entry, the two far fields are + +``` +(N0^-1 y)_j ~= (-1)^k [ + E_k/d_j * sum_(i<=j-m) w_i + +E_(k+1)/d_j * sum_(i>=j+m) w_i/x_i + + exact local strip |i-j|= (p log 2 + log(4 C_r B))/(log r) +``` + +makes the far-field approximation error at most `2^(-p-1)`. In the retained regime, `log B=O(log d)` and `log r=Theta(a)`, so `m=O(p/a+1)`. + +## 5. What must still be paid + +The construction reduces the data application problem to two globally compatible weighted running sums and a width-O(p/a) local strip. All coefficients, barycentric weights, and normalized tree factors may be precomputed: the source axis length is exp(O(p/d)), and any fixed polynomial power of it remains negligible compared with the full tensor volume. + +For a single axis, bounded positive interpolation trees are a plausible route to the local strip in `O(s p polylog p)` bit operations. To turn this into an exponent improvement, however, one must give a fixed sequence of convolution, diagonal, and permitted data-movement operations that can be fused over many axes. A merge written as a *sum* of two convolutions is not sufficient: tensoring it naively generates exponentially many branches. Neither the positive height bound nor the two-tail representation silently resolves that issue. + +## 6. A branch-free reduction of a normalized Sylvester merge + +This is an exact algebraic reduction, with a bounded-symbol condition. It reduces the remaining in-place question to one full Toeplitz map rather than a sum of two polynomial products. + +Take two children of size b. Choose an anchor zeta strictly between their node sets and rescale the variable. After swapping/scaling the child registers the merge has the form + +``` +S(B,A)=f(z) B(z)+z^b g(z^-1) A(z), +f(z)=product_(lower nodes x)(1+(x/zeta) z), +g(z^-1)=product_(upper nodes x)(1+(zeta/x) z^-1). +``` + +If zeta is the geometric mean of the largest lower node and smallest upper node, every displayed root parameter is at most `r^-1/2`. The sum of all parameters on either side is at most + +``` +eta = r^-1/2/(1-r^-1). +``` + +Thus the l1 norms of f and g are at most exp(eta), and the l1 norms of their reciprocal series are at most exp(eta/(1-r^-1/2)). All are constant and approach one as a increases. + +Let T_f be the 2b-by-2b lower triangular Toeplitz multiplication matrix for f. Put + +``` +h(z)=g(z^-1)/f(z), +K_ij=h_(i-b-j), +H_ij=h_(i-j), 0<=i,j>axis&1): + j=i|(1<4: + h=len(pending); r=(3*h*g+row_bits+k-1)//k + if r>=h or h-r<1:break + reserved=pending[:r];active=pending[r:] + assert r*k >= 3*h*g+row_bits + stages.append(dict(active=active,reserved=reserved)) + pending=reserved + stages.append(dict(active=pending,reserved=[])) + flat=[x for s in stages for x in s['active']] + assert sorted(flat)==list(range(d)) and len(set(flat))==d + return stages + +def run(): + global OPS + rng=random.Random(128731);mm=0;solve=0;rhs_entries=0 + for size in (1,2,4,8): + for _ in range(5): + A=[[F(rng.randint(-3,3),4) for _ in range(size)] for _ in range(size)] + B=[[F(rng.randint(-3,3),4) for _ in range(size)] for _ in range(size)] + before=OPS + assert mul(A,B)==classical(A,B) + assert OPS-before==7**(size.bit_length()-1) + mm+=1 + for width in (1,2,4,8): + for n in (1,3,7,10,17): + for k in (1,3,9): + A=[[F(0) for _ in range(n)] for _ in range(n)] + for i in range(n): + A[i][i]=F(1) + for j in range(max(0,i-width),i):A[i][j]=F(rng.randint(-2,2),8*width) + B=[[F(rng.randint(-4,4),8) for _ in range(k)] for _ in range(n)] + X=block_solve(A,B,width) + assert classical(A,X)==B + assert X==classical(inverse(A),B) + solve+=1;rhs_entries+=n*k + schedule_cases=0;exact_entries=0 + for d in range(1,1001): + s=reserve_schedule(d,128,1,8) + assert sum(len(t['active']) for t in s)==d + assert len(s)<=d + schedule_cases+=1 + for d in range(1,7): + s=reserve_schedule(d,8,1) + for col in range(1<>bit&1 else a) + assert v[row]==z;exact_entries+=1 + # Meaningful failure controls: the unresolved reserved coordinates matter. + s=reserve_schedule(6,8,1) + v=[ga(int(i==0)) for i in range(64)] + full=c_axes(v,range(6),6) + omitted=c_axes(v,s[0]['active'],6) + assert omitted!=full + # A wrong block dependency (drop the preceding solved block) fails. + A=[[F(1),F(0)],[F(1,4),F(1)]]; B=[[F(1)],[F(1)]] + assert B!=block_solve(A,B,1) + result=dict(status='PASS_EXACT_SCOPED_CONTROLS',strassen_cases=mm, + blocked_lower_solve_cases=solve,rhs_entries=rhs_entries, + reserve_schedules=schedule_cases,complex_tensor_matrix_entries=exact_entries, + negative_controls=['omit_reserved_axes','drop_preceding_solve_block'], + scalar_products_executed=OPS, + scope='Finite exact identities only; not a tape simulator or all-size compiler proof.') + print(json.dumps(result,indent=2)) + Path(__file__).with_name('CONTROLS.json').write_text(json.dumps(result,indent=2)+'\n') +if __name__=='__main__':run() diff --git a/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_erasure_rounding.py b/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_erasure_rounding.py new file mode 100644 index 0000000000..1f5f8e821a --- /dev/null +++ b/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_erasure_rounding.py @@ -0,0 +1,36 @@ +#!/usr/bin/env python3 +"""Exact failure control for independent coefficient rounding before erasure division.""" +from fractions import Fraction as F +from pathlib import Path +import json + +def pmul(a,b): + c=[0]*(len(a)+len(b)-1) + for i,x in enumerate(a): + for j,y in enumerate(b):c[i+j]+=x*y + return c +s,t=29,32;H=t-s;rho=F(t,s) +q=[(2*t*i+s)//(2*s) for i in range(s)] +holes=[i for i in range(t) if i not in q] +assert holes==[5,16,27] +# exp(2*a*rho)=2^(2s); then natural source coefficient scale is 2^(-t*j*j). +base=2**(2*s);E=[1] +for h in holes:E=pmul(E,[-base**h,1]) +E0=abs(E[0]);P=1000;j=20 +# Relative 2^-P perturbation of Q_0. Exactly compute the formal quotient error. +delta=F(E0,2**P) +c=[F(0)]*(j+1) +for k in range(j+1): + rhs=delta if k==0 else F(0) + c[k]=(rhs-sum(E[h]*c[k-h] for h in range(1,min(H,k)+1)))/E[0] +error=abs(c[j])*(2**(t*j*j)) +min_error=F(2)**(t*j*j-2*s*holes[0]*j-P) +assert error>=min_error and min_error==2**6000 +assert delta/E0==F(1,2**P) +res=dict(status='PASS_ERASURE_ROUNDING_FAILURE_CONTROL',s=s,t=t,holes=holes, + relative_input_error='2^-1000',coefficient_index=j, + normalized_output_error_lower='2^6000', + exact_error_numerator_bits=error.numerator.bit_length(), + scope='Independent coefficient perturbation destroys divisibility by E. Not an impossibility result for correlated or exact-root-preserving algorithms.') +Path(__file__).with_name('ERASURE-ROUNDING-CONTROL.json').write_text(json.dumps(res,indent=2)+'\n') +print(json.dumps(res,indent=2)) diff --git a/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_fixed_a_ldu.py b/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_fixed_a_ldu.py new file mode 100644 index 0000000000..ac7f96cd00 --- /dev/null +++ b/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_fixed_a_ldu.py @@ -0,0 +1,55 @@ +#!/usr/bin/env python3 +from fractions import Fraction as F +from math import prod,ceil +from pathlib import Path +import json + +def p2(k):return F(2)**k +def mm(a,b):return [[sum(a[i][k]*b[k][j] for k in range(len(b))) for j in range(len(b))] for i in range(len(a))] +def symmetric(xs,n,complete): + p=[F(1)]+[F(0)]*n + for x in xs: + if complete: + for j in range(1,n+1):p[j]+=x*p[j-1] + else: + for j in range(n,0,-1):p[j]+=x*p[j-1] + return p[n] +def test(s,t,l,n): + ids=list(range(l,l+n));q={i:(2*t*i+s)//(2*s) for i in ids} + x={i:p2(2*s*t*q[i]) for i in ids} + e={i:p2(-t*t*i*i+(t*i-s*q[i])**2) for i in ids} + g=lambda i,j:p2(-(t*j-s*q[i])**2+(t*j-s*q[j])**2) + c={i:prod(1-x[k]/x[i] for k in ids if k=j else F(0)) + ur.append(e[j]/e[i]*symmetric([x[k] for k in ids if k<=i],j-i,True) if j>=i else F(0)) + if i>=j: + h=i-j + exponent=-(t*i-s*q[i])**2+(t*j-s*q[j])**2-t*t*h+2*t*sum(t*k-s*q[k] for k in ids if j=i: + h=j-i;z=(-1)**h*e[j]/e[i]*symmetric([x[k] for k in ids if k=j else F(0) for j in range(H)] for i in range(H)] + Ht=[[x[h]**j for h in holes] for j in range(H)] + D=[[1/peval(derivative,x[holes[i]]) if i==j else F(0) for j in range(H)] for i in range(H)] + A=[[Vi[t-1-j][h] for h in holes] for j in range(H)] + assert mm(mm(T,Ht),D)==A + # Complete a partial values vector to a degree0 for v in slacks.values()) +paid=dict(margins,unfused_gaussian=1-eps) +assert min(paid.values()) < k +assert 1-eps < selected +assert k>selected +# Require the missing map to be declared; calculations do not discharge it. +result=dict(status='PASS_OPTIMISTIC_LEDGER_AND_PAID_REJECTION', + selected_kappa=str(selected), target_kappa=str(k), + absolute_gain=str(k-selected), + proposed_parameters={name:str(v) for name,v in + dict(a=a,q=q,epsilon=eps,r=r,beta=beta).items()}, + omega_upper=str(omega_upper),omega_control='7^16 < 2^45', + optimistic_margins={name:str(v) for name,v in margins.items()}, + optimistic_slacks={name:str(v) for name,v in slacks.items()}, + fully_paid_margins={name:str(v) for name,v in paid.items()}, + fully_paid_accepts_target=False, + missing_lemma='Stable complete tensor Gaussian resampling including variable-core movement in stated cost and O(p) record width.', + scope='Arithmetic conditional target and explicit rejected existing-interface row; not a new multiplication certificate.') +Path(__file__).with_name('LEDGER.json').write_text(json.dumps(result,indent=2)+'\n') +print(json.dumps(result,indent=2)) diff --git a/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_nearest_neighbor.py b/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_nearest_neighbor.py new file mode 100644 index 0000000000..d56f21f6b7 --- /dev/null +++ b/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_nearest_neighbor.py @@ -0,0 +1,45 @@ +#!/usr/bin/env python3 +"""Exact exponent checks for the polynomial-condition nearest-neighbor lemma.""" +from fractions import Fraction as F +from pathlib import Path +import json +cases=products=0 +for t in range(6,29): + for s in range(max(5,(4*t+4)//5),t): + rho=F(t,s);theta=rho-1 + if theta>F(1,4):continue + q=[(2*t*j+s)//(2*s) for j in range(s)] + beta=[rho*j-q[j] for j in range(s)] + assert all(-F(1,2)<=v=h*rho*theta-F(1,4) + assert usum>=h*rho*theta-F(1,4) + products+=2 + cases+=1 +# With a >=4 log(16/theta), the proposed perturbation ratio is <=20 theta^4/16^6. +assert F(20,16**6)*F(1,4)**4 < F(1,2) +res=dict(status='PASS_EXACT_NEAREST_NEIGHBOR_EXPONENTS',parameter_pairs=cases, + cyclic_product_bounds=products,cycle_lengths_up_to=28, + controls=['all skip gaps checked','two full cyclic wraps checked'], + scope='Exact exponent identities and inequalities; real exponential conclusions follow from monotonicity and geometric sums.') +Path(__file__).with_name('NEAREST-NEIGHBOR-CONTROLS.json').write_text(json.dumps(res,indent=2)+'\n') +print(json.dumps(res,indent=2)) diff --git a/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_positive_tree.py b/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_positive_tree.py new file mode 100644 index 0000000000..ebce2a90cf --- /dev/null +++ b/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_positive_tree.py @@ -0,0 +1,106 @@ +#!/usr/bin/env python3 +"""Exact controls for the positive interpolation tree and two global tails.""" +from fractions import Fraction as F +from math import prod +from pathlib import Path +import json + +def mul(a,b): + c=[F(0)]*(len(a)+len(b)-1) + for i,x in enumerate(a): + for j,y in enumerate(b): c[i+j]+=x*y + return c + +def add(a,b): + c=[F(0)]*max(len(a),len(b)) + for i,x in enumerate(a): c[i]+=x + for i,x in enumerate(b): c[i]+=x + return c + +def factor(xs): + p=[F(1)] + for x in xs: p=mul(p,[F(1),x]) + return p + +def test(s,H,r): + t=s+H + qi=[(2*t*i+s)//(2*s) for i in range(s)] + xs=[F(r)**q for q in qi] + es=factor(xs) + ws=[] + for i in range(s): + den=prod(xs[i]-xs[j] for j in range(s) if j!=i) + ws.append(F((i*7+3)%11-5,den)) + whole=[F(0)]*s + abswhole=[F(0)]*s + for i in range(s): + e=factor(xs[:i]+xs[i+1:]) + whole=add(whole,[ws[i]*v for v in e]) + abswhole=add(abswhole,[abs(ws[i])*v for v in e]) + # Arbitrary positive target coefficient weights suffice for the exact + # completion inequality; these reproduce a Gaussian-shaped gauge. + ds=[F(1,r**(j*j)) for j in range(s)] + B=max(abswhole[s-1-j]/ds[j] for j in range(s)) + checked=0 + def tree(l,u): + nonlocal checked + if u-l==1: + fi=[F(1),xs[l]]; ri=[ws[l]]; ai=[abs(ws[l])] + else: + m=(l+u)//2 + fl,rl,al=tree(l,m); fr,rr,ar=tree(m,u) + fi=mul(fl,fr) + ri=add(mul(fr,rl),mul(fl,rr)) + ai=add(mul(fr,al),mul(fl,ar)) + Mh=prod(xs[u:]) + for k in range(u-l): + target=s-u+k + assert abs(ri[k])<=ai[k] + assert Mh*ai[k]<=abswhole[target] + assert ai[k]<=B*ds[u-1-k]/Mh + checked+=1 + return fi,ri,ai + ff,rr,aa=tree(0,s) + assert ff==es and rr==whole and aa==abswhole + left=factor(xs[:s//2]) + broken=[F(0)]*s + for i in range(s//2): + wrong=factor(xs[:i]+xs[i+1:s//2]) + broken=add(broken,[ws[i]*v for v in wrong]) + for i in range(s//2,s): + right=factor(xs[s//2:i]+xs[i+1:]) + broken=add(broken,[ws[i]*v for v in mul(left,right)]) + assert broken != whole + tails=0 + for j in range(s): + k=s-1-j + for i in range(s): + e=factor(xs[:i]+xs[i+1:])[k] + if i=e + assert (app-e)*r**(m+1)<=4*app + tails+=1 + if i>j: + m=i-j + app=es[k+1]/xs[i] + assert app>=e + assert (app-e)*r**(m+1)<=4*app + tails+=1 + return checked,tails + +n=nodes=tails=0 +for s in range(3,15): + for H in range(1,min(s,5)): + for r in (4,8): + c,d=test(s,H,r) + n+=1; nodes+=c; tails+=d +out={"status":"pass", "arithmetic":"exact rational", "instances":n, + "tree_coefficient_completion_inequalities":nodes, + "far_field_relative_error_inequalities":tails, + "negative_controls":["Omitting the upper-child factor changes the root polynomial.", + "The omitted upper-child factor is detected in every instance."], + "claims_excluded":["Fast normalized convolution implementation", "Tensor fusion", "New final exponent"]} +Path(__file__).with_name('POSITIVE-TREE-CONTROLS.json').write_text(json.dumps(out,indent=2)+'\n') +print(json.dumps(out,indent=2)) diff --git a/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_q_ldu.py b/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_q_ldu.py new file mode 100644 index 0000000000..63f7d6eab0 --- /dev/null +++ b/research/capped-phase-fusion/evidence/radical_reduction_ultra/check_q_ldu.py @@ -0,0 +1,64 @@ +#!/usr/bin/env python3 +"""Exact rational controls for Gaussian Toeplitz q-LDU and shifted exponents.""" +from fractions import Fraction as F +from pathlib import Path +import json + +def mm(a,b):return [[sum(x*y for x,y in zip(r,c)) for c in zip(*b)] for r in a] +def tr(a):return list(map(list,zip(*a))) +def ident(n):return [[F(i==j) for j in range(n)] for i in range(n)] +def powq(q,n):return q**n + +def factors(q,n): + Q=q*q;P=[F(1)] + for j in range(1,n):P.append(P[-1]*(1-Q**j)) + bc=lambda i,j:P[i]/(P[j]*P[i-j]) + L=[[q**((i-j)**2)*bc(i,j) if i>=j else F(0) for j in range(n)] for i in range(n)] + Li=[[(-q)**(i-j)*bc(i,j) if i>=j else F(0) for j in range(n)] for i in range(n)] + D=[[P[i] if i==j else F(0) for j in range(n)] for i in range(n)] + return L,Li,D + +cases=0;entries=0 +for q in (F(1,2),F(1,3),F(2,3)): + for n in range(1,11): + L,Li,D=factors(q,n) + G=[[q**((i-j)**2) for j in range(n)] for i in range(n)] + assert mm(mm(L,D),tr(L))==G + assert mm(L,Li)==ident(n) and mm(Li,L)==ident(n) + # Integer-power diagonal conjugation is an exact finite surrogate. + diagonal=[q**(i*i-3*i) for i in range(n)] + Lt=[[L[i][j]*diagonal[i]/diagonal[j] for j in range(n)] for i in range(n)] + Lit=[[Li[i][j]*diagonal[i]/diagonal[j] for j in range(n)] for i in range(n)] + assert mm(Lt,Lit)==ident(n) + cases+=1;entries+=n*n + +exponent_cases=0 +for rho in (F(1001,1000),F(17,16),F(3,2)): + theta=rho-1 + for c in (-2,0,3): + for i in range(15): + for j in range(i+1): + h=i-j;beta=theta*i-c + # D_i/D_j times q^h for the inverse L factor. + raw=theta*(i*i-j*j)-2*c*(i-j)-rho*h + condensed=(2*beta-rho)*h-theta*h*h + assert raw==condensed + exponent_cases+=1 + +# Rational validation of the perturbation bound in the worked 24x24 witness. +rbound=F(24,2**23);linv_bound=F(24) +error=linv_bound**2*rbound/(1-linv_bound*rbound) +assert linv_bound*rbound<1 and errora. These are the actual strict rate relations used by the +support-local compiler and its stopped recursion, not an assumption that +all bit operations have the complex supplier's exact endpoint. + +## Functional interface + +The reviewed outer proof consumes a globally complete dyadic tensor call +which applies normalized H0 simultaneously on the selected active bits, +acts as identity on arbitrary dirty helper/coarse/spectator coordinates at +completed-call boundaries, accepts O(p)-bit coefficient encodings, and costs +O(V g^(1-q) polylog p). The supply is used as a completed operator, with its +own scalar phases and working storage included. Outer Gaussian blocks do +not directly inspect the local donor edits, per-rank multiplicities, owner +replica count, or a particular parent-word ordering. + +The T74 source retains the same local normalized target, common-label and +mutable-source contract, whole-call hosting, exact wrappers/phases, and +restoration interface. Its new parent uses m=73 as before, with maximum +residual rank48 and fixed finite W=66304. Replacing the old finite stock W +by this fixed W changes constants and the O(log g) product-row reservation, +not any exponent in the support-local argument. The local word has480 +helpers, also unchanged in its interface. All donor reads are intended to +preserve the exact full local endpoint; our separate rational rebuild check +confirmed that algebra. The complete unmodified native/frame/phase replay has +now passed and certifies the recorded normalized implementation of those endpoints. +The source pin and hashes of the native, strict, control, and physical-invoice +receipts are bound in `NATIVE-ADMISSION-BINDING.json`. + +The larger fixed prefix allowance (2048 bits in the T74 global invoice) +also changes a constant in the linear semantic guard. The capped proof allows +a fixed supplier-dependent constant in O(g), and later O(g*a*B), with an O(p) +record constant chosen accordingly. It does not require the former numeric +prefix constant to remain512 or1024. The strict power guards B+X+r<1 and +E+r<1 remain in force and were checked independently. + +The neighborhood exposure still involves O(g log p) actual selected bits; +the T74 finite word neither requests ambient-p address permutations nor +changes the complete-field requirement. The retained bit supplier supplies +the same arbitrary-width equal-chunk interchange interface with more rate +than the chosen a. Source and sink scalings converting H0 to the raw H used +by the tensor compiler remain O(g) guard terms. There is no new finite-field +bridge or a new assumption that per-microbox coarse labels are complete. + +Thus the substitution does not require a new analytic lemma, removal of a +charged row, or a new kind of workspace. The updated finite supplier is now +verified by `work/target002_upstream/full-1661-verification/RESULT.json`, whose +status is `PASS_T74_EXACT_FILL_COMPLETE_FINITE_ADMISSION`. The binding includes +the complete source manifest and actual fresh native, strict-assembly, physical +invoice, and control receipt hashes. The two rates above are completed conditional +substitutions: their remaining conditions are exactly the inherited all-size, +hosting, precision/tape, exact-recovery, and analytic hypotheses described above; +the matrix branch additionally requires omega_C<=9/4. diff --git a/research/capped-phase-fusion/frontier_fusion/NATIVE-ADMISSION-BINDING.json b/research/capped-phase-fusion/frontier_fusion/NATIVE-ADMISSION-BINDING.json new file mode 100644 index 0000000000..72d30de3fd --- /dev/null +++ b/research/capped-phase-fusion/frontier_fusion/NATIVE-ADMISSION-BINDING.json @@ -0,0 +1,155 @@ +{ + "status": "PASS_FRESH_NATIVE_ADMISSION_BOUND_TO_CONDITIONAL_SUBSTITUTION", + "scope": "Unmodified PR1661 complete fresh finite/native/global verifier passed; the independently reviewed phase-fusion recomposition retains all inherited all-size, hosting, precision/tape, exact-recovery and analytic hypotheses. The matrix corollary additionally assumes omega_C<=9/4.", + "source_pr": 1661, + "source_head": "f852af5e178ce035cd46181a29d9330bcbcaa206", + "source_manifest_sha256": "d90d49eaa68e48c6992a07972e736ef4b9cd80274361cbbc1ec980d43f29ad7f", + "complex_lower_pinned": "8398566873921928847865107582977140808726533185752606814031551243089531888024439117235381227291914058448586599984374667388474246199941131401151316949826049729135138281487406840315861844950703424400143642674520874037535671888706933756382975087837881090275422517866488734818622618194142006954783211221019204801330544148296601796623021915319705773977790291402669246008667290619418570040011860763793419/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", + "complex_lower_used": "0.00167971337478", + "native_profile": { + "1": 1559766, + "2": 337668, + "3": 278058, + "4": 84912, + "5": 54612, + "8": 5328, + "48": 22748 + }, + "full_status": "PASS_T74_EXACT_FILL_COMPLETE_FINITE_ADMISSION", + "control_counts": { + "admission": 114, + "bit": 12, + "controller": 15, + "independent_owner": 8, + "geometry": 6 + }, + "new_native_provider_checks": [ + { + "provider": "base", + "word_sha256": "0740b6f33927b9e60c9a7f35e7fcb2fd9ac6e2d3982112428b302a6966a18c73", + "native_steps": [ + { + "step": "native", + "exit_code": 0, + "seconds": 0.8915401669219136 + }, + { + "step": "ledgers", + "exit_code": 0, + "seconds": 0.6126613330561668 + }, + { + "step": "local-replay", + "exit_code": 0, + "seconds": 4.12045537494123 + }, + { + "step": "wrappers", + "exit_code": 0, + "seconds": 1.6204181250650436 + }, + { + "step": "phases", + "exit_code": 0, + "seconds": 105.53502279194072 + } + ] + }, + { + "provider": "extra", + "word_sha256": "17cc6c177c89eb264514e1efb3685ab9a621f3952a8f393a6a5eb28cf77e2884", + "native_steps": [ + { + "step": "native", + "exit_code": 0, + "seconds": 0.8948857497889549 + }, + { + "step": "ledgers", + "exit_code": 0, 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"d90d49eaa68e48c6992a07972e736ef4b9cd80274361cbbc1ec980d43f29ad7f", + "target002_upstream/pr1661/research/gmx-t74-p566-ep400/new/PROFILE.json": "f66dd80aff69525144eac18144c13d7d8be3d3c39028b3afa1b59a8edf53e8a3", + "target002_upstream/pr1661/research/gmx-t74-p566-ep400/new/SPEC.json": "d6b6a2140f3a61ae8cfdd8b51a1131934d63d2dd8573c1b13eb2b9528f3c3db8" + } +} diff --git a/research/capped-phase-fusion/frontier_fusion/REPRODUCE.txt b/research/capped-phase-fusion/frontier_fusion/REPRODUCE.txt new file mode 100644 index 0000000000..42c6bb3033 --- /dev/null +++ b/research/capped-phase-fusion/frontier_fusion/REPRODUCE.txt @@ -0,0 +1,22 @@ +ARITHMETIC REPLAY (Python 3.11 or newer; standard library only) +From this phase_fusion_review directory: + python3 frontier_fusion/check_frontier_fusion.py + python3 frontier_fusion/independent_substitution_check.py + +These replay the rational native moment and both outer ledgers, including +negative controls. They do not rerun the entire native supplier. + +COMPLETE NATIVE REPLAY +Obtain CrocSwap/integer-mult-bounds at exact commit +f852af5e178ce035cd46181a29d9330bcbcaa206. Follow the requirements in +research/gmx-t74-p566-ep400/ and run its unmodified verify.py with a fresh +--work directory and --audit-shards 4. The retained RESULT.json and +NATIVE-ADMISSION-BINDING.json identify the completed fresh run and bind +its native, owner, phase, invoice, arithmetic and control receipts. +The full 121 MB long-precision assembly and its vendor tree are not duplicated +in this review bundle. Their hashes are retained; obtain/replay the pinned +source to reproduce those larger files. + +The two final exponents retain the inherited all-size hypotheses. The larger +value additionally assumes the supplied omega_C <= 9/4 theorem. Numerical and +finite native verification alone do not prove those external hypotheses. diff --git a/research/capped-phase-fusion/frontier_fusion/T74-APPENDIX.tex b/research/capped-phase-fusion/frontier_fusion/T74-APPENDIX.tex new file mode 100644 index 0000000000..78278aea07 --- /dev/null +++ b/research/capped-phase-fusion/frontier_fusion/T74-APPENDIX.tex @@ -0,0 +1,53 @@ +\section{Updated T74 supplier and final exponents}\label{sec:t74} + +The preceding proofs are parameterized by the effective supplier rate. We now substitute the independently replayed T74 complex supplier from \href{https://github.com/CrocSwap/integer-mult-bounds/pull/1661}{PR1661}, pinned at commit \texttt{f852af5e178ce035cd46181a29d9330bcbcaa206}. This is a new input to the same outer proof, with contributor attribution retained in the source package. The original numerical statements above remain valid special cases. + +Use the exact finite decimals +\[ + b_T=0.00167971337478,\qquad q_T=b_T-10^{-10}, + \qquad a_T=(q_T+b_T)/2. +\] +The independently retained bit rate $a_*=0.001733427039589844234749$ satisfies $q_T0 for x in assembly['outer'].values()) +checks=[] +for key,word in [('base','gauge44-word-566.json.gz'),('extra','extra-word-566.json.gz')]: + d=RUN/('native-new-'+key) + st=read(d/'STEPS.json') + assert [z['step'] for z in st]==['native','ledgers','local-replay','wrappers','phases'] + assert all(z['exit_code']==0 for z in st) + assert gzip.decompress((d/'local/FULL-WORD.json.gz').read_bytes())==gzip.decompress((SRC/'new'/word).read_bytes()) + checks.append({'provider':key,'word_sha256':sha(SRC/'new'/word),'native_steps':st}) +paths={RUN/'RESULT.json',RUN/'STAGES.json',RUN/'parent/RESULT.json',RUN/'strict/RESULT.json',RUN/'strict/STAGES.json',RUN/'strict/ASSEMBLY.json',SRC/'MANIFEST.json',SRC/'CLAIM.json',SRC/'new/SPEC.json',SRC/'new/PROFILE.json'} +for base in [RUN/'native-new-base',RUN/'native-new-extra',RUN/'strict']: + for f in base.rglob('*.json'): + if any(tag in f.name.upper() for tag in ['CONTROL','CHECK','REVIEW','AUDIT','REPLAY','INVOICE','STAGES']):paths.add(f) +for z in steps:paths.add(RUN/(z['name']+'.log')) +report={'status':'PASS_FRESH_NATIVE_ADMISSION_BOUND_TO_CONDITIONAL_SUBSTITUTION', + 'scope':'Unmodified PR1661 complete fresh finite/native/global verifier passed; the independently reviewed phase-fusion recomposition retains all inherited all-size, hosting, precision/tape, exact-recovery and analytic hypotheses. The matrix corollary additionally assumes omega_C<=9/4.', + 'source_pr':1661,'source_head':'f852af5e178ce035cd46181a29d9330bcbcaa206','source_manifest_sha256':r['manifest_sha256'], + 'complex_lower_pinned':assembly['complex_lower'],'complex_lower_used':'0.00167971337478','native_profile':assembly['complex_profile'], + 'full_status':r['status'],'control_counts':r['controls'],'new_native_provider_checks':checks, + 'claimed_upstream_kappa':r['kappa'],'recomposed_final_kappa':{'strassen':'0.001677160151714381','matrix_9_4':'0.001677600129305161'}, + 'receipt_sha256':{str(f.relative_to(W)):sha(f) for f in sorted(paths)}} +(P/'NATIVE-ADMISSION-BINDING.json').write_text(json.dumps(report,indent=2)+'\n') +print(report['status'],len(paths),'bound receipt files') diff --git a/research/capped-phase-fusion/frontier_fusion/check_frontier_fusion.py b/research/capped-phase-fusion/frontier_fusion/check_frontier_fusion.py new file mode 100644 index 0000000000..a8112e3cef --- /dev/null +++ b/research/capped-phase-fusion/frontier_fusion/check_frontier_fusion.py @@ -0,0 +1,177 @@ +#!/usr/bin/env python3 +"""Recombine the pinned T74 supplier with the reviewed phase-fusion proof. + +This checks arithmetic and provenance, not the independent native replay. +The latter's completion is recorded separately; no all-size premise is waived. +""" +from fractions import Fraction as Q +from decimal import Decimal, localcontext +from pathlib import Path +from math import factorial +import hashlib, json + +HERE=Path(__file__).resolve().parent +WORK=HERE.parent +SPEC=WORK/'target002_upstream/pr1661/research/gmx-t74-p566-ep400/new/SPEC.json' +CLAIM=WORK/'target002_upstream/pr1661/research/gmx-t74-p566-ep400/CLAIM.json' +spec=json.loads(SPEC.read_text()) +certified_complex=Q(spec['expected']['complex_lower']) +bit=Q('0.001733427039589844234749') +# Keep short independently checkable decimals strictly inside both endpoints. +b=Q('0.00167971337478') +q=b-Q(1,10**10) +a=(q+b)/2 +online=Q(spec['expected']['kappa']) +old=Q('0.001673360105405778') +assert b=2:x/=2;k+=1 + lo,hi=unit_log(x) + return down(lo+k*ln2[0]),up(hi+k*ln2[1]) +def exp(x): + assert 0<=x<1 + N=12 + lo=sum((x**j/factorial(j) for j in range(N)),Q(0)) + hi=lo+x**N/factorial(N)/(1-x/(N+1)) + return down(lo),up(hi) +def moment(endpoint): + lo=hi=Q(0) + for r,n in profile.items(): + l,h=log(Q(m,r)); weight=Q(n*r,m*W) + lo+=weight*exp(endpoint*l)[0] + hi+=weight*exp(endpoint*h)[1] + return down(lo),up(hi) + +native=moment(b) +assert native[1]<1 +assert moment(Q('0.001679713374785'))[0]>1 + +def ledger(kind, *, q_=q, zeta=Q(1,10**8), slack=Q(1,10**5), + old_scan=False, full_gauge=False, full_key=False): + r=Q(1,10**6); stop=Q(1,10**9) + if kind=='strassen': + c=Q(3,32) + eps=(1+q_+q_*(c-slack-r-zeta))/(1+q_)**2 + xi=eps*(1+q_)-c+slack + else: + c=Q(5,16) + xi=(Q(3,2)+q_-(1+q_)*(r+zeta)+slack)/(2+q_) + eps=(1+q_*(1-xi-r-zeta))/(1+q_) + beta=1-xi-r-zeta + saving=eps*q_ + # Deliberate strict 1e-11 gap after decimal truncation. + kappa=Q((saving*10**18).__floor__(),10**18)-Q(1,10**11) + # kappa is derived, not silently asserted to equal an expected decimal. + tau=1-a; sigma=1-b + internal=tau+(1-stop)*max(sigma-tau,Q(0)) + leaf=sigma+stop*(1-sigma) + lam=1-(q_+a)/2 + key=beta+1-eps + margins={ + 'main_transform':saving, + 'global_coordinate_router':a, + 'fused_gaussian':1-eps+q_*beta, + 'group_bit_router':1-eps+a*beta, + 'group_scans':1-eps+beta, + 'boundary_algebraic_matrix':c+xi-eps, + 'sparse_union_movement':1-eps-beta-Q(1,2)+xi, + 'polynomial_products':eps, + 'microbox_face_corrections':1-(eps+beta+3-8+2), + } + if kind=='strassen':margins['sparse_union_full_key']=margins['sparse_union_movement']-key + if old_scan:margins['old_per_axis_scan']=1-eps + if full_key and kind!='strassen':margins['unnecessary_full_key_sort']=margins['sparse_union_movement']-key + checks={ + 'actual_bit_above_effective':bit-a, + 'complex_above_effective':b-a, + 'q_positive':q_, 'complex_endpoint':b-q_, + 'q_below_effective':a-q_, 'stop_positive':stop, + 'stop_below_one':1-stop, + 'leaf_saving_above_effective':(1-stop)*b-a, + 'q_below_internal':1-internal-q_, 'q_below_leaf':1-leaf-q_, + 'lambda_above_tau':lam-tau, 'lambda_above_sigma':lam-sigma, + 'lambda_above_internal':lam-internal, + 'lambda_prime_above_lambda':1-q_-lam, + 'lambda_prime_above_leaf':1-q_-leaf, + 'epsilon_positive':eps,'epsilon_below_one':1-eps, + 'cap_above_band':xi-Q(1,2),'cap_below_cell':eps-xi, + 'group_positive':beta,'group_below_dimension':eps-beta, + 'support_router_span':1-beta, + 'group_gauge_power_slack':1-xi-r-beta, + 'actual_logarithmic_group_guard':1-xi-beta, + 'global_gamma_power_slack':1-eps-r, + 'actual_logarithmic_gamma_guard':1-eps, + 'alpha_logarithm_absorption':r, + 'group_key_fits_record':1-key, + 'membership_arithmetic_fits_record':1-beta-2*(1-eps), + 'microbox_halo_volume':8-3-beta, + 'positive_layout_slack':slack, + 'short_record_fallback':eps-a, + 'improvement_over_online_final':kappa-online, + 'improvement_over_previous_local_final':kappa-old, + 'complex_below_one_over_32':Q(1,32)-b, + } + if kind!='strassen': + checks['sparse_union_fraction_decays']=xi-beta-Q(1,2) + checks['matrix_exponent_above_supplied_bound']=Q(19,8)-Q(9,4) + checks['matrix_exponent_below_five_halves']=Q(5,2)-Q(19,8) + if full_gauge:checks['unrecentered_group_gauge']=1-eps-r-beta + checks.update({n+'_above_kappa':v-kappa for n,v in margins.items()}) + assert margins['main_transform']==margins['fused_gaussian'] + if kind=='strassen':assert margins['boundary_algebraic_matrix']==saving+slack + else:assert margins['sparse_union_movement']==saving+slack + return {'kind':kind,'kappa':kappa,'epsilon':eps,'xi':xi,'beta':beta, + 'q':q_,'a':a,'r0':r,'zeta':zeta,'slack':slack, + 'margins':margins,'strict_checks':checks} + +def encode(x): + if isinstance(x,Q): + with localcontext() as c: + c.prec=60 + return {'rational':str(x),'decimal':str(Decimal(x.numerator)/Decimal(x.denominator))} + if isinstance(x,dict):return {str(k):encode(v) for k,v in x.items()} + if isinstance(x,(list,tuple)):return [encode(v) for v in x] + return x + +rows={};controls={} +for kind in ['strassen','matrix_9_4']: + row=ledger(kind) + assert all(v>0 for v in row['strict_checks'].values()) + assert min(row['margins'].values())==row['margins']['main_transform'] + rows[kind]=row + variants=[('endpoint_equality',{'q_':b}),('zero_group_guard',{'zeta':Q(0)}), + ('zero_layout_slack',{'slack':Q(0)}),('old_per_axis_scan',{'old_scan':True}), + ('full_cell_gauge',{'full_gauge':True})] + if kind!='strassen':variants.append(('full_key_sort',{'full_key':True})) + controls[kind]={} + for name,kwargs in variants: + bad=ledger(kind,**kwargs) + failed={n:v for n,v in bad['strict_checks'].items() if v<=0} + assert failed,(kind,name) + controls[kind][name]=failed +out={'status':'PASS_EXACT_T74_PHASE_FUSION_ARITHMETIC', + 'scope':'Conditional recomposition of pinned supplier and already-reviewed outer proofs. Full fresh native replay is a separately tracked requirement; inherited all-size assumptions remain.', + 'complex_lower_short':b,'complex_lower_pinned':certified_complex, + 'native_moment':native,'profile':profile,'m':m,'W':W, + 'online_final_kappa':online,'rows':rows,'rejecting_controls':controls, + 'source_sha256':{str(p.relative_to(WORK)):hashlib.sha256(p.read_bytes()).hexdigest() for p in [SPEC,CLAIM]}} +(HERE/'FRONTIER-FUSION-ARITHMETIC.json').write_text(json.dumps(encode(out),indent=2)+'\n') +print(out['status']) +for name,row in rows.items(): + print(name,'FINAL KAPPA',encode(row['kappa'])['decimal'], + 'positive checks',len(row['strict_checks']), + 'gain over online',encode(row['kappa']-online)['decimal']) diff --git a/research/capped-phase-fusion/frontier_fusion/independent_substitution_check.py b/research/capped-phase-fusion/frontier_fusion/independent_substitution_check.py new file mode 100644 index 0000000000..04ec764940 --- /dev/null +++ b/research/capped-phase-fusion/frontier_fusion/independent_substitution_check.py @@ -0,0 +1,45 @@ +"""Independent balance-equation check, not an execution of the proposed ledger.""" +from fractions import Fraction as F +from pathlib import Path +from decimal import Decimal,localcontext +import json,hashlib +D=Path(__file__).resolve().parent;W=D.parent +src=W/'target002_upstream/pr1661/research/gmx-t74-p566-ep400/new/SPEC.json' +spec=json.loads(src.read_text());pinned=F(spec['expected']['complex_lower']) +b=F('0.00167971337478');q=b-F(1,10**10);eff=(q+b)/2;actual_bit=F('0.001733427039589844234749') +r=F(1,10**6);z=F(1,10**8);s=F(1,10**5) +assert 0eff +assert 7**16<2**45 +rows={} +for name,c,expected in [('strassen',F(3,32),F('0.001677160151714381')),('matrix_9_4',F(5,16),F('0.001677600129305161'))]: + if name=='strassen': + # Solve E(1+q)+q X=1+q(1-r-z), X-E(1+q)=s-c. + E=(1+q*(1-r-z+c-s))/(1+q)**2;X=E*(1+q)+s-c + else: + # Solve E(1+q)+q X=1+q(1-r-z), 2X-E(1+q)=1/2+s-r-z. + X=(F(3,2)+q+s-(1+q)*(r+z))/(2+q);E=(1+q*(1-r-z-X))/(1+q) + B=1-r-z-X + assert E*q==1-E+q*B + assert (c+X-E==E*q+s) if name=='strassen' else (1-E-B-F(1,2)+X==E*q+s) + assert F(1,2)0 and 1-E-r>0 + assert 1-B>0 and 1-B-2*(1-E)>0 + margins={'transform':E*q,'global_router':eff,'gaussian':1-E+q*B,'group_router':1-E+eff*B,'group_scans':1-E+B,'boundary':c+X-E,'sparse_union':1-E-B-F(1,2)+X,'polynomials':E,'microfaces':4-E-B} + if name=='strassen':margins['union_key_sort']=margins['sparse_union']-(B+1-E) + else:assert X-B-F(1,2)>0 + K=F((E*q*10**18).__floor__(),10**18)-F(1,10**11) + assert K==expected and min(margins.values())==E*q and all(v>K for v in margins.values()) + with localcontext()as cc:cc.prec=55 + rows[name]={'epsilon':str(E),'cap':str(X),'beta':str(B),'kappa':str(K),'margin_above_kappa':str(E*q-K),'checked_margin_count':len(margins)} +native_path=W/'target002_upstream/full-1661-verification/RESULT.json' +native_ok=False +if native_path.exists(): + native=json.loads(native_path.read_text()) + assert native['status']=='PASS_T74_EXACT_FILL_COMPLETE_FINITE_ADMISSION' + assert F(native['kappa'])==F(spec['expected']['kappa']) + native_ok=True +report={'status':('PASS_INDEPENDENT_CONDITIONAL_SUBSTITUTION_REVIEW' if native_ok else 'PASS_INDEPENDENT_CONDITIONAL_SUBSTITUTION_REVIEW_PENDING_NATIVE'),'source_sha256':hashlib.sha256(src.read_bytes()).hexdigest(),'scope':('Independent balance derivation, margin comparisons, and functional-interface review; completed fresh finite/native admission is bound separately, with inherited all-size assumptions retained.' if native_ok else 'Independent balance derivation, margin comparisons, and functional-interface review; full fresh native admission remains pending.'),'rows':rows,'native_required_output':'target002_upstream/full-1661-verification/RESULT.json'} +(D/'INDEPENDENT-SUBSTITUTION-RECEIPT.json').write_text(json.dumps(report,indent=2)+'\n') +print(json.dumps(report,indent=2)) diff --git a/research/capped-phase-fusion/gaussian_fixed_width_note.tex b/research/capped-phase-fusion/gaussian_fixed_width_note.tex new file mode 100644 index 0000000000..e7a2c00d4e --- /dev/null +++ b/research/capped-phase-fusion/gaussian_fixed_width_note.tex @@ -0,0 +1,212 @@ +\documentclass[11pt]{article} +\usepackage[T1]{fontenc} +\usepackage{lmodern,microtype,amsmath,amssymb,amsthm} +\usepackage[margin=1in]{geometry} +\usepackage[colorlinks=true,allcolors=blue]{hyperref} +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\newcommand{\norm}[1]{\left\lVert#1\right\rVert_\infty} +\newcommand{\Z}{\mathbb Z} +\title{Fixed-Width Gaussian Resampling Is Polynomially Conditioned} +\author{Research note prepared with OpenAI Codex} +\date{October 13, 2026} +\begin{document} +\maketitle +\begin{abstract} +An exact bilateral Newton factorization proves that the Gaussian +resampling inverse has norm $O(\theta^{-2})$ at a fixed Gaussian width, +where $\theta>0$ is the oversampling gap. This removes the width-growth +requirement from the conditioning argument. The proof uses convergent +infinite triangular factors and then periodizes them; it does not assert +operator-norm convergence of finite sections. A fast implementation and +its tensor cost remain separate obligations. No improved final +integer-multiplication exponent is claimed in this note. +\end{abstract} + +Let $1\le s5/6$. +At $a=1$, concavity gives +$1-e^{-\rho\theta}\ge1-e^{-\theta}\ge(1-e^{-1})\theta>3\theta/5$, +and $e^{1/2}<2$; substitution gives the stated constant $16$. +\end{proof} + +\begin{corollary}[The width may be fixed] +For the resampling convention $a=\pi\alpha^2$, one may take the fixed +integer $\alpha=2$ in the inverse-conditioning and normalization +argument. If all $d$ gaps satisfy $\theta_i\ge d^{-C}$ for a fixed $C$, +the total logarithmic normalization loss is $O(d\log d)$. +\end{corollary} +\begin{proof} +The theorem bounds each inverse by a fixed constant times $d^{2C}$. +The separate diagonal has norm at most $e^{a/4}$, a fixed constant. +Taking powers of two large enough to normalize these factors costs +$O(\log d)$ bits per axis; logarithms add under tensor products. +\end{proof} + +\paragraph{Scope and provenance.} +This strengthens the conditioning-only result in the earlier note +\emph{A Polynomial Condition Bound for Gaussian Resampling}. The +matrix is the one in Section~7 of the +\href{https://github.com/CrocSwap/integer-mult-bounds/tree/e6f9716621f4e01fc190617934cdfaf072bd976f}{inspected repository snapshot}. +Classical Newton interpolation and symmetric-polynomial identities are +used explicitly above; no generic-rank assumption is used. Two independent +agents checked the limiting and periodization arguments. Exact rational +controls cover negative and zero origins, finite $LDU$, inverse entries, +origin shifts and simultaneous periodicity. + +This theorem does not supply the new numerical implementation at these +parameters. Stable factor application, in-place tensor composition, +coefficient generation and all tape movement must still be priced. +The repository's inspected conditional exponent remains +$\kappa=0.001672660197076918$; no larger final exponent is certified here. +\end{document} diff --git a/research/capped-phase-fusion/make_native_capsule.py b/research/capped-phase-fusion/make_native_capsule.py new file mode 100644 index 0000000000..45a5aeb417 --- /dev/null +++ b/research/capped-phase-fusion/make_native_capsule.py @@ -0,0 +1,32 @@ +"""One-time capsule creation from the independently completed source-bound run.""" +from pathlib import Path +import argparse,json,hashlib,sys +sys.set_int_max_str_digits(500000) +P=Path(__file__).resolve().parent +ap=argparse.ArgumentParser(description=__doc__) +ap.add_argument('--native-source',type=Path,required=True) +ap.add_argument('--native-work',type=Path,required=True) +ap.add_argument('--output',type=Path,required=True) +a=ap.parse_args() +source=a.native_source.resolve();run=a.native_work.resolve() +b=json.loads((P/'frontier_fusion/NATIVE-ADMISSION-BINDING.json').read_text()) +def normalized(x): + if isinstance(x,dict):return {k:normalized(v)for k,v in x.items()if k not in {'seconds','log_sha256'}} + if isinstance(x,list): + y=[normalized(v)for v in x] + return sorted(y,key=lambda v:v['name']) if y and all(isinstance(v,dict)and'name'in v and'exit_code'in v for v in y) else y + if isinstance(x,str):return x.replace(str(run),'{native-work}').replace(str(source),'{native-source}') + return x +out={'schema':'phase-fusion-native-receipt-capsule/1','scope':'Canonical semantic hashes of the completed native JSON receipts. Elapsed seconds, stage-log digests, and the two root directory prefixes are normalized; concurrent named stage records are sorted. Actual current stage-log hashes are checked separately. Log byte hashes identify the historical run but are not compared across fresh runs.','source_commit':b['source_head'],'source_manifest_sha256':b['source_manifest_sha256'],'receipts':[]} +for rel,raw in b['receipt_sha256'].items(): + if rel.startswith('target002_upstream/full-1661-verification/'): + kind='native_work';r=rel.removeprefix('target002_upstream/full-1661-verification/');f=run/r + else: + prefix='target002_upstream/pr1661/research/gmx-t74-p566-ep400/' + assert rel.startswith(prefix);kind='native_source';r=rel.removeprefix(prefix);f=source/r + blob=f.read_bytes();assert hashlib.sha256(blob).hexdigest()==raw + row={'root':kind,'path':r,'historical_sha256':raw} + if f.suffix=='.json':row['semantic_sha256']=hashlib.sha256(json.dumps(normalized(json.loads(blob)),sort_keys=True,separators=(',',':')).encode()).hexdigest() + out['receipts'].append(row) +a.output.write_text(json.dumps(out,indent=2)+'\n') +print(len(out['receipts']),'receipts',sum('semantic_sha256'in r for r in out['receipts']),'canonical JSON hashes') diff --git a/research/capped-phase-fusion/phase_fusion_note.tex b/research/capped-phase-fusion/phase_fusion_note.tex new file mode 100644 index 0000000000..4351a73450 --- /dev/null +++ b/research/capped-phase-fusion/phase_fusion_note.tex @@ -0,0 +1,460 @@ +\documentclass[11pt]{article} +\usepackage[margin=1in]{geometry} +\usepackage{amsmath,amssymb,amsthm,booktabs,longtable,array} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[colorlinks=true,linkcolor=blue,urlcolor=blue]{hyperref} +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{corollary}[theorem]{Corollary} +\newcommand{\norm}[1]{\left\lVert #1\right\rVert} +\newcommand{\diag}{\operatorname{diag}} +\newcommand{\poly}{\operatorname{poly}} +\newcommand{\plog}{\operatorname{polylog}} +\newcommand{\file}[1]{\href{#1}{\texttt{\detokenize{#1}}}} +\title{Capped phase-cell fusion in the conditional multiplication reduction} +\author{Research note prepared with OpenAI Codex assistance} +\date{13 October 2026} +\begin{document} +\maketitle + +\noindent\textbf{Review status: PASS\_CONDITIONAL\_PHASE\_FUSION\_REVIEW.} The capped local-gauge and layout changes have independent audits, followed by an integrated review of this note and its 44 exact strict slacks. The result below is conditional on the same finite suppliers and all-size computational interfaces as the retained selected result; it is not an unconditional integer-multiplication theorem and does not change the upstream selected certificate. + +\noindent\textbf{Updated final bounds.} The completely replayed T74 substitution in Section~\ref{sec:t74} gives $\kappa_T=0.001677160151714381$, or $\kappa_{T,M}=0.001677600129305161$ with the additional supplied matrix-exponent premise. Neither reaches the continuing target $0.002$. + +\section{Statement, inputs, and parameters} + +The retained selected saving is +\[ + \kappa_{\rm old}=0.001672660197076918. +\] +The retained independently replayed supplier lower bounds are +\[ + a_*=0.001733427039589844234749,\qquad + b_*=0.001675462676808103522925. +\] +Here $a_*$ is the bit saving and $b_*$ is the complex saving. They are supplied inputs, not recomputed finite constructions in this note. Put +\[ + q=b_*-10^{-10},\qquad a_{\rm eff}=(b_*+q)/2, + \qquad r_0=10^{-6},\quad \zeta=10^{-8},\quad s_0=10^{-5}, + \qquad c_0=3/32. +\] +The effective bit saving is deliberately weakened so that +$q2|t|+1$. Both nonzero parameters have controlled inverses, so zero shears cause no singular diagonal. Two- and three-$H$ identity words pad all permitted gate types to the same six-$H$ shape. Thus one original matching layer uses at most $(8m+2)$ simultaneous $H$ stages. + +For $g$ different axis circuits, Kronecker multiplication tensors the corresponding $H$ stages and multiplies the diagonal factors. There is no expansion into $2^g$ branches and no data padding. Coarse labels may select different switch and coefficient tables, while the stage schedule stays common. A tensor diagonal has coefficient $\prod_i d_i(x_i)$. In lexicographic order, cached prefix products are recomputed only on odometer carries; the total number of extra products is bounded by $L^g\sum_{k\geq1}k/L^k=O(L^g)$. Scanning the one-axis tables on a fixed number of tapes has the same amortized bound. Consequently each diagonal costs $O(V\plog p)$ rather than $O(Vg)$. + +Crucially, an $H$ stage is invoked \emph{globally across all complete coarse-label ranges}. A fixed coarse-label microbox does not contain complete high-coordinate bits to borrow. Each chosen local bit is placed in a complete $K$-bit neighborhood of its original axis, with +\[ + G=4\lceil\log_2p\rceil+6,\qquad K=128G. +\] +An individual neighborhood fits because $K+O(\log p)=o(\log s_i)$. The $gK$ selected positions can be exposed by a permutation supported on at most $2gK$ positions; the complement need not be stably compacted. Coarse labels are restored before the next diagonal is evaluated. + +The retained compact supplier's residual basis acts on its selected bits, not on the entire ambient address. Its control shuttles have width $O(hG)$ at a node with $h$ active bits. Reserve whole neighborhoods for the two front fields, one back field, and the full product row stock required by nested bit and complex calls. The completed active call is the desired kernel on active bits and identity on arbitrary dirty reserved coordinates. Thus these reserved selected bits can be deferred, rather than processed individually first. Their count is at most $h/16$ above a polylogarithmic tail. Padding adds whole rows only for the current phase and is deleted before the next one. + +The original sufficient wording $K/\log p\to\infty$ is replaced by the finite inequality actually used in its proof, $K\geq G+4\lceil\log_2p\rceil+10$. The present $K$ satisfies it. The support router implements all needed placements with support $O(h\plog p)$, and its short-record construction borrows only $\lceil\log_2p\rceil^2$ suffix bits. It does not move an entire long axis. + +Using supplier stopping exponent $10^{-9}$ gives a strict leaf saving above $a_{\rm eff}$ and $q$. The retained stopped recurrence and the geometric sum of deferred widths yield +\[ + O\bigl(Vg^{1-q}\plog p\bigr) +\] +for a complete global selected-bit call, including all exposure/restoration. The retained supplier implements $H_0=H/2$; multiplying by $2^g$ costs one shift scan and $O(g)$ guard bits. The full circuit therefore costs the same bound up to polylogarithms. The derivations and restrictions are in \file{TENSOR-COMPILER.md}, \file{SUPPLIER-SUPPORT-AUDIT.md}, and \file{SUPPORT-ROUTER.md}. + +\section{The two actual Gaussian operators} + +\subsection{Narrow compression and target-grid completion} +For one axis put $q_j=\lfloor\rho j+1/2\rfloor$ and $\beta_j=\rho j-q_j$. The retained narrow normalized Gaussian is $N=CTD$, where $C$ selects the $q_j$ target rows. Its off-diagonal row sum satisfies +\[ + \norm{N-I}_\infty\leq e^{-2a\theta}+5e^{-2a}. +\] +The two nearest directions must be bounded together; charging each by $e^{-2a\theta}$ would not give this result. With the chosen $a\geq4\log(16p)$ and $\theta\geq1/(4p)$, the row gap is at least $1/(8p)$ eventually. The globally fixed, truncated, structure-preserving rounded matrix $H$ can be chosen with $\norm{H-N}<1/(16p)$, diagonal exactly one, gap at least $1/(16p)$, row norm below two, and $\norm{H^{-1}}_\infty\leq16p$. Every local inverse uses a principal restriction of this same $H$. + +Let $E$ inject source coordinate $j$ at target coordinate $q_j$. On the full dyadic target grid define +\[ + H_{\rm aug}=EHC+(I-EC). +\] +It is permutation-similar to $H\oplus I$, so it inherits row dominance and inverse bounds. Its physical target bandwidth is at most $2w+2$, since $q$ increments by one or two. Its inverse on selected rows is $EH^{-1}C$. A diagonal normalization on selected rows, zero on holes, followed by the increasing $q$-image extraction gives exactly the retained compression approximation. + +\subsection{Broad expansion} +The broad forward matrix is distinct: +\[ + S_{kj}=\alpha^{-1}e^{-b_0(j-\lambda k)^2},\qquad + b_0=\pi/\alpha^2,\quad\lambda=1/\rho. +\] +Embed source values at the $q_j$ coordinates, zero elsewhere. Give every hole input column an identity column. The resulting square matrix $M$ agrees with $S$ on the intended zero-hole input. In selected/hole order, +\[ + M=\begin{bmatrix}\alpha^{-1}N_bD_b^{-1}&0\\ R&I\end{bmatrix}, + \quad b=\pi/(\alpha^2\rho^2),\quad + D_b=\diag(e^{b\beta_j^2}). +\] +The Gaussian row sum of $R$ is bounded. In particular $\norm{M}<3$ eventually; use $M/4$ as the completed forward map, making it contractive also on numerical errors in hole coordinates. Relative to the former $S/2$ normalization this contributes an extra $2^{-D}$ over $D$ axes. A single exact $2^D$ shift compensates it, increasing the retained scalar exponent $\Gamma$ by $D=o(p)$. + +On a phase cell $q_j=j+c$, a selected target block has entries +\[ + K_{ki}=\alpha^{-1}e^{-b_0(i-c-\lambda k)^2}. +\] +Its inverse is polynomially bounded even though an uncentered diagonal split can have a large range. The finite Gaussian LDU proof only uses gaps at least one and $|\beta|\leq1/2$. For +\[ + c_b=\prod_{m\geq1}(1-e^{-2b\rho m}) +\] +it gives +\[ + \norm{K^{-1}}_\infty + \leq\frac{\alpha e^{3b/4}c_b^{-4}} + {(1-e^{-b\rho\theta})^2}. +\] +Indeed $q'_k=k+\rho c$ has unit gaps and $\beta'_{q_j}=\beta_j$. Also +\[ + \log c_b^{-1} + =\sum_{h\geq1}\frac1{h(e^{2b\rho h}-1)} + \leq\frac{\pi^2}{12b\rho}, + \qquad c_b^{-4}\leq e^{\pi\alpha^2\rho/3}. +\] +Since $\alpha^2=O(\log p)$ and $\theta^{-1}=\poly(p)$, the bound is polynomial in $p$, uniformly over all contiguous cell subblocks. + +For a block anchor $m$, set $x=i-m$, $y=k-m$, and $\delta=(1-\lambda)m-c$, with $|\delta|\leq1/2$. The exact Toeplitz split is +\begin{align*} + K_{ki}&=D_{\rm row}(y)G(y,x)D_{\rm col}(x),\\ + G(y,x)&=e^{-b_0\lambda(x-y)^2},\\ + D_{\rm col}(x)&=e^{-b_0((1-\lambda)x^2+2\delta x)},\\ + D_{\rm row}(y)&=\alpha^{-1}e^{b_0\lambda(1-\lambda)y^2+ + 2b_0\delta\lambda y-b_0\delta^2}. +\end{align*} +The Gaussian Poisson formula bounds the symbol of $G$, with $\gamma=b_0\lambda$, between +\[ + \sqrt{\pi/\gamma}\,e^{-\pi^2/(4\gamma)} + \quad\hbox{and}\quad1+\sqrt{\pi/\gamma}. +\] +Thus every finite principal $G$ is SPD with polynomial condition number. Truncating at width $W$ and rounding symmetrically to sufficiently many $Cp$ bits preserve these bounds. The SPD shallow circuit applies. See \file{LOG-A-PHASE-SCHUR-AUDIT.md}, \file{BILATERAL-LDU.md}, and \file{FORWARD-SMALL-A-AUDIT.md} for the analytic details and exact local-identity controls. + +\section{Capping and recentering the phase interiors} + +All target holes and original physical period cuts are boundary coordinates. Selected target coordinates are consecutive inside an original phase cell, so no interior circuit is applied to a masked Toeplitz matrix. Use the maximum physical bandwidth $W$ for both directions. + +In each original $O(d)$-length cell, retain $W$ endpoints and repeatedly choose the largest dyadic interior of size at most $B$ that fits with a following $W$ separator. Once no full $B$ block remains, the dyadic-tail procedure halves the remaining length at every step until its size is $O(W)$, which is all boundary. There are $O(d/B+\log B)$ interiors and +\[ + O\bigl(W(d/B+\log B)\bigr) +\] +boundary coordinates per cell. Since $B=o(d)$, the boundary density is $O((W/B)\plog p)$. Each eliminated interior touches only its two neighboring $O(W)$ boundary groups. Schur fill therefore has half-bandwidth $O(W)$, regardless of the number of capped blocks. + +For the narrow exact cell weights +\[ + \omega_j=e^{-a(1/4-\beta_j^2)/\theta}, +\] +one step changes their logarithm by $2a\beta_j+a\theta=O(a)$. On a capped block use the exact rounded-weight ratio $\widehat\omega_j/\widehat\omega_m$ around one anchor. The common scalar cancels from the similarity, and its logarithmic range is $O(aB)$. The retained global rounding changes this by $O(1)$, because its $Cp$ precision dominates the original $O(ad)=o(p)$ range. This is symbolic recentering of the same matrix, not a new local rerounding. For the broad split, $|x|,|y|\leq B$ and $B\leq d$ give diagonal ranges +\[ + O\bigl((B+B^2/d)/\alpha^2\bigr)=O(B/\alpha^2)\subset O(aB). +\] +The number of blocks does not multiply circuit depth: they run as a direct sum on the $L$ existing coordinates. + +\section{Five factors, sparse windows, and batched boundaries} + +For either square operator write $Q=\left[\begin{smallmatrix}A&B_1\\C_1&D_1\end{smallmatrix}\right]$ in interior/boundary order, and $F=D_1-C_1A^{-1}B_1$. Exact multiplication gives +\begin{align} + Q^{-1}&=\diag(A^{-1},I)E_{-B_1}\diag(A,F^{-1}) + E_{-C_1}\diag(A^{-1},I),\\ + Q&=\diag(A,I)E_{C_1}\diag(A^{-1},F)E_{B_1}\diag(A,I). +\end{align} +The off-diagonal factors use the original banded $B_1,C_1$. The forward formula uses $F$ only as a matrix application; it does not require a forward Schur inverse or a forward LU-prefix assertion. For the narrow inverse, row diagonal dominance survives elimination. Its Schur row gap stays at least $1/\poly(p)$ and its row norm is bounded, so the retained no-pivot boundary solver applies. + +Batch independent right-hand sides in a dyadic width between $W$ and $2W$. A banded triangular solve becomes block updates +\[ + X_j=L_{jj}^{-1}(Y_j-L_{j,j-1}X_{j-1}). +\] +Precomputed diagonal inverses and two square block products implement each update. The seven-product recursion gives exponent $\omega=\log_2 7<45/16$, the latter inequality checked exactly by $7^{16}<2^{45}$. The same batching handles boundary/interior coupling products. Intermediate Strassen additions grow only polynomially in $W$. Round after a completed block update and bound its residual. Each local window has polynomial length $L$, the no-pivot factors and exact partial right-hand sides have polynomial bounds, and the completed system inverse has polynomial norm. Thus the uniform local residual is $\poly(p)2^{-P}$; there is no factor equal to the global number of windows. The total boundary cost is +\[ + O\bigl(V(d/B)W^{\omega-1}\plog p\bigr). +\] + +For a group let $J_i$ be its boundary set, and let $E_i$ also include every interior coordinate incident to $B_1$ or $C_1$. Then $|E_i|/L=O((W/B)\plog p)$ and each sparse factor fixes $E_i^c$. The union +\[ + U=\{x\in[L]^g:\exists i,\ x_i\in E_i\} +\] +has fraction $O(gW/B\plog p)$. It is closed under the sparse factors, including a possibly dense inverse on $J_i$. It is \emph{not} closed under a dense $A$ stage. Use two sparse windows, with scatter/regather around the intervening dense stage. + +For one sparse axis, $U_i=E_i\times[L]^{g-1}$ is a complete rectangle wholly present in $U$. Scan $U$, extract $U_i$, number $E_i$ in increasing order by a stream counter, transpose only this polynomial-size field, perform its boundary operation, and merge back. Only this one field is temporarily padded, and the padding is deleted before the next axis. Each sparse window costs $O(Vg^2W/B\plog p)$ per group. Across $d/g$ groups this is +\[ + O\bigl(V(dg/B)W\plog p\bigr). +\] +Every group also has a fixed number of full scans costing $O(V)$. Membership calculations use $O(g(\log s)^2\plog p)=o(p)$ bit operations per $p$-bit record. Detailed fixed-tape extraction and batching are in \file{PHASE-LAYOUT-AUDIT.md}. + +\section{Core faces, ordered embedding, and precision} + +The increasing map $j\mapsto q_j$ permits one product scan for a whole group. Enumerate the dyadic target rectangle. For target $k$, put +\[ + j=\max\{0,\lceil s(2k-1)/(2t)\rceil\}. +\] +It lies in the image exactly when $j0.000428$. The old core-face cost has exponent $\epsilon_M+\beta_M-3<-1.75$. + +The main and grouped Gaussian savings are equal, while the sparse-union saving has an exact $s_U$ surplus: +\begin{align*} + \epsilon_Mq&=1-\epsilon_M+q\beta_M,\\ + 1-\epsilon_M-\beta_M-1/2+\xi_M&=\epsilon_Mq+s_U. +\end{align*} +Their minimum equals +\[ + 0.001673360115405778453817047951720376\ldots>\kappa_M+10^{-11}. +\] +The new boundary saving is $5/16+\xi_M-\epsilon_M>0.0639$, so it is not binding. Global routing, group scans, packed products and core-face repairs have strictly larger savings. + +The optional full-key surcharge in the earlier conservative table is absent here: the actual sparse-union algorithm extracts complete rectangles $U_i$, transposes one field, and performs an ordered merge. It does not sort union records. In contrast, all sparse \emph{microhalo} radix keys remain paid in their negligible correction volume. Adding a full-key sort on the union is an explicit rejecting control; omitting it follows from the layout proof, not from the matrix exponent. The new parameter range and q-image ordering are independently checked in \file{MATRIX-LAYOUT-SIGNOFF.md}. + +The exact ledger \file{check_matrix_corollary.py} verifies 47 positive strict slacks and six rejecting controls, recorded in \file{MATRIX-COROLLARY-ARITHMETIC.json}. The gain over the upstream selected saving is $6.9990832886\times10^{-7}$. The gain over the separate Strassen result is $4.37755816907\times10^{-7}$. The strict minimum gap absorbs all remaining fixed logarithms as before. +\end{proof} + +\noindent\textbf{Corollary review status: PASS\_CONDITIONAL\_MATRIX\_PHASE\_FUSION\_REVIEW.} The additional matrix transfer, new-parameter layout and exact arithmetic passed their independent audits and integrated root review. Corollary~\ref{cor:matrix} retains its explicit additional matrix-exponent premise. The first theorem and its separate reviewed arithmetic receipt remain unchanged. + +\section{Updated T74 supplier and final exponents}\label{sec:t74} + +The preceding proofs are parameterized by the effective supplier rate. We now substitute the independently replayed T74 complex supplier from \href{https://github.com/CrocSwap/integer-mult-bounds/pull/1661}{PR1661}, pinned at commit \texttt{f852af5e178ce035cd46181a29d9330bcbcaa206}. This is a new input to the same outer proof, with contributor attribution retained in the source package. The original numerical statements above remain valid special cases. + +Use the exact finite decimals +\[ + b_T=0.00167971337478,\qquad q_T=b_T-10^{-10}, + \qquad a_T=(q_T+b_T)/2. +\] +The independently retained bit rate $a_*=0.001733427039589844234749$ satisfies $q_T0 for x in assembly['outer'].values()),'native strict inequalities failed') + words={} + for key,word in [('base','gauge44-word-566.json.gz'),('extra','extra-word-566.json.gz')]: + d=run/('native-new-'+key);st=read(d/'STEPS.json') + need([s['step'] for s in st]==['native','ledgers','local-replay','wrappers','phases'] and all(s['exit_code']==0 for s in st),'native word coverage failed: '+key) + need(gzip.decompress((d/'local/FULL-WORD.json.gz').read_bytes())==gzip.decompress((source/'new'/word).read_bytes()),'native word payload differs: '+key) + words[key]=sha(source/'new'/word) + caps=read(package/'frontier_fusion/NATIVE-ADMISSION-BINDING.json') + need(result['controls']==caps['control_counts'],'cold control coverage differs') + capsule=read(package/'NATIVE-RECEIPT-CAPSULE.json') + def native_normalized(x): + if isinstance(x,dict):return {k:native_normalized(v)for k,v in x.items()if k not in {'seconds','log_sha256'}} + if isinstance(x,list): + y=[native_normalized(v)for v in x] + return sorted(y,key=lambda v:v['name']) if y and all(isinstance(v,dict)and'name'in v and'exit_code'in v for v in y) else y + if isinstance(x,str):return x.replace(str(run),'{native-work}').replace(str(source),'{native-source}') + return x + bound={} + for z in capsule['receipts']: + root=run if z['root']=='native_work' else source + f=root/z['path'];need(f.is_file(),'missing native receipt: '+z['path']) + bound[z['root']+'/'+z['path']]=sha(f) + if 'semantic_sha256' in z: + digest=hashlib.sha256(json.dumps(native_normalized(read(f)),sort_keys=True,separators=(',',':')).encode()).hexdigest() + need(digest==z['semantic_sha256'],'native semantic receipt differs: '+z['path']) + files=[run/'RESULT.json',run/'STAGES.json',run/'strict/RESULT.json',run/'strict/ASSEMBLY.json'] + return {'status':'PASS_FRESH_NATIVE_RUN_BOUND_TO_PINNED_PHASE_FUSION','scope':'Existing complete cold native run checked against the pinned source and the independently replayed outer inputs; all-size assumptions remain.','source_commit':PIN,'source_manifest_sha256':sha(source/'MANIFEST.json'),'native_words':words,'complex_lower':assembly['complex_lower'],'profile':assembly['complex_profile'],'controls':result['controls'],'canonical_native_receipts_checked':46,'all_native_receipt_sha256':bound,'run_receipt_sha256':{str(f.relative_to(run)):sha(f) for f in files}} + +def main(): + ap=argparse.ArgumentParser();ap.add_argument('--package',type=Path,default=Path(__file__).resolve().parent);ap.add_argument('--native-source',type=Path);ap.add_argument('--native-work',type=Path);ap.add_argument('--output',type=Path);a=ap.parse_args() + need((a.native_source is None)==(a.native_work is None),'provide both --native-source and --native-work') + package=a.package.resolve() + if a.output:need(not a.output.resolve().is_relative_to(package),'--output must be outside the immutable package') + out=fast(package) + if a.native_source:out['cold_native_binding']=cold_binding(package,a.native_source,a.native_work) + if a.output:a.output.write_text(json.dumps(out,indent=2)+'\n') + print(out['status'],len(out['checks']),'exact control replays') + if 'cold_native_binding' in out:print(out['cold_native_binding']['status']) +if __name__=='__main__':main() From 2569871ff906d2a2abfc86e97126ab9afe2e4a5a Mon Sep 17 00:00:00 2001 From: Vlado Vojdanovski Date: Tue, 13 Oct 2026 23:18:51 -0700 Subject: [PATCH 2/3] Simplify submission authorship and source presentation --- .../capped-phase-fusion/PACKAGE-MANIFEST.json | 6 +++--- research/capped-phase-fusion/README.md | 17 +++++------------ research/capped-phase-fusion/TASK.json | 2 +- .../evidence/radical_reduction_ultra/README.md | 2 +- 4 files changed, 10 insertions(+), 17 deletions(-) diff --git a/research/capped-phase-fusion/PACKAGE-MANIFEST.json b/research/capped-phase-fusion/PACKAGE-MANIFEST.json index 029d57cb07..c3977299d4 100644 --- a/research/capped-phase-fusion/PACKAGE-MANIFEST.json +++ b/research/capped-phase-fusion/PACKAGE-MANIFEST.json @@ -27,7 +27,7 @@ "PROOF-AND-COVERAGE.md": "db7509d754bb7fc3381419e0fed2674f22a9d57d34e8420fd5afa3c688fde9aa", "PUBLICATION-CHECKS.json": "6e2fb68e6768c3aa4224aacb4900263586fc8114815a51df40f3f9fdf6124217", "PUBLICATION-SOURCES.json": "6082583794168b1c2320187214b86c93a5a6e76fbbddfcb3bdbb8c5bd34c0586", - "README.md": "61b371fe72353cd1d80cf2a1df5cf4e10df18ca78d2265acf43d263222895103", + "README.md": "91af88b8fc53a22e1e968908f0e449cdfe2154c83f0dddb8a504e661c0c6965c", "REVIEW-RECEIPT.json": "97b20e50d3722fcde3831124a107df6c15c0ffdbb9e057b7867f41aaa2481800", "ROOT-EXACT-REPLAY.json": "d36114abe9d7a83c327f49f5ea153c6dcada4a7421f5e74a9a5ceb392cecb994", "ROOT-MATRIX-EXACT-REPLAY.json": "d6bd4fcb8a528655181b4aa3fd1bad7e6647a4ca5aa76ee7f2cdd800b698b225", @@ -36,7 +36,7 @@ "SUPPLIER-SUPPORT-AUDIT.md": "bd5b06eca043c5c871227264f6a954473c55cd16ec85bb420ff538fb8ec1d8d6", "SUPPORT-ADVERSARIAL-AUDIT.md": "b1290f8720bf075ea2ad58d6070f3e1fc95c0155f8e33b165bc13439bbb9222d", "SUPPORT-ROUTER.md": "6ddb322e11883c1177ba6cd1a77d23a1dac827d7290b32eaf642457b1d762a54", - "TASK.json": "494271a60db34e6651bfe164a0516c685081e899475ebf8f9fabe98a6a8014b5", + "TASK.json": "47b82ef23e20991010fae838ad4da9bdac3f947a8fea916a028fd6260f95f5f8", "TENSOR-COMPILER.md": "4d3b8466598a864a234b978e25976648d4e761840aad9153dd18d73d3dfe4d15", "VERIFIER-NEGATIVE-CONTROLS.json": "eeb887cdc0c76426c75ddbfc3d4af7ccbbe19d17962c6ad7a1549dafe8dabb5a", "VERIFY.json": "cb50aa523a0863a457181ca815e69a1b3b4d91ed1006a79263f4a8f18457c669", @@ -148,7 +148,7 @@ "evidence/radical_reduction_ultra/PARTIAL-FUSION.md": "811a3a0c88a90708add5b12fb0a05c25623f0af155010d5dcacd56021c55cba5", "evidence/radical_reduction_ultra/POSITIVE-TREE-CONTROLS.json": "7b31811402ddda4668fe8375f6c90a6047511d50836dbb7cfc7ffcf24fe19a11", "evidence/radical_reduction_ultra/Q-LDU-CONTROLS.json": "370a07aa1a56c656ec1f44d7651dfbe999ad0c19dede9036257cc2f29dc39452", - "evidence/radical_reduction_ultra/README.md": "7fb6138ac39689c6ffd9da2fab3e4f0e81008423554b8dfffea2d139929bd130", + "evidence/radical_reduction_ultra/README.md": "4d65ea3c8b6a0d5faef819a58c8b8277681d11bed54686e9a7f78de29736e68e", "evidence/radical_reduction_ultra/REFINED-PHASE-FUSION.md": "b4367eac012624f803b3d2e3a72339e050a0071bbec37e6d33b2cf322bd70bc5", "evidence/radical_reduction_ultra/STABLE-POSITIVE-TREE.md": "ec2f8a1c4273b81217545d2710bcffdd09b5c9ba9349e887b5f7bf6f5bddfdc7", "evidence/radical_reduction_ultra/SYLVESTER-CONTROLS.json": "198af9f58c2950f17d8b1861ba4566bb2f6a0108a889a56db328fe67f8faa1a2", diff --git a/research/capped-phase-fusion/README.md b/research/capped-phase-fusion/README.md index 75daa3cfe9..f7f3348ee3 100644 --- a/research/capped-phase-fusion/README.md +++ b/research/capped-phase-fusion/README.md @@ -118,24 +118,17 @@ snapshots. The copied upstream CLAIM.json is immutable and contains older pendin language; the later completed native RESULT records this contribution's replay. Hosted CI and independent human review are separate from the local checks. -## Sources, overlap and credit +## Sources and overlap -Prepared by **Vlado Vojdanovski with substantial OpenAI Codex assistance**, -including separate GPT-6.1 Sol Ultra research and audit agents. New material is +Prepared by **Vlado Vojdanovski with substantial AI assistance**. New material is Apache-2.0. The original manuscript and source-specific notices retain their licenses. This is not an official OpenAI release or endorsement. The main research snapshot was `e6f9716621f4e01fc190617934cdfaf072bd976f`; the publication base is recorded in SOURCES.json. Exact T74 source pins are in -PUBLICATION-SOURCES.json and the source manifest. Credit **Rohan Arun** for -PR1319's T74 exact fill and PR1297's four cold kernels; **mdumitrean** for PR1625's -pivot-566 helper-donor kernel; **Jeremy Kothe** for the endpoint and outer-witness -lineage; **Midhat / midhatn** for PR1661's endpoint-grid refinement; **philippeVV** -for the fine-Mercator repair; **Jeremy Diviney, Thomas DiFiore, Hao Yan, eumemic, -icekylinx / Zhihao Chen, DreamingOfClouds**, and the credited predecessor chain -for the retained construction, recurrence, bit supplier and verification. +PUBLICATION-SOURCES.json and the source manifest. The [unaltered T74 NOTICE](target002_upstream/pr1661/research/gmx-t74-p566-ep400/NOTICE) -is included. Douglas Colkitt maintains the community repository; the original +is included. The original OpenAI #109 framework and the Harvey–van der Hoeven reduction remain inputs. The in-place algebra builds on Giorgi–Grenet–Roche, @@ -143,7 +136,7 @@ The in-place algebra builds on Giorgi–Grenet–Roche, and Dumas–Grenet, [*Fast in-place accumulation*](https://arxiv.org/abs/2302.13600). SHALLOW-TOEPLITZ.md distinguishes their results from the new parity-split, low-displacement and prefix-stability arguments. The Gaussian factorization -also relates to Chafik Boukhalfa's +also relates to [PR489](https://github.com/CrocSwap/integer-mult-bounds/pull/489), which already gives a branch-free local Gaussian inverse but leaves joint evaluation, routing and boundary composition open. The present package supplies a different diff --git a/research/capped-phase-fusion/TASK.json b/research/capped-phase-fusion/TASK.json index 62b7e778be..8fc25b910b 100644 --- a/research/capped-phase-fusion/TASK.json +++ b/research/capped-phase-fusion/TASK.json @@ -48,7 +48,7 @@ "outcome": "completed", "evidence": "phase_fusion_note.tex; frontier_fusion/FRONTIER-FUSION-ARITHMETIC.json; frontier_fusion/NATIVE-ADMISSION-BINDING.json; PUBLICATION-CHECKS.json", "author": "Vlado Vojdanovski", - "ai_assistance": "Substantial OpenAI Codex assistance; separate GPT-6.1 Sol Ultra derivation and audit agents. No independent human or proof-assistant review claimed.", + "ai_assistance": "Substantial AI assistance. No independent human or proof-assistant review claimed.", "open_questions": [ { "question": "Can a new finite supplier materially exceed the present complex rate?", diff --git a/research/capped-phase-fusion/evidence/radical_reduction_ultra/README.md b/research/capped-phase-fusion/evidence/radical_reduction_ultra/README.md index 61f7e8558c..7d0b3a41bf 100644 --- a/research/capped-phase-fusion/evidence/radical_reduction_ultra/README.md +++ b/research/capped-phase-fusion/evidence/radical_reduction_ultra/README.md @@ -2,7 +2,7 @@ **Status: two scoped construction refinements, one explicit conjectural route; no new final multiplication exponent.** The selected conditional exponent remains `0.001672660197076918`. The proposed target `0.00167498` below depends on a tensor-resampling lemma that is not proved here. This package does not change a selected certificate. -Base checkout: `work/integer-mult-current`, commit `e6f9716621f4e01fc190617934cdfaf072bd976f`. Prepared with substantial GPT-6.1-sol assistance. Apache-2.0; original sources and contributor credit are retained by reference. +Base checkout: `work/integer-mult-current`, commit `e6f9716621f4e01fc190617934cdfaf072bd976f`. Prepared with substantial AI assistance. Apache-2.0; original sources and contributor credit are retained by reference. ## 1. Where the present loss occurs From 85ddaa312e463cfaca98d49a2a262994551cec4e Mon Sep 17 00:00:00 2001 From: Vlado Vojdanovski Date: Wed, 14 Oct 2026 18:40:57 -0700 Subject: [PATCH 3/3] Add source-bound T74 admission and original outer proof review --- research/capped-phase-fusion/OUTER-AUDIT.md | 2 + .../capped-phase-fusion/PACKAGE-MANIFEST.json | 36 +- research/capped-phase-fusion/README.md | 19 +- .../SUPPLIER-SUPPORT-AUDIT.md | 17 + research/capped-phase-fusion/VERIFY.json | 2 +- .../capped-phase-fusion/phase_fusion_note.tex | 33 +- .../review20261014/CONTROL-REPLAY.json | 1096 +++++++++++++++++ .../review20261014/NATIVE-SUPPLEMENT.json | 213 ++++ .../review20261014/RECURRENCE-CONTROLS.json | 83 ++ .../review20261014/REVIEW-RESPONSE.md | 70 ++ .../review20261014/check_recurrence.py | 111 ++ .../review20261014/check_review.py | 69 ++ .../review20261014/external/SOURCE-PINS.json | 50 + .../o1-admission/t74_final_binding.py | 67 + .../o1-admission/t74_semantic_regen.py | 58 + .../h-call-support/sim_deferred.py | 153 +++ .../outer-audit/pr1673-fusion/a_qembed.py | 126 ++ .../outer-audit/pr1673-fusion/b_diag_tape.py | 69 ++ .../pr1673-fusion/c_fused_group.py | 137 +++ .../independent/ORIGINAL-OUTER-CONTROLS.json | 696 +++++++++++ .../ORIGINAL-OUTER-PROOF-REVIEW.md | 300 +++++ .../independent/SOURCE-PINS.json | 31 + .../independent/check_original_outer.py | 338 +++++ .../source/notes/batched-complex-rows.tex | 141 +++ .../source/notes/compact-control-layout.tex | 118 ++ .../source/notes/compact-control-movement.tex | 167 +++ .../source/notes/endpoint-gauge-complex.tex | 225 ++++ .../rad20/reports/review-bulk-resampling.md | 247 ++++ .../reports/review-phase-cell-inverse.md | 335 +++++ .../review20261014/native_validation.py | 181 +++ research/capped-phase-fusion/verify.py | 2 + 31 files changed, 5180 insertions(+), 12 deletions(-) create mode 100644 research/capped-phase-fusion/review20261014/CONTROL-REPLAY.json create mode 100644 research/capped-phase-fusion/review20261014/NATIVE-SUPPLEMENT.json create mode 100644 research/capped-phase-fusion/review20261014/RECURRENCE-CONTROLS.json create mode 100644 research/capped-phase-fusion/review20261014/REVIEW-RESPONSE.md create mode 100644 research/capped-phase-fusion/review20261014/check_recurrence.py create mode 100644 research/capped-phase-fusion/review20261014/check_review.py create mode 100644 research/capped-phase-fusion/review20261014/external/SOURCE-PINS.json create mode 100644 research/capped-phase-fusion/review20261014/external/o1-admission/t74_final_binding.py create mode 100644 research/capped-phase-fusion/review20261014/external/o1-admission/t74_semantic_regen.py create mode 100644 research/capped-phase-fusion/review20261014/external/outer-audit/h-call-support/sim_deferred.py create mode 100644 research/capped-phase-fusion/review20261014/external/outer-audit/pr1673-fusion/a_qembed.py create mode 100644 research/capped-phase-fusion/review20261014/external/outer-audit/pr1673-fusion/b_diag_tape.py create mode 100644 research/capped-phase-fusion/review20261014/external/outer-audit/pr1673-fusion/c_fused_group.py create mode 100644 research/capped-phase-fusion/review20261014/independent/ORIGINAL-OUTER-CONTROLS.json create mode 100644 research/capped-phase-fusion/review20261014/independent/ORIGINAL-OUTER-PROOF-REVIEW.md create mode 100644 research/capped-phase-fusion/review20261014/independent/SOURCE-PINS.json create mode 100644 research/capped-phase-fusion/review20261014/independent/check_original_outer.py create mode 100644 research/capped-phase-fusion/review20261014/independent/source/notes/batched-complex-rows.tex create mode 100644 research/capped-phase-fusion/review20261014/independent/source/notes/compact-control-layout.tex create mode 100644 research/capped-phase-fusion/review20261014/independent/source/notes/compact-control-movement.tex create mode 100644 research/capped-phase-fusion/review20261014/independent/source/notes/endpoint-gauge-complex.tex create mode 100644 research/capped-phase-fusion/review20261014/independent/source/references/semantic-bulk/rad20/reports/review-bulk-resampling.md create mode 100644 research/capped-phase-fusion/review20261014/independent/source/references/semantic-bulk/rad20/reports/review-phase-cell-inverse.md create mode 100644 research/capped-phase-fusion/review20261014/native_validation.py diff --git a/research/capped-phase-fusion/OUTER-AUDIT.md b/research/capped-phase-fusion/OUTER-AUDIT.md index 974b8db67f..ec0959255d 100644 --- a/research/capped-phase-fusion/OUTER-AUDIT.md +++ b/research/capped-phase-fusion/OUTER-AUDIT.md @@ -1,5 +1,7 @@ # Phase-cell fusion: outer assembly audit +Historical uncapped assembly snapshot. Its prospective status and weaker numerical case below are retained as provenance. For the current capped/T74 theorem, mixed-width recurrence and completed outer review, use [phase_fusion_note.tex](phase_fusion_note.tex) and [the 14 October response](review20261014/REVIEW-RESPONSE.md). The current proof pays the complete depth-dependent product row stock and a polylogarithmic number of scans per group. + Status: assembly audit in progress. The exact arithmetic in this directory is a **conditional target**, not a replacement of the selected certificate. The remaining proof interfaces are recorded explicitly below. ## Conditional transfer statement diff --git a/research/capped-phase-fusion/PACKAGE-MANIFEST.json b/research/capped-phase-fusion/PACKAGE-MANIFEST.json index c3977299d4..f16046b896 100644 --- a/research/capped-phase-fusion/PACKAGE-MANIFEST.json +++ b/research/capped-phase-fusion/PACKAGE-MANIFEST.json @@ -22,24 +22,24 @@ "MATRIX-TRANSFER-MANIFEST.json": "35d98d7579a16dae36530b4e5f4d0b8f925a069daa3f6fe1ea5276916167e0c2", "NATIVE-RECEIPT-CAPSULE.json": "a04d6bba3d15c11a219536b66a23012797484ae106ec1952ecd7b76f077ddb01", "OUTER-ARITHMETIC.json": "050c61d90236c593fd0827d7edc297e05604884e4ec093659aee46a9ee336020", - "OUTER-AUDIT.md": "94799bfa2794eb867d6ba68593c3e92988ade083e74d895e8d07ae3b8d0670dc", + "OUTER-AUDIT.md": "0068df9954b58a562e5d148e634e29abc5945ffa2f69919747513b99242cfba6", "PHASE-LAYOUT-AUDIT.md": "54aafc7afd09fe1a5312e69fad09c87422690543ab75c6ae35f6cbcc26b3eb8a", "PROOF-AND-COVERAGE.md": "db7509d754bb7fc3381419e0fed2674f22a9d57d34e8420fd5afa3c688fde9aa", "PUBLICATION-CHECKS.json": "6e2fb68e6768c3aa4224aacb4900263586fc8114815a51df40f3f9fdf6124217", "PUBLICATION-SOURCES.json": "6082583794168b1c2320187214b86c93a5a6e76fbbddfcb3bdbb8c5bd34c0586", - "README.md": "91af88b8fc53a22e1e968908f0e449cdfe2154c83f0dddb8a504e661c0c6965c", + "README.md": "d3fdbf5642ab684755bdecab12af247244a25b54765d6866bf5af5c342140aba", "REVIEW-RECEIPT.json": "97b20e50d3722fcde3831124a107df6c15c0ffdbb9e057b7867f41aaa2481800", "ROOT-EXACT-REPLAY.json": "d36114abe9d7a83c327f49f5ea153c6dcada4a7421f5e74a9a5ceb392cecb994", "ROOT-MATRIX-EXACT-REPLAY.json": "d6bd4fcb8a528655181b4aa3fd1bad7e6647a4ca5aa76ee7f2cdd800b698b225", "SHALLOW-TOEPLITZ.md": "b384b93f3fb4a8c97da2e617ebc14d55cebb0a739070e7f8dbb66f63e24e9310", "SOURCES.json": "14ce6fdc7c83e18ccf4fee19ff51062726c960ca44462a45af60dd686167d456", - "SUPPLIER-SUPPORT-AUDIT.md": "bd5b06eca043c5c871227264f6a954473c55cd16ec85bb420ff538fb8ec1d8d6", + "SUPPLIER-SUPPORT-AUDIT.md": "c41a2571b9c4c1d7920633f10c3ed2c74da39ca3bc8d6a910e6ffed380327413", "SUPPORT-ADVERSARIAL-AUDIT.md": "b1290f8720bf075ea2ad58d6070f3e1fc95c0155f8e33b165bc13439bbb9222d", "SUPPORT-ROUTER.md": "6ddb322e11883c1177ba6cd1a77d23a1dac827d7290b32eaf642457b1d762a54", "TASK.json": "47b82ef23e20991010fae838ad4da9bdac3f947a8fea916a028fd6260f95f5f8", "TENSOR-COMPILER.md": "4d3b8466598a864a234b978e25976648d4e761840aad9153dd18d73d3dfe4d15", "VERIFIER-NEGATIVE-CONTROLS.json": "eeb887cdc0c76426c75ddbfc3d4af7ccbbe19d17962c6ad7a1549dafe8dabb5a", - "VERIFY.json": "cb50aa523a0863a457181ca815e69a1b3b4d91ed1006a79263f4a8f18457c669", + "VERIFY.json": "4b09d63087b783cffb71f0af96f166fb9900151319171ec1a4e1e60d9da394d2", "check_capped_outer.py": "b4555c77de7a3ed4abc708a8d93363ef7c9a914be1091f80ab3ddea83f71230a", "check_matrix_corollary.py": "30db2d8e0f40fcc5c91e25f74cdc54e46aa4a0353c725345f9bcf5744e0b617f", "evidence/gaussian_fast_ultra/BILATERAL-CONTROLS.json": "eb3ba38d18bac4aceaffba39ef48dbb55bd15e6abcb68cdf66920d9d13ec00ab", @@ -173,7 +173,31 @@ "frontier_fusion/independent_substitution_check.py": "579bbde41e90792818c2726b38dcab26cf5a2695d3348731e814175146b0a737", "gaussian_fixed_width_note.tex": "6f3edb7b238f2b6af4f3e2d98dee90fe6f4ba063a5ccd120422e45c941712aab", "make_native_capsule.py": "8f342ca212b8d601a11f704042c577c1b088e0079bda04b3065467c2f3c36ea0", - "phase_fusion_note.tex": "b5957f38c98a68f2584a42cedf91d753180dab438129fdf749b441074f00d4e1", + "phase_fusion_note.tex": "afa44dcbff3a9ef3f81021668550ef260820b0156321b74eb7f947f0bddb1bce", + "review20261014/CONTROL-REPLAY.json": "45d8dd874be87cc52632beb112aa2d3a5629700744dfc0c41b21979fb6b9177c", + "review20261014/NATIVE-SUPPLEMENT.json": "7b775f692243af829e843dafb526540e10e8a84596a91c3c22c0eaa701a9bd5d", + "review20261014/RECURRENCE-CONTROLS.json": "a9afc22817099fb3b9cc2d7e9a53c346048521adad4cf48e4d53c82acbf54a69", + "review20261014/REVIEW-RESPONSE.md": "c6acd1303011cba1261ad1dd142dd60cbb6076de7588212adf6219e63ce96a2b", + "review20261014/check_recurrence.py": "494add49675aa1f032ccce8242ce66b4079139940de412ddfed640806d117dfd", + "review20261014/check_review.py": "f490f1f911f09e4bee8e78fefc69af36c0d1bfa29196e735f0cf71a77e81a578", + "review20261014/external/SOURCE-PINS.json": "6f23b4a3463e84a9f0c60847a062ac1f7ccb9d011344bb6f153d36d3b9a57b1a", + "review20261014/external/o1-admission/t74_final_binding.py": "f33025e6aa7769d0a688ca5c3be92110c69a602c0649e5e25f84ca0b8ed3b8f3", + "review20261014/external/o1-admission/t74_semantic_regen.py": "48f7cd7bb9da005f3667e35dfd30f278669766e2b7894b0af471ceaedd0fceeb", + "review20261014/external/outer-audit/h-call-support/sim_deferred.py": "fcbbfbaa191f15db937572da4ccf560b280540eb3f21de5f119946fdf700e40f", + "review20261014/external/outer-audit/pr1673-fusion/a_qembed.py": "bc2d8f40197e54e31c957da2007b1c2f6eb459f9f1f1a67528bc4f50bd35c772", + "review20261014/external/outer-audit/pr1673-fusion/b_diag_tape.py": "ebe4e49161984059b3f914b7b946e1aa0b21d123d57dcff2c8f2f94b9a3d5edb", + "review20261014/external/outer-audit/pr1673-fusion/c_fused_group.py": "ffe34fb718c1663bb06f3e833b546c8c253333397e1877e7dfab7001a896aba6", + "review20261014/independent/ORIGINAL-OUTER-CONTROLS.json": "885d75534b06ca1e5476fa9736b0f3b52506c1710c2e1d2d9ca0526085a9e8b4", + "review20261014/independent/ORIGINAL-OUTER-PROOF-REVIEW.md": "bbd941eb6a145d1c85252daee18155fa4203b6056fcaa6dfe8b4675f427b92be", + "review20261014/independent/SOURCE-PINS.json": "82a59ead584e647958686c3f99289723259f3aa1c176e389da3e3cf484d32dab", + "review20261014/independent/check_original_outer.py": "9260ce7328e23423ebc8cd40ab2d5d3d6cfab446cff36c4c03016bb0ffbcfeba", + "review20261014/independent/source/notes/batched-complex-rows.tex": "64c70926a8705874f1c9b42a2f703e42cad4e90f927f544458cfdf729540236c", + "review20261014/independent/source/notes/compact-control-layout.tex": "2bc28e50f3eb092195294cafa81213e520f6d42e7935e9905ad8a6199e2a7c87", + "review20261014/independent/source/notes/compact-control-movement.tex": "b963c3d2f4e1b7059aa289bb4b4ac89b688920bcf42f5b20d7698cdef0132cc6", + "review20261014/independent/source/notes/endpoint-gauge-complex.tex": "dfee33b0281e5551bb6927f757ff0240abade60ae1800f715afd88853ae0191e", + "review20261014/independent/source/references/semantic-bulk/rad20/reports/review-bulk-resampling.md": "12aefaa9a0eee367fb71e8f910ed5f0094e1203d2caa29836b0c37887c446c29", + "review20261014/independent/source/references/semantic-bulk/rad20/reports/review-phase-cell-inverse.md": "113927dd4fb77a55a457bd5f6da8cbfc7262d0fbc4e9d708b591a1d632e6caf6", + "review20261014/native_validation.py": "da08513969aae2f3ffb4214c2cb4adf5ebf457b4eb5c92bd38665cd422c75e55", "target002_upstream/full-1661-verification/RESULT.json": "11c57817a4d43a489010a13af7621d37eeedc398308c036e63a2a97119443861", "target002_upstream/full-1661-verification/STAGES.json": "c2edb01a61c03e04bf48e605422ff83c1d304e4d586268959d4ea24b743640fb", "target002_upstream/pr1661/research/gmx-t74-p566-ep400/CLAIM.json": "70aacd45f9e6fac8675687cf50da32e0ad4e479d9a5c28f2afd1550311ae8849", @@ -181,5 +205,5 @@ "target002_upstream/pr1661/research/gmx-t74-p566-ep400/NOTICE": "3918a42010a435782de0fe8c38fab8f7df5d48a375ecf603c2bad1b818a16d6f", "target002_upstream/pr1661/research/gmx-t74-p566-ep400/inputs/PR589-EXPECTED.json": "b9f0e5ca038abe837b0007c8fe006f366a6f02009244ac57705a8a200e2301f4", "target002_upstream/pr1661/research/gmx-t74-p566-ep400/new/SPEC.json": "d6b6a2140f3a61ae8cfdd8b51a1131934d63d2dd8573c1b13eb2b9528f3c3db8", - "verify.py": "93ef0f46e132485da6194587199b8f039e25b1b9f099f93bbdaeac5c0d66de7f" + "verify.py": "5a55639612613309eda2c6eff8f49ec5b921b3b0b58fb27a9905765b3c8cf772" } diff --git a/research/capped-phase-fusion/README.md b/research/capped-phase-fusion/README.md index f7f3348ee3..a8c26bf532 100644 --- a/research/capped-phase-fusion/README.md +++ b/research/capped-phase-fusion/README.md @@ -22,6 +22,8 @@ The proof's earlier numerical specializations remain valid historical cases. “Independent” in this package means separate agent implementations/reviews, not an independent human referee or a proof-assistant verification. +The [14 October review response](review20261014/REVIEW-RESPONSE.md) addresses the inherited supplier admission and the outer proof separately. It includes a complete written outer review, pinned external checks reused from PR2044, fresh semantic regeneration and mutation controls on our actual supplier, and the corrected mixed-rank recurrence. The bound is unchanged. + ## Exact paid inputs and result | Quantity | Value | @@ -88,7 +90,11 @@ The first command after installation is the bounded outer-proof evidence replay, not a cold native-supplier replay. The native run already completed with all 12 top-level stages passing, 319680 owners, prefix 1441 charged at 2048, physical row coefficient 19545 below 20161, and rejected controls: 114 admission, 12 bit, -15 controller, 8 independent owner, 6 geometry. The retained compact RESULT +23 controller, 8 independent owner, 6 geometry. The historical compact RESULT +and binding incorrectly label the controller count as 15; the actual controller +audit rejects 23 distinct mutations (7 global plus 8 for each of 2 providers). +The historical bytes are preserved; the new supplement records the correction. +The retained compact RESULT and [native binding](frontier_fusion/NATIVE-ADMISSION-BINDING.json) record it. To reproduce the full native dependency, obtain the exact source commit @@ -104,7 +110,16 @@ python3 -B research/capped-phase-fusion/verify.py \ --native-work /absolute/new-native-work ``` -The second command binds a completed external native run to this package's +After the second command, replay the new admission supplement on the same completed run: + +```sh +python3 -B research/capped-phase-fusion/review20261014/native_validation.py \ + --native-source /absolute/pr1661/research/gmx-t74-p566-ep400 \ + --native-work /absolute/new-native-work \ + --output /absolute/new-native-supplement.json +``` + +The package verifier binds a completed external native run to this package's source and mathematical result. It compares regenerated proof-relevant content; fresh timings and log bytes need not match the historical run. The full 121 MB assembly and vendor tree are not duplicated here. The receipt capsule adds canonical hashes for 46 JSON receipts. The 58 hashes diff --git a/research/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md b/research/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md index 72404c42d7..ae676a6504 100644 --- a/research/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md +++ b/research/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md @@ -102,6 +102,23 @@ Let tau=1-a. The retained stopped branching argument gives O(V h^(1-q) polylog p Therefore the entire selected-bit call, including support-local exposure/restoration, costs O(V g^(1-q) polylog p). The old V p^(1-a) term was the full-span exposure estimate. It is not needed for this support-local call family. +For the T74 specialization, "stopped branching" specifically means the mixed +weighted-tree argument in `notes/endpoint-gauge-complex.tex` (the rows/stopping +paragraph) and `notes/batched-complex-rows.tex`. Its children have unequal +widths `r floor(e/73)`, with weights `H_r/66304`. The uniform `e/73` tree cited +in some historical compact-control notes does not by itself justify using +the mixed moment endpoint. The mixed proof bounds weighted widths by depth, +not by assuming equal leaf depth; see +`review20261014/REVIEW-RESPONSE.md` for its explicit derivation. + +The actual T74 stock is 2644 bits and its row coefficient is 19545, below the +retained 20161 allowance. The row reservation must use the complete +depth-dependent product stock. A conservative bound is +`R(h)<=3h/128+O(log(h+1)+loglog(p+2))<=h/16` above a fixed polylogarithmic +tail. A one-row-chunk estimate is not valid for this supplier. Processing +that tail individually adds only polylogarithmic work per group. No power +of g or p is changed by these fixed supplier-dependent constants. + ## 7. Precision, normalization and validity boundary The compact address operations only permute coefficient encodings. Their temporary binary combinations are address-control arithmetic, not coefficient values. The accepted whole-child semantic guard remains linear in selected width. Completed deferred phases act on disjoint selected bits, and the sum of their widths is g; only one current phase contributes unfinished internal excess. Thus its arithmetic guard is O(g), independently of the total address width and of the number of coarse labels. diff --git a/research/capped-phase-fusion/VERIFY.json b/research/capped-phase-fusion/VERIFY.json index 964863373c..2104db75e9 100644 --- a/research/capped-phase-fusion/VERIFY.json +++ b/research/capped-phase-fusion/VERIFY.json @@ -14,7 +14,7 @@ "{package}/verify.py" ], "timeout_seconds": 600, - "checks": "Exact source-bound outer ledger, tensor/Toeplitz/Gaussian/layout identities and deliberate rejecting controls; not a cold native replay." + "checks": "23 exact replays: source-bound outer ledger, tensor/Toeplitz/Gaussian/layout identities, two independent outer/schedule reviews and deliberate rejecting controls. Native supplement requires a separate source-bound complete native run." } ] } diff --git a/research/capped-phase-fusion/phase_fusion_note.tex b/research/capped-phase-fusion/phase_fusion_note.tex index 4351a73450..758d7987aa 100644 --- a/research/capped-phase-fusion/phase_fusion_note.tex +++ b/research/capped-phase-fusion/phase_fusion_note.tex @@ -14,7 +14,7 @@ \newcommand{\plog}{\operatorname{polylog}} \newcommand{\file}[1]{\href{#1}{\texttt{\detokenize{#1}}}} \title{Capped phase-cell fusion in the conditional multiplication reduction} -\author{Research note prepared with OpenAI Codex assistance} +\author{Vlado Vojdanovski, with substantial AI assistance} \date{13 October 2026} \begin{document} \maketitle @@ -83,7 +83,7 @@ \section{Statement, inputs, and parameters} \] The spare power $r_0$ is not the growth rate of $u$: it only provides the eventual bound $u\leq p^{r_0}$. Let $W=O(\sqrt p\,\plog p)$ dominate both physical Gaussian bandwidths, and let $A=C_Ap^3$ be a sufficiently large fixed-constant locality radius. The dense tensor path uses the dyadic $L$-point cores, not a simultaneous padding of $(L+2A)^g$. -Global one-axis coefficient preprocessing is polynomial in $s_i,t_i,p$ and hence $n^{o(1)}$. Local pages regenerated for each coarse block have polynomial cost in $p$ and are charged to that block's superpolynomial $L^g$ payload. These are two different charges. +Global one-axis coefficient preprocessing is polynomial in $s_i,t_i,p$ and hence $n^{o(1)}$. Local pages regenerated for each coarse block have polynomial cost in $p$ and are charged to that block's superpolynomial $L^g$ payload. These are two different charges. Partition the $D$ axes into balanced groups of between $g/2$ and $g$ axes, by evenly splitting into $\lceil D/g\rceil$ consecutive groups. Since $\beta<\epsilon$, all groups have $\Theta(g)$ axes eventually; the final group therefore has the same setup amortization. \section{Dimension-preserving local linear circuits} @@ -135,6 +135,26 @@ \section{Uniform tensor compilation and the actual supplier call} \] for a complete global selected-bit call, including all exposure/restoration. The retained supplier implements $H_0=H/2$; multiplying by $2^g$ costs one shift scan and $O(g)$ guard bits. The full circuit therefore costs the same bound up to polylogarithms. The derivations and restrictions are in \file{TENSOR-COMPILER.md}, \file{SUPPLIER-SUPPORT-AUDIT.md}, and \file{SUPPORT-ROUTER.md}. +The recurrence used here is the \emph{mixed weighted-tree} recurrence of +\file{../../notes/endpoint-gauge-complex.tex} and +\file{../../notes/batched-complex-rows.tex}, rather than a uniform-depth +rank-one expansion. For T74 its children have widths +$r\lfloor e/73\rfloor$, with multiplicities $H_r$ and normalized volume +weight $1/66304$. The strict moment +$\Psi(1-b_T)=\sum_r(H_r/66304)(r/73)^{1-b_T}<1$ bounds weighted widths +at every depth, including unequal stopping depths. Internal work has +exponent $1-a_T>1-b_T$, so its moment is also strict; leaf work has +exponent $1-(1-10^{-9})b_T<1-a_T<1-q_T$. +Integer remainders have fewer than 73 axes and belong to the fixed +per-node overhead. Product row stock uses the maximal-child depth, +not $\log_{73} e$. Its finite invoice has stock 2644 bits and row +coefficient $19545<20161$. Using the full depth-dependent product stock +gives $R(h)\leq3h/128+O(\log(h+1)+\log\log(p+2))\leq h/16$ +above a fixed polylogarithmic tail. Processing that tail individually +adds only polylogarithmic work per group. The complete derivation, +source binding and independently executable controls are in +\file{review20261014/REVIEW-RESPONSE.md}. + \section{The two actual Gaussian operators} \subsection{Narrow compression and target-grid completion} @@ -150,6 +170,13 @@ \subsection{Narrow compression and target-grid completion} \] It is permutation-similar to $H\oplus I$, so it inherits row dominance and inverse bounds. Its physical target bandwidth is at most $2w+2$, since $q$ increments by one or two. Its inverse on selected rows is $EH^{-1}C$. A diagonal normalization on selected rows, zero on holes, followed by the increasing $q$-image extraction gives exactly the retained compression approximation. +For an explicit contraction constant with the new bound $16p$, take +$j_{\rm scale}=\lceil\log_2(128p)\rceil$ in the retained scalar normalization. +This preserves even the stronger norm allowance $1/8$. Relative to the +historical $32p$ choice, its extra two scaling bits per axis are compensated +by one exact global $2^{2D}$ shift. The added $O(D)=o(p)$ scalar exponent +and linear scan cost are already within the stated normalization bill. + \subsection{Broad expansion} The broad forward matrix is distinct: \[ @@ -251,7 +278,7 @@ \section{Five factors, sparse windows, and batched boundaries} \[ O\bigl(V(dg/B)W\plog p\bigr). \] -Every group also has a fixed number of full scans costing $O(V)$. Membership calculations use $O(g(\log s)^2\plog p)=o(p)$ bit operations per $p$-bit record. Detailed fixed-tape extraction and batching are in \file{PHASE-LAYOUT-AUDIT.md}. +Every group also has a polylogarithmic number of full scans costing $O(V\plog p)$, including the dense circuit's diagonal scans. Membership calculations use $O(g(\log s)^2\plog p)=o(p)$ bit operations per $p$-bit record. 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External five-factor examples also test a denser union; no other PR outer result is imported. Written proof and retained all-size interfaces are separate." +} diff --git a/research/capped-phase-fusion/review20261014/NATIVE-SUPPLEMENT.json b/research/capped-phase-fusion/review20261014/NATIVE-SUPPLEMENT.json new file mode 100644 index 0000000000..4928914b9d --- /dev/null +++ b/research/capped-phase-fusion/review20261014/NATIVE-SUPPLEMENT.json @@ -0,0 +1,213 @@ +{ + "status": "PASS_PR1673_PINNED_T74_SEMANTIC_AND_FINAL_ADMISSION_SUPPLEMENT", + "native_source_commit": "f852af5e178ce035cd46181a29d9330bcbcaa206", + "external_reviewer": { + "repository": "philippeVV/integer-mult-bounds", + "commit": "d1e01310b92e9a9af90b89efe4f3f9aad466b8b1", + "scripts_reused": [ + "t74_final_binding.py", + "t74_semantic_regen.py" + ] + }, + "native_replay_mode": "Existing complete cold run, unmodified and source/capsule bound; not rerun here.", + "new_work": "Fresh owner/template regeneration, whole-receipt final binding, recompression and semantic rejection controls, geometry mutations.", + 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template hash shared": true, + "profile: ind == asm == owner == audit == PLAN": true, + "provider counts agree": true, + "phase semantic hashes agree": true, + "artifact semantic hashes agree with PLAN": true, + "provider content binding gauge44/CANONICAL-WORD.json.gz": true, + "provider content binding gauge44/LOCAL-PROGRAMS.json.gz": true, + "provider content binding gauge44/LOCAL-CHECKS.json": true, + "provider content binding gauge44/INDEPENDENT-LOCAL-REPLAY.json": true, + "provider content binding gauge44/RESIDUAL-ADAPTERS.json": true, + "provider content binding gauge44/CLIFFORD-RECIPES.json": true, + "provider content binding gauge44/CLIFFORD-REVIEW.json": true, + "provider content binding extra/CANONICAL-WORD.json.gz": true, + "provider content binding extra/LOCAL-PROGRAMS.json.gz": true, + "provider content binding extra/LOCAL-CHECKS.json": true, + "provider content binding extra/INDEPENDENT-LOCAL-REPLAY.json": true, + "provider content binding extra/RESIDUAL-ADAPTERS.json": true, + "provider content binding extra/CLIFFORD-RECIPES.json": true, + "provider content binding extra/CLIFFORD-REVIEW.json": true, + "admission controls 114, none accepted": true, + "bit controls 12": true, + "independent controls 8": true, + "geometry: 6 rejected": true, + "strictly improves parent kappa": true + }, + "semantic_checks": { + "pinned assignment_sha256 == sha(pinned container)": true, + "regenerated assignment_sha256 == sha(regenerated container)": true, + "pinned replica_lift.owners_file_sha256 == sha(pinned container)": true, + "regenerated replica_lift.owners_file_sha256 == sha(regenerated container)": true, + "pinned assignment_content_sha256 == payload sha of regenerated": true, + "template content pin": true, + "PLAN differs only in container-hash fields": true, + "PLAN semantic hash (container fields removed) equal": true, + "all non-PLAN semantic hashes equal": true, + "SPEC geometry equal": true, + "control: altered role changes semantic hash": true + }, + 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{ + "name": "low_row", + "reason": "strict arithmetic invariant" + } + ] + }, + "geometry_runtime_spec_and_engines_bound": true, + "geometry_discovery_environment_cleared": true, + "input_sha256": { + "native_source_manifest": "d90d49eaa68e48c6992a07972e736ef4b9cd80274361cbbc1ec980d43f29ad7f", + "controller": "37b1b8d81353c99db83edbf3d9d1510906e0ae0a3301165894190d416f1c6d60", + "owner": "9d83217650339b8386ae5a968ca62fb4a6e5a2c0e37364a3ad531573c75e2af6", + "invoice": "321d98f8342fb61418aacee7a99a7e54ae284b32a5f142fa692ebe6825d05ea0", + "assembly": "36f8c00c200cc375512ad0e881e91b42daba10c962fce37465040cb717b5cc60", + "geometry_generator": "7edb4732b34e5814b27650a55c938626cb50d68f06397744dd4fe451437e90e0", + "mixed_assembler": "8cca983c2d3edade5b24ab55aab0418400b35822b8dcca2935b80c58339cb521", + "outer_ledger": "82656563e7c8725f95c4aea7e273805a3d3faf34c495a330f1c91a036765414a", + "geometry_runtime_spec": "d6b6a2140f3a61ae8cfdd8b51a1131934d63d2dd8573c1b13eb2b9528f3c3db8", + 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"exit_code": 0, + "expected_exit_code": 0 + }, + { + "name": "changed-role-control", + "exit_code": 1, + "expected_exit_code": 1 + }, + { + "name": "changed_geometry_SPEC", + "status": "REJECTED_BY_INPUT_BINDING" + }, + { + "name": "changed_geometry_engine", + "status": "REJECTED_BY_INPUT_BINDING" + }, + { + "name": "fresh-geometry-controls", + "exit_code": 0, + "expected_exit_code": 0 + } + ], + "controls_rejected": [ + "swapped_native_directories", + "nonimproving_rate", + "changed_owner_role", + "changed_geometry_SPEC", + "changed_geometry_engine" + ], + "harmless_recompression_accepted": true, + "scope": "Finite admission evidence for exactly the pinned PR1661 supplier used by PR1673. No new native word or exponent; inherited all-size/common-frame/mutable-source/hosting/precision/tape/recovery/analytic hypotheses remain. The external T37 regression is separate cited evidence, not a new local regression run.", + "cold_source_binding_status": "PASS_FRESH_NATIVE_RUN_BOUND_TO_PINNED_PHASE_FUSION" +} diff --git a/research/capped-phase-fusion/review20261014/RECURRENCE-CONTROLS.json b/research/capped-phase-fusion/review20261014/RECURRENCE-CONTROLS.json new file mode 100644 index 0000000000..82e6be0633 --- /dev/null +++ b/research/capped-phase-fusion/review20261014/RECURRENCE-CONTROLS.json @@ -0,0 +1,83 @@ +{ + "status": "PASS_EXACT_MIXED_RECURRENCE_AND_ORIGINAL_OUTER_COST_CONTROLS", + "native": { + "m": 73, + "W": 66304, + "physical": 66794, + "profile": { + "1": 1559766, + "2": 337668, + "3": 278058, + "4": 84912, + "5": 54612, + "8": 5328, + "48": 22748 + }, + "mass": 4816512, + "deficit": 23680, + "halving": [ + 5329, + 4608 + ], + "physical_stock_bits": 2644, + "row_coefficient": 19545, + "coefficient_to_chunks_conversion": 39 + }, + "rate_order": { + "sigma": "49916014331261/50000000000000", + "tau": "49916014333761/50000000000000", + "target": "49916014336261/50000000000000", + "leaf_exponent": "49916014331344985668739/50000000000000000000000", + "stop": "1/1000000000" + }, + "deferred_reserve": { + "all_size_proof_uses": "Full depth-dependent product stock and a polylogarithmic tail.", + "geometric_sum_factor_upper": "4/3" + }, + "cost_exponents": { + "main_transform": "83473287399679562907734886383739407/83613520665520456012648670700000000", + "global_routing": "49916014333761/50000000000000", + "Gaussian_groups": "83473287399679562907734886383739407/83613520665520456012648670700000000", + "group_bit_routing": "83473287399288327721793771926771907/83613520665520456012648670700000000", + "polylog_full_group_scans": "756617269395650000000000000/836135206655204560126486707", + "boundary_Strassen": "83472451264472907703174759897032407/83613520665520456012648670700000000", + "sparse_union": "57329162212927511587386237425157407/83613520665520456012648670700000000", + "full_union_key_surcharge": "434476576852166893071/556488729597100000000", + "polynomial_products": "1270900071331668732986707/836135206655204560126486707", + "core_face": "-1595294276193517897592460121/836135206655204560126486707", + "halo_keys": "-65216331051652/16694661887913", + "halo_metadata_fallback": "-4022911058828245349112446828/836135206655204560126486707" + }, + "strict_cost_margins": { + "main_transform": "8361352859074664449126597542766633/836135206655204560126486707000000000000000000", + "global_routing": "2553173065619/1000000000000000000", + "Gaussian_groups": "8361352859074664449126597542766633/836135206655204560126486707000000000000000000", + "group_bit_routing": "12273704718485809018801597542766633/836135206655204560126486707000000000000000000", + "polylog_full_group_scans": "78115604609506981936423528286520667542766633/836135206655204560126486707000000000000000000", + "boundary_Strassen": "8361360427904904675929316196597542766633/836135206655204560126486707000000000000000000", + "sparse_union": "261441251875881866062561154034946597542766633/836135206655204560126486707000000000000000000", + "full_union_key_surcharge": "1210788320227746915486995941049/5564887295971000000000000000000", + "polynomial_products": "833461973933825313203436821286520667542766633/836135206655204560126486707000000000000000000", + "core_face": "2430027150198674879528883649286520667542766633/836135206655204560126486707000000000000000000", + "halo_keys": "81882993317900247538590987823147/16694661887913000000000000000000", + "halo_metadata_fallback": "4857643932833402331048870356286520667542766633/836135206655204560126486707000000000000000000" + }, + "strict_side_margins": { + "one_axis_neighborhood": "1270900071331668732986707/836135206655204560126486707", + "total_support": "15132345387913/16694661887913", + "capped_gauge": "1/100000000", + "global_absolute_gauge": "1270063936125013528426580513293/836135206655204560126486707000000", + "membership_record": "755346369324318331267013293/836135206655204560126486707", + "halo_fraction": "81910992939565/16694661887913" + }, + "controls": { + "uniform_depth_two_is_insufficient": true, + "old_controller_count_15_rejected": true, + "one_row_chunk_not_accepted_as_T74_stock": true, + "former_axis_scan_unpaid": true, + "unrecentered_group_guard_fails": true, + "zero_complex_gap_rejected": true + }, + "mixed_depth_at_73_squared": 9, + "scope": "Exact rational input/outer/stock checks for the published PR1673 Strassen specialization. Weighted-tree proof and complete functional hypotheses are stated in REVIEW-RESPONSE.md. No unconditional multiplication theorem is claimed." +} diff --git a/research/capped-phase-fusion/review20261014/REVIEW-RESPONSE.md b/research/capped-phase-fusion/review20261014/REVIEW-RESPONSE.md new file mode 100644 index 0000000000..79f50dd912 --- /dev/null +++ b/research/capped-phase-fusion/review20261014/REVIEW-RESPONSE.md @@ -0,0 +1,70 @@ +# PR1673: supplier admission and original outer proof + +This responds to [the review of head 2569871](https://github.com/CrocSwap/integer-mult-bounds/pull/1673#issuecomment-6172528048). The conditional headline remains **κ = 0.001677160151714381**. This update adds the missing evidence and makes the recurrence, row stock and normalization arguments explicit; it does not substitute a new supplier or claim an unconditional theorem. + +The reviewer identified two separate obligations: the inherited T74/#1319 supplier admission, and a written audit of the new outer schedule, including precision, movement and the recursive interfaces. Positive arithmetic margins alone settle neither obligation. + +## Existing review reused, with an exact source boundary + +Relevant independent work already appears in PR2044. We reuse philippeVV's checks from `philippeVV/integer-mult-bounds@d1e01310b92e9a9af90b89efe4f3f9aad466b8b1`, rather than presenting those implementations as new. The six unmodified files and their Git blob SHA1 and SHA256 are recorded in [external/SOURCE-PINS.json](external/SOURCE-PINS.json). + +The applicable external records are: + +- [Supplier admission and semantic regeneration](https://github.com/CrocSwap/integer-mult-bounds/pull/2044#issuecomment-6181046525). +- [The original PR1673 fused-group, diagonal-tape and embedding audit](https://github.com/CrocSwap/integer-mult-bounds/pull/2044#issuecomment-6181466068). +- [Support-local complete H calls](https://github.com/CrocSwap/integer-mult-bounds/pull/2044#issuecomment-6181619058). +- [The mixed-width recurrence and corrected product-stock invoice](https://github.com/CrocSwap/integer-mult-bounds/pull/2044#issuecomment-6181752567). + +Those comments concern several packages and later outer regimes. We import only the stated supplier/schedule checks and the applicable interface arguments. No later fine-prime exponent, precision choice, hybrid bound or claimed priority is imported. The optional NumPy deferred-call example is retained as a source, but the registered replay does not execute it or rely on it as a proof. The new written review and exact checker below concern PR1673's original capped/Strassen regime only. + +## 1. Admission of the actual pinned supplier + +The supplier is still PR1661 at `f852af5e178ce035cd46181a29d9330bcbcaa206`, with source-manifest SHA256 `d90d49eaa68e48c6992a07972e736ef4b9cd80274361cbbc1ec980d43f29ad7f`. Its 169 source files, retained parent and new provider words are unchanged. The publication already contains a binding to a complete cold native run: all 12 top-level stages and both five-stage native provider replays passed, without native-reuse flags. That run is reused here after checking its source manifest, stage logs, complete native word payloads and the capsule's 46 canonical JSON receipts. We do **not** describe this reuse as a newly repeated cold native computation. + +The [new admission supplement](NATIVE-SUPPLEMENT.json) comes from [native_validation.py](native_validation.py). It then performs fresh work against that exact completed run: + +| Obligation | New binding or replay | +| --- | --- | +| Controller | Checks actual audit status, provider counts and 23 distinct rejected mutations: `7 + 8*2`. One owner is bound to each complete invocation; all charts and provider bindings agree. | +| Owners, calls and charts | Independent counts agree with owner audit, controller, invoice and dispatch: 319680 roles, 666 calls, 11142 charts. Rank mass, deficit, live/physical widths, idle children, provider counts and global profile agree. | +| Invoice and assembly | Every invoice strict condition holds; owner/controller hashes bind the invoice, the assembly binds that invoice, and all 47 assembly inequalities are positive. Prefix 1441 is below charged 2048; physical row coefficient 19545 is below 20161. | +| Native provider contents | Seven canonical word/program/adapter/review families for each of the two provider directories bind semantically to the PLAN. Swapping the base and extra directories rejects. | +| Regenerated geometry | Reruns the actual owner and parent generators, then compares seven artifact families by canonical decompressed mathematical content. Every owner role, time and chart is included. | +| Gzip/container distinction | Recompresses the complete owner payload at a different compression level and timestamp, reruns the actual parent generator, and accepts the unchanged semantics. Only the two PLAN container references may differ, and each must match its actual file. A changed owner time rejects. Frozen source pins are never rewritten. | +| Geometry controls | Reruns all six original mutations: understated prefix, old call count, old idle charge, old stock, missing call and understated physical row. Records the actual owner, invoice, assembly and generator hashes consumed. | +| Geometry runtime | The consumed runtime SPEC must equal the pinned new SPEC byte for byte. The three consumed engine files must match the assembly's recorded hashes. Changed-SPEC and changed-engine fixtures reject. Discovery/record environment variables are cleared. | +| Ancestry and improvement | Retains 114 admission, 12 bit and 8 independent-owner rejecting controls from the source-bound cold run. A deliberately nonimproving parent comparison rejects. | + +The historical top-level RESULT and binding label the controller controls as 15. That label was incorrect for two providers; 15 is the one-provider count. The actual controller list already contains 23 distinct rejected controls. Historical receipt bytes remain untouched; this supplement and the README correct the description. An external T37 regression is cited in the external review above; it was not newly run locally or counted as our native replay. + +The independent final binding has **44** conjunctive checks. Its success and the new negative fixtures strengthen finite admission evidence; they do not discharge the inherited all-size arithmetic, common-frame, mutable-source or hosting assumptions. + +## 2. Written proof of the original outer + +The [complete independent written review](independent/ORIGINAL-OUTER-PROOF-REVIEW.md) gives the argument in eight parts: scales and address model; complete global H calls; local linear circuits and tensorization; both rounded Gaussian maps; capped five-factor elimination and two sparse windows; embeddings, halos and precision; the complete cost ledger; and the finite/all-size boundary. Its [independent exact checker](independent/check_original_outer.py) imports no published parameter checker. The [receipt](independent/ORIGINAL-OUTER-CONTROLS.json) binds the actual main note and 22 supporting sources. Six unmodified inherited notes/reports are bundled with a separate commit/hash pin for package-only replay. + +The main note and supplier audit now make these previously implicit points explicit: + +1. **Mixed widths require a weighted tree.** The uniform tree in `notes/compact-control-layout.tex` does not cover children of rank 48. Use `notes/endpoint-gauge-complex.tex` and `notes/batched-complex-rows.tex`. At width e, a rank-r child has width `r floor(e/73)` and relative volume `1/66304`; fewer than 73 remainder axes contribute fixed per-node overhead. Write + + `Ψ(t) = sum_r (H_r/66304) (r/73)^t`, `σ=1-b`, `τ=1-a`, `δ=10^-9`. + + The strict moment `Ψ(σ)<1` sums weighted σ-mass geometrically over all depths. Since `σ<τ<1-q`, internal work sums as `O(V h^τ polylog p)`. Stopped leaves have exponent `σ+δ(1-σ)<τ`. Unequal stopping depths are therefore covered. The checker independently encloses the T74 moment using rational Taylor bounds and rejects a rate above the endpoint. + +2. **Reserve the full product row stock.** The longest child is `48 floor(e/73)` and `2*48^2<73^2`, so depth is `O(log e)`. The row divisor includes complex stock and nested bit stock at their actual depths. Even a conservative product over all nested depths has `log2 Q_rows=O(log^2(hG))`. With `G=Θ(log p)`, `K=128G` and `h<=p`, its complete-chunk cost is `O(log(h+1)+loglog(p+2))`. Thus the two front fields, back field and full row stock reserve + + `R(h)<=3h/128+O(log(h+1)+loglog(p+2))<=h/16` + + above a fixed polynomial in log p. Handle the tail elementarily; deferred widths then sum geometrically. The finite invoice's 2644 stock bits and coefficient 19545 are checked separately. Neither a one-chunk row prefix nor a particular constant chunk threshold is assumed. + +3. **Scans and setup must be paid at their actual multiplicity.** A group has a polylogarithmic number of diagonal/full scans. Their normalized power remains `epsilon-beta`, with that polylogarithm included. Balanced groups have between g/2 and g axes, so there is no singleton final group whose setup amortization fails. Coefficient pages are regenerated sequentially from actual origins and amortized over their complete payload. + +4. **Normalization uses the new inverse bound.** The narrow rounded inverse bound is 16p. Choosing `j_scale=ceil(log2(128p))` keeps the literal stronger contraction bound. Its extra two bits per axis are compensated by one `2^(2D)` shift and O(D) extra precision, already within the paid scalar scan and `o(p)` guard. + +The supporting exact tests include 34 positive inequalities and six rejecting variants in each of the two published Strassen specializations, both five-factor orientations on 24 varied rational systems, exact q-image/order cases, sparse-union closure and dense leakage, halo volume identities, and mixed-depth/product-stock examples. [CONTROL-REPLAY.json](CONTROL-REPLAY.json) additionally reruns the pinned external product-diagonal tape, group embedding and five-factor models, together with [our recurrence/cost controls](check_recurrence.py). The latter include the conservative metadata fallback and all 12 cost rows. These are finite corroborations of the written argument. + +## Reproduction and limits + +The top-level `verify.py` now registers **23** bounded exact replays and runs them in a temporary copy. Python 3.12 with SymPy 1.14.0 is the registered environment. The native supplement additionally needs the exact external native source and a completed full run; commands are in the package README. It reruns owner/parent generation and geometry, and binds rather than repeats the expensive native run. + +This update supplies reproducible finite admission checks and the written original-outer argument requested in the review. It remains a conditional result: the inherited all-size compiler, exact common-frame mutable-source realization, whole-call hosting, semantic guard with C1=1, fixed-tape/prime-packing interfaces, paid arbitrary-width equal-chunk interchange, exact recovery and analytic transfer are still assumptions. The separate `omega_C<=9/4` corollary has an extra premise and is outside the new original-outer signoff. Nothing here asserts human review, proof-assistant certification, a practical cutoff or a stronger κ. diff --git a/research/capped-phase-fusion/review20261014/check_recurrence.py b/research/capped-phase-fusion/review20261014/check_recurrence.py new file mode 100644 index 0000000000..bffbca680b --- /dev/null +++ b/research/capped-phase-fusion/review20261014/check_recurrence.py @@ -0,0 +1,111 @@ +"""Exact mixed-width recurrence, product stock and outer cost controls. + +These calculations corroborate REVIEW-RESPONSE.md's parameterized proof. +They do not replace its all-size supplier/semantic/tape hypotheses. +""" +from fractions import Fraction as Q +from pathlib import Path +import hashlib, json, sys + +P=Path(__file__).resolve().parent +PACKAGE=P.parent + + +def need(ok, message): + if not ok: + raise RuntimeError(message) + + +def rational(value): + return Q(value['rational']) if isinstance(value,dict) else Q(value) + + +def main(): + ledger=json.loads((PACKAGE/'frontier_fusion/FRONTIER-FUSION-ARITHMETIC.json').read_text()) + row=ledger['rows']['strassen'] + # The short endpoint has its own independent exact enclosure in this + # source-bound receipt. No native rounded-weight optimization is used. + E,X,B=[rational(row[k])for k in ('epsilon','xi','beta')] + q,a=[rational(row[k])for k in ('q','a')] + b=rational(ledger['complex_lower_short']) + need(all(rational(v)<1 for v in ledger['native_moment']),'native moment is not strict') + kappa=rational(row['kappa']) + stop=Q(1,10**9) + sigma,tau=1-b,1-a + H={1:1559766,2:337668,3:278058,4:84912,5:54612,8:5328,48:22748} + m,W,physical=73,66304,66794 + mass=sum(r*n for r,n in H.items()) + need(m*W-mass==23680,'wrong native rank deficit') + need(m*m>2*max(H)**2,'halving guard fails') + need(00,'written outer cost is unpaid') + sides={'one_axis_neighborhood':1-E,'total_support':1-B, + 'capped_gauge':1-B-X-Q(1,10**6), + 'global_absolute_gauge':1-E-Q(1,10**6), + 'membership_record':1-B-2*(1-E), + 'halo_fraction':8-B-3} + need(min(sides.values())>0,'written side condition fails') + # A mixed-width tree is not a uniform m-ary width tree: this exact + # control would have depth one in the latter interpretation. + def depth(e): + d=0 + while e>=m: + e=max(H)*(e//m);d+=1 + return d + need(depth(m*m)>2,'control did not distinguish mixed-width recursion') + controls={ + 'uniform_depth_two_is_insufficient':depth(m*m)>2, + 'old_controller_count_15_rejected':7+8*2!=15, + 'one_row_chunk_not_accepted_as_T74_stock':q0!=1, + 'former_axis_scan_unpaid':1-E1, + 'zero_complex_gap_rejected':not (b1 else P/'RECURRENCE-CONTROLS.json' + target.write_text(json.dumps(out,indent=2)+'\n') + print(out['status'],len(costs),'cost rows',len(controls),'controls') + + +if __name__=='__main__':main() diff --git a/research/capped-phase-fusion/review20261014/check_review.py b/research/capped-phase-fusion/review20261014/check_review.py new file mode 100644 index 0000000000..cf3c41e075 --- /dev/null +++ b/research/capped-phase-fusion/review20261014/check_review.py @@ -0,0 +1,69 @@ +"""Replay independent finite schedule evidence plus the repaired recurrence. + +External scripts are retained byte-for-byte and Git-object pinned. Their +finite models check identities and failure modes, not an all-size machine. +""" +from pathlib import Path +import hashlib, json, os, subprocess, sys, tempfile + +P=Path(__file__).resolve().parent + + +def need(ok,message): + if not ok:raise RuntimeError(message) + + +def main(): + pins=json.loads((P/'external/SOURCE-PINS.json').read_text()) + for rec in pins['files']: + blob=(P/'external'/rec['path']).read_bytes() + need(hashlib.sha256(blob).hexdigest()==rec['sha256'],'external script changed') + need(hashlib.sha1(b'blob '+str(len(blob)).encode()+b'\0'+blob).hexdigest()==rec['git_blob_sha1'], + 'external script git object differs') + env=dict(os.environ,PYTHONDONTWRITEBYTECODE='1',PYTHONHASHSEED='0') + env.pop('PYTHONOPTIMIZE',None) + need(__debug__,'assertions required') + results={} + paths={ + 'recurrence':P/'check_recurrence.py', + 'product_diagonal':P/'external/outer-audit/pr1673-fusion/b_diag_tape.py', + 'ordered_embedding':P/'external/outer-audit/pr1673-fusion/a_qembed.py', + 'five_factor_group':P/'external/outer-audit/pr1673-fusion/c_fused_group.py', + } + with tempfile.TemporaryDirectory(prefix='phase-fusion-independent-controls-')as td: + scratch=Path(td) + for name,script in paths.items(): + output=scratch/(name+'.json') + cp=subprocess.run([sys.executable,'-B',str(script),str(output)], + env=env,cwd=scratch,capture_output=True,text=True,timeout=180) + need(cp.returncode==0,name+' failed\n'+cp.stdout[-1500:]+cp.stderr[-1500:]) + need(output.is_file(),name+' left no fresh output') + result=json.loads(output.read_text()) + if name=='recurrence': + need(result['status']=='PASS_EXACT_MIXED_RECURRENCE_AND_ORIGINAL_OUTER_COST_CONTROLS',name) + need(result==json.loads((P/'RECURRENCE-CONTROLS.json').read_text()), + 'retained recurrence receipt differs from fresh calculation') + need(len(result['cost_exponents'])==12 and all(result['controls'].values()),name) + elif name=='product_diagonal': + need(len(result)>=20 and all(x['coef_moves_per_record']+x['suffix_moves_per_record']<7 + and x['mults_per_record']<3 for x in result),'diagonal cost control failed') + elif name=='ordered_embedding': + need(result['exhaustive_axes_checked']>1000 and len(result['group_embedding'])==6,name) + need(all(x['one_pass_moves'] 0 for v in asm['outer'].values()) +C['assembly: template hash shared'] = asm['template_sha256'] == own['template_sha256'] == inv['template_sha256'] == aud['template_sha256'] == aud['complex_template_sha256'] +C['profile: ind == asm == owner == audit == PLAN'] = ind['profile'] == asm['complex_profile'] == own['paid_profile'] == aud['complex_profile'] == plan['profile'] +C['provider counts agree'] = ind['provider_counts'] == asm['provider_counts'] == inv['provider_counts'] == aud['provider_counts'] +C['phase semantic hashes agree'] = own['provider_phase_semantic_sha256'] == aud['provider_phase_semantic_sha256'] == inv['provider_phase_semantic_sha256'] +C['artifact semantic hashes agree with PLAN'] = own['provider_artifact_semantic_sha256'] == aud['provider_artifact_semantic_sha256'] == {k: v['artifact_semantic_sha256'] for k, v in plan['provider_registry'].items()} +for pid, loc in (('gauge44', Path(L44)), ('extra', Path(LEX))): + for n in ('CANONICAL-WORD.json.gz', 'LOCAL-PROGRAMS.json.gz', 'LOCAL-CHECKS.json', 'INDEPENDENT-LOCAL-REPLAY.json', + 'RESIDUAL-ADAPTERS.json', 'CLIFFORD-RECIPES.json', 'CLIFFORD-REVIEW.json'): + C['provider content binding %s/%s' % (pid, n)] = rd(g / 'base/providers' / pid / n) == rd(loc / n) +C['admission controls 114, none accepted'] = len(adm['rejected_controls']) == 114 and not adm.get('discovery_accepted_controls') +C['bit controls 12'] = len(bit['rejected_controls']) == 12 +C['independent controls 8'] = len(ind['controls_rejected']) == 8 +C['geometry: 6 rejected'] = len(geo['rejected']) == 6 +GAPS = {'geometry receipt carries no input hash of the strict owner/invoice it mutated': not any(k for k in geo if 'sha' in k)} +C['strictly improves parent kappa'] = Fraction(asm['kappa']) > Fraction(PK) +ok = all(C.values()) +out = dict(status='PASS_T74_T16_PARITY_FINAL_BINDING' if ok else 'FAIL', strict=str(W), kappa=asm['kappa'], kappa_decimal=asm['kappa_decimal'][:40], checks=C, informational_gaps=GAPS) +Path(OUT).write_text(json.dumps(out, indent=1) + '\n'); print(out['status'], out['kappa_decimal']) +for k, v in C.items(): + if not v: print('FAILED:', k) +sys.exit(0 if ok else 1) diff --git a/research/capped-phase-fusion/review20261014/external/o1-admission/t74_semantic_regen.py b/research/capped-phase-fusion/review20261014/external/o1-admission/t74_semantic_regen.py new file mode 100644 index 0000000000..08c8a7c55a --- /dev/null +++ b/research/capped-phase-fusion/review20261014/external/o1-admission/t74_semantic_regen.py @@ -0,0 +1,58 @@ +"""Semantic regenerated-artifact check for a T74 package (PR1319 family). + +Regenerated owners (reconstruct.py) and parent (lift_parent.py) are compared to the pinned package files by +semantic hash = sha256 of canonical JSON (sort_keys, compact) of the decompressed payload, NOT by container bytes. +PLAN.json is the only file allowed to differ, and only in fields that are container (gzip-byte) hashes; each such field +must equal sha256 of the corresponding regenerated container, and every content-hash field in PLAN must equal the +semantic payload hash of the regenerated file. Container-byte agreement is reported separately (portability witness). +usage: t74_semantic_regen.py PKG OWNERS_DIR PARENT_DIR OUT.json""" +import gzip, hashlib, json, sys +from pathlib import Path +P, O, Q, out = map(Path, sys.argv[1:5]) +def payload(p): b = p.read_bytes(); return gzip.decompress(b) if p.suffix == '.gz' else b +def obj(p): return json.loads(payload(p)) +def sem(p): return hashlib.sha256(json.dumps(obj(p), sort_keys=True, separators=(',', ':')).encode()).hexdigest() +def raw(p): return hashlib.sha256(p.read_bytes()).hexdigest() +def paysha(p): return hashlib.sha256(payload(p)).hexdigest() +def diff(x, y, pre=''): + if isinstance(x, dict) and isinstance(y, dict): + return [d for k in sorted(set(x) | set(y)) for d in (diff(x.get(k), y.get(k), pre + k + '.') if x.get(k) != y.get(k) else [])] + return [pre.rstrip('.')] +rows, ok = {}, True +for d, names in ((O, ['OWNERS.json.gz', 'ORDER.json', 'DISPATCH.json', 'PROFILE.json']), + (Q, ['GLOBAL-TEMPLATE.json.gz', 'CONSTRUCTION.json', 'PLAN.json'])): + for n in names: + r = dict(container_equal=raw(d / n) == raw(P / n), semantic_equal=sem(d / n) == sem(P / n), + semantic_sha256=sem(P / n), regenerated_semantic_sha256=sem(d / n)) + if n == 'PLAN.json': r['paths_differing'] = diff(obj(d / n), obj(P / n)) + rows[str(d.name) + '/' + n] = r +pinned, regen = obj(P / 'PLAN.json'), obj(Q / 'PLAN.json') +CONTAINER = {'assignment_sha256': 'OWNERS.json.gz', 'replica_lift.owners_file_sha256': 'OWNERS.json.gz'} +CONTENT = {'assignment_content_sha256': (O / 'OWNERS.json.gz', paysha)} +def get(x, path): + for k in path.split('.'): x = x[k] + return x +checks = {} +for path, n in CONTAINER.items(): + checks['pinned ' + path + ' == sha(pinned container)'] = get(pinned, path) == raw(P / n) + checks['regenerated ' + path + ' == sha(regenerated container)'] = get(regen, path) == raw(O / n) +for path, (f, h) in CONTENT.items(): + checks['pinned ' + path + ' == payload sha of regenerated'] = get(pinned, path) == h(f) == get(regen, path) +checks['template content pin'] = get(pinned, 'replica_lift.parent_template_content_sha256') == get(regen, 'replica_lift.parent_template_content_sha256') +pd = rows[Q.name + '/PLAN.json']['paths_differing'] +checks['PLAN differs only in container-hash fields'] = set(pd) <= set(CONTAINER) +def strip(x): + x = json.loads(json.dumps(x)); x.pop('assignment_sha256', None); x['replica_lift'].pop('owners_file_sha256', None); return x +checks['PLAN semantic hash (container fields removed) equal'] = strip(pinned) == strip(regen) +checks['all non-PLAN semantic hashes equal'] = all(r['semantic_equal'] for k, r in rows.items() if not k.endswith('PLAN.json')) +checks['SPEC geometry equal'] = obj(Q / 'SPEC.json')['geometry'] == obj(P / 'SPEC.json')['geometry'] +# negative control: alter one role, semantic hash must change +roles = obj(P / 'OWNERS.json.gz'); roles[0] = dict(roles[0], time=roles[0]['time'] + 1) +checks['control: altered role changes semantic hash'] = hashlib.sha256(json.dumps(roles, sort_keys=True, separators=(',', ':')).encode()).hexdigest() != sem(P / 'OWNERS.json.gz') +ok = all(checks.values()) +res = dict(status='PASS_T74_SEMANTIC_REGENERATED_ARTIFACTS' if ok else 'FAIL', python=sys.version.split()[0], package=str(P), + container_bytes_all_equal=all(r['container_equal'] for r in rows.values()), checks=checks, files=rows) +Path(out).write_text(json.dumps(res, indent=1) + '\n'); print(res['status'], 'containers_equal=%s' % res['container_bytes_all_equal']) +for k, v in checks.items(): + if not v: print('FAILED:', k) +sys.exit(0 if ok else 1) diff --git a/research/capped-phase-fusion/review20261014/external/outer-audit/h-call-support/sim_deferred.py b/research/capped-phase-fusion/review20261014/external/outer-audit/h-call-support/sim_deferred.py new file mode 100644 index 0000000000..e604c0f706 --- /dev/null +++ b/research/capped-phase-fusion/review20261014/external/outer-audit/h-call-support/sim_deferred.py @@ -0,0 +1,153 @@ +"""Exact small model of SUPPLIER-SUPPORT-AUDIT sec. 2-3/6 (PR1673 as vendored in PR2044): +support-local exposure + deferred reservation phases + whole-row padding/role split, for one +simultaneous H call on g selected bits taken one per axis from a d-axis binary tensor. + +The supplier itself is modelled by its *contract* (completed invocation = H on active selected +bits, identity on reserved/spectator/row coordinates), but internally it is adversarial: inside each +role stream it applies an arbitrary role-dependent bijection to the dirty front/back control +chunks (which hold UNPROCESSED deferred selected bits) and undoes it. We check: + 1. end result == H^{(x)g} applied directly in the native layout (exact integers); + 2. every exposure permutation has moved support <= 2hK; + 3. padded zero rows are zero at completion (so deletion is exact); + 4. mutation controls fail: skip deferred phase; supplier also hits row label; no restore of + exposure before the next phase; preliminary C on reserved chunks (old exposition) applied twice. +Usage: python3 sim_deferred.py OUT.json +""" +import json, sys, itertools, random +import numpy as np + +def bitperm(a, n, perm): + """new address bit perm[i] = old address bit i.""" + t = a.reshape([2] * n) # numpy axis k <-> address bit n-1-k + # new array axis for bit perm[i] must come from old axis for bit i + src_axes = [None] * n + for i in range(n): + src_axes[n - 1 - perm[i]] = n - 1 - i + return np.ascontiguousarray(t.transpose(src_axes)).reshape(-1) + +def hbit(a, j): + v = a.reshape(-1, 2, 1 << j) + x0, x1 = v[:, 0, :].copy(), v[:, 1, :].copy() + out = np.empty_like(v); out[:, 0, :] = x0 + x1; out[:, 1, :] = x0 - x1 + return out.reshape(-1) + +def support(perm): + return sum(1 for i, p in enumerate(perm) if i != p) + +def run(seed, d, ell, K, k, g_axes, Q, mutate=None): + rng = random.Random(seed) + n = d * ell + data = np.array([rng.randrange(-50, 51) for _ in range(1 << n)], dtype=np.int64) + # axis a occupies address bits [a*ell, (a+1)*ell); axis 0 holds the untouched suffix + sel = {a: a * ell + k for a in g_axes} # selected bit per axis + nbhd = {a: [a * ell + k + i for i in range(K)] for a in g_axes} + assert all(k + K <= ell for _ in g_axes) + suffix = list(range(0, ell)) # reserved untouched suffix (record grouping) + assert 0 not in g_axes + # reference + ref = data.copy() + for a in g_axes: ref = hbit(ref, sel[a]) + cur = data.copy() + pending = list(g_axes) + log = [] + phase = 0 + while pending: + h = len(pending) + if h <= 3: # tail: process individually (O(V) each) + for a in pending: cur = hbit(cur, sel[a]) + log.append({'phase': phase, 'h': h, 'mode': 'individual'}); pending = [] + break + # reservation: 1 row chunk, 1 front chunk (two toy H-fields), 1 back chunk + rng.shuffle(pending) + fc = max(1, -(-h // 4)) # front chunks grow with h (toy 2hG/K) + row, fronts, back = pending[0], pending[1:1 + fc], pending[1 + fc] + active = pending[2 + fc:] + assert active + order = [back] + active + fronts + [row] + front = fronts[0] + # exposure: S -> T (top hK bits), T\S -> S\T, identity elsewhere + T = list(range(n - h * K, n)) + S = [b for a in order for b in nbhd[a]] + assert len(set(S)) == len(S) and not set(S) & set(suffix) + perm = list(range(n)) + for s, t in zip(S, T): perm[s] = t + rest_dom = [t for t in T if t not in S]; rest_img = [s for s in S if s not in T] + for x, y in zip(rest_dom, rest_img): perm[x] = y + assert sorted(perm) == list(range(n)) + sup = support(perm) + assert sup <= 2 * h * K, (sup, h, K) + assert all(perm[b] == b for b in suffix) + x = bitperm(cur, n, perm) + pos = {a: n - h * K + K * i for i, a in enumerate(order)} # selected offset rho=0 in chunk + # rows: top K bits = row chunk; pad row count 2^K up to multiple of Q + R = 1 << K; Rp = -(-R // Q) * Q + body = 1 << (n - K) + M = np.zeros((Rp, body), dtype=np.int64); M[:R] = x.reshape(R, body) + roles = [M[w::Q].copy() for w in range(Q)] # u = Q*g' + w -> role w + fb_bits = [pos[f] + i for f in fronts for i in range(K)] + [pos[back] + i for i in range(K)] + for w in range(Q): + st = roles[w].reshape(-1) + nb = n - K + # adversarial dirty use: role-dependent bijection on front/back chunk values + vals = 1 << len(fb_bits) + sig = list(range(vals)); random.Random(seed * 1000 + w + phase).shuffle(sig) + idx = np.arange(st.size, dtype=np.int64) + addr_body = idx % body + def field(addr): + f = 0 + for i, b in enumerate(fb_bits): f |= ((addr >> b) & 1) << i + return f + def setfield(addr, f): + for i, b in enumerate(fb_bits): + addr = (addr & ~(1 << b)) | (((f >> i) & 1) << b) + return addr + F = field(addr_body) + sigarr = np.array(sig, dtype=np.int64) + newaddr = (idx - addr_body) + setfield(addr_body, sigarr[F]) + scr = np.empty_like(st); scr[newaddr] = st + # active H on selected bits (contract); mutation: also touch row label + for a in active: scr = hbit(scr, pos[a]) + if mutate == 'row_label_touched' and phase == 0: + scr = hbit(scr.reshape(-1), (scr.size // body - 1).bit_length() + n - K - 1) if False else scr + scr = scr.reshape(-1, body); scr[[0, 1]] = scr[[1, 0]]; scr = scr.reshape(-1) # swap two rows of role stream + # undo dirty bijection + back_st = scr[newaddr] + roles[w] = back_st.reshape(-1, body) + M2 = np.zeros_like(M) + for w in range(Q): M2[w::Q] = roles[w] + pad_zero = bool(np.all(M2[R:] == 0)) + x = M2[:R].reshape(-1) + if mutate == 'preliminary_on_reserved' and phase == 0: + for a in [row, back] + fronts: x = hbit(x, pos[a]) + inv = [0] * n + for i, p in enumerate(perm): inv[p] = i + if mutate == 'no_restore' and phase == 0: + cur = x # forget inverse exposure + else: + cur = bitperm(x, n, inv) + log.append({'phase': phase, 'h': h, 'active': len(active), 'deferred': h - len(active), 'exposure_support': sup, + 'bound_2hK': 2 * h * K, 'rows': R, 'padded_rows': Rp, 'padding_zero_at_completion': pad_zero}) + pending = [a for a in pending if a not in active] + if mutate == 'drop_deferred': pending = [] + phase += 1 + return bool(np.array_equal(cur, ref)), log, all(e.get('padding_zero_at_completion', True) for e in log) + +def main(): + out = {'positive': [], 'controls': {}} + configs = [dict(d=9, ell=2, K=2, k=0, g_axes=list(range(1, 9)), Q=3), + dict(d=7, ell=3, K=2, k=1, g_axes=list(range(1, 7)), Q=3), + dict(d=5, ell=4, K=3, k=1, g_axes=[1, 2, 3, 4], Q=5), + dict(d=10, ell=2, K=1, k=1, g_axes=list(range(1, 10)), Q=3)] + for ci, c in enumerate(configs): + for seed in range(3): + ok, log, padok = run(seed + 10 * ci, **c) + assert ok and padok, (c, seed, log) + out['positive'].append({'config': c, 'seed': seed, 'exact_equal': ok, 'phases': log}) + for m in ['drop_deferred', 'no_restore', 'preliminary_on_reserved', 'row_label_touched']: + ok, log, padok = run(1, **configs[0], mutate=m) + out['controls'][m] = {'exact_equal': ok, 'padding_zero': padok} + assert not (ok and padok), m + json.dump(out, open(sys.argv[1], 'w'), indent=1) + print('positive cases', len(out['positive']), 'all exact; controls', {k: v for k, v in out['controls'].items()}) + +main() diff --git a/research/capped-phase-fusion/review20261014/external/outer-audit/pr1673-fusion/a_qembed.py b/research/capped-phase-fusion/review20261014/external/outer-audit/pr1673-fusion/a_qembed.py new file mode 100644 index 0000000000..033ffdb051 --- /dev/null +++ b/research/capped-phase-fusion/review20261014/external/outer-audit/pr1673-fusion/a_qembed.py @@ -0,0 +1,126 @@ +"""Part A: q-image group embedding at fine theta (PR1786 regime), on a stream/tape model. + +Independent of the PR's code. Checks + (1) the membership predicate of phase_fusion_note.tex:258-262 for every target k, + for all s in [t/2, t) at small t, and for fine-theta primes s (theta in [1/(4P^2), 1/(2P^2-1)]) at larger t; + (2) q increments are 1 or 2, holes = t-s, phase-cell lengths ~ 1/theta (so B=p^xi < cell holds when xi<2); + (3) a one-pass group embedding of g axes with prefix/suffix spectators on a 2-tape stream equals the + composition of g per-axis embeddings, and its head travel is (input + output) records, independent of g. +""" +import itertools, json, math, random, sys + +def qmap(j, s, t): return (2 * t * j + s) // (2 * s) # floor(t j / s + 1/2) +def cdiv(a, b): return -((-a) // b) +def inv(k, s, t): + j = max(0, cdiv(s * (2 * k - 1), 2 * t)) + return j if (j < s and 2 * t * j < s * (2 * k + 1)) else None + +def is_prime(n): + if n < 2: return False + for p in (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37): + if n % p == 0: return n == p + d, r = n - 1, 0 + while d % 2 == 0: d //= 2; r += 1 + for a in (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37): + x = pow(a, d, n) + if x in (1, n - 1): continue + for _ in range(r - 1): + x = x * x % n + if x == n - 1: break + else: return False + return True + +def check_axis(s, t): + img = [qmap(j, s, t) for j in range(s)] + assert all(b - a in (1, 2) for a, b in zip(img, img[1:])), (s, t) + assert img[0] == 0 and img[-1] < t + pre = [inv(k, s, t) for k in range(t)] + hit = {k: j for k, j in enumerate(pre) if j is not None} + assert sorted(hit) == img and all(img[j] == k for k, j in hit.items()), (s, t) + holes = t - s + # cell lengths: maximal runs with q_j - j constant + runs, cur = [], 1 + for a, b in zip(img, img[1:]): + if b - a == 1: cur += 1 + else: runs.append(cur); cur = 1 + runs.append(cur) + return holes, runs + +out = {} +# (1) exhaustive small t +n_axes = 0 +for e in range(1, 11): + t = 1 << e + for s in range(max(1, t // 2), t + 1): + check_axis(s, t); n_axes += 1 +out['exhaustive_axes_checked'] = n_axes + +# (1b,2) fine-theta primes: theta = t/s - 1 in [1/(4P^2), 1/(2P^2-1)] +fine = [] +for e, P in [(16, 8), (18, 12), (20, 16), (21, 30)]: + t = 1 << e + lo = math.ceil(t / (1 + 1 / (2 * P * P - 1))); hi = math.floor(t / (1 + 1 / (4 * P * P))) + primes = [s for s in range(lo, hi + 1) if is_prime(s)] + if not primes: fine.append(dict(t=t, P=P, primes=0)); continue + s = primes[len(primes) // 2] + holes, runs = check_axis(s, t) + theta = t / s - 1 + inner = runs[1:-1] + fine.append(dict(t=t, P=P, s=s, theta=theta, primes_in_window=len(primes), holes=holes, + min_inner_cell=min(inner), max_cell=max(runs), one_over_theta=1 / theta, + cells_at_least_2P2_minus_2=min(inner) >= 2 * P * P - 2)) +out['fine_theta_axes'] = fine + +# (3) one-pass group embedding with prefix/suffix spectators on a stream model +def stream_group_embed(data, shape, grp, sizes): + """data: flat list in lexicographic order of `shape`; axes in grp (consecutive) go s->t. + One sequential read pass + one sequential write pass. Returns output and (reads, writes).""" + a0, a1 = grp[0], grp[-1] + 1 + pre = math.prod(shape[:a0]); suf = math.prod(shape[a1:]) + reads = writes = 0; outl = []; pos = 0 + for _ in range(pre): + for k in itertools.product(*(range(t) for s, t in sizes)): + if all(inv(kk, s, t) is not None for kk, (s, t) in zip(k, sizes)): + outl.extend(data[pos:pos + suf]); pos += suf; reads += suf + else: + outl.extend([0] * suf) + writes += suf + assert pos == len(data) + return outl, reads, writes + +def ref_embed(arr, shape, axis, s, t): + """per-axis embedding via an explicit index map (reference, not a stream).""" + newshape = list(shape); newshape[axis] = t + out = [0] * math.prod(newshape) + for idx in itertools.product(*(range(n) for n in shape)): + src = 0 + for i, n in zip(idx, shape): src = src * n + i + j = list(idx); j[axis] = qmap(idx[axis], s, t) + dst = 0 + for i, n in zip(j, newshape): dst = dst * n + i + out[dst] = arr[src] + return out, newshape + +rng = random.Random(1673) +grp_res = [] +for trial in range(6): + g = rng.choice([2, 3, 4]) + sizes = [] + for _ in range(g): + t = rng.choice([8, 16]); s = rng.randint(t // 2 + 1, t - 1); sizes.append((s, t)) + pre_n, suf_n = rng.choice([1, 2, 3]), rng.choice([1, 2, 3]) + shape = [pre_n] + [s for s, t in sizes] + [suf_n] + data = [rng.randint(1, 10**6) for _ in range(math.prod(shape))] + outl, r, w = stream_group_embed(data, shape, list(range(1, g + 1)), sizes) + ref, sh = data, shape + for i, (s, t) in enumerate(sizes): ref, sh = ref_embed(ref, sh, 1 + i, s, t) + assert outl == ref + S = math.prod(s for s, t in sizes); T = math.prod(t for s, t in sizes) + seq_moves = 0; cur = list(shape) + for i, (s, t) in enumerate(sizes): # g per-axis passes would each read+write the whole current tensor + seq_moves += math.prod(cur); cur[1 + i] = t; seq_moves += math.prod(cur) + grp_res.append(dict(g=g, sizes=sizes, input=len(data), one_pass_moves=r + w, per_axis_pass_moves=seq_moves, + ratio=(r + w) / len(data), T_over_S=T / S)) +out['group_embedding'] = grp_res +print(json.dumps(out, indent=1)) +json.dump(out, open(sys.argv[1], 'w'), indent=1) diff --git a/research/capped-phase-fusion/review20261014/external/outer-audit/pr1673-fusion/b_diag_tape.py b/research/capped-phase-fusion/review20261014/external/outer-audit/pr1673-fusion/b_diag_tape.py new file mode 100644 index 0000000000..1529522597 --- /dev/null +++ b/research/capped-phase-fusion/review20261014/external/outer-audit/pr1673-fusion/b_diag_tape.py @@ -0,0 +1,69 @@ +"""Part B: tape-model count for the tensor-diagonal scan of TENSOR-COMPILER.md sec. 6 / note line 120. + +Model: fixed number of tapes. Coefficient tape holds g one-axis tables (N cells each) in axis order; a suffix-product +tape holds P_1..P_{g-1}. A head move is one cell. The data stream is read once sequentially (not counted below). +The record layout is a list of fields, fastest first: group axes ('a',k) appear in order k=0..g-1, spectator fields +('s',M) may be interleaved anywhere (inner, between, outer). The diagonal is d(x)=prod_k d_k(x_k) (exact integers), +checked against the naive product for every record. We report coefficient/suffix head moves and multiplications +per record, and compare with the naive method (g reads + g-1 products per record). +""" +import json, math, random, sys + +def run(N, g, layout, seed=0): + rng = random.Random(seed) + tables = [[rng.randint(2, 9) for _ in range(N)] for _ in range(g)] + sizes = [N if f[0] == 'a' else f[1] for f in layout] + V = math.prod(sizes) + pos = {f[1]: i for i, f in enumerate(layout) if f[0] == 'a'} + hc = 0; hs = 0; cm = 0; sm = 0; mults = 0 + def readc(k, i): + nonlocal hc, cm + c = k * N + i; cm += abs(hc - c); hc = c; return tables[k][i] + def touchs(j): # suffix cell j (1..g-1) stored at cell j-1 + nonlocal hs, sm + c = j - 1; sm += abs(hs - c); hs = c + x = [0] * len(layout) + # initial suffix products + P = [1] * (g + 1) + for j in range(g - 1, 0, -1): + P[j] = readc(j, 0) * P[j + 1]; mults += 1; touchs(j) + maxd = 0 + for r in range(V): + # record value + i0 = x[pos[0]] + d0 = readc(0, i0) + val = d0 * P[1] if g > 1 else d0; mults += 1 + if g > 1: touchs(1) + naive = math.prod(tables[k][x[pos[k]]] for k in range(g)) + assert val == naive + # odometer increment over all fields + f = 0 + while f < len(layout): + x[f] += 1 + if x[f] < sizes[f]: break + x[f] = 0; f += 1 + if f == len(layout): break + # field f incremented, fields < f wrapped. Lowest group axis whose value changed from the top: + changed = [layout[i][1] for i in range(f + 1) if layout[i][0] == 'a'] + if not changed: continue + top = max(changed) # highest group axis changed (incremented or wrapped) + if top == 0: continue # only axis 0 changed: next record reads table 0 sequentially + for j in range(top, 0, -1): # recompute P_top..P_1 + P[j] = readc(j, x[pos[j]]) * P[j + 1]; mults += 1; touchs(j) + maxd = max(maxd, top) + return dict(N=N, g=g, layout=[f"{f[0]}{f[1]}" for f in layout], V=V, + coef_moves_per_record=cm / V, suffix_moves_per_record=sm / V, mults_per_record=mults / V, + naive_mults_per_record=g - 1, naive_table_reads_per_record=g, + bound_2sum_kN_over_Nk=2 * sum(k * N / N**k for k in range(1, g)) + 1) + +res = [] +for N in (2, 4, 8): + for g in range(2, 21): + if N**g > 1 << 18: break + res.append(run(N, g, [('a', k) for k in range(g)])) +# spectators interleaved (inner, between, outer) +res.append(run(2, 10, [('s', 3)] + [('a', k) for k in range(10)])) +res.append(run(2, 10, [('a', 0), ('a', 1), ('s', 5)] + [('a', k) for k in range(2, 10)] + [('s', 2)])) +res.append(run(4, 6, [('a', 0), ('s', 2), ('a', 1), ('a', 2), ('s', 3), ('a', 3), ('a', 4), ('a', 5)])) +for r in res: print(json.dumps(r)) +json.dump(res, open(sys.argv[1], 'w'), indent=1) diff --git a/research/capped-phase-fusion/review20261014/external/outer-audit/pr1673-fusion/c_fused_group.py b/research/capped-phase-fusion/review20261014/external/outer-audit/pr1673-fusion/c_fused_group.py new file mode 100644 index 0000000000..715500bbaf --- /dev/null +++ b/research/capped-phase-fusion/review20261014/external/outer-audit/pr1673-fusion/c_fused_group.py @@ -0,0 +1,137 @@ +"""Part C: one fused group in the DENSE-UNION regime of PR2044 (g*mu >= 1, U ~ whole microbox), exact rationals. + +For each of g axes take a random banded (bandwidth W) row-dominant rational matrix Q_i on n coordinates, a capped +carving (W endpoints, B-interiors, W separators, all boundary), and the five-factor forward schedule + diag(A,I) . E_C . diag(A^-1,I) diag(I,F) . E_B . diag(A,I) (note line 231, execution right to left) +with three dense tensor stages (each ONE simultaneous global call, regardless of g) and two sparse windows. +Each sparse window is executed (a) as PR1673's per-axis scan of U (PHASE-LAYOUT-AUDIT.md:84-99) and +(b) as PR2044's flag-class rounds (UNION-SCHEDULE.md), with record-visit counts. The final tensor is compared with +the direct per-axis application of Q_1 x ... x Q_g. Counts are reported per record of the microbox. +""" +import itertools, json, math, random, sys +from fractions import Fraction as Q + +def eye(n): return [[Q(int(i == j)) for j in range(n)] for i in range(n)] +def mm(a, b): return [[sum(x * y for x, y in zip(r, c)) for c in zip(*b)] for r in a] +def inv(a): + n = len(a); m = [r[:] + e for r, e in zip(a, eye(n))] + for j in range(n): + k = next(i for i in range(j, n) if m[i][j]); m[j], m[k] = m[k], m[j] + v = m[j][j]; m[j] = [x / v for x in m[j]] + for i in range(n): + if i != j and m[i][j]: + v = m[i][j]; m[i] = [x - v * y for x, y in zip(m[i], m[j])] + return [r[n:] for r in m] +def sub(h, R, C): return [[h[i][j] for j in C] for i in R] + +def carve(n, B, W): + J = set(range(W)) | set(range(n - W, n)); blocks = []; c = W; stop = n - W + while stop - c >= B + W: + blocks.append(list(range(c, c + B))); J.update(range(c + B, c + B + W)); c += B + W + J.update(range(c, stop)) + return blocks, sorted(J) + +def axis_factors(n, W, B, rng): + H = [[Q(0)] * n for _ in range(n)] + for i in range(n): + H[i][i] = Q(1) + for j in range(max(0, i - W), min(n, i + W + 1)): + if j != i: H[i][j] = Q(rng.randint(-3, 3), 16 * W) + blocks, J = carve(n, B, W); I = [i for i in range(n) if i not in J] + A, Bm, C, D = sub(H, I, I), sub(H, I, J), sub(H, J, I), sub(H, J, J) + Ai = inv(A); F = [[d - x for d, x in zip(rd, rx)] for rd, rx in zip(D, mm(mm(C, Ai), Bm))] + def full(blocksmap): + M = eye(n) + for (R, Cc), blk in blocksmap: + for ii, i in enumerate(R): + for jj, j in enumerate(Cc): M[i][j] = blk[ii][jj] + return M + dA = full([((I, I), A)]); dAi = full([((I, I), Ai)]); dF = full([((J, J), F)]) + EB = full([((I, J), Bm)]); EC = full([((J, I), C)]) + for i in I: EB[i][i] = Q(1) + for j in J: EC[j][j] = Q(1) + # sanity: product equals H (execution right to left) + M = eye(n) + for f in [dA, EB, dF, dAi, EC, dA]: M = mm(f, M) + assert M == H + # E_i: coordinates moved or read by the sparse factors + E = sorted(set(J) | {i for i in I for j in J if Bm[I.index(i)][J.index(j)] or C[J.index(j)][I.index(i)]}) + return dict(H=H, dense=[dA, dAi, dA], win1=mm(dF, EB), win2=EC, E=E, J=J, blocks=len(blocks)) + +def apply_axis(T, M, ax, n, g): + out = dict(T) + for other in itertools.product(range(n), repeat=g - 1): + idx = [other[:ax] + (i,) + other[ax:] for i in range(n)] + old = [T[k] for k in idx] + for i, k in enumerate(idx): out[k] = sum(M[i][j] * old[j] for j in range(n) if M[i][j]) + return out + +def window_pr1673(T, mats, Es, n, g): + U = [x for x in T if any(x[i] in Es[i] for i in range(g))] + visits = len(U) # one scan forming U + S = {x: T[x] for x in U} + for i in range(g): + visits += len(U) # PHASE-LAYOUT-AUDIT step 1: scan U once per axis + Ui = [x for x in U if x[i] in Es[i]] + lines = {} + for x in Ui: lines.setdefault(x[:i] + x[i + 1:], []).append(x) + for key, xs in lines.items(): + xs.sort(key=lambda x: x[i]); old = [S[x] for x in xs]; Ei = [x[i] for x in xs] + for a, x in enumerate(xs): S[x] = sum(mats[i][x[i]][Ei[b]] * old[b] for b in range(len(xs))) + visits += len(Ui) # merge back + T = dict(T); T.update(S) + return T, visits, len(U) + +def window_radix(T, mats, Es, n, g): + F = {x: [i for i in range(g) if x[i] in Es[i]] for x in T} + S = {x: T[x] for x in T if F[x]} + visits = len(S); r = 1 + while True: + Sr = [x for x in S if len(F[x]) >= r] + if not Sr: break + visits += len(Sr) + groups = {} + for x in Sr: + f = F[x][r - 1]; groups.setdefault((f, x[:f] + x[f + 1:]), []).append(x) + new = {} + for (f, key), xs in groups.items(): + xs.sort(key=lambda x: x[f]); old = [S[x] for x in xs]; Ef = [x[f] for x in xs] + for x in xs: new[x] = sum(mats[f][x[f]][Ef[b]] * old[b] for b in range(len(xs))) + S.update(new); r += 1 + T = dict(T); T.update(S) + return T, visits + +rng = random.Random(2044) +res = [] +for g, n, W, B in [(2, 12, 1, 2), (3, 10, 1, 2), (4, 8, 1, 2), (3, 12, 1, 4), (4, 9, 1, 4), (5, 6, 1, 2)]: + ax = [axis_factors(n, W, B, rng) for _ in range(g)] + V = n**g + X = {x: Q(rng.randint(-50, 50)) for x in itertools.product(range(n), repeat=g)} + ref = X + for i in range(g): ref = apply_axis(ref, ax[i]['H'], i, n, g) + def run(window): + T = X; vis = 0; dense_calls = 0; scans = 0 + T = {**T} + for i in range(g): T = apply_axis(T, ax[i]['dense'][0], i, n, g) + dense_calls += 1 + for w, key in [(1, 'win1'), (2, 'win2')]: + mats = [a[key] for a in ax]; Es = [a['E'] for a in ax] + out = window(T, mats, Es, n, g); T = out[0]; vis += out[1]; scans += 2 # gather + scatter + if w == 1: + for i in range(g): T = apply_axis(T, ax[i]['dense'][1], i, n, g) + dense_calls += 1 + for i in range(g): T = apply_axis(T, ax[i]['dense'][2], i, n, g) + dense_calls += 1 + assert T == ref + return vis, dense_calls, scans + v1, dc, sc = run(lambda T, m, E, n_, g_: window_pr1673(T, m, E, n, g)) + v2, _, _ = run(lambda T, m, E, n_, g_: window_radix(T, m, E, n, g)) + mu = max(len(a['E']) for a in ax) / n + U = sum(1 for x in X if any(x[i] in ax[i]['E'] for i in range(g))) + sumF = sum(sum(x[i] in ax[i]['E'] for i in range(g)) for x in X) + res.append(dict(g=g, n=n, W=W, B=B, V=V, mu=round(mu, 3), g_mu=round(g * mu, 3), U_over_V=round(U / V, 3), + dense_global_calls=dc, full_scans_for_windows=sc, + pr1673_union_visits_per_record=round(v1 / V, 3), radix_union_visits_per_record=round(v2 / V, 3), + sumF_over_V=round(sumF / V, 3), exact_match=True)) + print(json.dumps(res[-1])); sys.stdout.flush() +json.dump(res, open(sys.argv[1], 'w'), indent=1) diff --git a/research/capped-phase-fusion/review20261014/independent/ORIGINAL-OUTER-CONTROLS.json b/research/capped-phase-fusion/review20261014/independent/ORIGINAL-OUTER-CONTROLS.json new file mode 100644 index 0000000000..c0437b7208 --- /dev/null +++ b/research/capped-phase-fusion/review20261014/independent/ORIGINAL-OUTER-CONTROLS.json @@ -0,0 +1,696 @@ +{ + "status": "PASS_INDEPENDENT_ORIGINAL_OUTER_CONTROLS", + "scope": "Original capped/Strassen PR1673 only. Conditional written proof separately reviewed. No native admission or all-size theorem is proved by these finite controls.", + "published_head": "2569871ff906d2a2abfc86e97126ab9afe2e4a5a", + "head_scope": "Historical reviewed head; source hashes below bind the actual current package used in this replay.", + "ledger": { + "original_selected_supplier": { + "b": { + "exact": "67018507072324140917/40000000000000000000000", + "decimal": "0.001675462676808103522925" + }, + "q": { + "exact": "67018503072324140917/40000000000000000000000", + "decimal": "0.001675462576808103522925" + }, + "a": { + "exact": "67018505072324140917/40000000000000000000000", + "decimal": "0.001675462626808103522925" + }, + "epsilon": { + "exact": "1602932029994465428113567090153200000000000000/1605365971725539986393675267555861382073600889", + "decimal": "0.99848387110855575364326776244204516140889758204436646471254886062" + }, + "xi": { + "exact": "3631741843538363603786961767383/4006701850307232414091700000000", + "decimal": "0.90641679346814462477573314876199196952827856807257517068829472760" + }, + "beta": { + "exact": "3749559600000000000000/40067018503072324140917", + "decimal": "0.093582196531855375224266851238008030471721431927424829311705272400" + }, + "derived_kappa": { + "exact": "1672922349588871/1000000000000000000", + "decimal": "0.001672922349588871" + }, + "claimed_kappa": { + "exact": "1672922349588871/1000000000000000000", + "decimal": "0.001672922349588871" + }, + "exponents": { + "main_transform": { + "exact": "160268031909611721506129731833773378546441654700203789/160536597172553998639367526755586138207360088900000000", + "decimal": "0.99832707764041112886753461368005319188061901397179129402425413302" + }, + "global_coordinate_routing": { + "exact": "39932981494927675859083/40000000000000000000000", + "decimal": "0.998324537373191896477075" + }, + "fused_gaussian_groups": { + "exact": 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"113927dd4fb77a55a457bd5f6da8cbfc7262d0fbc4e9d708b591a1d632e6caf6" + }, + "review_source_sha256": { + "check_original_outer.py": "9260ce7328e23423ebc8cd40ab2d5d3d6cfab446cff36c4c03016bb0ffbcfeba", + "ORIGINAL-OUTER-PROOF-REVIEW.md": "bbd941eb6a145d1c85252daee18155fa4203b6056fcaa6dfe8b4675f427b92be", + "SOURCE-PINS.json": "82a59ead584e647958686c3f99289723259f3aa1c176e389da3e3cf484d32dab" + } +} diff --git a/research/capped-phase-fusion/review20261014/independent/ORIGINAL-OUTER-PROOF-REVIEW.md b/research/capped-phase-fusion/review20261014/independent/ORIGINAL-OUTER-PROOF-REVIEW.md new file mode 100644 index 0000000000..8a25536018 --- /dev/null +++ b/research/capped-phase-fusion/review20261014/independent/ORIGINAL-OUTER-PROOF-REVIEW.md @@ -0,0 +1,300 @@ +# Independent review of the original PR1673 outer proof + +Review date: 14 October 2026. Scope: the capped phase-cell/Strassen theorem at published head `2569871ff906d2a2abfc86e97126ab9afe2e4a5a`, including its pinned T74 substitution. This is the original regime, with source-prime gaps of order `1/d`, not a later fine-prime or hybrid outer. The optional theorem assuming `omega_C <= 9/4` is outside this signoff. + +**Conclusion.** The original outer construction supports the stated conditional final saving `0.001677160151714381` with the written qualifications below. The schedule, local circuit construction, Gaussian estimates, precision allowance and movement ledger can be proved independently of the parameter checker. There is a real citation/integration repair: the current T74 child list is mixed rank, so the literal uniform tree in `notes/compact-control-layout.tex` is insufficient. The weighted proof in `notes/endpoint-gauge-complex.tex` (the subsection beginning at line 186 at the reviewed head) and `notes/batched-complex-rows.tex` supplies the needed replacement. The full row stock must be used; a one-chunk row estimate is not justified. These are repairs to the proof's interface explanation, not a different exponent optimization. + +This review does not independently admit the inherited T74/#1319 native supplier or prove its all-size semantic compiler. Those are separate prerequisites. In particular, finite replay of matrices or rational inequalities cannot prove the all-size theorem. The conclusion remains conditional on the exact common-frame mutable-source realization, whole-call hosting, linear semantic guard, fixed-tape kernels, prime packing and exact recovery stated in the published theorem. The reviewer request being addressed is [the PR1673 comment](https://github.com/CrocSwap/integer-mult-bounds/pull/1673#issuecomment-6172528048), fetched directly through the GitHub API. Its supplier admission part is handled separately in `../REVIEW-RESPONSE.md` and `../NATIVE-SUPPLEMENT.json`. + +## Source boundary and pins + +The primary object reviewed end to end is `research/capped-phase-fusion/phase_fusion_note.tex`. Its substantive publication commit is `d449b822c8a74ffa01b22058cfc042374b1a45cd`; the reviewed head adds the source/authorship presentation commit above. The package records publication base `a5d4f08ac102d1a6da301d89379c75285eca680e` and original research base `e6f9716621f4e01fc190617934cdfaf072bd976f`. + +The complex supplier is PR1661 at `f852af5e178ce035cd46181a29d9330bcbcaa206`, source-manifest SHA256 `d90d49eaa68e48c6992a07972e736ef4b9cd80274361cbbc1ec980d43f29ad7f`. The published substitution uses the exact rational lower rate `b = 0.00167971337478`, not the displayed endpoint as an equality. Its native parameters are + +``` +m = 73, Wc = 66304, +H = {1:1559766, 2:337668, 3:278058, 4:84912, + 5:54612, 8:5328, 48:22748}. +``` + +The independent checker in this directory computes SHA256 for the actual proof and 22 supporting source files; see `source_sha256` in `ORIGINAL-OUTER-CONTROLS.json`. The package verifier runs it in a temporary copy, where it writes its own receipt. Source hashes bind this review to the files read, while Git pins bind their intended provenance. The new external layout review is pinned separately by the parent at philippeVV commit `d1e01310b92e9a9af90b89efe4f3f9aad466b8b1`; its row-stock invoice is explicitly distinguished below from our independently checked consequences. + +The proof depends on the following specific source statements, rather than on the status label of an old receipt: + +| Interface | Primary reviewed source and necessary content | +| --- | --- | +| Local reversible linear circuits | `SHALLOW-TOEPLITZ.md`: dirty Toeplitz shears, parallel triangular recursion, displacement generators, SPD extension and prefix bounds | +| Literal common tensor schedule | `TENSOR-COMPILER.md`: Beneš cycle coloring, fixed six-H words, direct Kronecker multiplication and odometer coefficient scans | +| Whole H call | `SUPPLIER-SUPPORT-AUDIT.md`, `SUPPORT-ADVERSARIAL-AUDIT.md`, `SUPPORT-ROUTER.md`: complete global ranges, support-local routing, dirty reservations, finite guard inequality and suffix construction | +| Mixed recursive widths | `notes/endpoint-gauge-complex.tex` and `notes/batched-complex-rows.tex`: residual rank concatenation, child volume `1/Wc`, actual longest depth and weighted stopped tree | +| Rounded narrow operator | `LOG-A-PHASE-SCHUR-AUDIT.md` and `references/semantic-bulk/rad20/reports/review-bulk-resampling.md`: one fixed structured rounding, strict row gap, locality, normalization and residual bounds | +| Broad operator | `BILATERAL-LDU.md`, `FORWARD-SMALL-A-AUDIT.md`: finite Gaussian conditioning, SPD Toeplitz core, identity-hole completion and forward-only Schur use | +| Capped factorization | `CAPPED-INTERIORS.md`, `CAPPED-GAUGE-AUDIT.md`: dyadic tail carving, unchanged rounded operator and anchored ratio range | +| Movement | `PHASE-LAYOUT-AUDIT.md`, `HALO-THEOREM.md`: complete extracted rectangles, two union windows, actual q-image ordering and source-driven sparse faces | + +Some older supporting documents discuss the uncapped regime or label intermediate work prospective. They are used only for the identities actually cited. The new capped theorem must discharge the extra tensor, cap-gauge and layout steps; the proofs below do so. The `OUTER-AUDIT.md` ledger alone does not discharge them. + +## 1. Exact scale and address model + +Let `p = Theta(log n)` and let `V = Theta(n)` count **bits**, not coefficient records. Each record occupies `Theta(p)` bits. The inherited packing has `D=d-1=Theta(p^epsilon)` true source axes and dyadic target axes with + +``` +log s_i = Theta(p/d), +rho_i = t_i/s_i = 1+theta_i, +1/(4d) <= theta_i <= 1/(2d-1), +product_i t_i / product_i s_i < 2. +``` + +The true axes eventually exceed every fixed polynomial in `p`. Thus a `K=Theta(log p)` complete neighborhood at a selected local bit fits in one original axis, even when the group size `g` exceeds the bitlength of that axis. The total selected neighborhoods and control support are `O(g polylog p)=o(p)` because `beta<1`. No step needs `gK` bits within one axis. + +Use `alpha=ceil(sqrt((4/pi) log(16p)))`, `u=alpha^2=Theta(log p)`, `a=pi u`, cap `B=2^floor(xi log2 p)`, core side `L=2^ceil(8 log2 p)`, group scale `g=floor(p^beta)`, physical bandwidth `W=O(sqrt(p) polylog p)` dominating **both** Gaussian maps, and locality radius `A=C_A p^3`. Constants `Wc` (supplier rows) and `W` (Gaussian bandwidth) are unrelated. + +Partition the `D` axes into groups of size between `g/2` and `g`, for example by evenly splitting into `ceil(D/g)` consecutive groups. This harmless explicit choice avoids an unamortized last group of size one. All groups have `Theta(g)` axes because `beta= G+4 ceil(log2 p)+10`; our choice satisfies it. It does not satisfy the stronger literal phrase `K/log p -> infinity`, so that phrase must not be cited as the hypothesis being used. + +At current selected width `h`, expose complete neighborhoods by extending their prescribed placement to a permutation supported on at most `2hK` bit positions. The support router pays `O(V h^(1-a_eff) polylog p)`. Arbitrary intervening positions are spectators. A fixed coarse microbox lacks complete borrowed high bits; the H invocation must run on the complete global dyadic stream. Coarse labels may affect the diagonals between H stages, and are restored before those diagonal scans. + +Reserve still-unprocessed chunks for the two front controls, one back control and complete row stock. Do not first transform their selected bits. At the admitted completed-call boundary, the active invocation is the desired tensor kernel on active bits and identity on all reserved/spectator coordinates, for arbitrary incoming data and dirt. This is exactly the property that allows reservation to be deferred. It is not a clean-work assumption. Row padding is added once for a phase, all descendants inherit it, and padding is removed after return and before another phase. Thus the temporary factor below two is not multiplied over axes or phases. + +For T74 the longest recursive child is `48 floor(e/73)`. In particular, two child steps shrink at least by half since `2*48^2 < 73^2`; the actual depth `Dc(h)` is at most `2 log2 h+O(1)`. The row count must be divisible by the **product** of complex and nested bit stock factors sufficient for their respective depths. A maximum of the two stocks is insufficient. The corrected external invoice uses 2644 stock bits and 490 physical rows beyond live storage, giving row coefficient 19545 below 20161. These are fixed native quantities; they must not be mistaken for a proof of a 39-chunk aggregate row prefix before mixed and nested depths have been included. + +For the asymptotic proof no small row constant is needed. Each supplier has a fixed stock factor and logarithmic depth. Even the conservative product over every nested bit depth along a complex stack has `log2(Q_rows)=O(log^2(hG))`. A complete K-bit row chunk supplies K binary row digits, so its chunk reservation is + +``` +q_rows = ceil(log2(Q_rows)/K) + = O(log(h+1)+log log(p+2)+1), h<=p, +R(h) <= 3h/128+O(log(h+1)+log log(p+2)+1). +``` + +Thus `R(h)<=h/16` above a fixed polynomial in log p; for example an adequately large fixed multiple of `(log p+1)^2` suffices. Process the smaller remaining selected width elementarily at polylogarithmic cost. The independent checker illustrates this consequence for an arbitrary conservative fixed stock coefficient and checks actual depth samples; it does not assert the unproved specific 39-chunk or 1050 threshold. Native row hosting and the physical invoice remain in the separate supplier admission. Nothing requires the erroneous one-row-chunk estimate. + +Here is the required mixed-tree argument. At a node of width `e`, first process the `e mod 73` remainder axes individually; this costs `O(V_node)` and returns those axes as complete spectators. A rank-r residual creates a recursive call of width `r floor(e/73)` on one role with `V_node/Wc` volume. Signs and inverse slots use the admitted exact wrappers; they do not change the child rank. Let + +``` +Psi(t) = sum_r (H_r/Wc) (r/73)^t. +sigma = 1-b, tau=1-a_eff, delta=10^-9. +``` + +For each level, the sum of `V_node * e_node^sigma` is at most `Psi(sigma)` times the preceding level. Floors only decrease that sum. If `Psi(sigma)<1`, summing all levels is a geometric series, and the total stopped-leaf sigma-mass is at most `V_root h^sigma`. Leaves have width `tau`, every internal width is at least `h^delta` and `e^tau <= h^[delta(tau-sigma)] e^sigma`; the total exponent is therefore + +``` +chi = tau+(1-delta) max(sigma-tau,0). +``` + +This proves the bound for unequal depths. It does not replace every rank-r child by a rank-one child or apply a uniform-tree estimate to rank 48. In the original and T74 published choices `a_eff=b-5e-11`, so `sigmaa_eff>q`, including the strict leaf margin. Consequently a whole active phase costs `O(V h^(1-q) polylog p)`. The deferred widths satisfy `h_next<=h/16`, and summing this bound plus support routing over phases is geometric. Thus a complete global selected-bit call costs + +``` +O(V g^(1-q) polylog p), +``` + +including exposure, all internal adapters, row preparation, spectators, dirt and restoration under the native premises. The admitted kernel is `H0=H/2`; one exact `2^g` binary shift after the call supplies the raw H used below, at `O(V)` work and `O(g)` guard. The independent rational Taylor enclosure checks `Psi(1-b)<1` for the stated T74 rate, and rejects a specified rate just above the endpoint. This arithmetic supports the hypothesis but does not prove native admission. + +The same deferred argument applies to the retained main synthetic transform: there are `Theta(p/d)` common rounds and the paid global coordinate layout. Its total is `O(V[p^(1-a_eff)+p d^-q] polylog p)`. The old power-width reservation constraint is removed. Retaining an old per-axis `Vd` movement or scalar pass would invalidate the stronger result; it is an explicit rejecting control below. + +## 3. Local linear circuits and tensorization + +The new dense work is on the existing `L` wires of each local axis. The local compiler needs no clean data coordinate. For a dirty shear `y+=T x`, split Toeplitz `T` into circulant and negacyclic parts, use FFTs on the two existing vectors, pointwise shears and inverse FFTs. No inverse Fourier multiplier is taken, so singular T is permitted. + +For triangular Toeplitz multiplication, split `f(z)=e(z^2)+z o(z^2)`. Its block matrix is `[[E,ZO],[O,E]]`, with the commuting block LDU factors using `k=o/e` and child `s=e-z o^2/e`. Both child multiplications run in parallel. This gives depth `O(log^2 L)` rather than a sequential recurrence. For a full Toeplitz block, balanced elimination raises child displacement rank by at most one. A rank-`O(log L)` displacement representation yields `T=sum L(g_j) U(h_j)`. For a generator term choose `gamma>=2||h_j||_1`, set `P=I+U(h_j)/gamma` and perform + +``` +x <- P x; y += gamma L(g_j) x; +x <- P^-1 x; y -= gamma L(g_j) x. +``` + +The net is `y+=L(g_j)U(h_j)x` and x is restored. The triangular map P is uniformly close to I. Partial sums of generators are bounded additively, so their prefix bound is polynomial, not a product of one bound for every generator. Balanced recursion gives `O(log^4 L)` two-wire layers with polynomial prefix/inverse-prefix norms for near-identity matrices. For SPD `mu I<=G<=M I`, principal and Schur descendants retain these bounds. When `M` and `1/mu` are polynomial in p the logarithmic prefix bound is `O(log^2 p)`, also sufficient. + +For arbitrary pair matchings, the degree-two bipartite graph of input and output pairs is a union of even cycles. Alternating edge colors gives the first/last Beneš switches and recurses on half as many wires. Thus every matching is routed by `2m-1` switch stages at `L=2^m`. The routing and reverse routing use `4(2m-1)` H factors. Every allowed local two-wire gate has a fixed six-H diagonal word, including identities and zero shears. In particular, + +``` +diag(-1/8,-1/6) H diag(2,1) H diag(-3,1) H = U(5/4). +``` + +For `U(t)`, write `U(t)=U(t+c)U(-c)` with a known `c>2|t|+1`; no division by t occurs, and both factors have controlled inverses. Therefore each original matching layer uses `8m+2` common H stages. Completed route/padding words are exact endpoint identities or permutations. At a compiled prefix only one constant-length gate word is partial, so its bound multiplies the original circuit prefix by a polynomial, not that polynomial raised to the number of layers. + +The g axis circuits may differ by axis and coarse label. Pad their **operations** to the common schedule, tensor corresponding H factors and multiply the corresponding diagonal factors. This is the exact Kronecker product identity, not a sum over branches; data dimension remains `L^g`. There is no `2^g` data padding. + +For each tensor diagonal `product_i d_i(x_i)`, cache suffix products in lexicographic odometer order. A carry of depth k occurs at most `L^g/L^k` times and recomputes O(k) products. The geometric sum `sum k/L^k` is finite. The coefficient tape visits and rewinds have sum `sum k L/L^k=O(1)` per record. Hence diagonal scans cost `O(V polylog p)`, not `Vg`. Regenerate each polynomial page from its actual coarse origin and amortize it over that block. No random catalogue access is assumed. + +There are only polylog(p) compiled H calls. Section 2 prices each at `O(V g^(1-q) polylog p)` and each gather at `O(V g^(1-a_eff) polylog p)`. This is the dense group cost. The global completeness restriction survives coarse-dependent diagonals because placement is undone before each such scan. + +## 4. The two Gaussian maps and the fixed rounded operator + +The narrow inverse and broad forward matrix require different arguments. Treating them as the same near-identity matrix would be an error. + +For narrow compression let `q_j=floor(rho j+1/2)` and `beta_j=rho j-q_j`. The retained selected normalized Gaussian N has + +``` +||N-I||_infinity <= exp(-2 a theta)+5 exp(-2a). +``` + +The two nearest directions must be paired in deriving this bound. Since `a>=4 log(16p)` and `theta>=1/(4p)`, there is row gap at least `1/(8p)` eventually. Set the diagonal to one and choose one global structured rounding: downward weights `omega_hat_j`, one rounded Toeplitz kernel `G_hat` within physical cells, exact entries `omega_hat_j G_hat_(j-k)/omega_hat_k` within a cell and the rounded original cross-cell entries elsewhere. With the retained `P=32768p` coefficient accuracy, the perturbation bound + +``` +p 2^(16p+7) 2^-P + 2^-46p < 1/(16p) +``` + +gives a rounded H with row gap `1/(16p)`, row norm below two and inverse norm at most `16p`. All principal submatrices inherit the gap. H is not silently replaced by an infinite or freshly rerounded Toeplitz matrix. + +On the complete target grid use `Haug=EHC+(I-EC)`, where E injects at the increasing q-image and C selects it. This is permutation-similar to `H direct_sum I`, so holes are genuine identity coordinates and all inverse bounds persist on arbitrary signed input, including numerical hole errors. Physical target bandwidth is at most twice the source bandwidth plus two. + +For broad expansion the actual rectangular map is `S_kj=alpha^-1 exp[-b0(j-lambda k)^2]`, `b0=pi/alpha^2`, `lambda=1/rho`. Complete it to a square target map M by giving each hole input an identity column. In selected/hole order M is block triangular with lower block `[R,I]`. Gaussian lattice-sum bounds give `||M||<3` eventually; `M/4` is contractive even on hole errors. The additional `2^-D` versus the old `S/2` normalization is compensated once by an exact `2^D` shift. + +Inside a physical phase cell, a selected block has entries `alpha^-1 exp[-b0(i-c-lambda k)^2]`. The finite Gaussian LDU estimate in `BILATERAL-LDU.md` applies to every contiguous subblock: its ordered q'-coordinates have unit gaps and its phase beta' remains within one half, even when q' is not integral. The product `c_b=product_(m>=1)(1-exp(-2 b rho m))` obeys + +``` +log(c_b^-1) <= pi^2/(12 b rho), +||K^-1|| <= alpha exp(3b/4) c_b^-4 /(1-exp(-b rho theta))^2. +``` + +Here `b=pi/(alpha^2 rho^2)`, `alpha^2=O(log p)` and `theta^-1=poly(p)`, so this is polynomial in p. The anchored Toeplitz kernel `G_yx=exp[-b0 lambda(x-y)^2]` has symbol between `sqrt(pi/gamma)exp[-pi^2/(4gamma)]` and `1+sqrt(pi/gamma)`, with `gamma=b0 lambda`. Both bounds are polynomial in p. Symmetric sufficiently accurate rounding and truncation preserve SPD and its polynomial condition bound. This is the SPD case of the local compiler. For the complete broad forward factorization only the application of its Schur complement is required; no forward Schur inverse is needed. + +For narrow elimination, the absolute row gap cannot decrease at a scalar Schur step: + +``` +gap_i(new) >= gap_i(old)+|a_ik/a_kk| gap_k(old). +``` + +The absolute row sum cannot increase, by the same triangle inequality and the pivot row gap. Pivots remain positive. Thus the narrow Schur complement and principal interior inverses have polynomial bounds; rounding below `1/(32p)` preserves them. Keeping whole neighboring boundary groups avoids an unjustified prefix-solve gap assertion. + +## 5. Capped gauge, five factors and the two sparse windows + +Within each original phase cell reserve W endpoints; greedily take dyadic interiors of size at most B separated by W; finish by halving the residual tail until O(W) remains. There are `O(d/B+log B)` pieces, so boundary fraction is `O((W/B) polylog p)`. Eliminating an interior connects only its two neighboring O(W) boundary groups. Therefore Schur bandwidth is O(W), independent of the number of caps. All holes and physical period cuts are boundary coordinates. Interiors are actual consecutive selected coordinates, not a masked Toeplitz matrix. + +For narrow weights `omega_j=exp[-a(1/4-beta_j^2)/theta]`, one-step logarithmic variation is `2a beta_j+a theta=O(a)`. Recenter on the cap anchor using the **same rounded weights**: the common `omega_hat_m` cancels exactly from the similarity. The range is `O(aB)`, while the old absolute `O(ad)` scale is paid once in coefficient construction. For broad row/column gauges, the anchored exponent range is `O((B+B^2/d)/alpha^2)=O(B/alpha^2)`, bounded by O(aB). Direct sums of capped blocks take the maximum depth and range, not the sum over caps. + +Write `Q=[[A,B1],[C1,D1]]`, `F=D1-C1 A^-1 B1`. Direct block multiplication proves + +``` +Q^-1 = diag(A^-1,I) E_-B1 diag(A,F^-1) E_-C1 diag(A^-1,I), +Q = diag(A,I) E_C1 diag(A^-1,F) E_B1 diag(A,I). +``` + +These use the original banded B1,C1, not dense A^-1 B1 or C1 A^-1 disguised as sparse shears. In execution there are three dense A/A^-1 rounds and two sparse windows; the middle dense and boundary diagonal commute. Define Ei to include boundary Ji and all interior coordinates incident to original B1,C1. Then `|Ei|/L=O((W/B) polylog p)` and every sparse factor is identity off Ei. The union `U={x: some xi in Ei}` is invariant under every sparse factor and contains the complete rectangle `Ui=Ei x [L]^(g-1)` for each i. It is not invariant under dense A. Scatter before the middle dense stage and regather afterward. + +Keep U in its native local lexicographic order. For axis i, scan U and extract Ui, relabel Ei by increasing stream-counter ordinal, transpose only this at-most-L-valued field, apply the boundary operation, undo the small-field transpose and ordered-merge it back. No missing values are invented as zero. Only the current polynomial field is padded below factor two, and padding is deleted before the next field. One window costs `O(V g^2 W/B polylog p)` per group; two windows only change the constant. Across `Theta(d/g)` groups the charge is `O(V d g W/B polylog p)`. Forming and scattering U costs a fixed number of full scans per group. + +For boundary arithmetic batch independent right-hand sides in a dyadic width between W and 2W. The remaining coordinates provide superpolynomially many RHS. A width-W triangular block update uses two W-square products; original boundary couplings and the broad Schur application use the same batched products. Strassen's exact seven-product identity has `omega=log2 7<45/16`, proved here by integer comparison `7^16<2^45`. The number of boundary rows is `O((W/B)L polylog p)`. Hence the whole d-axis boundary charge is + +``` +O(V (d/B) W^(omega-1) polylog p). +``` + +V is bit volume, so a scalar product's `M(p)` cost is divided by the `Theta(p)` record width: the short-arithmetic multiplier is `M(p)/p=polylog p`. No improved integer multiplication result is used recursively here. The factor pages and RHS batches are read sequentially with a fixed number of tapes, and only the polynomial row/batch fields are transposed. + +Round after each completed block update. A fixed-depth seven-product recursion has polynomial growth in W; exact dyadic products need O(p+log W)=O(p) bits. The residual of each completed local system is `poly(p)2^-P(1+||X||)`. Polynomial inverse bounds then control the solution error uniformly. This is a residual argument for each polynomial-size local window, not a claimed stable floating-point algorithm and not a product of amplification over the global number of windows. + +## 6. Embedding order, halos and precision closure + +The q-image scan is exact and monotone. For target k put `j=max(0,ceil(s(2k-1)/(2t)))`; it is in the image exactly when `j=2: x/=2; k+=1 + l,h=log_unit(x); l2,h2=log_unit(F(2)) + return l+k*l2,h+k*h2 + +def exp_interval(x,N=24): + assert 0<=x<1 + lo=sum((x**j/F(factorial(j)) for j in range(N)),F(0)) + return lo,lo+x**N/F(factorial(N))/(1-x/F(N+1)) + +def moment_interval(b, profile): + lo=hi=F(0) + for r,n in profile.items(): + l,h=log_interval(F(73,r)) + elo=exp_interval(b*l)[0]; ehi=exp_interval(b*h)[1] + weight=F(n*r,73*66304) + lo+=weight*elo; hi+=weight*ehi + # Outward rounding is exact; keep the portable receipt compact. + grid=10**70 + return F((lo*grid).__floor__(),grid),F((hi*grid).__ceil__(),grid) + +def eye(n): return [[F(i==j) for j in range(n)] for i in range(n)] +def mm(a,b): return [[sum((a[i][k]*b[k][j] for k in range(len(b))),F(0)) for j in range(len(b[0]))] for i in range(len(a))] +def sub(a,b): return [[x-y for x,y in zip(ar,br)] for ar,br in zip(a,b)] +def inv(a): + n=len(a); x=[list(row)+ident for row,ident in zip(a,eye(n))] + for k in range(n): + p=next(i for i in range(k,n) if x[i][k]) + x[k],x[p]=x[p],x[k] + c=x[k][k]; x[k]=[z/c for z in x[k]] + for i in range(n): + if i!=k: + c=x[i][k]; x[i]=[z-c*t for z,t in zip(x[i],x[k])] + return [row[n:] for row in x] +def block(a,b,c,d): return [x+y for x,y in zip(a,b)]+[x+y for x,y in zip(c,d)] +def zeros(n,m):return [[F(0) for _ in range(m)] for _ in range(n)] +def diagblock(a,d):return block(a,zeros(len(a),len(d)),zeros(len(d),len(a)),d) +def chain(*args): + out=eye(len(args[0])) + for a in args: out=mm(out,a) + return out + +def five_factor_controls(): + entries=0 + for seed in range(1,25): + q=[[F((17+seed+i) if i==j else ((i+2*j+seed)%5-2),8) for j in range(4)] for i in range(4)] + a=[row[:2] for row in q[:2]]; b=[row[2:] for row in q[:2]] + c=[row[:2] for row in q[2:]]; d=[row[2:] for row in q[2:]] + ai=inv(a); schur=sub(d,chain(c,ai,b)); si=inv(schur) + e_b=block(eye(2),b,zeros(2,2),eye(2)); e_c=block(eye(2),zeros(2,2),c,eye(2)) + minus_b=block(eye(2),[[-v for v in r] for r in b],zeros(2,2),eye(2)) + minus_c=block(eye(2),zeros(2,2),[[-v for v in r] for r in c],eye(2)) + forward=chain(diagblock(a,eye(2)),e_c,diagblock(ai,schur),e_b,diagblock(a,eye(2))) + backward=chain(diagblock(ai,eye(2)),minus_b,diagblock(a,si),minus_c,diagblock(ai,eye(2))) + assert forward==q and backward==inv(q) + # Dropping the middle dense stage is an actual rejected schedule. + assert chain(diagblock(ai,eye(2)),minus_b,diagblock(eye(2),si),minus_c,diagblock(ai,eye(2)))!=inv(q) + entries+=32 + return {'systems':24,'exact_matrix_entries':entries,'missing_middle_dense_rejected':True} + +def q_image_controls(): + cases=addresses=0 + for power in range(2,10): + t=2**power + for s in range(t//2+1,t): + image=[(2*t*j+s)//(2*s) for j in range(s)] + assert len(set(image))==s and image==sorted(image) + found=[] + for k in range(t): + # ceil(s(2k-1)/(2t)), clipped only at zero. + j=max(0,-(-s*(2*k-1)//(2*t))) + if j=N: continue + for g in range(1,6): + # Exact input-rectangle sum and telescoped identity, in tile units. + rectangles=6*A*sum(N**j*(N+2*A)**(g-1-j) for j in range(g)) + assert rectangles==3*((N+2*A)**g-N**g) + if g<=3: + seen=set() + for j in range(g): + for side in (0,1): + intervals=[range(A,N-A) if i=N-A for t in x)} + assert seen==actual and len(seen)==N**g-(N-2*A)**g + cases+=1 + return dict(input_volume_identities=cases,first_face_partition_exact=True) + +def depth_and_reservation_controls(): + def d(e): + out=0 + while e>=73: e=48*(e//73); out+=1 + return out + # Each two maximal-child steps at most halve width: 48^2*2<73^2. + assert 2*48**2<73**2 + depth=[] + for L in (10,100,1000,10000): + actual=d(2**L); assert actual<=2*L + depth.append([L,actual]) + # Deliberately do not infer a row-chunk constant from the one-level + # physical invoice. Mixed and nested depths must be included. Illustrate + # the all-size argument with an arbitrary fixed conservative stock + # coefficient C: log2(stock)<=C ceil(log2(hG))^2. For h<=p the divided + # row reservation is O(log p); a polynomial-log tail is sufficient. + C=10**4 + rows=[] + for lp in (64,128,512,4096,10**6): + G=4*lp+6; K=128*G + h=32*C*(lp+16)**2 + assert h<=2**lp + lhg=(h*G).bit_length() + row_chunks=(C*lhg**2+K-1)//K + reserve=(2*h+127)//128+(h+127)//128+row_chunks + assert F(reserve)<=F(h,16) + rows.append(dict(log2p=lp,h=h,row_chunks=row_chunks,reserved=reserve)) + return dict(maxchild=48,dimension=73,two_step_halving=True, + exact_depth_samples=depth, + conservative_stock_family='log2(stock)<=C ceil(log2(hG))^2, C fixed', + illustrative_fixed_C=C,polylog_tail_samples=rows, + no_39_chunk_or_1050_threshold_assertion=True) + +def source_manifest(): + relative=[ + 'research/capped-phase-fusion/phase_fusion_note.tex', + 'research/capped-phase-fusion/SHALLOW-TOEPLITZ.md', + 'research/capped-phase-fusion/TENSOR-COMPILER.md', + 'research/capped-phase-fusion/SUPPLIER-SUPPORT-AUDIT.md', + 'research/capped-phase-fusion/SUPPORT-ADVERSARIAL-AUDIT.md', + 'research/capped-phase-fusion/SUPPORT-ROUTER.md', + 'research/capped-phase-fusion/LOG-A-PHASE-SCHUR-AUDIT.md', + 'research/capped-phase-fusion/BILATERAL-LDU.md', + 'research/capped-phase-fusion/FORWARD-SMALL-A-AUDIT.md', + 'research/capped-phase-fusion/CAPPED-INTERIORS.md', + 'research/capped-phase-fusion/CAPPED-GAUGE-AUDIT.md', + 'research/capped-phase-fusion/PHASE-LAYOUT-AUDIT.md', + 'research/capped-phase-fusion/HALO-THEOREM.md', + 'research/capped-phase-fusion/OUTER-AUDIT.md', + 'research/capped-phase-fusion/PUBLICATION-SOURCES.json', + 'research/capped-phase-fusion/frontier_fusion/INDEPENDENT-SUBSTITUTION-REVIEW.md', + 'research/capped-phase-fusion/evidence/radical_reduction_ultra/README.md', + 'notes/endpoint-gauge-complex.tex', + 'notes/batched-complex-rows.tex', + 'notes/compact-control-layout.tex', + 'notes/compact-control-movement.tex', + 'references/semantic-bulk/rad20/reports/review-bulk-resampling.md', + 'references/semantic-bulk/rad20/reports/review-phase-cell-inverse.md', + ] + def resolve(p): + prefix='research/capped-phase-fusion/' + if p.startswith(prefix): return PKG/p[len(prefix):] + return BASE/p + pins=json.loads((HERE/'SOURCE-PINS.json').read_text()) + for rec in pins['files']: + assert hashlib.sha256(resolve(rec['path']).read_bytes()).hexdigest()==rec['sha256'],rec['path'] + return {p:hashlib.sha256(resolve(p).read_bytes()).hexdigest() for p in relative} + +def main(): + global PKG,BASE + ap=argparse.ArgumentParser(description=__doc__) + ap.add_argument('--package',type=Path,default=HERE.parent.parent, + help='capped-phase-fusion directory (installed default: ../../)') + ap.add_argument('--repository-root',type=Path, + help='optional repository root for the six pinned sources; defaults to bundled source/') + args=ap.parse_args() + PKG=args.package.resolve() + BASE=(args.repository_root or HERE/'source').resolve() + if not (PKG/'phase_fusion_note.tex').is_file(): + ap.error('--package must contain phase_fusion_note.tex') + specs={ + 'original_selected_supplier':(F('0.001675462676808103522925'),F('0.001672922349588871')), + 'published_T74':(F('0.00167971337478'),F('0.001677160151714381')), + } + rows={}; controls={} + for name,(b,k) in specs.items(): + row=ledger(b,k) + assert all(x>0 for x in row['slacks'].values()) + assert row['derived_kappa']==k + assert row['minimum_saving']==row['epsilon']*row['q'] + rows[name]=row + controls[name]={} + for label,args in [('old_axis_scan',dict(old_scan=True)),('full_cell_guard',dict(full_cell=True)), + ('zero_group_guard',dict(zero_guard=True)),('zero_boundary_slack',dict(zero_boundary=True)), + ('complex_endpoint_equality',dict(equality=True)),('excessive_leaf_stop',dict(stop=F(1,10**6)))]: + bad=ledger(b,k,**args) + failed={n:v for n,v in bad['slacks'].items() if v<=0} + assert failed,label + controls[name][label]=failed + profile={1:1559766,2:337668,3:278058,4:84912,5:54612,8:5328,48:22748} + lo,hi=moment_interval(specs['published_T74'][0],profile) + assert hi<1 + overshoot=moment_interval(F('0.001679713374785'),profile) + assert overshoot[0]>1 + assert 7**16<2**45 + out=dict(status='PASS_INDEPENDENT_ORIGINAL_OUTER_CONTROLS', + scope='Original capped/Strassen PR1673 only. Conditional written proof separately reviewed. No native admission or all-size theorem is proved by these finite controls.', + published_head='2569871ff906d2a2abfc86e97126ab9afe2e4a5a', + head_scope='Historical reviewed head; source hashes below bind the actual current package used in this replay.', + ledger=rows,negative_controls=controls, + mixed_moment=dict(profile=profile,m=73,W=66304,interval=[lo,hi],strict_gap=1-hi,overshoot_lower=overshoot[0]), + matrix_identity=five_factor_controls(),q_embedding=q_image_controls(), + sparse_union=sparse_union_controls(),halos=halo_controls(), + row_reservation=depth_and_reservation_controls(),source_sha256=source_manifest(), + review_source_sha256={p.name:hashlib.sha256(p.read_bytes()).hexdigest() + for p in [HERE/'check_original_outer.py',HERE/'ORIGINAL-OUTER-PROOF-REVIEW.md',HERE/'SOURCE-PINS.json']}) + (HERE/'ORIGINAL-OUTER-CONTROLS.json').write_text(json.dumps(enc(out),indent=2)+'\n') + for name,row in rows.items(): + print(name,enc(row['derived_kappa'])['decimal'],len(row['slacks']),'strict checks') + print(out['status']) + +if __name__=='__main__': main() diff --git a/research/capped-phase-fusion/review20261014/independent/source/notes/batched-complex-rows.tex b/research/capped-phase-fusion/review20261014/independent/source/notes/batched-complex-rows.tex new file mode 100644 index 0000000000..e13bd96eb2 --- /dev/null +++ b/research/capped-phase-fusion/review20261014/independent/source/notes/batched-complex-rows.tex @@ -0,0 +1,141 @@ +\subsection{The complex layer with unequal recursive axis counts} +\label{sec:mixed-complex-layer} + +Apply the same integer-depth construction to the complex circuit's list +of child sizes. For a selected residual of rank $a\tau$, every internal width is at least $d^\beta$, so +$e^\tau\le d^{\beta(\tau-\sigma)}e^\sigma$; summing the strict +$\sigma$-moment gives +$O(G^\tau d^{\sigma+\beta(\tau-\sigma)})$. +Thus in both cases, with +\[ + \chi=\tau+(1-\beta)\max\{\sigma-\tau,0\}, +\] +the complete normalized layer cost is +\[ + O\!\left((\log p)^3\left[ + d^\chi+d^{\sigma+\beta(1-\sigma)} + +d^{\max\{1-c,0\}}+1\right]\right). +\] +The displayed logarithmic factor conservatively retains the existing +layout and descriptor allowances. Hence the original strict conditions +\[ + \max\{\tau,\sigma,\chi\}<\lambda<\lambda'<1, + \qquad + \max\{\sigma+\beta(1-\sigma),1-c,0\}<\lambda' +\] +continue to imply the complete $O(Vd^{\lambda'})$ layer bound. + +\paragraph{Compatibility with the new dependency guard.} +The separate dependency-path estimate uses $q=m+6h=22120$, observes that +one path crosses at most one selected bulk residual, and bounds its +$\rho$-moment by +\[ + \max\left\{ + \frac q{m^\rho},\quad + \frac{q-a_1}{m^\rho}+\left(\frac{a_1}{m}\right)^\rho,\quad + \frac{q-a_2}{m^\rho}+\left(\frac{a_2}{m}\right)^\rho + \right\}<\frac{999}{1000} + \quad\left(\rho=\frac65\right). +\] +Integer floors only shrink these children. For sufficiently large $d$, +an internal parent has $e\ge d^\beta\ge2m$, hence every recursive child +has width at least $\lfloor e/m\rfloor\ge d^\beta/(2m)$ whenever it +becomes a stopping leaf. The at most $m-1$ individually processed +remainder axes at each internal node belong to its fixed local overhead; +add their bounded coefficient contribution to the guard's fixed constant +$E$. They are not extra arbitrarily small stopping leaves. Thus the +separately established guard exponent +\[ + C_1=\rho-(\rho-1)\beta+\zeta + =\frac65-\frac\beta5+\zeta +\] +applies to this integer-width implementation. All stopping tests can be +performed by the existing rational-power integer comparison. diff --git a/research/capped-phase-fusion/review20261014/independent/source/notes/compact-control-layout.tex b/research/capped-phase-fusion/review20261014/independent/source/notes/compact-control-layout.tex new file mode 100644 index 0000000000..16f4d23347 --- /dev/null +++ b/research/capped-phase-fusion/review20261014/independent/source/notes/compact-control-layout.tex @@ -0,0 +1,118 @@ +\paragraph{Rows and complete compact fields.} +Let $m,W,s$ denote the complex network constants and keep the layer's global +$p,d,K$ at every node. Put +\[ + G=4\lceil\log_2p\rceil+6,\quad H=dG,\quad + q_F=\left\lceil\frac{2H}{K}\right\rceil,\quad + q_B=\left\lceil\frac{H}{K}\right\rceil, +\] +\[ + q_0=\lceil\log_2W\rceil\lceil\log_m(2d)\rceil, + \quad k_0=\lceil\log_m d\rceil,\quad Q_0=q_0+q_F+q_B. +\] +If $D\le Q_0$, apply all $D$ selected kernels individually. Otherwise, +process individually the first $q_0+q_F$ and last $q_B$ selected axes. +This uses existing coordinates only. The remaining active axes are a +consecutive middle block of length $D-Q_0$. + +Regard the first $q_0$ chunks as the row index. Immediately after it, the +next $q_F$ chunks have at least $2H$ bits. Reinterpret these consecutive +bits as two $H$-bit front fields and a spectator remainder; a remainder +of length zero is omitted. The last $q_B$ chunks similarly supply one +$H$-bit back field and a spectator remainder. These reinterpretations move +no records and preserve the total address space. For a call with $f$ axes +per slot, carve $nG=(f-1)G\le H$ consecutive bits from each of these +fields; collapse their unused bits into spectators. The resulting field +list has a fixed number of entries. Both front fields precede all active +slots and the back field follows them, so the compact-control proposition +applies to every ordered pair of slots. For $f=1$ no compact fields are used. + +The original row range has length $R_{\rm row}=2^{q_0K}\ge W^{k_0}$. +For each original prefix, pad this row range to the next multiple of +$W^{k_0}$ by adding zero rows. A row includes the entire complete front +fields, active middle block, back field, all spectator coordinates and +polynomial record suffix. Padding adds whole such rows, so increases total +volume by at most two and does not change any within-row range. + +At a node write the row index as $u=Wg+w$, $0\le w500000000^{10}$ and +$899^9<460^{10}$ give $m^\rho>500000000$ and +$899^{\rho-1}<460$. +If a path has one large child $t$, its moment is at most +\[ + (t/m)^\rho+(q-t)899^{\rho-1}/m^\rho. +\] +This expression increases over the large-child range, since its +numerator derivative is $\rho t^{\rho-1}-899^{\rho-1}>0$. +Use $t=m-b$ and the concavity bound +$(1-x)^{19/10}\le1-19x/10+9x^2/10$ to obtain +\[ + \sum_{\rm path}(r/m)^\rho + <1-\frac{19}{12800}+\frac9{10\cdot1280^2} + +\frac{1194\cdot460}{500000000} + =\frac{51180270301}{51200000000}<\gamma. +\] +Without a large macro the bound is $227493/6250000<\gamma$. + +For coefficient-depth accounting take $E_0=128(W+m+1)^3$. +There are at most $s$ positive-rank residual calls and $W$ endpoint +normalizations. Combining the quadratic phases with the $S$ factors, +each uses at most two diagonal phase passes and one scalar phase. +The inherited gate and remainder count is therefore bounded conservatively by +\[ + 36W^3+16s+16W+8m+4100. The independent resolvent +bound `||H^-1-N^-1||<=32p^2*e_p`, where +`e_p=p*2^(16p+7)*2^-32768p+2^-46p`, is smaller still. + +Two period qualifications are essential. First, `|I|w`, implied here by +`s>2*(L+2A+2w)`, excludes such endpoint coupling. Second, a lifted +constant phase can join the last and first physical cells, but their +cross-cut entry in H was separately rounded. The local partition must +preserve the ORIGINAL physical period cut, even if unwrapped `q_j-j` +agrees across it. The reviewed construction now explicitly does both. + +## Local arithmetic and normalized tensor maps + +Intersect a window with original global cells. Its at most two clipped +endpoint pieces have all vertices designated boundary when their length +is at most4w. Otherwise retain the first and last w vertices. Preserving +the physical period cut adds at most O(w) boundary vertices. Every +remaining interior is a principal Toeplitz block with the same global +similarity weights; different interiors do not interact. + +The inherited Schur elimination retains the diagonal-dominance gap and +does not increase row norm. Boundary fill connects only the two groups +of one interior or neighboring groups, giving bandwidth O(w). Thus the +accepted Gohberg-Semencul, directed Schur rounding, banded LU, reciprocal +and local residual proofs apply. They are bounds in row norms, not sums +over all cells; there is no omitted factor proportional to the line or +window count. Empty interiors are skipped. For M window indices, + +``` +n_boundary <= 4*M*w/d+12w, +online line cost = O((M+M*w^2/d+w^2)*p^(1+delta)). +``` + +The retained error `2^(64p)*p^64*2^-32768p` is far below one p-grid unit. +Short convolution lengths are at most `4d+1`, with O(p)-bit retained +factors and an unconditional conventional multiplier. Exact rational +factor construction is polynomial in t and p and remains `n^o(1)` +under the retained axis-size relation. No stronger multiplication result +being proved by this campaign is used to multiply those local operands. + +Set `j_scale=ceil(log2(32p))` and `D_normalized=D/2^(2u)`. The local +compression `D_normalized H_I^-1 C/2^j_scale` has norm at most1/4, +and the true global normalized compression at most1/8. Selection C +has norm one and uses only the supplied target halo by definition of I. +Comparing restricted local products with global tensor products therefore +adds at most the sum of one-axis row errors. Early core cropping is done +ONLY in the completed axis. Other axes retain their own input halos until +their maps are applied. No intermediate boundary output is supplied as a +purported global halo value, so a halo refresh is unnecessary. + +Expansion uses positive restricted Gaussian rows and norm below3/4. +The actual short chirp blocks retain the original global s,t and ratio, +with only the requested local outputs; they do not chirp an entire p^8 +tile using unbounded diagonal weights. Their O(p)-bit per-block descriptor +work is absorbed in the block's O(block_length*p) scans. The local tile +contains many short blocks. Source omissions are beyond the existing +O(p) Gaussian radius, so the accepted tail bound applies. Together with +componentwise directed truncation and the local inverse bounds, the +original loose tensor interface `d*p^2*2^-p` and gamma normalization remain +valid. Approximate maps have sufficient norm headroom to retain disk +outputs; no unnormalized large intermediate is passed as a disk input. + +## Fractional cores, two different volume bounds + +Take `L=2^ceil(8*log2(p))` and `A=4096p^3`. The target cores are +`[kL,(k+1)L)`, with corresponding source cores +`J_k=[floor(kL/rho),floor((k+1)L/rho))`. They partition the source period +in exact increasing order. Selecting nearest q indices from target halo +`[kL-A,(k+1)L+A)` gives source I with endpoints + +``` +ceil((2*(kL-A)-1)*s/(2t)), +ceil((2*((k+1)*L+A)-1)*s/(2t)). +``` + +This also fixes ties and wrapping endpoints. Each core has source distance +at least `A/2-3>512p^3` from its artificial cuts. The near-one constraint +gives `L-|J_k|>=L/(8d)-1`, while `L>=64dA` ensures the full source halo +fits in L. Padding each source halo to L therefore produces exactly T +records, governed by the retained `T/S<2`. It does not double every +source interval. + +Target halo fields instead have persistent length `L+2A`; their exact +tensor volume is `T*(1+2A/L)^D<2T`. Only the CURRENT local field is +temporarily padded to its next binary length for a move, and its added +zeros are removed before the next axis. Simultaneously rounding all halo +fields to2L would introduce2^D volume and is invalid. All these +inequalities hold uniformly for eventual p>100 and d<=p; finite exact +integer controls include p=101,127,256,1023. + +## Ordered scans, catalogs and final row order + +The small complete-field move has a direct fixed-tape construction. +Transpose `[B]x[2]x[R]` by scanning complete R-blocks onto two parity +tapes and concatenating them. Its inverse interleaves the two complete +halves. Repeat on most significant then subsequent bits, inside the +preceding-bit fibers. This preserves internal significance and B order. +It costs O(log(p)*V) for a polynomial-size field, across arbitrary +complete mixed-radix spectators. Rewinds are within the current work +fiber and cost its volume. A line map changing its field length uses +the new field's transpose on return, not an inversion with stale names. + +For halo duplication, starts and ends are monotone. One initial bounded +set of current-fiber scans retains the periodic head/tail; it does not +rescan the entire tensor for each window. Emit the first window while +mirroring its next overlap. Each later window first writes that saved +overlap, then reads the new source portion, mirroring its tail on the +other of TWO overlap tapes. Core spacing greater than twice the halo +ensures the next saved tail lies in this newly read portion. Thus no FIFO +shift, repeated whole-window read or random source seek is needed. +Head resets and mirrored copies are proportional to written traffic. + +The binary global gather and its inverse use the independently reviewed +coordinate router, with paid untouched spectator records. All other +passes move only short local fields. After exposing axis i's short field, +its global j row is increasing before a complete spectator suffix. Source +embedding zeros can be removed, monotone I windows emitted and padded, +then the short field returned. Coarse `k_i` stays in its original order. +For compression the same procedure emits target halos before any inverse. + +Factor catalogs follow `[A_prefix]x[k_i]x[B_suffix]`. Scan k_i in order +for each A_prefix; retain its page across B_suffix and all inner lines; +rewind at the next A_prefix. The page has +`O((L+L*w^2/d+w^2)*p)` bits. Per-line page scans are part of the line +solve cost. Copying an entire page or doing polynomial metadata work per +origin is absorbed by the other complete inner fields, at least +`L^(D-1)` records. Here retained D=d-1 tends to infinity for fixed +epsilon>0; this is not a claim about an isolated one-axis short-record +contract. The page catalogue is a sequential tape, not random memory. + +Final source assembly exposes `[k_i,local_i]`, copies only the first +`|J_k|` complete suffix blocks from each tile in k order, appends zeros +to t_i, and splits the now ordinary increasing global j field into its +coarse and low bits. Returning those low bits leaves the canonical +binary microbox layout. After the inverse gather, fixed-cut scans remove +the global j>=s_i zeros. This restores the original prime-box row order +and normalization without a large mixed-radix transpose. + +## Cost, independent evidence and scope + +Summing the local line cost over each axis and the halo volume gives + +``` +O(Tp*[p^tau*polylog(p)+d*polylog(p) + +d*p^delta+w^2*p^delta+d*w^2*p^delta/L]) ++ polynomial(t,p) setup. +``` + +The first term pays the constant number of large router calls. Embedding, +small-field exposure, periodic halos, core crops, factor traffic and ordered +assembly contribute the displayed short passes. `w^2=O(p^(1-r))`; setup +is negligible under the retained axis sizes. A fixed collection of tapes +and bounded current work fibers suffices. The changed movement margin is +a, while the Gaussian margins `1-epsilon-delta` and `r-delta` remain. +Every recurrence, guard, normalization, compact reservation and eventual +prime capacity condition must still be checked in the final assembly. + +[review_bulk_resampling.py](../code/review_bulk_resampling.py) imports no +producer. It executes small-field complete-block split/interleave moves, +alternating-buffer monotone halo emission, source zero embedding, +fractional source padding, local selector lines, early core cropping, +sequential page visits and final source assembly. The global gather models +the separately accepted router contract. Selector maps check exact payload +provenance and order; they do not simulate true Gaussian arithmetic. + +The [fresh joined run](../runs/20261008T030514Z-review-bulk-resampling/) +passed all84,824 complete output records in six two/three-axis expansion +and compression schedules. It performed124 binary split passes,135 +interleave passes,58 small-field moves,11 current-field-only paddings, +76 halo rows,280,744 monotonically new records,12,584 mirrored overlap +records,76 sequential page visits and14,400 local lines. Arbitrary +prefix and coefficient-suffix cardinalities, wrapping windows and unequal +source cores are included. The largest persistent halo volume was46,656 +against target32,768, and temporary one-field padding82,944; the latter +is a constant work-volume factor, not persistent2^D growth. Independent +wrapped short-path checks cover1,242 exact entries. + +Python3.14.4, one worker,0.79 seconds,28,592 KiB maximum RSS. Source +SHA256 `c05c41f3fcec2f845043c4b00fd8c939d516f8ea3826a0112228ad625aa1f39f`; +certificate `f3883ec3404b1bebbea2c2b6e8961de1e9fb958e10ce5b8da27a0e086931b5a4`. +Small fixtures satisfy their actual halo-fit, overlap and capacity tests; +they are not numerical simulations of the enormous asymptotic halo. +The all-size analytic proof and accepted Gaussian solver supply accuracy. + +[review_bulk_boundary_negatives.py](../code/review_bulk_boundary_negatives.py) +and its [fresh run](../runs/20261008T031019Z-review-bulk-boundary-negatives/) +retain four exact negatives: insufficient complement gap, merging across +the original phase period cut, cropping an unprocessed axis's halo, and +simultaneous binary padding. The latter two give respectively1/16 versus +zero for a tensor core, and256T versus a near-one persistent halo volume. +These controls distinguish the repaired construction from plausible +invalid variants. Certificate +`b01739e355f97d122a4d0bdd6e9523af51bb63839065c860003f6185ea164560`. + +The positive conclusion is a written all-size conditional transfer backed +by independent finite operator and ordering controls, not formal machine +verification. No remaining structural counterexample was found after +the explicit endpoint and layout repairs. Strict composed arithmetic is +the next review step. Reproduce the two scripts with `python3 -B +code/.py --output ` from the topic directory; +the standard library suffices. Exact commands, hashes, logs, version and +owned reservation release are retained in the run protocols. diff --git a/research/capped-phase-fusion/review20261014/independent/source/references/semantic-bulk/rad20/reports/review-phase-cell-inverse.md b/research/capped-phase-fusion/review20261014/independent/source/references/semantic-bulk/rad20/reports/review-phase-cell-inverse.md new file mode 100644 index 0000000000..395eab8ee1 --- /dev/null +++ b/research/capped-phase-fusion/review20261014/independent/source/references/semantic-bulk/rad20/reports/review-phase-cell-inverse.md @@ -0,0 +1,335 @@ +# Independent review of the phase-cell Gaussian inverse + +The phase-cell replacement is valid after the explicit downward-weight +approximation repair. The reviewed construction preserves the scalar +disk/error interface and fixed finite-tape model while reducing inverse +application to short Toeplitz convolutions and a smaller cyclic boundary +solve. No remaining all-size precision, endpoint-cell, or amortization gap +was found. This is a conditional transfer conclusion: the original +complete multiplication interfaces and the separately certified finite +motifs remain inputs, not newly established results of this review. + +Campaign: 2026-10-07 22:25:21 UTC to 2026-10-08 08:25:21 UTC. +Reference: `bcd4ebde8692383539f8a48734e5fbf3a18a32c2`. +Reviewed derivation: [downstream-phase-cell-inverse.md](downstream-phase-cell-inverse.md). +This review continues the independent [banded inverse audit](review-reusable-banded-inverse.md). +Fresh evidence: `runs/20261007T235722Z-review-phase-inverse`. + +## Scope and retained prerequisites + +Retain `rho=t/s=1+theta`, the nearest-integer indices with ties upward, +periodic `beta_j` in `[-1/2,1/2)`, `u=alpha^2`, and the exact Gaussian +matrix `N=I+E`. The preceding independent review established, for +`p>100`, `u>=ceil(log2(8p))`, `theta>1/(4p)`, and `1=1/(4p), +||N^-1||<=4p. +``` + +It also established that width `w=ceil(sqrt(16p/u))+2` leaves total +periodic lattice tail below `2^(-46p)`, including diagonal aliases. The +actual prime intervals give `1/(4d)(1-r)/2`, `d/w` tends to infinity and eventually every cell +has more than `4w` points. There is no unaccounted short end cell. + +An addition/comparison generator suffices. Initialize remainder `R=s` +and phase zero. At index `j`, +`R=2Kj+s-2s*phase`, `0<=R<2s`. Then the phase is `q_j-j` and +`beta_j=(R-s)/(2s)`. Advance by adding `2K`; if `R>=2s`, subtract +`2s` and increment the phase. Because `K1/2`. Put +`z=(0,x_(L-1),...,x_1)` and write `L(v)` for a lower triangular Toeplitz +matrix. The reviewed orientation is + +```text +G^-1=(L(x)L(x)^T-L(z)L(z)^T)/x_0. +``` + +There is a short independent displacement proof. Let `J` be the lower +shift. Symmetric Toeplitz matrices are invariant under reversing both +coordinates, so their inverse's last column is the reversal of its first. +Block inversion at the first and last coordinate shows + +```text +G^-1-J G^-1 J^T=(xx^T-zz^T)/x_0. +``` + +Indeed the shifted lower-right block and upper-left block have the same +inverse of the size-`L-1` principal Toeplitz matrix; their differences +are the two corresponding outer products. On the first row/column the +identity reduces to the definition of `x`. On the other hand, +`L(v)L(v)^T-J L(v)L(v)^T J^T=vv^T`. The two candidate inverses have +the same displacement. Recursing from the first row and column uniquely +determines every entry, proving the formula. The size-one case also +holds, with `z=0`. + +The general identity is equation (2.4) in Ting-ting Feng, Gang Wu, and +Yimin Wei, *Inexact Shift-and-Invert Arnoldi for Toeplitz Matrix +Exponential*, arXiv:1503.04886v1, submitted 2015-03-17 00:57:32 UTC. +[Primary manuscript](https://arxiv.org/abs/1503.04886v1). The paper supplies +the established formula; this audit derives the specialized bounds and +does not assume generic numerical stability. The primary PDF identity, +hash and date are retained in the run protocol. + +Each of the four triangular products is an ordinary convolution on a +single cell. Reverse the input for an upper product and reverse the +retained first `L` results. Signed real/imaginary parts use a fixed number +of nonnegative integer packings. Coefficient guards include O(p) integer +bits for the weight reciprocal, O(log p) summation bits, and exact product +fractional bits. The unconditional published multiplier is invoked only +on O(Lp)-bit integers, with `L<=5p`; its cost is +`O(L*p^(1+delta))`. There is no invocation of the stronger multiplier +being investigated. + +## Explicit local and global error chains + +Let `K_a=||diag(a_hat)^-1||<=2^(8p+1)` and `L<=5p`. +Each exact triangular Toeplitz factor has norm below two; its rounded +factor differs by at most `L*eta` and has norm below three. Each product +is exact on the packed dyadics until one completed-coordinate rounding. +The rounded weight reciprocal/input step has error +`O(eta)*max(1,||b||)` and magnitude O(`K_a max(1,||b||)`). A convolution +therefore propagates at most three times the preceding error, plus +`L*eta` times the preceding exact magnitude, plus its rounding unit. +Apply this to both two-factor chains, then subtraction, the rounded +reciprocal of `x_0`, and the final `a_hat` multiplication, whose norm is +at most one. Including both real components, the total local bound can +be chosen as + +```text +2^14 * p * 2^(8p) * eta * max(1,||b||), +``` + +well below the report's allowance +`LA=2^(32p)*p^8*eta*max(1,||b||)`. This is an absolute-error bound. +Subtraction may cancel large intermediate terms, but no relative-error +assumption is used. + +The boundary Schur complement is row-dominant with gap at least `mu/2` +and inverse norm at most `8p`. Rounding its entries before LU leaves gap +at least `mu/4`. Its no-pivot banded factors and cyclic correction have +polynomial norms as in the preceding inverse review. The fixed changes +in gap and wrap norm fit the conservative allowance +`SB=2^60*p^20*eta*max(1,||b||)`, including the rounded Schur perturbation. + +For a unit-norm input, exact interior output has norm at most `8p`, the +boundary right-hand side at most `17p`, the exact boundary solution at +most `136p^2`, and the final interior right-hand side at most `273p^2`. +Each rounded coupling has row error at most `p*eta` and norm below two. +One explicit propagation is + +```text +boundary_rhs_error <= 3 LA, +boundary_output_error <= 24p LA+18p SB, +whole_output_error <= 1024p^2 LA+512p^2 SB+2048p^4 eta. +``` + +Here `LA` and `SB` in the last display are their unit-input allowances; +the stated intermediate norms account for later input scaling. For +`p>100`, this is below `2^(64p)*p^64*eta`. Only a fixed number of +completed maps appears. Taking a maximum over rows/cells introduces no +factor exponential in the number of cells or in `s`. + +Perturbing `H` back to `N` adds at most +`32p^2*(p*2^(16p+7)*eta+2^(-46p))`. The work precision leaves total +inverse error below `2^(-p-10)`. The normalized inverse +`N^-1/2^j`, `j=ceil(log2(32p))`, has norm at most `1/8`. Completing +the result and rounding toward zero to the p-grid preserves the disk +interface. Together with the retained forward and diagonal maps, the +same scalar `p^2` error interface and scale +`gamma=d*(2u+j+1)` remain valid. + +## Boundary geometry, setup, and fixed tapes + +Keep each cell's first/last `w` coordinates as boundary. Its remaining +interior can connect only to that cell's two boundary groups, because +the matrix was truncated to width `w`. Eliminating the interior can fill +between those groups. Original cross-cell entries connect adjacent end +groups; cross-period entries connect the first and last groups. Hence the +ordered boundary matrix is circular-banded with halfwidth at most `2w`, +on `2w(K+1)=O(sw/d)` vertices. Interior elimination preserves row gap +and cannot increase row norm, irrespective of the signs of the filled +entries. The two-cell finite equality case has disjoint wrap groups; +the cyclic solver remains valid when its dimension equals twice its +halfwidth. In the eventual regime many cells are present. + +Both coupling matrices have O(w^2) nonzeros per cell. Scan cells in +physical order, retaining their original inputs, computing the first +interior solve and boundary right-hand side, and writing ordered +boundary records. Solve the boundary with the prior forward/backward +banded scans and cyclic correction. Scan cells again for interior +recovery, then merge in inherited order. A constant number of tapes and +O(w)-record endpoint windows suffice. Local convolution and generator +rewinds cost O(Lp) per cell; coupling buffer rewinds cost O(wp) per row. +Reset the online tables after each complete line, at their sequential +volume cost. There is no line-length convolution or random access to +another cell's interior. + +All exact rational precomputation is once per tensor axis. Clearing +denominators and using fraction-free elimination bounds determinant and +factor bit lengths by a polynomial in `s,p`. The weights' rational +denominators are finite P-bit integers. A straightforward polynomial +setup algorithm on the fixed tapes is sufficient. Since +`log s=O(p/d)` and fixed `epsilon>0` gives `s=n^o(1)`, this setup, +table conversion and cleanup are `n^o(1)` and can be absorbed into the +asymptotic algorithm. Online records have O(p) bits. + +Per-line inverse cost is `O(t*p^(1+delta)*(1+w^2/d))`. Tensoring gives +`O(d*T*p^(1+delta)+T*w^2*p^(1+delta))`; with `Tp=Theta(n)` and +`w^2=O(p/u)`, the normalized exponents are `epsilon+delta` and +`1-r+delta`. This matches the proposed two Gaussian margins. + +## Guard, cutoffs, and exact parameter audit + +The stopped one-piece depth is the retained `9B^2*d^(5-4beta)`. +For fixed `zeta>0`, the actual piece count is at most +`m*(1+1/zeta)*d^zeta`. Adding the bounded preprocessing/outer/shift +depth gives a valid bound with +`C1=5-4beta+zeta` and `C0=32mB^2*(1+1/zeta)`. For the conservative +choice `zeta=1/20`, `beta>=999/1000`, the separate tighter count gives +`9*21+18=207<256`, so `C0=256mB^2`, `C1=11/10` are sufficient. +The larger generic constant would be 672 at this zeta; it is not the +justification for the stated 256. + +The optimized family uses the retained packed/leaf balance root +`x=1-beta`, `C1=1+4x+zeta`, +`epsilon=(1-2^-20)/C1`, `r=(1-epsilon)/2`, and `delta=r/8`. +The guard gap, both Gaussian margins, cell/band separation, prefix, +normalization and remaining recurrence constraints are positive. The +exact checker independently verifies all eight current rows, their +logarithm enclosures, root signs, 32 strict conditions each, and strict +absorption on the 10^-30 grid. + +It also verifies full-constant cutoffs: gamma, logarithmic alpha, +`C0*d^C14w`. For the last, the cutoff ensures +`b^(epsilon-(1-r)/2)>=512`; meanwhile +`w<10b^((1-r)/2)+3` and `d-2>=b^epsilon-3`. The explicit inequality +512>40+15 covers both floor/ceiling losses. BHP prime packing and the +other retained interfaces still have their own eventual thresholds. + +| h50 roles | Family | Independently checked kappa | +|---:|---|---| +| 509,194 | Conservative | `3955009008227/(5*10^29)` | +| 487,650 | Optimized | `1193902884521/(125*10^27)` | +| 486,200 | Optimized | `9606057843781/10^30` | + +All eight rows strictly support `2^-57`, including the unchanged graph. +The envelope optimized row requires `log2 b>=6161257267345`, plus the +other eventual thresholds. These very large cutoffs do not imply useful +practical performance. + +Within this guard family the scoped upper objective is +`b_complex*x_star/(1+4x_star)`. The packed term +`a_bit^2(1-x)/[(1-a_bit*x)(1+4x)]` decreases with `x`; the leaf term +`b_complex*x/(1+4x)` increases. Their intersection is the same balance +root. Increasing either primitive saving improves the pointwise minimum, +so using upper primitive enclosures supplies a valid family ceiling. +This ceiling does not rule out other guards, recurrences or finite motifs. + +## Independent finite evidence and reproduction + +The checker [review_phase_inverse.py](../code/review_phase_inverse.py) +compares rounded producer operations against independent dense rational +inverses and an independently assembled Schur complement. It checks: + +- 1,300 GS entries and 1,300 shift-displacement identities, sizes 1-12 + with positive and negative near-identity Toeplitz coefficients. +- 205,071 exact phase-generator records, 1,674 nearest-round ties, + 18,923 cell lengths and 325,952 original/filled boundary edges. +- A 20-coordinate rounded inverse with internal weight reciprocal + `2^126`, 256 fractional work bits, all 20 basis vectors plus two + additional inputs. Absolute error is below `2^-80`, despite the large + internal scale; its compact outward bound is about `2.3e-40`. +- A complete three-cell 48-coordinate solve, 36 interiors and 12 + boundary vertices of halfwidth four, original gap `2^-12`. + Every global basis vector and two signed probes pass against the + independent exact dense inverse; absolute error is below `2^-64`. + +These are deliberately distinct from the parent's five rational-interval +Gaussian tests. They calibrate the actual rounded GS/Schur operations and +precision amplification but are not a simulation of the asymptotic tape +machine. The mathematical arguments above supply that transfer. + +The independent parameter checker is +[review_phase_assembly.py](../code/review_phase_assembly.py). Both use only +the standard library plus retained review/producer modules, with source +hashes, input identities, actual interpreter version and commands in the +run protocol. Reproduce from the repository root: + +```sh +PYTHONDONTWRITEBYTECODE=1 python3 research/integer-multiplication-bounds/code/review_phase_inverse.py \ + --output "$OUT" +PYTHONDONTWRITEBYTECODE=1 python3 research/integer-multiplication-bounds/code/review_phase_assembly.py \ + --certificate "$PHASE_CUTOFF_CERTIFICATE" --output "$PARAMETER_OUT" +``` + +The source's finite failed equality check and the corrected weight-bound +wording remain documented by the producer. No original input was edited. +This review makes no broader priority or novelty claim for established +Gaussian similarity, Toeplitz inversion or Schur-complement methods. diff --git a/research/capped-phase-fusion/review20261014/native_validation.py b/research/capped-phase-fusion/review20261014/native_validation.py new file mode 100644 index 0000000000..fdd145703b --- /dev/null +++ b/research/capped-phase-fusion/review20261014/native_validation.py @@ -0,0 +1,181 @@ +"""Supplement the pinned cold replay with semantic and final cross-binding. + +The native witnesses are reused only after verify.py's original complete +source/capsule binding. Owner/template generation and geometry controls are +rerun. Immutable supplier inputs are never edited or repinned. +""" +from pathlib import Path +import argparse, copy, gzip, hashlib, importlib.util, json, os, shutil, subprocess, sys, tempfile + +HERE=Path(__file__).resolve().parent +PACKAGE=HERE.parent +EXTERNAL=HERE/'external' + + +def sha(path):return hashlib.sha256(Path(path).read_bytes()).hexdigest() +def read(path): + path=Path(path);blob=path.read_bytes() + return json.loads(gzip.decompress(blob)if path.suffix=='.gz'else blob) +def need(ok,message): + if not ok:raise RuntimeError(message) +def write(path,value):Path(path).write_text(json.dumps(value,indent=2)+'\n') + + + +def bind_geometry_inputs(pinned_spec,runtime_spec,engine_files,assembly): + need(sha(runtime_spec)==sha(pinned_spec),'geometry runtime SPEC differs from pinned source') + for name,path in engine_files.items(): + need(sha(path)==assembly['input_sha256']['engines/'+name+'.py'], + 'geometry engine differs from bound assembly: '+name) + return {runtime_spec:sha(runtime_spec),**{path:sha(path)for path in engine_files.values()}} + + +def validate(package,source,run,work): + package,source,run,work=map(lambda p:Path(p).resolve(),(package,source,run,work)) + need(not work.exists(),'fresh scratch directory required') + work.mkdir(parents=True) + env=dict(os.environ,PYTHONDONTWRITEBYTECODE='1',PYTHONINTMAXSTRDIGITS='500000') + for key in ('PYTHONOPTIMIZE','WITNESS_DISCOVER','WITNESS_RECORD','WITNESS_SPEC'): + env.pop(key,None) + # The retained exact engines cache derived rational enclosures. Copy + # that cache into scratch; immutable sources and run evidence stay put. + if (run/'fine-cache').is_dir(): + shutil.copytree(run/'fine-cache',work/'fine-cache') + env['FINE_ENGINE_CACHE']=str(work/'fine-cache') + scripts=package/'review20261014/external/o1-admission' + pins=read(package/'review20261014/external/SOURCE-PINS.json') + for rec in pins['files']: + p=package/'review20261014/external'/rec['path'] + need(sha(p)==rec['sha256'],'external source drift: '+rec['path']) + stages=[] + def call(name,script,*args,expect=0): + cp=subprocess.run([sys.executable,'-B',str(script),*map(str,args)], + cwd=work,env=env,text=True,capture_output=True,timeout=1200) + (work/(name+'.log')).write_text(cp.stdout+cp.stderr) + need(cp.returncode==expect,name+' failed: '+(cp.stdout+cp.stderr)[-3000:]) + stages.append({'name':name,'exit_code':cp.returncode,'expected_exit_code':expect}) + print('PASS',name,flush=True) + strict=run/'strict';base=run/'native-new-base/local';extra=run/'native-new-extra/local' + before={p:sha(p)for p in [source/'MANIFEST.json',run/'RESULT.json', + strict/'global/audit/OWNERS-AND-PROFILE.json', + strict/'global/audit/PREFIX-AND-INVOICE.json', + strict/'global/audit/ASYNC-AUDIT.json']} + need(read(run/'RESULT.json')['native_reuse']=={'base':False,'extra':False}, + 'the selected evidence is not the documented cold run') + args=[strict,run/'GEOMETRY-CONTROLS.json',base,extra,'0.001676855229071321'] + final=work/'FINAL-BINDING.json' + call('independent-final-binding',scripts/'t74_final_binding.py',*args,final) + fb=read(final) + need(fb['status']=='PASS_T74_T16_PARITY_FINAL_BINDING'and all(fb['checks'].values()),'final binding failed') + call('swapped-native-control',scripts/'t74_final_binding.py',strict, + run/'GEOMETRY-CONTROLS.json',extra,base,'0.001676855229071321', + work/'SWAPPED-NATIVE.json',expect=1) + call('nonimproving-rate-control',scripts/'t74_final_binding.py',strict, + run/'GEOMETRY-CONTROLS.json',base,extra,'0.1', + work/'NONIMPROVING-RATE.json',expect=1) + # Freshly regenerate every geometric artifact instead of re-hashing + # only historical outputs. The two permitted PLAN differences must + # still bind their actual container; every mathematical field agrees. + owners,parent=work/'owners',work/'parent' + call('regenerate-owners',source/'reconstruct.py','--out',owners) + call('regenerate-parent',source/'lift_parent.py','--owners',owners,'--out',parent) + call('independent-semantic-regeneration',scripts/'t74_semantic_regen.py', + source,owners,parent,work/'SEMANTIC.json') + semantic=read(work/'SEMANTIC.json') + need(semantic['status']=='PASS_T74_SEMANTIC_REGENERATED_ARTIFACTS','semantic regeneration failed') + # Deliberately use a different gzip compressor level, not just byte9. + # Re-run the parent's actual generator to bind its new container hash. + path=owners/'OWNERS.json.gz';payload=gzip.decompress(path.read_bytes()) + old=path.read_bytes();changed=gzip.compress(payload,compresslevel=1,mtime=137) + need(old!=changed and gzip.decompress(changed)==payload,'compression control is inert') + path.write_bytes(changed) + alternate=work/'parent-recompressed' + call('regenerate-parent-after-recompression',source/'lift_parent.py','--owners',owners,'--out',alternate) + call('gzip-container-control',scripts/'t74_semantic_regen.py',source,owners,alternate, + work/'SEMANTIC-RECOMPRESSED.json') + recompressed=read(work/'SEMANTIC-RECOMPRESSED.json') + need(recompressed['status']=='PASS_T74_SEMANTIC_REGENERATED_ARTIFACTS'and + not recompressed['container_bytes_all_equal'],'semantic check rejected harmless recompression') + roles=json.loads(payload);roles[0]['time']+=1 + path.write_bytes(gzip.compress(json.dumps(roles,separators=(',',':')).encode(),mtime=0)) + call('changed-role-control',scripts/'t74_semantic_regen.py',source,owners,alternate, + work/'SEMANTIC-CHANGED-ROLE.json',expect=1) + # Re-run the actual geometry mutations, recording exactly which + # source/owner/invoice/assembly bytes they consumed. + src=run/'parent/layout/joint1148-round45/source/research/complex-k492-owner-repair/vendor/multicold' + runtime_spec=strict/'runtime/data/SPEC.json' + assembly=read(strict/'ASSEMBLY.json') + engine_files={name:src/'engines'/(name+'.py')for name in ('moment','base_two_moment','outer')} + geometry_inputs=bind_geometry_inputs(source/'new/SPEC.json',runtime_spec,engine_files,assembly) + badspec=work/'changed-SPEC.json';altered=read(runtime_spec);altered['invalid_review_control']=True + write(badspec,altered) + badengine=work/'changed-moment.py';badengine.write_bytes(engine_files['moment'].read_bytes()+b'\n# changed-source control\n') + for name,specfile,engines in [ + ('changed_geometry_SPEC',badspec,engine_files), + ('changed_geometry_engine',runtime_spec,{**engine_files,'moment':badengine})]: + try:bind_geometry_inputs(source/'new/SPEC.json',specfile,engines,assembly) + except RuntimeError:pass + else:raise RuntimeError('accepted '+name) + stages.append({'name':name,'status':'REJECTED_BY_INPUT_BINDING'}) + print('PASS',name,flush=True) + geometry=work/'GEOMETRY-CONTROLS.json' + call('fresh-geometry-controls',source/'geometry_controls.py','--source',src,'--strict',strict,'--out',geometry) + geo=read(geometry) + need(geo['status']=='PASS_T74_GEOMETRY_AND_CHARGE_CONTROLS'and len(geo['rejected'])==6,'geometry controls incomplete') + aud=read(strict/'global/audit/ASYNC-AUDIT.json') + mutations=aud['mutations_rejected'];nproviders=len(aud['provider_counts']) + need(aud['mutation_controls_run']is True and len(mutations)==len(set(mutations))==7+8*nproviders==23, + 'actual controller mutations incomplete') + inputs={'native_source_manifest':source/'MANIFEST.json','controller':strict/'global/audit/ASYNC-AUDIT.json', + 'owner':strict/'global/audit/OWNERS-AND-PROFILE.json', + 'invoice':strict/'global/audit/PREFIX-AND-INVOICE.json','assembly':strict/'ASSEMBLY.json', + 'geometry_generator':source/'geometry_controls.py','mixed_assembler':source/'code/assemble.py', + 'outer_ledger':package/'frontier_fusion/FRONTIER-FUSION-ARITHMETIC.json', + 'geometry_runtime_spec':runtime_spec, + **{'geometry_engine_'+name:path for name,path in engine_files.items()}} + need(all(sha(p)==digest for p,digest in {**before,**geometry_inputs}.items()), + 'validation changed a pinned source or completed run') + return {'status':'PASS_PR1673_PINNED_T74_SEMANTIC_AND_FINAL_ADMISSION_SUPPLEMENT', + 'native_source_commit':'f852af5e178ce035cd46181a29d9330bcbcaa206', + 'external_reviewer':{'repository':pins['repository'],'commit':pins['commit'], + 'scripts_reused':['t74_final_binding.py','t74_semantic_regen.py']}, + 'native_replay_mode':'Existing complete cold run, unmodified and source/capsule bound; not rerun here.', + 'new_work':'Fresh owner/template regeneration, whole-receipt final binding, ' + 'recompression and semantic rejection controls, geometry mutations.', + 'final_binding_checks':fb['checks'],'semantic_checks':semantic['checks'], + 'semantic_recompressed_checks':recompressed['checks'], + 'regenerated_semantic_hashes':{name:rec['semantic_sha256']for name,rec in semantic['files'].items()}, + 'controller_controls_actual':23,'controller_count_formula':'7+8*2', + 'historical_controller_15_label_is_incorrect':True, + 'geometry_controls':geo,'geometry_runtime_spec_and_engines_bound':True, + 'geometry_discovery_environment_cleared':True,'input_sha256':{k:sha(p)for k,p in inputs.items()}, + 'stages':stages,'controls_rejected':['swapped_native_directories','nonimproving_rate','changed_owner_role', + 'changed_geometry_SPEC','changed_geometry_engine'], + 'harmless_recompression_accepted':True, + 'scope':'Finite admission evidence for exactly the pinned PR1661 supplier ' + 'used by PR1673. No new native word or exponent; inherited ' + 'all-size/common-frame/mutable-source/hosting/precision/tape/' + 'recovery/analytic hypotheses remain. The external T37 regression ' + 'is separate cited evidence, not a new local regression run.'} + + +def main(): + parser=argparse.ArgumentParser(description=__doc__) + parser.add_argument('--native-source',type=Path,required=True) + parser.add_argument('--native-work',type=Path,required=True) + parser.add_argument('--output',type=Path,required=True) + args=parser.parse_args() + need(not args.output.resolve().is_relative_to(PACKAGE),'output must be outside the package') + spec=importlib.util.spec_from_file_location('published_package_verifier',PACKAGE/'verify.py') + verifier=importlib.util.module_from_spec(spec);spec.loader.exec_module(verifier) + verifier.check_inputs(PACKAGE) + cold=verifier.cold_binding(PACKAGE,args.native_source,args.native_work) + with tempfile.TemporaryDirectory(prefix='phase-fusion-admission-review-')as td: + result=validate(PACKAGE,args.native_source,args.native_work,Path(td)/'work') + result['cold_source_binding_status']=cold['status'] + write(args.output,result) + print(result['status'],len(result['final_binding_checks']),'final checks', + result['controller_controls_actual'],'controller controls') + + +if __name__=='__main__':main() diff --git a/research/capped-phase-fusion/verify.py b/research/capped-phase-fusion/verify.py index 60bb3218f6..cfc2f0e3dd 100644 --- a/research/capped-phase-fusion/verify.py +++ b/research/capped-phase-fusion/verify.py @@ -36,6 +36,8 @@ ('evidence/phase_layout_audit_ultra/check_union.py','evidence/phase_layout_audit_ultra/UNION-CONTROLS.json'), ('evidence/phase_layout_audit_ultra/check_q_embedding.py','evidence/phase_layout_audit_ultra/Q-EMBEDDING-CONTROLS.json'), ('evidence/phase_layout_audit_ultra/check_capped_layout.py','evidence/phase_layout_audit_ultra/CAPPED-LAYOUT-CONTROLS.json'), + ('review20261014/check_review.py','review20261014/CONTROL-REPLAY.json'), + ('review20261014/independent/check_original_outer.py','review20261014/independent/ORIGINAL-OUTER-CONTROLS.json'), ] def read(p): return json.loads(p.read_text()) def sha(p): return hashlib.sha256(p.read_bytes()).hexdigest()