[petalas]

Summary

Scoped analytic proof: sharpen Rafael Guevara's PR803 degree-controlled cyclic inverse envelope by evaluating its Schur kernel integral exactly.

- ||1/P||_W <= (1+4m)(1+4n)/delta
+ ||1/P||_W <= (1+pi*m)(1+pi*n)/delta
+           <= (1+22m/7)(1+22n/7)/delta

Here P has degree n, has m interior roots counted with multiplicity, and satisfies |P|>=delta>0 on the unit circle. The rational envelope is strictly smaller for every positive degree. The same bound applies to P(U)^-1 under the full integer-power norm hypothesis ||U^k||<=1 for all integers k.

The sufficient binary norm-bound ceilings decrease from 19 to 18 at radix 3, 441 to 440 at radix 9, 11,999 to 11,998 at radix 25 and 347,859 to 347,858 at radix 71, using the same cited PR621 circle certificate. These are upper-bound ceilings, not actual norms or working-precision savings.

The refinement was produced by Nick Petalas. Credit Rafael Guevara/PR803 for the inverse-envelope argument, Ali Romman/PR621 for the circle certificate, and the retained classical Hardy, Hilbert, Blaschke, Dolzhenko-Spijker and Baranov-Zarouf lineage. The constant pi and the positive integral proving pi<22/7 are classical. The new contribution is their explicit strict refinement of this repository's envelope; no historical priority is asserted for those tools.

Evidence

The written proof is research/hardy-inverse-constant/PROOF.md. It cites PR803's factorization, rational arc-length and operator-transfer implications as premises, without auditing or replaying its result.

Local CPython 3.12.15 checks passed:

• Registered normal and optimized commands, each timed at 0.09 seconds; complete final package replay 0.50 seconds.
• PASS_HARDY_INVERSE_CONSTANT: the exact polynomial division/integral certificate, 8,385 degree pairs, 4,096 exact logarithm brackets, 14 prime-power budgets and 7 rejected controls. Normal and optimized generated receipts agree byte for byte.
• External receipt overwrite preservation, rejection of checkout output, and the pre-read two-MiB limit control passed.
• Metadata/source pins, AST compilation and staged whitespace checks passed. The publication-byte/source-history preflight passed for all seven declared files at commit f09d7c81dbc743fae3df338eab6b01086e28bbf0.
• A separate read-only review found no mathematical issue in the new step. Its low resource-ordering finding was fixed and checked. Independent human and formal proof-kernel review are unclaimed.

From the worktree root, reproduce with the exact interpreter used locally:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/hardy-inverse-constant
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/hardy-inverse-constant/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/hardy-inverse-constant --check-source-history

Other machines may substitute their CPython 3.12 executable. Only the standard library is required. Receipts are generated from the exact source and package hashes; none is hand-written.

Limitations

This is a scoped proof, not a finite certificate of a complete supplier or multiplication theorem. The all-degree analytic argument and cited premises remain distinct from its finite arithmetic checks. No optimal degree factor, rounded execution, preparation, routing, dirty restoration, supplier admission or new kappa is claimed. No Lean build or complete selected replay was run. Later PR1025's improved circle gap is an independent compatible input.

Open next questions

• Substitute the smaller envelope into one actual PR628 or PR764 rounded precision recurrence, retaining every preparation, error and restoration charge.
• Seek a direct two-sided Wiener estimate with smaller degree dependence; the present proof still multiplies two analytic-factor bounds.

Related open work

No open PR studying this degree-controlled circle-gap inverse envelope was found in the live PR-body snapshot and focused searches. Merged #803 is the predecessor; merged #832 gives a separate Sobolev estimate, #852 treats weighted-cycle power caps, and #1025 improves the gap input. Their results retain their credit.

• Merged #4914 proves the sharp factor-two inverse bound for mixed quadratics, improving Research: degree-controlled cyclic inverse bound by root reflection (no new kappa) #803 on that subclass; this PR sharpens the general all-degree Research: degree-controlled cyclic inverse bound by root reflection (no new kappa) #803 envelope and does not improve Prove the sharp mixed-quadratic inverse factor two (no new κ) #4914's sharp subclass result.

The completion refresh read this account's other open PRs created since the CLAIM: #5083 studies scalar-rank-capped GL2 swap words, #5084 studies dyadic norm-six decoder compression, and #5085 studies partial Pauli-map transvection counts. None studies this cyclic-polynomial inverse envelope; no numerical comparison or additional computation was needed.

CI notes

Both hosted checks passed on exact head f09d7c81dbc743fae3df338eab6b01086e28bbf0 in run 39924099160: planning in 4m17s and the registered package replay in 1m18s. The replay runs both warning-strict normal and optimized commands. Local registered replay and publication-byte/source-history preflight also passed. These bounded checks do not imply independent human or formal proof-kernel acceptance of the analytic theorem.

Merge Danger

Two-way. The change adds only the isolated six-file research package; existing sources and the selected frontier are unchanged.