[petalas]

Summary

Scoped written proof with exact finite corroboration, produced by Nick Petalas. This answers Open next question 2 of PR4832: the least direct higher-moment coefficient for every integer split and every real 0<t<1, sigma>1.

split_t(R,J) <= K(t,sigma) R^sigma J^(t-sigma)
K(t,sigma) = (1+s^t)/(1+s)^sigma
unique maximizing remainder: t s^(t-1)+(t-sigma)s^t-sigma=0

retained complete strictification bound:
alpha^v + K(t,1)   gamma_u^v + 2 delta^v
alpha^v + K(t,t+u) gamma_u^v + 2 delta^v     for u>1-t

The new coefficient is strictly smaller. At t=1/2, sigma=17/10, the exact coefficient is (5/4)(16/17)^(17/10), attained by every J=16n,R=17n. Exact rational comparisons give 9/8<K<8/7<6/5<K(1/2,1).

Only research/direct-higher-moment-split/ is added. The universal theorem rests on the written quotient/remainder proof and integer sharpness argument. It is neither a finite search failure nor a complete multiplication theorem. No new kappa, source-word improvement, selected-frontier change or particular smaller compiler is claimed. Advance CLAIM.

Credit Nick Petalas PR4832 for the question and linear-rank predecessor; zxv PR4433 for the higher-moment splitting/completion framework, PR4480 for net-tax results and PR4483 for the different direct compiler; Cinna Davis PR4412/4414/4416 for chronological product/padding; Jacob Alan Sussman for the integer compiler; and their retained native/helper/controller/outer lineage. Predecessor results are cited premises, not audit deliverables. Elementary calculus is classical.

Evidence

Before, the linear-rank coefficient was transferred through a separate higher-moment comparison. The new proof keeps the full denominator during optimization, proves strict quotient monotonicity and a unique maximizing remainder, and characterizes exact integer equality.

Local CPython 3.12.15 checks pass:

• 30 rational parameter pairs, 7,650 literal integer splits, 6,720 strict quotient comparisons, 30 near-maximizing integer witnesses, the exact attaining example, and 13 rejecting controls. All acceptance decisions use rational/integer arithmetic.
• Warning-strict direct normal and optimized checks and both registered scientific commands. Each direct command takes about 0.08 seconds locally; the complete registered launcher takes about 0.40 seconds. These observations establish the replay budget, not a runtime improvement.
• Metadata/source pins, AST compilation and function-length inspection, whitespace checks, committed-byte comparison and declared source-history ancestry at f9e68f40b0da7236a7fad415f7172218c0666d27.
• Official eight-file package/input export and input-coverage integrity check. Normal and optimized replay pass from the read-only Git-free export with empty PATH. Its nine files, including the export manifest, retain their bytes and read-only modes. Three actual mathematical verifier corruptions reject in both modes.

Reproduce from the repository root:

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

Any CPython 3.12 installation can replace the absolute interpreter path. Standard library only. Expected markers are PASS_DIRECT_HIGHER_MOMENT_SPLIT, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Direct replay prints generated receipts bound to all six package files and both source-document hashes. No generated receipt is treated as an oracle.

Limitations and unrun obligations

This sharp constant concerns the full split charge. It does not optimize the net splitting tax, remove the unsplit overcount or change the full completion charge. No numerical supplier inventory, least tensor power or favorable physical construction is certified. The finite verifier uses bounded rational parameter domains; the written proof supplies the all-real statement.

Native semantics, arbitrary-input physical realization, restoration, ownership, chronological compiler closure, stock, precision, tape access, exact recovery and the five framework assumptions remain inherited. No actual caller, selected cold replay, native suite, Lean or Docker run was performed. Independent human and formal proof-kernel review are unclaimed. All registered package obligations ran locally.

Open next questions

• Apply the new coefficient to one fully charged source-bound inventory that still needs strictification. Preserve its native/completion/compiler/outer hypotheses and reuse its exact moment enclosures before claiming a smaller sufficient tensor power.
• Determine the sharp direct coefficient for the net splitting tax. Start from (q+s^t-(q+s)^t)/(q+s)^sigma and inspect PR4480; the quotient reduction here does not settle that different objective.

Related open work

The completion refresh inspected all 13 other account open PRs created since the CLAIM at 2026-11-02T19:33:27Z. None studies this direct split coefficient or computes the same sharp constant. No disagreement on a common computed number was identified, and no extra computation checked another contribution.

• PR6384 proves a rank-sensitive capped-flow mirror-capacity criterion. This is adjacent compiler accounting; its source-capacity condition and toy mirror ratios differ from this split-charge coefficient. No shared numerical optimum is compared.
• PR6391 proves an exact signed whole-idle inventory floor for overlapping stage outputs. Its signed subset-inventory optimization concerns a different boundary; this result keeps PR4433's completion charge unchanged and sharpens only the original-paid split term.

PR4832, PR4433, PR4480 and PR4483 are merged predecessors. Entry searches included 332 open PR bodies, live handoff claims, this account's recent work, distinctive formulas across tracked main READMEs and merged PR titles/bodies. The overlap helper's sparse local inventory covered 116 of 8,917 note paths; complete tracked-README Git searches were separate. Arbitrary historical diffs and unpublished work remain limits.

CI notes

Both hosted checks, plan and replay (research/direct-higher-moment-split), passed at exact head f9e68f40b0da7236a7fad415f7172218c0666d27 in run 41511979921. The completed checks were read once. Separate run metadata confirms status completed, conclusion success, and the matching head SHA. Planning took 4m33s and package replay took 1m49s.

Hosted CI certifies successful execution of the registered finite replay and metadata/source checks. It does not provide independent mathematical acceptance or proof-kernel verification of the written theorem. All registered checks and publication/export checks also passed locally. The completion refresh of Related open work found no conflicting computation of this coefficient. The stated scope and remaining obligations are unchanged.

Merge danger

Door: two-way. Blast radius: research. One isolated opt-in package; source inputs and selected records are unchanged.