Conditional κ = 0.00267987 with a complete p6 native controller

research/pr4359-p6-fresh/ENDPOINT.md

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+# Conditional p6 outer saving 0.00267987
+
+The source-bound p6 native construction in `NATIVE.md` and its finite tensor
+construction in `TENSOR.md` support the proposed conditional bound
+
+```
+T(n) = O(n (log n)^(1-kappa)),       kappa = 0.00267987.
+```
+
+This improves the exponent saving by `0.00037987` over PR4359's `0.0023`
+and by `0.00020232` over the preceding all-fresh p3 point `0.00247755`.
+These are differences between exponent savings, not measured runtimes.
+
+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 new p6 geometry and native
+ownership are checked independently in this package; finite checks do not
+constitute a full formal multiplication-theorem proof. The selected frontier
+is unchanged. Apache-2.0; substantial OpenAI assistance.
+
+The native source and p6 Laurent factorization are credited to zxv's
+PR4359. The integer mixed flow is Jacob Alan Sussman's Theorem 1G. The
+outer slack family and executable ledger are retained from PR4405, commit
+`873d1b3c96a83b701a5c0bba8e37a861e6672fef`, with its attribution to
+@Vlado-V's PR1673, @jeremykothe2's PR1758, @chafreaky's PR1786 and their
+cited predecessors.
+
+## The new native invoice and its strictification
+
+The exact all-fresh p6 invoice is
+
+```
+targets = {5:364, 6:234, 7:728, 8:1209, 9:2769,
+           25:448, 41:448, 57:224, 105:224},
+paid    = {1:32331, 2:9269, 3:2912, 4:1326, 5:260,
+           8:1224, 16:448, 32:448, 48:224}.
+```
+
+Its target and paid rank-one moments are 108769 and 108257, leaving surplus
+512. Its paid ranks overlap and exceed some target ranks, so the raw rule
+does not directly meet the source compiler's strict separation condition.
+
+At the exact saving `C_test=0.00267988`, put `tau=1-C_test` and write
+`P_tau,T_tau` for these two moments. The exact certificate proves
+
+```
+alpha = P_tau/T_tau < 1 - 1/10^8,
+gamma = 108257/((19/4)^C_test*T_tau) < 9999/10000.
+```
+
+Use the chronological native tensor construction with `v=150000` copies,
+then split each paid rank into consecutive disjoint pieces of size at most
+
+```
+J = floor((19/4)^v).
+```
+
+Since `J >= (19/4)^v/2`, the normalized paid moment is below
+`alpha^v+4*gamma^v`. The certificate checks
+`(9999/10000)^10000 < 1/2`, and consequently
+
+```
+alpha^v+4*gamma^v
+  < 1/(1+150000/10^8) + 4/2^15 < 1.
+```
+
+Every paid rank is now at most `J<5^v`, while productive target ranks range
+from `5^v` to `105^v`. Thus the finite mixed word meets strict separation
+and the tested superpotential. Its positive rank-one surplus is
+`108769^v-108257^v`; splitting preserves total rank. No claim about uniqueness
+or global optimality of a raw mixed root is required.
+
+## Explicit uniform rank
+
+Choose the outer target and slack family
+
+```
+kappa   = 0.00267987,
+d       = (C_test-kappa)/7 = 1/700000000,
+uniform = kappa+6*d = 375183/140000000,
+t       = 1-C_test,
+z       = 1-uniform,
+z0      = (t+z)/2,
+delta   = z0-t = 1/1400000000.
+```
+
+The new rank parameters have `m=105^v`, `k_min=5^v` and
+`r_max<=floor((19/4)^v)`. The exact integer inequality
+
+```
+19^32 < 4^32*5^31
+```
+
+proves that the source compiler's exit exponent is below `31/32`. Set
+`h=1/32`. For ambient rank `x>=m`, its exit bound gives
+
+```
+m E(x)/x < m^h*r_max*x^(-h)
+          < 2^B*x^(-h),            B=(79/32)*v=740625/2.
+```
+
+Here `(19/4)^4<2^9` follows from `19^4<2^17`, and `105<2^7`. The exact
+complement coefficient is `eta=z0*m^(z0-1)`. Since `z0>1/2`,
+
+```
+-log2(eta) < 1+7*v*(1-z0) = 11259493/4000.
+```
+
+Therefore the entirely rational sufficient choices
+
+```
+H0 = ceil((B*delta+1+7*v*(1-z0))/(h*delta)),
+H1 = floor(H0*(1-z0)/(z-z0))+1
+```
+
+give
+
+```
+H0 = 126106333450000,
+H1 = 473129651134046950001,
+N  = 2^473129651134046950001.
+```
+
+The verifier checks `H0>=7*v`, covering every finite base rank. Above
+`2^H0` the exit coefficient is below the complement coefficient. Below it,
+the positive rank-one surplus gives `L(x)<=x`. The same strong induction as
+the source compiler then yields `L(x)<=A*x^z0` for all positive integer
+ranks, with `A=(2^H0)^(1-z0)`. The strict final power gap is
+`1/1400000000`, proving `P_N/B_N<N^(1-uniform)`.
+
+This specifies a finite exact uniform program at a very large rank. Neither
+the tensorized mixed word nor its uniform compilation has been materialized.
+The integer batching, scalar word and owned banks are defined by the finite
+algorithms, and their full constants must be computed from this new recipe.
+The one-less-integer controls only test these sufficient formulas; they do
+not establish a globally minimal template rank.
+
+## Outer transfer and replacement constants
+
+Use the same positive-slack family with
+
+```
+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 `a` is strictly below both the new complex saving and
+the admitted BIT saving. The preserved ledger recomputes all nine outer
+cost rows and checks all 54 named predicates plus three additional
+functional predicates. The limiting main-transform margin is exactly
+`262265511/61250000000000000 > 0`. Their exact receipt is generated by
+`endpoint.py`.
+
+This uses the current predicate set audited in PR4405. Its comparison to the
+older 63-row ledger showed that the 13 omitted rows were historical endpoint
+comparisons, with all 50 shared predicates and all nine cost formulas
+agreeing. The larger kappa preserves those historical comparisons.
+
+Apply the source precision and stock theorem to the new p6 word and its
+tensor/compiled word counts, rather than carrying over the p3 raw scalar
+or movement counts. All fixed constants, rational denominators and owned
+storage come from the new recipe. The positive precision, stopping,
+support-reservoir and fine-prime spare powers checked by the ledger absorb
+these fixed constants after an effective input threshold. The unchanged BIT
+template retains its separately compiled and delayed-cover construction.
+
+## Replay and next questions
+
+```
+python3 -I -O -B research/pr4359-p6-fresh/endpoint.py
+```
+
+The integrated command checks the new native invoice and geometry, the
+rational-cutoff tensor moment, the explicit uniform rank, and every outer
+predicate. Mutation controls reject closing the precision or uniform gap,
+asking for the unfunded tested endpoint as kappa, and reducing the BIT
+saving below the effective saving. Package source authentication is a
+separate complete verifier stage.
+
+Further work includes improving the native controller family, reducing the
+tensor power and sufficient rank, independently formalizing native tensor
+closure, and reviewing the inherited all-size outer hypotheses. The bounded
+controller search is not an optimality proof.

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