Prove the two-edge half-line operator limit of broad Gaussian inverses

research/gaussian-half-line-operator-limit/PROOF.md

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+# Two boundary layers of a finite Gaussian inverse
+
+JoyboyBrian, with substantial OpenAI Codex assistance. Scoped written theorem;
+finite controls are corroboration, not formal verification. Apache-2.0.
+
+## Statement and embedding
+
+Fix alpha>0, put b=pi/alpha². Let 0<theta<1, h=sqrt(theta),
+rho=1+theta, N=floor(1/theta), n=N-1 and c=1/(2rho). On 0<=k,i<N set
+K[k,i]=alpha^-1 exp(-b(i-k/rho-c)²). No band truncation is made.
+Let S=diag((-1)^i) and A=S K^-1 S. Write
+P(t)=product_(r>=1)(1-exp(-2tr)), and a=alpha exp(b/4)/P(b)^3.
+On H=L²(0,infinity) direct-sum L²(0,infinity), let T have kernel
+
+    T(x,y)=exp(-b(y²-x²)) 1{x<=y}.
+
+Partition the finite indices at floor(N/2). U_h maps e_i to the normalized
+indicator h^-1/2 1_[hi,h(i+1)) in the first summand on the left half;
+on the right half it maps e_i to the analogous indicator indexed by n-i
+in the second summand. U_h is an isometry. Its adjoint averages over cells;
+the orthogonal complement of its image is mapped to zero. Then
+
+    || U_h (h A) U_h* - a(T direct-sum T) || -> 0.
+
+In particular,
+
+    sqrt(theta) ||K^-1||₂ -> alpha exp(b/4)/(2 sqrt(b) P(b)^3).
+
+This is for each fixed alpha. Uniformity for alpha growing with theta is
+NOT asserted. The section is N=floor(1/theta), beta_0=-1/2; changing section
+length or starting phase can change the limit. The result determines the
+finite matrix boundary layers, not merely a formal differential expression.
+
+## Exact predecessor identity and full coefficient
+
+Credit the finite Gaussian q-LDU identity to the lineage retained by Vlado
+Vojdanovski/#1673 and Jose Manuel/#4193, reproduced by JoyboyBrian/#5103 and
+Nick Petalas/#5150. Set gamma=b/rho, q=exp(-gamma), Q=q²,
+P_j=product_(r=1)^j(1-Q^r). For G[i,j]=q^((i-j)²),
+
+ G^-1[i,j]=(-1)^(i+j) sum_(k=max(i,j))^(N-1)
+              q^(2k-i-j) P_k/(P_i P_(k-i) P_j P_(k-j)).
+
+K=Dr G Dc with the exact positive diagonals in #5103. For i<=j, d=j-i,
+define the FULL positive coefficient (not just its first summand)
+
+ F_ij = sum_(r=0)^(N-1-j) Q^r P_(j+r)/(P_i P_(d+r) P_j P_r).
+
+Then
+
+ A_ij = alpha exp(E_ij) F_ij,
+ E_ij = b beta_j²/rho² - gamma theta(j²-i²),
+ beta_j=theta j-1/2.                                      (1)
+
+For i>j the same coefficient after reflection i'=n-i,j'=n-j satisfies
+
+ E_ij = b beta_j²/rho²
+        - gamma theta[(j'+delta)²-(i'+delta)²],
+ delta=1/theta-n in [1,2).                               (2)
+
+The coefficient is invariant under simultaneous reflection because G and
+G^-1 are. These are exact formulas. In particular all entries of A are
+nonnegative. The predecessor entry lemma gives F_ij<=P(gamma)^-3.
+Both exponents are <=b/4. The upper triangle is attached to the left edge;
+the strictly lower triangle is attached to the right edge. We assign the
+diagonal only to the upper triangle.
+
+Here is the coefficient limit with an explicit error. For gamma>=b/2 put
+r0=exp(-b) and p0=exp(-r0/(1-r0)²). Then all finite products and P(gamma)
+are >=p0, because -log(1-z)<=z/(1-z) and geometric summation apply.
+Also |P_j-P(gamma)|<=r0^(j+1)/(1-r0). For i>=m, d>=m and N-j>=m+1,
+
+ |F_ij-P(gamma)^-3| <= E_m,
+ E_m=5 p0^-5 r0^m/(1-r0)².                              (3)
+
+Indeed replace P_(j+r), P_i, P_(d+r), P_j by P(gamma) in each term.
+The numerator and three inverse factors have telescoping error bounded by
+4 p0^-5 r0^(m+1)/(1-r0). Sum Q^r<=1/(1-r0), and bound the omitted tail
+by p0^-4 r0^(N-j)/(1-r0). The limiting infinite sum is
+P(gamma)^-2 sum Q^r/P_r =P(gamma)^-3: the sum telescopes since
+Q^r/P_r=1/P_r-1/P_(r-1) for r>=1. This is why the prefactor contains
+THREE infinite-product factors. Dropping all but r=0 does not recover it.
+Finally logarithmic differentiation of the convergent product gives
+
+ |P(gamma)^-3-P(b)^-3| <= L_b h²,
+ L_b=6 b p0^-3 r0/(1-r0)³,                              (4)
+
+using b-gamma<=b h². Termwise differentiation is justified by the same
+summable bound 2 sum k r0^k/(1-r0).
+
+## Uniform tails, including the middle and cross-edge entries
+
+Set C=alpha exp(b/4)p0^-3. For i<=j the matrix entry is at most
+C exp[-gamma h²(j²-i²)]. For i>j use the reflected bound with delta>=0,
+which is no larger than C exp[-gamma h²(j'²-i'²)].
+Fix R>=1, h<=1/2, J=floor(R/h), and suppose 2J<=N. Remove the left
+upper-triangular J by J corner and the right strictly-lower J by J corner.
+For the upper remainder j>=J, j²-i²=(j-i)(j+i)>=(j-i)J. Every row sum
+and every column sum of h times this remainder is bounded by
+
+ C h sum_(d>=0) exp(-gamma h² J d)
+ <= C(h+1/(gamma h J)) <= C(h+2/(gamma R)).
+
+The inequality 1/(1-exp(-z))<=1+1/z follows from exp(z)>=1+z.
+The lower remainder has the identical bound by reflection. Schur's test
+therefore bounds their SUM in operator norm by 2C(h+4/(bR)). This bound
+includes the middle and cross-edge blocks; no unproved localization is used.
+The two retained corners act in different summands of H, so are a direct
+sum, not a sum on a shared output. In particular their norms do not add
+in quadrature.
+
+For the continuum operator, remove its [0,Jh]² triangle. The same row and
+column integral bound gives norm <=1/(b Jh)<=2/(bR). After multiplying
+by a<=C and allowing both summands, the continuum tails cost <=4C/(bR).
+Together the discrete and continuum tail errors are safely bounded by
+
+    4C(h+4/(bR)).                                       (5)
+
+This also proves boundedness of T, by a compact-window bound plus its tail.
+
+## Compact-window comparison and a quantitative convergence regime
+
+Suppose J>=m>=1 and N-J>=m+1 as well. On each corner the integral kernel
+of U_h(h A)U_h* is A_ij on its h by h cells: the h in h A cancels the
+normalization h^-1 from the two indicators. On cells with i>=m and
+j-i>=m, (3)-(4) control the coefficient. Everywhere both the discrete
+kernel and the proposed limiting kernel have magnitude <=C.
+
+For x in [hi,h(i+1)), y in [hj,h(j+1)), 0<=i<=j<J, compare E_ij
+(or its reflected form) with b/4-b(y²-x²). Direct expansion, using
+|beta_j|<=1/2, gamma=b/(1+h²), delta<2, R>=1 and h<=1/2, gives
+
+ |E_ij-[b/4-b(y²-x²)]| <= D,
+ D=12b(R+1)²h.                                        (6)
+
+For example the beta²/rho² discrepancy is at most b[h(R+2)+h²],
+the gamma discrepancy at most b h²(R+2)², the reflected delta term
+at most4bRh, and rounding x,y to cell endpoints at most4b(R+1)h.
+Their sum is bounded by (6). Each exponential argument is <=b/4, so
+the mean-value theorem gives an absolute exponential error <=exp(b/4)D.
+
+The exceptional cells i<m or j-i<m have total area <=2mhR in each
+corner. The strict diagonal convention and cells straddling x=y are
+included in j-i<m. Their kernel difference is bounded by2C. Thus the
+Hilbert-Schmidt bound for both exceptional regions is at most
+8C sqrt(mh(R+1)), using the triangle inequality generously. The remaining
+two squares of side Jh<=R contribute at most
+
+ 2R alpha exp(b/4) [E_m+L_b h²+p0^-3 D].                 (7)
+
+Combining (5)-(7) gives the following explicit finite error bound:
+
+ ||U_h(hA)U_h* - a(T direct-sum T)||
+ <= 4C(h+4/(bR)) + 8C sqrt(mh(R+1))
+       + 2R alpha exp(b/4)[E_m+L_b h²+p0^-3 D].          (8)
+
+All constants and hypotheses have been displayed. One explicit convergence
+regime is h<=1/64, R=h^-1/10, m=ceil(h^-1/5), J=floor(R/h), N=floor(h^-2).
+Here 2J<=N, J>=m, N-J>=m+1; these follow from R/h=h^-11/10,
+m<=2h^-1/5 and h<=1/64. Every term of (8) tends to zero, including
+R r0^m. Alternatively first choose R, then m, then h. Constants can be
+very large for large alpha; this is not a uniform growing-alpha estimate.
+
+## Exact half-line norm and the endpoint domain
+
+Let L=-d/dx+2bx on its maximal domain of locally absolutely continuous
+u with u,Lu in L²(0,infinity). No boundary condition is imposed on u at0.
+For compactly supported smooth f, u=Tf is locally absolutely continuous,
+satisfies Lu=f and lies in L². The homogeneous solution exp(bx²) is not
+in L², so this inverse is unique. Boundedness extends T to all L² and
+identifies it with L^-1 by closedness (or the distributional equation).
+
+The adjoint is L*=d/dx+2bx on its maximal L² graph domain with g(0)=0.
+Integration by parts first on compact intervals proves the boundary term;
+cutoffs and the graph norm give the stated domains. More explicitly the
+Dirichlet form identity below controls both g' and xg. Truncation and
+mollification with the zero trace show that smooth compactly supported
+functions on [0,infinity) vanishing at0 form a graph core. At infinity
+the boundary term vanishes by those cutoffs. These facts also follow by
+solving the first-order distributional adjoint equation and imposing its
+boundary pairing with arbitrary u(0).
+
+For that core, integration by parts gives
+
+ ||L*g||² = integral [|g'|²+4b²x²|g|²-2b|g|²]
+           >=4b ||g||².                                (9)
+
+A direct proof of the inequality, requiring no oscillator spectrum, is the
+ground-state identity with v=x exp(-bx²):
+
+ integral [|g'|²+(4b²x²-6b)|g|²]
+   = integral v² |(g/v)'|² >=0.
+
+Use a cutoff at0 first; g(0)=0 makes the limiting boundary term zero.
+The identity extends by the graph core. Duality then gives
+||Tf||<=||f||/(2sqrt(b)): for every core g,
+<Tf,L*g>=<f,g>, and the range of L* is dense (ker L=0).
+Density and (9) bound this functional on that range, giving the asserted
+bound without a boundary assumption on Tf itself.
+Equality is attained at f(x)=x exp(-bx²), since
+
+ T f(x)=exp(-bx²)/(4b),
+ ||exp(-bx²)||² / ||x exp(-bx²)||²=4b.
+
+Therefore ||T||=1/(2sqrt(b)). This is a half-line Dirichlet condition on
+the ADJOINT, derived from the original Volterra inverse, not an assumed
+Neumann condition on a full-line oscillator. The direct sum has the same
+norm as either summand. Norm continuity in (8) completes the theorem.
+
+## Relationship to the formal prediction and limitations
+
+Ryanson Jonathan/#4408 observed theta^-1/2 growth; #4435 proposed
+exp(pi alpha²/4)/(2alpha sqrt(pi)) by a two-frequency approximation and
+explicitly left the boundary condition and rigorous upper argument open.
+JoyboyBrian/#5103 proved a full-cell upper order; Nick Petalas/#5150 proved
+a matching edge lower order. Those results and classical q-binomial
+identities are credited dependencies, not new deliverables here.
+The exact fixed-alpha coefficient above keeps the entire theta product.
+No replacement of it by the formal large-alpha expression is made, and no
+joint theta/alpha limit is inferred. The new result is the complete
+operator-norm boundary-layer limit with its domain and explicit remainder.
+
+The exact-real theorem does not supply coefficient preparation, an inverse
+application algorithm, rounded prefixes, integer width, physical movement
+or storage, native supplier admission, outer transfer or integer recovery.
+No selected frontier or multiplication theorem changes.
+
+## Open next questions
+
+1. Obtain useful uniform constants for the production growing-alpha regime;
+   (8) is explicit but too coarse to assert that regime.
+2. Classify sections whose endpoint distance or length scales like sqrt(theta),
+   where a different limiting interval or boundary layer can occur.
+3. Bind an actual rounded complete Gaussian caller, preserving all preparation,
+   prefix, movement and storage charges, before substituting any new guard.

research/gaussian-half-line-operator-limit/README.md

@@ -0,0 +1,15 @@
+# Gaussian half-line inverse limit
+
+JoyboyBrian, with substantial OpenAI Codex assistance. Scoped written theorem with exact finite corroboration; no new kappa.
+
+For fixed alpha>0 and the original full cell N=floor(1/theta), the checkerboard inverse scaled by sqrt(theta) converges in operator norm, under an explicit two-edge step embedding, to two copies of a half-line Volterra operator. [PROOF.md](PROOF.md) gives the full infinite-product coefficient, boundary domain and quantified remainder. This resolves the boundary-layer implication left formal in Ryanson Jonathan #4435 and open after Nick Petalas #5150, beyond JoyboyBrian #5103's matching-order upper estimate. All finite q-LDU identities retain Jose Manuel/#4193, Vlado Vojdanovski/#1673 and classical lineage credit.
+
+The exact limit is alpha exp(b/4)/(2 sqrt(b) P(b)^3), b=pi/alpha², P(b)=product(1-exp(-2br)). A growing-alpha limit is not inferred. The complete cell is untruncated; the first source fixture is alpha3,theta1/400,N400.
+
+Run `python3.12 -B scripts/research_package.py --package research/gaussian-half-line-operator-limit`. Both normal and optimized commands use standard-library exact rational elimination and unchanged source exponent AST, without fitting a numerical norm. RESULT and SOURCES are mandatory and the entire receipt has typed canonical JSON equality.
+
+No coefficient-preparation or inverse-application algorithm, rounded-prefix guarantee, tape/movement/storage tariff, native admission, outer transfer or integer recovery is supplied. These are obligations, not zero charges.
+
+## Open next questions
+
+See PROOF.md: uniform growing-alpha estimates, other section/endpoint scalings, and an actual complete rounded caller remain separate.

research/gaussian-half-line-operator-limit/RESULT.json

@@ -0,0 +1,40 @@
+{
+  "exact": {
+    "matrices": 12,
+    "entries": 261,
+    "nonfirst_tail": "181711281244526291631416372864043841938044799092738477785088/190471383981728519572397642029361552234543366836814853515625",
+    "product_interval": [
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+    ]
+  },
+  "geometry": {
+    "source_pairs": 7906,
+    "offsets": [
+      "1",
+      "3/2",
+      "4/3"
+    ],
+    "first_fixture": {
+      "alpha": 3,
+      "theta": "1/400",
+      "N": 400
+    }
+  },
+  "operators": {
+    "controls": [
+      "cell-isometry",
+      "kernel-h",
+      "missing-h",
+      "orthogonal-corners",
+      "adjoint-ground-state",
+      "nonzero-adjoint-trace"
+    ]
+  },
+  "remainders": {
+    "exponent_comparisons": 216,
+    "geometric_tail_parameters": 9,
+    "regime_subsequence_cases": 6
+  },
+  "claim": "finite corroboration of written fixed-alpha operator-norm theorem; no physical caller or growing-alpha claim"
+}