[petalas]

Summary

Give a closed Lucas-number formula for PR5221's resonant characteristic-five proper-subset disjointness-repair rank. Claim type: scoped written arithmetic proof with exact finite corroboration, using the credited Jordan and endpoint theorem as premises.

- rank(I-A_h) = 2^h - 1 - sum of five recursively updated multiplicities
+ rank(I-A_h) = (4*2^h - 2*L_h - 5)/5,  h>=4 divisible by four

The new arithmetic step proves total_blocks=(2^h+2*L_h)/5 for every even h. L_0=2, L_1=1 and L_(n+2)=L_(n+1)+L_n. The inherited rank and additive correction count are unchanged. This is a closed-form contribution, with no new kappa or physical supplier.

This contribution was produced by Nick Petalas. Credit Brian / JoyboyBrian PR5221 at ad83478022fed6c1d6a6014096bd8b443e8c9d9a; Ali Romman PR4962, Nick Petalas PR5173, mikey1201 PR172 and icekylinx PR144 for the retained tensor/endpoint/compiler lineage. The modular Jordan and Fibonacci ingredients are classical, including David Benson's Section 5.1. The new specialization and its exact premise boundary are in research/quinary-repair-lucas-formula/PROOF.md. Independent human and formal proof-kernel review are unclaimed. Apache-2.0.

Advance claim. Canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030 remained unchanged at submission. The coframe card is explicitly changed to this historical arithmetic side result. Live handoffs, open bodies, main and merged formula/object searches were checked; the complete overlap snapshot covered 505 open PRs and 6,496 Discussion records. Arbitrary historical diffs and unpublished work remain limits.

Evidence

Before, the source presented this rank through a five-state recurrence. After, the written proof evaluates that recurrence exactly by a Fibonacci two-step calculation and the weighted dimension identity.

Local CPython 3.12.15, standard library only:

• All 513 Lucas values through index 512 agree between iterative fast doubling and a separate binomial sum. Literal replacement bookkeeping agrees with 512 parity/multiplicity formulas, 257 even block counts and 128 resonant rank substitutions. The inherited examples evaluate to 9 at h=4 and 185 at h=8.
• Fifteen invalid inputs and five named formula mutations are rejected. Each direct run emits a generated receipt bound to all six package files and the recurrence transcript. No handwritten receipt is used.
• Warning-strict normal and optimized direct commands pass in 0.145034 s and 0.147856 s locally. These are timeout measurements, not a speedup claim. Both registered commands and metadata/source pins pass.
• Committed bytes, modes and full declared source-history preflight pass at 0bba898293725033cde78c557c8a48cf53314926. Official six-file export and input-coverage checks pass. Normal and optimized read-only Git-free replays with empty PATH produce identical receipts; all seven export files remain unchanged.
• In-memory syntax compilation, function-size inspection and staged whitespace checks pass. No separate static analyzer is claimed.

From the worktree root, reproduce with the exact local interpreter path or substitute another CPython 3.12:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -X dev -B research/quinary-repair-lucas-formula/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -X dev -B research/quinary-repair-lucas-formula/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/quinary-repair-lucas-formula
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/quinary-repair-lucas-formula --check-source-history --check-modes

Expected markers: PASS_QUINARY_REPAIR_LUCAS_FORMULA, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The new proof evaluates an integer recurrence; it cites rather than audits PR5221's Jordan rules, deletion argument and all-field rank theorem. Its verifier runs no predecessor scientific suite or matrix-rank census. Bounded arithmetic checks corroborate the written proof and do not replace its quantifiers.

Nonresonant dimensions, a nonadditive physical repair, native operators, routing, dirty restoration, precision, stock, tape, compiler and outer admission remain outside scope. All five selected framework assumptions remain. No selected cold replay, Lean, Docker or complete multiplication theorem is claimed. No local mathematical check is deferred solely to CI. The bounded managed-allocation deviation is documented in README.md.

Open next questions

Can a nonadditive physical repair avoid the additive rank obstruction? Start with an arbitrary-input identity and the complete preparation, routing and dirty-restoration invoice. No physical caller is supplied and no continuation is reserved.

Related open work

• PR6469 improves the runtime of PR5221's bounded rank evaluator. This PR gives an all-dimension closed-form specialization without extending that evaluator, its corpus or its implementation question. No runtime comparison or extra predecessor computation is made.

The completion refresh inspected all 12 account open PRs created since the advance CLAIM, including this PR. The other 11 concern different objects or quantities; none was found to study this resonant characteristic-five repair rank or its Lucas specialization. No conflicting common-instance value was identified, and no extra scientific computation checked another contribution.

CI notes

Both hosted checks, plan and replay (research/quinary-repair-lucas-formula), passed at exact head 0bba898293725033cde78c557c8a48cf53314926 in run 41609088474. The completed checks were read once, and the run metadata confirms the matching head, completed status and successful conclusion. Hosted replay certifies the registered normal and optimized bounded commands; all local checks listed above also passed. This does not establish independent human or formal proof-kernel acceptance of the cited premises or written all-dimensional argument.

Merge Danger

Door: two-way. Blast radius: isolated research. Six new opt-in package files; shared code, predecessor packages, frozen evidence and the selected frontier are unchanged. Archival merge and scientific acceptance remain separate.