[petalas]

Summary

Scoped mathematical proof with exact finite corroboration, produced by Nick Petalas. A midpoint convexity argument gives a strictly larger uniform rational curvature coefficient for every integer rank k>=2:

theta_k(a) = 2 k^(1-a) - (k-1)^(1-a) - (k+1)^(1-a)
0 < ell <= a <= u < 1

previous coefficient: ell(1-u)/[(k+1)(1+uk)]
new coefficient:      ell(1-u)/[k(1+u(k-1))]

This answers Open next question 2 of Dugongue's PR3863, tightening its sufficient positive-cost allowance. For proposed 2+2 -> 1+3 exchanges on [17/10000,18/10000], 1,181 copies now suffice to fund one additional rank-one child, reducing the preceding sufficient target of 1,775 by 594. The exact new per-exchange credit is 84847/100180000; after that charge the guaranteed raw slack is 24307/100180000.

This is an original accounting refinement. The predecessor's capped-mass identity and paid-overhead transfer are cited, rather than audited or replayed. No physical exchange, true minimum exchange count, supplier admission, new kappa or selected-frontier change is claimed. Research stopped at the first verified improvement. Only the five-file research/midpoint-rank-credit/ package is added.

Credit Dugongue / PR3863 for the question, triangular integral and paid-overhead framework; the ranked-pair verification and Avi Eisenberg / ikeboy, eumemic, Chafik Boukhalfa, Dominik Scholz and their retained lineage for the capped-mass identity and all-exponent sign theorem; MathProof for the reserve precedent. Jensen's inequality and elementary convexity are classical. Predecessor files and credit lines remain unchanged. Independent human and formal proof-kernel review are unclaimed.

Evidence

The written proof applies Jensen to the symmetric triangular density, then uses the concave tangent bound k^a<=1+a(k-1). The new coefficient extends termwise to complete capped-mass profiles and actual added nonnegative rank fees, with unchanged normalization and common costs.

Local CPython 3.12.15 checks pass at head 0c9261aa5492b789f4dc8959d7f2ad901036330b:

• 288 exact rational curvature comparisons, 99 independent square/cube comparisons, 465 eligible profile/interval pairs and 1,395 complete power-moment comparisons. The profile enumeration considers all 81 bounded count profiles and all 6,561 ordered pairs for each interval. Fifteen domain and semantic controls pass.
• Both warning-strict normal and optimized commands in VERIFY.json, and the complete registered package launcher with source metadata validation. Direct command times are 18.98 and 19.19 seconds, below their 240-second limits. These timings establish the replay budget, not a performance improvement.
• Exact publication-byte and declared source-history preflight for all five files; warning-strict execution, AST/function-size inspection and whitespace checks. No standalone Python linter is installed.

Exact reproduction from the repository root:

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

Any CPython 3.12 executable may replace the absolute interpreter path. Standard library only. Expected markers are PASS_MIDPOINT_RANK_CREDIT, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. Direct JSON stdout is generated from the checks and binds all five package-file hashes. No receipt is handwritten or used as an oracle. Replay writes nothing and imports no predecessor executable.

Limitations and unrun obligations

The all-rank result rests on the written proof; finite power enclosures corroborate the implementation. The evaluator enforces rank at most 64, counts at most one million and rational endpoint denominators at most one million. These bounds do not restrict the written theorem.

The count 1,181 is minimal for this sufficient rational inequality. It does not exclude funding the charge with fewer exchanges under the true moment. The proposed exchanges need 2,362 available rank-two children; no particular current source inventory or legal source-level realization is asserted. Every physical fee, target debit and interface change still needs its own complete bill.

No native or physical construction, arbitrary-dirty restoration, ownership/chronology proof, all-size compiler, stock, precision/tape or analytic recovery obligation is discharged. No selected cold replay, Lean or Docker suite was run. This is not a complete multiplication theorem. All registered package obligations ran locally.

The scalar-to-moment card is explicitly changed to an accounting side result. Canonical main was refreshed before submission and remains 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8. The selected binding positive control passed. Live claims, account work, open/merged PRs and main formula/object searches were checked before the claim and commitment. Cached overlap covered 406 open PRs and 6,350 Discussion records, but only 114 of 8,917 local HEAD notes in the sparse checkout; separate main-tree searches supplement it. These are bounded searches, not a novelty guarantee.

Open next questions

• Can a legal chronological exchange attain this stronger paid target? Start with one compatible source pair, its arbitrary-dirty endpoint identity and every added fee. Native compatibility, ownership, chronology, restoration and compiler/precision/stock remain blockers.
• What is the least actual exchange count over the stated interval? Start by bounding the derivative of the literal two-rank power difference, then certify its minimum. Global monotonicity and true-threshold optimality are unproved here; no follow-on search is reserved.

Related open work

No open PR found in the scoped searches states this midpoint coefficient or computes a competing sufficient exchange count for PR3863's interval. Adjacent open scopes are distinct:

• Prove fixed-prefix P104 tails are optimal for every concave tariff #6394 proves fixed-prefix P104 tail optimality for every concave tariff; it changes neither this curvature coefficient nor PR3863's positive-fee target.
• Prove the sharp direct higher-moment integer split charge (no κ) #6390 sharpens an integer higher-moment splitting coefficient; it prices a different completion operation and supplies no competing exchange threshold.
• Draft: Conditional κ = 7.10436534421701e-4: concave-descent frame retiming of 25 gates on #251's source527 banks #260 studies source527 concave-descent retiming and its physical invoice; this package supplies no retimed word and computes no kappa for that construction.

Merged #5760 constructs majorization transfer plans and #5762 prices two-fee regions for all increasing-concave tariffs. This result concerns a specified power interval and quantifies the strictly positive curvature allowance. It does not change their all-tariff claims. No extra computations were run to check any of these contributions.

The completion refresh inspected the 12 other account open PRs created since the advance CLAIM at 2026-11-02T23:56:36Z: #6456 and #6459 through #6469. None states this curvature coefficient or computes a competing PR3863 sufficient exchange threshold. No disagreement on a common computed quantity was identified, and no additional computation checked those contributions.

• Prove sharp one-channel capacity change for paired cut edits (no κ) #6462 bounds unit-channel capacity changes under paired cut edits; its capacity is distinct from the power-moment credit priced here.
• Bound the stopped-recursion cost coefficient by 1,209,000 (no κ) #6463 bounds a stopped-recursion cost constant under a fixed critical moment; it changes no discrete curvature coefficient or paid exchange threshold.
• Prove sharp dyadic channel loss under frozen snapshots (no κ) #6465 bounds dyadic channel loss under frozen snapshots; it imposes a different physical insertion constraint and computes no competing raw moment allowance.

CI notes

Both hosted checks, plan and replay (research/midpoint-rank-credit), passed at exact head 0c9261aa5492b789f4dc8959d7f2ad901036330b in Registered research packages run 41549947220. The completed check list was read once with gh pr checks 6457 --repo CrocSwap/integer-mult-bounds; run metadata separately confirms completed status, success and the matching head. Planning took 4m39s and package replay took 2m45s.

Hosted replay executes both registered warning-strict normal and optimized finite checks and source metadata validation. All local scientific and publication checks listed above also passed. No failed or waived check remains. These checks certify execution of the declared finite evidence; they are not independent mathematical acceptance, formal proof-kernel review or supplier admission.

Merge danger

Door: two-way. Blast radius: isolated research package. Shared code, frozen evidence, predecessor credit and the selected frontier are unchanged.

Advance CLAIM.