[petalas]

Summary

Prove a closed formula for the number of shortest unsigned one-source dirty-connector words over every finite characteristic m. The nonzero target coefficients have equal counts if and only if m is prime. This is a scoped proof with finite corroboration, with no new kappa or selected-frontier change.

B: z += Bx; A: y += Az; target: y += c ABx
minimum length = 2m, for nonzero c modulo m
number of optimum words = (1/m) sum_{d|m} R_d(c) binom(2m/d,m/d)
R_d(c) = sum_{e|gcd(d,c)} e mu(d/e)
prime p: every nonzero c has [binom(2p,p)-2]/p words

The new result is the complete multiplicity law and prime/composite criterion. The elementary call floor is shared context from Riquelme Soares's #4189 and concurrent #4231. Ross Richardson's #943/#934 connector contract retains credit. Classical finite Fourier inversion and Ramanujan sums supply the counting method. Only research/unsigned-connector-residue-count/ is added.

Evidence

Before: #4189 gives six F3 optimal words. After: the written all-characteristic proof recovers that case, gives every coefficient count, and distinguishes composite characteristics. At m=4, coefficients 1,2,3 have 16,18,16 optimum words. Zero-target length-2m words include two pure words in addition to the balanced count; the zero target itself has empty optimum word.

Local CPython 3.12 checks pass:

• All 527 residues for m=2,...,32, using independent chronological DP and the divisor formula; all 250,950 position subsets for m=2,...,10.
• Literal dirty coefficient execution of 17,574 balanced words; all 21,840 unrestricted length-2m words at m=2,...,7, including 4,716 restored words.
• 496 attaining witnesses, 3,968 noncommuting two-coordinate dirty-input checks and 3,968 inverse roundtrips; eight semantic/domain controls.
• Both registered ordinary/optimized commands, with warnings as errors, take about 0.5 seconds each locally. These are individual check durations, not performance claims.
• Registered package replay, eight-file committed-byte/source-history preflight, official export integrity and both Git-free exported modes pass. All nine export files remain unchanged. On-disk predecessor-input corruption is rejected by its SHA256 pin.

Exact reproduction from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/unsigned-connector-residue-count/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/unsigned-connector-residue-count
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/unsigned-connector-residue-count --check-source-history

Any CPython 3.12 path may replace the local interpreter path. Expected marker: PASS_UNSIGNED_CONNECTOR_RESIDUE_COUNT moduli=31 residues=527. The generated JSON receipt binds all package files and the live predecessor input. Final committed-export receipt SHA256: 5695bc7e1d369028089d97e887cfce8929f7371f4777f8c34b0d69011852b854.

Claim: #1166 (comment) . Main remains 12fa8c5c844be21bfefb38a19503fbd867a11244, priority revision 7. The card's first working control and selected binding baseline pass. The task metadata explicitly changes the assignment to this scheduling side result.

Limitations and unrun obligations

The proof is uniform in finite characteristic, compatible linear maps and row size in the stated unsigned complete-call interface. Finite executable corroboration stops at m=32. Calls must include their full implementation costs. A concrete supplier consumer, physical routing, admission, coefficient preparation, dispatch, precision and outer multiplication transfer remain uninstantiated. No selected cold replay or formal-kernel build was run. Independent human and formal proof-kernel review are unclaimed. No pending executable is imported.

Open next questions

1. Can marginal spectra constrain simultaneous readout positions in a multi-source connector? Begin with one fixed path and retain correlations between coefficient demands.
2. Can an actual recursive caller choose among its minimum-length words to reduce fully charged routing? Pin one caller and compare complete access/head-return bills before further construction.

Related open work

• Research: exact F3 minimum for the one-source additive connector #4189, head 5fb3936dfc8082f438f21c72dd6ca49ff4366ae7: the unsigned F3 one-source predecessor. The formula gives (20-2)/3=6, agreeing with its six optimum words; minimum 2*3=6 also agrees, checked numerically. Its census is cited, not independently audited.
• Classify the unsigned five-source connector in every characteristic (36c for c ≥ 3; no new κ) #4231, inspected head 8a956cbc80dc55697fd3e214812ce567b5eca48f: concurrent unsigned five-source characteristic classification. Its one-source m*(1+1)=2m floor agrees for all 31 tested characteristics. It does not give the new multiplicity distribution.
• Audit dirty direct-call supplier gap #3919: one-source F2 direct-product alphabet comparison. Its old-alphabet minimum 4 agrees with 2*2=4; the new formula counts its two optimum unsigned words. A fused direct-product call is outside this alphabet.
• Prove the exact 108-call unsigned F3 five-source connector optimum #4225: five-source F3 binary demand. Its one-source projections have length 6, agreeing with 2*3=6; its full 108-call batch is a different quantity, and no new batch optimum is asserted here.
• Prove the exact unequal-price five-source binary connector envelope #4226 (closed as a duplicate of merged Research: exact five-source connector optimum for all 36 positive tariffs #2063, research/connector-all-tariff-optimum, which proves the same envelope): unequal-price five-source F2 envelope. Its 74-call batch and this one-source multiplicity law concern different demands; there is no overlapping five-source count to compare.
• research: audit minimal joint-output connector charge #3897: a paid joint-output operation for one source and two F2 outputs. Restricting its original-alphabet count to one output gives 2+2=4, agreeing with the one-output floor here, checked numerically. Joint-output pricing is excluded.
• Add five-source arity-four connector cost bound #3909: an added paid arity-four source operation for a five-source F2 batch. The operation alphabet and five-source tariff differ; no shared one-source word-count number is claimed.

CI notes

Hosted run 39488429762 is green on head bd9b426837cddd37338dd284e49cc2303de57ca0: planning passed in 3m14s and the package replay in 1m50s. Both hosted scientific modes print the same source-bound receipt digest as the local committed export, 5695bc7e1d369028089d97e887cfce8929f7371f4777f8c34b0d69011852b854. The committed-inputs-unchanged step also passed. All registered scientific commands and source checks passed locally; there is no unrun registered local obligation assigned to CI.

Merge danger

Door: two-way. Blast radius: isolated. Only the new research package and its opt-in replay are added.

Produced by Nick Petalas. Apache-2.0. Predecessor credit is retained above and in the package.

[rcfc-jreed]

Independent scoped audit of #4236 unsigned connector residue count at bd9b4268: 12/12

Code-disjoint checks; does not import package verify.py (local fork with patched PACKAGE / input root; counts.py via path). Package bytes from head bd9b426837cddd37338dd284e49cc2303de57ca0 via GitHub contents API. CLAIM.

No packaging defect in scope. Stdout receipt (no RESULT.json by design).

What passes:

1. Pins: base 12fa8c5c…; VERIFY ready; TASK completed; three SOURCES digests match.
2. Fork: PASS_UNSIGNED_CONNECTOR_RESIDUE_COUNT moduli=31 with receipt_sha256=….

Not covered: Lean formalization; new κ; merge/close.

This is a finite pin/seal/math replay, not a formal proof.

James Reed (@rcfc-jreed), with substantial AI assistance (Claude, in Cursor).