Conditional κ = 0.00268647 via paid target padding after native composition

research/pr4359-p7-padding/ENDPOINT.md

@@ -0,0 +1,198 @@
+# Conditional p7 outer saving 0.00268647
+
+The native p7 construction in `NATIVE.md`, together with the tensor and
+target-padding construction in `TENSOR.md`, supports the proposed conditional
+bound
+
+```
+T(n) = O(n (log n)^(1-kappa)),       kappa = 0.00268647.
+```
+
+This raises the exponent saving by `0.00038647` over PR4359's `0.0023`, and
+by `0.00000660` over PR4414's p6 point `0.00267987`. These compare exponent
+savings, not measured running times. The selected frontier is unchanged.
+
+The result remains conditional on the documented all-column native,
+whole-job adapter, spectator, finite compiler, storage, precision, analytic,
+fixed-tape, and recovery hypotheses inherited from zxv's PR4359 at
+`da637c87c89482bc4c86be36bd0071b3d400dbed`. The p7 Laurent factorization,
+geometry and native ownership are checked in this package. Finite checks do
+not constitute a formal proof of the full multiplication theorem.
+
+The native source is credited to zxv's PR4359 and integer mixed flow to
+Jacob Alan Sussman's Theorem 1G. The outer slack family and executable
+ledger are retained from PR4405 at
+`873d1b3c96a83b701a5c0bba8e37a861e6672fef`, with its attribution to
+@Vlado-V's PR1673, @jeremykothe2's PR1758, @chafreaky's PR1786, and their
+cited predecessors. Apache-2.0; substantial OpenAI assistance.
+
+## Native invoice and why target padding is needed
+
+The exact all-fresh p7 invoice is
+
+```
+targets = {5:420, 6:270, 7:840, 8:1395, 9:3195,
+           25:448, 41:448, 57:448, 121:224},
+paid    = {1:37305, 2:10695, 3:3360, 4:1530, 5:300,
+           8:1464, 16:448, 32:448, 48:448}.
+```
+
+It has 7,688 productive fields, target rank-one moment 131723, and paid
+rank-one moment 131115, leaving surplus 608. Its raw paid ranks exceed its
+smallest target ranks. Moreover, a rank-one tail bound with cutoff below
+the raw minimum target rank does not suffice near this proposed saving.
+
+The new construction tensors the native word `v=6240` times, completes
+every productive endpoint of rank smaller than `11^v` to rank `11^v` on
+unused disjoint axes, and splits every paid call, including those endpoint
+completions, into consecutive pieces of size at most `10^v`. Endpoint
+completion occurs only after the whole native tensor word has finished.
+It uses the same arbitrary productive array, without new productive credit
+for copied inputs or work storage. The ambient rank `121^v` has room for
+each completion.
+
+At `C_test=0.00268648`, write `tau=1-C_test` and let `P_tau,T_tau` denote
+the raw moments. The certificate proves
+
+```
+P_tau/T_tau < 1 - 1/10^8,
+131115/(10^C_test*T_tau) < 4991/5000,
+7688*11/(10^C_test*T_tau) < 2/3.
+```
+
+The first bound controls the original tensor word. The second pays for
+splitting its large calls. The third pays for completing small productive
+endpoints and splitting those completion calls. For a completion of rank
+`D<11^v`, writing `D=q*J+r` with `J=10^v` gives, including `q=0`,
+
+```
+q*J^tau+r^tau <= D*J^(-C_test)+J^tau
+              < 2*11^v/J^C_test.
+```
+
+Thus the complete normalized paid moment is bounded by
+`alpha^v+2*gamma^v+2*padding^v`, while the target moment only increases.
+The exact block bound `(4991/5000)^390<1/2` gives 16 decay blocks. The
+completion tail satisfies `(2/3)^2<1/2`, giving 3120 blocks. Bounding both
+tails by `2^-16` yields the simpler sufficient inequality
+
+```
+1/(1+6240/10^8) + 4/2^16 < 1.
+```
+
+Every paid rank is now at most `10^v`, and target ranks lie between `11^v`
+and `121^v`. Rank-one surplus is unchanged: each endpoint completion adds
+the same rank to the paid and target totals, and paid splitting preserves
+rank. Hence it remains `131723^v-131115^v>0`.
+
+## An explicit uniform rank
+
+Choose
+
+```
+kappa   = 0.00268647,
+d       = (C_test-kappa)/7 = 1/700000000,
+uniform = kappa+6*d = 376107/140000000,
+t       = 1-C_test,
+z       = 1-uniform,
+z0      = (t+z)/2,
+delta   = z0-t = 1/1400000000.
+```
+
+The source compiler now has `m=121^v`, `k_min=11^v`, and `r_max<=10^v`.
+The exact integer inequality `10^26<11^25` gives exit exponent below
+`25/26`, so use `h=1/26`. For ambient rank `x>=m`,
+
+```
+m E(x)/x < m^h*r_max*x^(-h)
+          < 2^B*x^(-h),              B=(111/26)*v=26640.
+```
+
+This follows from `10<2^4` and `121<2^7`. The exact complement coefficient
+is `eta=z0*m^(z0-1)`. Since `z0>1/2`,
+
+```
+-log2(eta) < 1+7*v*(1-z0) = 147931769/1250000.
+```
+
+The entirely rational sufficient formulas
+
+```
+H0 = ceil((B*delta+1+7*v*(1-z0))/(h*delta)),
+H1 = floor(H0*(1-z0)/(z-z0))+1
+```
+
+therefore give
+
+```
+H0 = 4307773805920,
+H1 = 16201843136005340321,
+N  = 2^16201843136005340321.
+```
+
+The verifier checks `H0>=7*v`, covering the entire finite base range.
+Below `2^H0`, the rank-one surplus gives `L(x)<=x`; above it the exit
+coefficient is below the complement coefficient. Strong induction yields
+`L(x)<=A*x^z0`, with `A=(2^H0)^(1-z0)`, for every positive integer rank.
+The strict final power gap is `1/1400000000`, proving
+`P_N/B_N<N^(1-uniform)`.
+
+This is a specified finite exact program with very large constants. Neither
+the tensorized and padded word nor the uniform compiled word has been
+materialized. Their complete instructions, integer batch factors, paid
+leaves, scalar operations, and owned banks are defined by the finite
+recipes. The one-less-integer controls only concern these sufficient
+formulas; no minimal-rank claim is made. The initial sufficient choice
+`v=30000`, paid cutoff `7^v` and target floor `8^v` gave
+`H1=47613655221424910001`; the final parameters reduce both the tensor
+power and this rank exponent while preserving the same tested saving.
+
+## Outer parameters and changed constants
+
+Use the same strict outer family:
+
+```
+b=kappa+5*d,  a=kappa+4*d,  q=kappa+3*d,
+epsilon=1-d, beta=1-3*d, r0=d/10, stop=d, layout=d.
+```
+
+Retain the ordinary BIT saving `0.0027`, atom exponent `0.003`, Strassen
+exponent `45/16`, theta power `2`, halo power `48`, and core power `128`.
+The effective saving remains strictly below both the new complex saving
+and the admitted BIT saving. The frozen ledger recomputes all nine outer
+cost rows, all 54 named predicates, and three additional functional
+predicates at this point. The limiting main-transform margin is exactly
+`524529867/122500000000000000 > 0`.
+
+The inherited 63-row ledger was compared with this current predicate set
+in PR4405: all 50 common predicates and all nine costs agreed, and the
+13 omitted rows were historical endpoint comparisons. The larger kappa
+preserves those comparisons; this package reruns the current 54 plus three
+functional predicates.
+
+Apply the source precision and stock theorem using the new p7 word,
+tensor/padding recipe, compiler counts, scalar word, and owned storage.
+No p3 or p6 raw scalar constant is substituted for the new word. Its new
+fixed constants and denominators are absorbed after an effective threshold
+by the positive semantic, support, stopping, and fine-prime spare powers.
+The unchanged BIT template retains its separately compiled and covered
+construction.
+
+## Replay and remaining work
+
+```
+python3 -I -O -B research/pr4359-p7-padding/endpoint.py
+```
+
+The integrated command checks the native p7 invoice and geometry, the
+tensor/cutoff/padding moment, the specified uniform rank, and every outer
+predicate. Mutation controls reject closing the precision or uniform gap,
+requesting the unfunded tested endpoint as kappa, and reducing BIT below
+the required effective saving. Complete source authentication is a separate
+package-verifier stage.
+
+Next questions include independently formalizing the native tensor and
+endpoint-completion construction, improving the controller family, and
+reducing the tensor power and sufficient rank. The bounded controller
+search does not establish global optimality. The inherited all-size
+analytic and native hypotheses remain conditional premises.