Composite cyclic inverses: exact certificates and modular lifting

[at1as]

Ali Romman's #621 proves uniform cyclic-kernel conditioning for prime-power radices and leaves other composites outside its claim. This package adds an integral seed-certificate lifting mechanism for specified composite families, together with six stronger fixed-radix certificates. No new κ is claimed.

If A f_s+B f_s*=d is an integral Bézout identity and p is prime with p not dividing s*d, the written proof establishes nonvanishing for r=s*p^a for every a>=1 and every primitive root choice. Reduction modulo p sends the two integral kernels to z^(r-s) f_s and f_s*; modular coprimality then lifts back to characteristic zero. Independently checked seeds give:

Family	Condition on prime p
r=2*p^a	p not in {2,3}
r=3*p^a	p not in {3,43}
r=4*p^a	p not in {2,37}

The proof handles reciprocal degree loss explicitly. #621's inherited norm/cofactor and Fourier arguments supply quantitative circle and inverse-norm bounds uniform in cyclic length, integer stride and unit scalar gauge. Excluded primes limit this modular route; they are not characteristic-zero counterexamples. This is not a theorem for all composite radices.

Separate exact certificates at 6, 10, 12, 14, 15 and 18 cover all 30 primitive embeddings symbolically and give sharper explicit rational bounds. The standard-library checker reconstructs cyclotomic polynomials and tail-product kernels without importing the SymPy generator or either predecessor checker. The lifting diagnostics reuse that independent integer engine and work over potentially reducible modular coefficient rings.

Reproduce

python3 -B -I -O research/composite-cyclic-bezout/verify.py --check research/composite-cyclic-bezout/EXPECTED.json
python3 -B -I -O research/composite-cyclic-bezout/verify_lifting.py --check research/composite-cyclic-bezout/LIFTING-EXPECTED.json
python3 -B -I research/composite-cyclic-bezout/test_verify.py
python3 -B -I research/composite-cyclic-bezout/test_lifting.py

Validation and scope

• Six original identities / 39 controls; three exact lifting seeds / 54 finite modular instances / three excluded-prime common-factor witnesses / 173 controls.
• Both public receipts and all 16 focused tests pass on macOS arm64 with Python 3.9.6, 3.11.13 and 3.13.4.
• Both optional SymPy 1.14.0 generator modes reproduce the certificate bytes exactly.
• Separate Codex agents reviewed the all-exponent proof and checker. Finite replay tests implementation; the written proof establishes all exponents. No independent human review or proof-assistant formalization is claimed.
• Prove uniform prime-power cyclic inverse bounds (no κ claim) #621's replay, source integrity and the selected Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543+Conditional κ = 0.001635527380375392: coordinated owner repair on PR623 #626 cold composition replay pass. The initial root run stopped on Apple clang's unsupported OpenMP flag; after a temporary adapter used the installed libomp, the failed stage and every remaining target passed, including 84 isolated test modules and all patch checks. Coverage spans the initial run and resumed tail; details are in VALIDATION.md. At head 8e5a8d4, GitHub reports action_required for all six workflow runs, with no jobs or check results created; the API does not state the reason. The scoped workflow run remains distinct from the completed local checks.

This supplies no paid inversion implementation, combined precision guard, restoration/tape schedule or multiplication improvement. upstream/, supplier code and the selected frontier are unchanged.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. #621 source head: 520774462b19b29af41554b74def31f9d01e997e. Assignment seed 82; one focused PR. Local claims and results are recorded because no pinned coordination issue exists. The refreshed 533-PR queue through #714 includes distinct rounded-inversion work (#628/#663/#673) and prime-power CRT auditing (#668); no overlapping composite seed-lifting result was identified.

Credit to Ali Romman for the kernel-conditioning problem, reciprocal method, norm/cofactor estimate and Fourier transfer; OpenAI and classical q-Pochhammer/Newton sources retain attribution in NOTICE.md. Prepared for Jason Willems (@at1as) with substantial OpenAI Codex assistance. “Independent” describes the separate arithmetic implementation, not independent human authorship. Apache-2.0; no literature-priority claim.

[Mister-Meeseeks]

Thank you @at1as. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Composite cyclic certificates and modular seed lifting.

Review and limits: Covers the stated seed/prime families and finite composite radices. Excluded primes are limits of the modular route, not characteristic-zero counterexamples; no all-composite theorem or kappa gain.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 8e5a8d4da5bf6d83dc0b39ec90fea15e3cf627d6. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.