[petalas]

Summary

Scoped written mathematical proof with exact finite corroboration. Retaining each inner source's paid rank gives a strictly weaker sufficient mirror-capacity condition for the conditional capped-flow compiler. The new implication includes complete leaf funding and a critical-root shifted-logarithmic corollary.

- Compare every mirror cell with the uniform source charge eta_1 S_J.
+ Retain the paid letter and compare with sum_r eta_r S_(J,r).
  eta_r(t) = t (r/Lambda)^(1-t), Lambda > m
  Keep the occupation/root and leaf-funding obligations.

The new abstract integer profile W16=1,W64=2; P1=2,P4=6,P36=1 crosses at exponent 1/2, with both moments exactly 20 and a positive tilted-mean gap. Its old uniform ratio is 3/28 > 1/16, while its new weighted ratio is 51/52 < 1. Exact tail factors are 279/400 and 18/25. The written proof specifies why the new test includes every strictly passing old test when the ambient scale is chosen sufficiently close to m.

The contribution is produced by Nick Petalas. Proof and precise hypotheses, verifier, advance CLAIM. Credit Dillon Amburgey's PR4532 for the open question and critical-root tail/majorant argument, zxv PR4436/4483 for the capped flow and occupation estimates, Ryanson Jonathan's corrected PR4459 for the mirror identity, Jacob Alan Sussman's integer compiler, and their retained native, helper, Cinna Davis and outer lineage. These are cited premises, not audit deliverables. Concavity and weighted summation are classical.

Evidence

Before, the uniform mirror sufficient test excludes the displayed profile. After, the rank-sensitive source charge certifies its mirror condition, with its leaf funding kept explicit. This is a compiler side lemma; no native word realizing the profile is asserted.

At commit 9a8eac9de8bf60cd573edbb6ea835e0febd49e89, local CPython 3.12.15 checks pass:

• All toy mirror cells; two enumerations of each capped base measure; 49 capped-flow fixtures, 369 source-term capacity checks and 1,220 funding cases. Nonlinear square-root comparisons use exact squared rational inequalities.
• Nine semantic controls, a separate one-unit underfunding control, and actual source-rank, leaf-funding, moment and capacity mutations rejected in normal and optimized execution.
• Warning-strict direct normal and optimized runs, both registered commands, metadata/pins, whitespace and AST/function checks. The verifier has 15 functions, at most 28 lines each, with no recursion, while loop or assertion-dependent check.
• Both direct registered commands take approximately 0.03 seconds locally; the complete registered launcher takes 0.29 seconds. These are check-budget observations, not speedup claims. Each registered timeout is 60 seconds.
• PASS_RESEARCH_COMMIT_BYTES for all six package files, including ancestral source-history checks. Official committed export integrity and normal/optimized replay from a read-only Git-free export with empty PATH pass. All seven exported files retain their bytes. Generated stdout receipts bind the six package files by SHA256.

Exact reproduction from the repository root:

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

Any CPython 3.12 interpreter may replace the absolute path. Only the standard library is required. Expected markers are PASS_RANK_SENSITIVE_MIRROR, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The universal conclusion rests on the written proof and the cited conditional capped-flow/recurrence contracts. Finite enumeration corroborates only the declared toy and funding cases. The positive tilted-mean gap is an independent hypothesis; it is not inferred from the weighted mirror condition.

The profile is abstract. No native realization, physical supplier, explicit useful compiler cutoff, arbitrary-input native semantics, owned restoration, chronological widths, adapters, movement, precision, stock, tape execution or analytic recovery is supplied. The full distribution-dependent measure question in PR4532 remains open. No new kappa, selected-frontier change or discharge of any of the five selected framework assumptions is claimed. Independent human and formal proof-kernel review, selected cold replay, native suites, Lean and Docker are unrun. All registered package checks ran locally.

Open next questions

1. Can an actual native profile pass this test while failing the old one, with its complete paid/target and physical bill? First bind its per-paid-rank source cells and mean gap; native realization and full implementation costs are missing here.
2. Can integration against the actual capped base measure weaken the new pointwise test further? First prove a uniform mass bound on a failing weighted cell. Finite sampled caps are insufficient.

Related open work

No open PR studying this rank-sensitive capped-flow mirror condition or the displayed toy profile was found in the live PR/Discussion searches, main-tree README searches and precommit refresh. Sparse local-note and unpublished coverage remain incomplete. PR4532, PR4483 and PR4459 are merged predecessors cited above.

• PR4192 studies physical sequential-arena charges in a uniform Benes compiler. This is adjacent implementation work; the new abstract source-capacity lemma neither proves those physical charges nor changes its compiler.

• PR6380 optimizes a fixed nineteen-leaf query-tree invoice. This result changes a sufficient capped-flow capacity test, without optimizing that tree or comparing its numerical invoice.

• PR6390 sharpens the direct higher-moment integer split charge. Its split/completion coefficient differs from this source-capacity condition; no shared numerical optimum is computed or compared here.

The completion refresh inspected all 14 account open PRs created since the CLAIM, including this PR. The thirteen others are PR6380--6383 and PR6385--6393. None studies the displayed toy profile or computes its weighted mirror ratio. No conflicting same-instance number was identified, and no additional computation checked another contribution. This refresh retains the adjacent distinctions above.

CI notes

Both hosted checks, plan and replay (research/rank-sensitive-mirror), passed at exact head 9a8eac9de8bf60cd573edbb6ea835e0febd49e89 in run 41511353741. The completed checks were read once. Separate run metadata confirms status completed, conclusion success, and the matching head SHA. Planning took 4m19s; the package replay took 1m41s.

Hosted CI certifies successful execution of the registered finite replay and metadata/source checks. It does not supply independent scientific acceptance or a formal proof-kernel check 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 the same toy profile or mirror ratio. The limitations above remain unchanged.

Merge Danger

Door: two-way. Blast radius: isolated.

Six new files in research/rank-sensitive-mirror/. Shared implementations, frozen predecessors and selected records retain their bytes.