[petalas]

Summary

Scoped written proof with exact finite corroboration. The adaptive compiler's tail/ordering gates have a sharp entropy screen, exact for a two-rank target:

q=T(t)/P(t)>=1, 0<x<p<1
p = target tilted mean of normalized log rank
x = paid tilted mean in the same normalization

two target ranks: gates feasible iff log q < D(x||p)
D(x||p)=x log(x/p)+(1-x)log((1-x)/(1-p))

For general finite target profiles this is a sharp sufficient test from range and mean. At equality, positive mass at an interior target rank still gives a strict passing point; a two-rank profile fails. PROOF.md covers the remaining mean/range regimes.

The exact new integer fixture W_4=1,W_256=32,P_16=c,t=1/2, with 1<=c<=128, passes precisely when c>=7. Its optimized scalar numerator is exactly 24. At c=7, a=1,b=31,M=2^132,s=1/2,lambda=3/2 gives u=2049/2056<1, v=64*(6/7)^32<1, and the required forward ordering.

Dillon Amburgey's #4532 retains credit for the exact scalar gate equivalence used as a cited premise. The new contribution is the closed-form screen, strict equality case and histogram transition, rather than a replay of that predecessor. Exponential convexity and binary entropy are classical methods; global literature priority for them is unclaimed. Credit zxv #4436/#4483, Jacob Alan Sussman's compiler, Ryanson Jonathan #4459 and the retained native/helper/geometry/outer lineage. Produced by Nick Petalas. Apache-2.0.

Evidence

Before, the cited scalar criterion asks whether some positive lambda satisfies T(t-lambda) exp(lambda mu_P)<P(t). After, the two-rank case is a closed-form strict entropy comparison with an exact equality classification.

Local CPython 3.12.15 checks pass:

• All 8,439 histogram cases in the declared q>=1 domain agree between the entropy and separate monomial critical-value classifiers. All 5,444 passing cases have a literal rational passing point; 2,995 fail, with 14,975 literal failing-point controls.
• All 288 three-position laws satisfy the chord bound, with 256 strict interior cases and 32 endpoint equalities. All 128 fixture integers, the original three gates at the rational witness, eleven controls and three corrupted verifier copies behave as required.
• Warning-strict direct normal/optimized runs take 0.115s each in the observed local runs. The full registered replay takes 0.454s; each registered command is comfortably below five minutes. These are local budget observations, not performance guarantees.
• Metadata/pins, AST/function bounds, exact committed bytes and source-history preflight pass for all six package files at f7c2625f5488b09e6d19b4bed7efe8b4e89a6782.
• Official export integrity and optimized read-only Git-free replay pass. All seven exported files, including the export manifest, remain unchanged. Normal, optimized and exported checks produce generated source-bound receipt c53cbd91601cc5528224d14bab34a26401f2911db4844603f3d87fb5d1364ef7.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/adaptive-tail-entropy/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/adaptive-tail-entropy/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/adaptive-tail-entropy
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/adaptive-tail-entropy --check-source-history

Any CPython 3.12 executable may replace the absolute local interpreter path. Standard library only. Expected markers include PASS_ADAPTIVE_TAIL_ENTROPY, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The universal claim rests on the written proof and cited scalar equivalence; finite replay corroborates its declared domains.

Limitations and claim correction

The advance CLAIM incorrectly predicted mirror slack for the histogram fixture. The progress correction explicitly reports that its source/exit windows are disjoint. The delivered result covers tail and ordering feasibility only, and the verifier records mirror failure. No full compiler admission follows.

These are abstract histograms. Native realizability, full mirror/recurrence funding, chronological widths, owned restoration, adapters, precision, movement, stock and outer transfer remain separate obligations. Failure of the sufficient entropy screen is inconclusive for general target profiles. No physical supplier, complete multiplication theorem, new kappa or selected-frontier change is claimed. No native, selected scientific cold replay, Lean or Docker run was performed. Independent human and formal proof-kernel review are unclaimed. All registered obligations ran locally.

Open next questions

1. Apply the screen to one actual changed native profile with pinned target/paid histograms and complete mirror cells. Native semantics, width, funding and the physical bill remain blockers.
2. Bound the strict chord deficit from a specified positive interior mass to improve the screen above the endpoint threshold. The present proof supplies only a qualitative equality improvement.

No further research extension is included.

Related open work

• #5243 screens a hypothetical changed p16/a7 paid profile at a fixed conditional compiler point. This contribution derives an analytic tail/ordering criterion for arbitrary finite profiles and an abstract integer fixture; it does not compute or compare that PR's kappa or establish its missing physical pair.
• The live open-PR/topic searches and latest handoff refresh found no open PR studying this entropy threshold or the same fixture transition. Sparse notes, arbitrary diffs and unpublished work remain coverage limits. The merged Prove adaptive gate criterion and critical-root leaf bounds (no κ) #4532/Remove tensor padding (199.32× smaller uniform-rank exponent) #4483 predecessors are credited above.

The completion refresh read this account's nine other open PRs created since the CLAIM at 2026-10-23T13:54:54Z: #5245 qubit commuting repair, #5246 Hermitian hyperplane intersections, #5247 downset inertia, #5248 symplectic partial-spread completion, #5249 P112 diagonal-completion rank, #5250 matrix-pencil determinant images, #5251 a projective design, #5252 finite etale tensor determinant images and #5253 characteristic-two odd-type fitting rank. None studies the adaptive tail entropy threshold or this fixture transition. No additional computations were run to validate those results or compare unrelated quantities.

CI notes

Both hosted checks passed at exact head f7c2625f5488b09e6d19b4bed7efe8b4e89a6782 in Registered research packages run 39967569746: planning in 4m44s and the focused registered replay in 1m14s. Run metadata confirms completed status, success and the matching head SHA. The replay executes both registered warning-strict normal and optimized commands.

All local checks and publication/export checks listed above also passed. Hosted replay certifies execution of the declared finite checks and controls; it does not establish independent human or formal proof-kernel acceptance of the written theorem. The mirror correction and remaining obligations are unchanged.

Merge Danger

Two-way door. Blast radius: one isolated research package. Shared code, frozen predecessor inputs and selected artifacts are unchanged.