[petalas]

Summary

Scoped algebraic proof with exact finite corroboration: minimum diagonal output rescaling for every unit-weighted reversal over Z/MZ, every M>=2 and every positive length.

p defective reversal pairs + c defective centers
minimum nonidentity coordinate scales on both banks = 2(p+c)
number of attaining repairs = 2^p rho^c
rho = number of units whose square is 1 modulo M

The proof classifies every attaining repair, gives the complete support generating polynomial, and characterizes when one common scalar can perform the repair. It resolves the even-length gap in Wentao Li's merged #4095 and permits multiple defective pairs and a defective center. The optimum counts output-coordinate scales at one callee boundary. It is not a native-call or Boolean-gate tariff. No new kappa or selected-frontier change.

This contribution was produced by Nick Petalas. Credit Wentao Li #4095/#4073/#4070 for the dual-scale setting and restricted classifications, Jerry Chen #2950 for the weighted ordinary-wrapper contract, and the retained Nick Petalas/David Arias/Jose Manuel wrapper and physical-caller lineage. Predecessor code and receipts are not imported or replayed.

Evidence

Before, the cited composite-modulus theorem covers odd lengths and a single defective end, leaving even lengths open. After, the new orbit proof gives an exact minimum and full support distribution for arbitrary unit weights and all lengths/moduli.

The independent finite map enumeration checks 319 weight lists and all 622,195 ordered two-bank scale assignments in 23 fixed domains. Every support histogram, minimum count and common-scalar optimum matches. Six further fixtures enumerate all constructed minimum repairs; the literal six-operation wrapper passes 202 module-basis columns. Ten boundary controls pass.

Both warning-strict normal and optimized registered commands pass locally on CPython 3.12.15, each timed at 0.42 seconds. Package metadata/replay, six-file committed-byte/source-history preflight, official export integrity and optimized Git-free replay pass at head f38ce4e79a14e71704e61ddaea4410c4bdedf88b. Source and typed-receipt mutations reject; all seven exported files remain unchanged. The generated receipt binds all package sources by SHA256.

From the repository root, using only the Python standard library:

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

Any CPython 3.12 installation can replace the displayed interpreter path. Expected markers are PASS_WEIGHTED_REVERSAL_RESCALING weights=319 assignments=622195 wrapper_columns=202 controls=10 no_new_kappa, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Advance CLAIM. Canonical main refreshed unchanged at 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7. Live PRs and recent claims were rechecked before committing, alongside distinctive main-tree and merged searches. The earlier complete remote snapshot covered 367 open PRs and 3,821 Discussion records; sparse local-note coverage was incomplete. These searches do not establish exhaustive novelty.

Limitations and unrun obligations

The result fixes the weighted reversal and permits independently chosen diagonal unit output scales. It does not optimize changed wrappers, nondiagonal maps, nonunit weights, auxiliary banks or input-dependent normalization. The finite checker uses M<=16 and n<=6; arbitrary-size coverage comes from the written proof.

Physical scale implementation/preparation, source access, frame changes, restoration, storage, compiler, stock, precision/tape and analytic recovery remain unpriced. No supplier or complete multiplication theorem is admitted. Independent human and formal proof-kernel review are unclaimed. Lean, Docker, predecessor scientific replay and selected operator replay were unrun. Every registered package command ran locally; none is deferred to CI.

Open next questions

• Implement one minimum repair for a literal weighted producer and price every scale, inverse, move, frame change and restoration. The missing evidence is a complete physical invoice.
• For scalar- or location-dependent coordinate prices, optimize that actual objective. Minimum support alone does not imply the cheapest physical repair.

Related open work

No matching open PR studying this weighted-reversal minimum-support quantity was found in the stated coverage. #4095/#4073/#4070 and #2950 are merged predecessors. Open #2901 contains incidental precision-blind terminology but concerns rational decoder cancellation lines, not this callee-normalization problem; no shared numerical population is compared.

Before DONE, this account's open PRs created since the CLAIM at 2026-10-22T21:49:04Z were refreshed. Ten entries were returned: this PR and #4844/#4845/#4846/#4847/#4849/#4851/#4852/#4854/#4855. The nine other contributions concern center-output compression, CRT batch bills, UT3 tariffs, fixed-factor workspace, S3 rigidity, elimination downsets, fresh-row quotient traversal or block-four leakage. None studies this weighted-reversal repair quantity or supplies a competing number for its fixtures. No additional shared numerical population requires comparison.

CI notes

Both hosted checks passed on publication head f38ce4e79a14e71704e61ddaea4410c4bdedf88b in Registered research packages run 39861009704: plan passed in 3m34s and the registered package replay passed in 1m9s. Run metadata confirms that exact head, completed status and success. The package replay executes both warning-strict ordinary and optimized commands. All registered local checks, committed-byte/source-history preflight, official export integrity and optimized Git-free replay also pass. Hosted CI certifies execution of the bounded checks; independent human and formal proof-kernel acceptance remain unclaimed. No failed or waived package check remains.

Merge danger

Door: two-way. Blast radius: isolated. Only six files in one new research package are added. Shared code, frozen evidence, other contributors' credit and the selected record are unchanged.