Research audit: Sobolev refinement of PR621's cyclic inverse bound (no new kappa)

[ThiagoCAltoe]

Change and mathematical scope

PR621's uniform circle gap and derivative bound imply the valid cyclic inverse estimate delta^-1 + 2 M delta^-2. This independently prepared audit gives an elementary periodic Sobolev/Fourier proof of

||P(tau Z_L^w)^-1||_infinity^2 <= delta^-2 + (44/7) M delta^-3.

The finite Fourier-polynomial argument establishes absolute summability before matrix transfer. It covers every cyclic length, integer stride (including zero and noncoprime strides), unit scalar and constant polynomial. The bound improves the inherited expression when M/delta > 4/7, including the prime-power kernels covered by PR621. For fixed positive M, its inverse-gap dependence is delta^-3/2.

Concurrent work: Rafael Guevara (@alephnan)'s #803, inspected at d213ac12d4a1ab8f717d942e844df318ceb395fb, proposes the stronger degree-controlled bound (1+4 degree(P))^2 / delta for the same conditioning step, using root reflection, arc-length and Hardy arguments. This submission preserves a separate Sobolev proof and exact regression evidence; it does not claim priority or the best currently proposed conditioning bound. No PR803 code is imported and its pending result is not independently certified here. #820 separately audits radix-9 coefficient enclosures and factor residuals. The publication overlap refresh covered the recent queue through #824.

Relative to PR621's original estimate, the radix-9 ceiling of log2 of the analytic upper bound falls from 870 to 651. These are bound ceilings, not actual required precision or runtime speedups. The logarithmic asymptotic order remains O(r^2 phi(r)). No new kappa, paid inverse algorithm, optimality or general-inequality novelty is claimed.

Only research/cyclic-fourier-transfer-audit/ is added: 12 files, including the proof, exact oracle, verifier, deterministic receipt, tests, provenance and notices.

Original research base: 66849e1475c930fd7732b1a0067ffee7e1e95f60.
Publication base: 7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261.
The four mathematical predecessor files from PR621 remain byte-identical. The selected-result context pin advances with main; its historical hash is retained. The selected construction, kappa and upstream/ are unchanged and are not audited by this contribution.

Validation

Python 3.10+; standard library only:

python3 -I -O -B research/cyclic-fourier-transfer-audit/verify.py --source-root .
python3 -I -O -B research/cyclic-fourier-transfer-audit/test_contracts.py

Fresh Windows / CPython 3.12.10 isolated optimized replay passed:

• 648 exact Gaussian-rational inverse cases, comparing dense Gauss-Jordan with independent finite-geometric factor formulas and checking both inverse identities.
• 200 exact Fourier sequences and all 26 prime-power budget comparisons through radix 64.
• Eight contract tests, including corrupted source/receipt rejection, logarithmic boundaries, constant mode and the non-improvement region.
• All five source pins against the downloaded publication-base snapshot.
• Staged whitespace validation and remote comparison: exactly 12 added files; every published Git blob matches the validated local bytes.

The deterministic receipt matches the original Linux result. The proof, verifier, oracle and expected receipt are unchanged from the original package. Its two separate AI-assisted mathematical reviews remain documented in REVIEW.md; they are not independent human review.

The finite inverse fixtures are generic rational polynomials, not actual cyclotomic-kernel executions. The written proof supplies the universal analytic statement, conditional on PR621's circle theorem. Full repository/supplier suites, a fresh selected-record checker, remote CI and Lean were not run or claimed in this publication replay.

Prepared for Thiago Costa (@ThiagoCAltoe) with substantial OpenAI Codex assistance in derivation, implementation, documentation, verification and review. Credit Ali Romman for PR621, the pinned OpenAI kernel, classical q-calculus, standard Fourier/Sobolev arguments, and Rafael Guevara's concurrent work. Apache-2.0; source pins and prior notices retained.

[Mister-Meeseeks]

100-PR batch review at d42f42714018064078263aa77b4b4b4b2af874f8.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Conditional Sobolev/Fourier inverse bound with an all-length written transfer and exact finite controls; circle-gap premise inherited.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.