Research: degree-controlled cyclic inverse bound by root reflection (no new kappa)

[alephnan]

PR621 proves a uniform circle gap for the prime-power cyclic kernels, but its Fourier-to-operator step pays an additional inverse-gap power. This contribution applies classical root reflection, the Dolzhenko-Spijker arc-length inequality, and Hardy coefficient control to obtain a degree-only multiplier.

For a degree-n polynomial P with |P(z)| >= delta > 0 on the unit circle, the note proves

||1/P||_W <= (1+4n)^2 / delta
||P(tau Z_L^w)^-1||_infinity <= (1+4n)^2 / delta.

The bound permits interior, exterior, repeated, and zero roots and is uniform in cyclic length, integer stride, and unit phase. A degree-one family shows that the inverse-gap exponent one is necessary; the quadratic degree factor is not claimed optimal. The proof includes elementary derivations with explicit constants and distinguishes the analytic argument from finite tests.

For PR621's notation, the constant changes from A(1+4X^2) to A(1+4n)^2, where A=delta^-1 and X=n H^(nD/2). The dominant logarithmic term is halved asymptotically. At radix 71, the exact ceiling of log2 of the sufficient norm bound changes from 695,757 to 347,859. These are norm overestimates, not working-precision or runtime measurements. The previous unpublished local weighted-Fourier audit gave 521,801; its proof and notice are retained for comparison.

Source and analytic review follow-up

Revision a0f15ce84f0951f69b39795f229cb854307ca52b addresses the continuation at the reviewed head d213ac12d4a1ab8f717d942e844df318ceb395fb. The source and analytic audit maps the proof obligations to the expanded argument and exact classical-source locators. It checks normalization, Hardy-factorization limits, the degree-2d level polynomial, both Fourier directions, and absolute convergence of the operator inverse.

The theorem and hypotheses are unchanged. PR692's attribution is corrected to Ryan Pinto (@Ambassators). The six immutable source/context inputs, executable scripts, finite receipt, and historical evidence are preserved. The existing exact verifier and all eight package controls pass on the revised package; these are regression checks, not proof of the analytic theorem. This is author/AI review; independent analytic acceptance, the inherited PR621 foundation and full multiplication obligations remain separately scoped.

Reproduce

Python 3.10+ and the standard library:

python -B research/rational-circle-bound/verify.py
python -B research/rational-circle-bound/test_package.py

Expected markers: PASS_RATIONAL_CIRCLE_BOUND and PASS_PACKAGE_CONTROLS.

Executed on Windows 11 / Python 3.12.2, including a relocated offline copy under isolated optimized Python:

• 126 exact reflection families and 756 boundary identities.
• 546 dense-versus-finite-geometric cyclic inverse comparisons, covering 9,516 inverse entries and including zero/noncoprime strides.
• 6,565 finite Hilbert-matrix inequalities and 128 exact near-circle parameter instances.
• 14 exact prime-power budget comparisons, 21 mathematical/admission controls, and eight package fault tests.
• A clean patch application with byte comparisons and an optimized replay of the applied artifact. Package attributes preserve source bytes across Windows checkouts.

The finite cyclic tests use supplied rational root families, not newly computed cyclotomic kernel inverses. No proof assistant, independent human review, full repository replay, second-platform replay, or remote CI result is claimed. The unchanged PR621 foundation replay was not repeated.

Scope and attribution

Only research/rational-circle-bound/ is added. Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. The source package was prepared in a partial pinned-input worktree; the published commit uses the complete upstream tree and that upstream commit as its parent. The selected frontier and upstream implementations are unchanged.

Credit Ali Romman for PR621's kernel setting, nonvanishing theorem and circle certificate; Dolzhenko, Spijker, Hardy and the classical Blaschke theory for the analytic tools; and Baranov-Zarouf for the inspected normalized arc-length statement and historical attribution. Source notices and six immutable source/context files are preserved. The standard finite-geometric oracle also appears in Ryan Pinto's PR692; it is not claimed as a new solver here. PR628 and PR764 address distinct rounded-scan questions.

This is an application relative to the inspected repository work, not a literature-priority claim. Preparation, rounding/error propagation, scratch restoration, routing, tape/compiler/hosting and full multiplication transfer remain open. The new norm bound has not been integrated into PR628/PR764 precision recurrences. No new kappa, complex saving, or BIT saving is claimed.

The pre-publication overlap check inspected PRs through #799. Concurrent #796 (Hao Yan / FanPUYen, head 22902155c7875139e43e4fdf15723142d6709bdd) gives nonarchimedean root valuations and explicitly distinguishes them from complex root moduli and conditioning; it is a separate audit. The local claim records the continuation of the earlier OS-seed-71 mathematical audit and the absence of a pinned coordination issue.

Prepared for Rafael Guevara (@alephnan) with substantial OpenAI Codex assistance in derivation, implementation, verification and writing. Apache-2.0. No sub-agents were used.

[Mister-Meeseeks]

100-PR batch review at d213ac12d4a1ab8f717d942e844df318ceb395fb.

This remains open after scoped triage. Next review step: Analytic Wiener-norm theorem needs source/analytic proof audit; finite tests alone cannot certify it.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

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

[alephnan]

Added the requested source and analytic proof audit in a0f15ce8.

The audit note records exact classical-source locators and measure normalization. The expanded proof explicitly justifies the Hardy-factorization limits, degree-2d level polynomial, two-sided Fourier convolution, and operator-series inverse. The bound and its hypotheses are unchanged.

I also corrected PR692's attribution to Ryan Pinto (@Ambassators). The original finite evidence and all six pinned source/context inputs are preserved byte for byte. The unchanged verifier and all eight package controls pass on the revision; these regression checks do not certify the analytic theorem.

This is an author/AI source and proof audit, with independent analytic acceptance still for the reviewer. No new kappa, PR621 foundation replay, or full multiplication-theorem audit is claimed.

[Mister-Meeseeks]

100-PR batch review at a0f15ce84f0951f69b39795f229cb854307ca52b.

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

Reviewed scope: Revised all-frequency analytic argument: Hardy coefficient bound, degree-controlled rational arc length, Blaschke reflection and absolutely convergent operator substitution. Classical normalization checked against Baranov–Zarouf §1.2.2; finite regression is separate evidence. Improves a sufficient norm constant, not a paid inversion algorithm or κ.

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.