[WilliamLawyer]

This research audit gives the exact worst-case infinity norm of a linear inverse on a balanced weighted cycle with bounded local weight magnitudes. It supplies rational attaining weights and a scoped negative result: exact unit cycle products, unit-circle spectrum, and arbitrarily small local weight deviations do not preserve a length-independent unit-weight inverse estimate.

Status: research note / scoped negative result with independent exact verification.

For P(z)=z-33/32, cycle length 256 and every weight within 1/16 of one, the simple two-block witness has inverse norm strictly between 1936 and 1937. The exact maximum over the admitted class lies between 2164 and 2165. The inherited unit-weight estimates would be 800, or 160 using the interior-root count. This does not contradict PR803: those matrices violate its explicit all-integer-power norm-one assumption. No new kappa or rounded implementation is claimed.

The written proof gives the exact cycle-local power factor K, transfer with one factor K, a sharp extremal inverse formula, the constant/linear/exponential local-error regimes for a linear polynomial, and a sufficient strict perturbation guard. The universal arguments are separate from the finite checks. The analytic Wiener bound from Rafael Guevara's PR803 is an explicit inherited premise; the cycle criterion and counterexample/extremal formulas have their own elementary proofs.

Reproduce offline with Python 3.10+ and its standard library:

python -B research/weighted-cycle-power-audit/verify.py
python -B -m unittest discover -s research/weighted-cycle-power-audit -p test_package.py -v

Expected: PASS: exact weighted-cycle power audit and 11 tests passing. Evidence covers:

• 1,377 generated weighted-permutation cases and 9,990 exact signed-power samples, plus 18 admission/failure controls.
• 72 dense inverses checked against 504 independent affine basis solves; 96 sharp-inverse attainments; 108 complex-phase inverses; three exact length-256 extremal solves; ten near-unit threshold certificates.
• 48 sharp power-cap witnesses / 2,600 direct path products and 240 polynomial perturbation cases (155 admitted, 85 inconclusive).

Normal and optimized replays from a clean export of staged Git blobs passed, as did the source/tamper/mutation checks and the selected-record check at the starting base. Full make verify and supplier cold replays were not run: only the new research/weighted-cycle-power-audit/ package is added, with no existing generator, selected certificate, schedule or upstream/ changes. This is not proof-assistant formalization or a full multiplication-theorem audit.

Exact base: 7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261. PR803 pin: d213ac12d4a1ab8f717d942e844df318ceb395fb. Ryan Pinto's PR692 geometric-resolvent pin: c5b53a5e80372a1fae9830d8dcf21aaabce3f729. Source proofs, original notices and metadata are preserved byte for byte. DOMAIN.json, SOURCES.json, OVERLAP.json and VALIDATION.json record the domain, pins, review scope and reproducible evidence.

The final refresh also inspected Rohan Guliani's concurrent PR846 at 4bdb3e755ea7118170a7a34764df955880dec282: its arbitrary-stride algorithm assumes a strict contraction margin and is distinct. The newly accepted #799 frontier on f567884aeb8851b6e794af1184119a3bfbe1e631 is recorded as context, without changing this package's starting base. No pinned coordination issue was present; the claim is local.

Prepared for William Lawyer (@WilliamLawyer) with substantial OpenAI Codex assistance in the derivation, implementation, verification and writing. Apache-2.0. Predecessor and classical-method credit is retained in NOTICE.md; no literature-priority or independent human-review claim is made.

[Mister-Meeseeks]

100-PR batch review at aa26b2ad761aa3bac76670e4749f53d2dafb3fea.

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

Reviewed scope: Weighted-cycle counterexamples and sharp power/resolvent bounds under explicit magnitude hypotheses; no defect in the unit-phase premise.

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.