[petalas]

Summary

Scoped written analytic proof with exact finite corroboration. A zero-mean extension of the sampled cyclotomic logarithm improves Chafik Boukhalfa's PR #1025 averaged-height bound and its sufficient uniform circle-gap exponent for every prime-power radix r>=3.

- sum_k log H_k <= phi(r) [4(log r)^2 + 8 log r + 2]
+ sum_k log H_k <= phi(r) [2(log r)^2 + 3 log r]

- d = 1 + 2ell + (r-1)phi(r) min(r,8ell^2)
+ d = 1 + 2ell + (r-1)phi(r) min(r,3ell^2)

Here ell=ceil(log2 r), logarithms in the height bound are natural, and |P_k(z)|>=2^-d for every primitive embedding and |z|=1. At r=4096 the sufficient exponent decreases from 9,661,317,145 to 3,622,993,945. These are analytic bound values, not executed word lengths.

The new proof sets the unused residue-zero logarithm to -log r, proves zero Fourier mean and strictly negative remaining coefficients, and obtains exact Fourier mass log r. It then uses the cited primitive-unit average and resultant/cofactor implication. No predecessor verifier is imported or replayed. The first verified improvement ended the investigation.

The result was produced by Nick Petalas. Credit Chafik Boukhalfa / #1025 for averaged heights and the precision framework, Ali Romman / #621 for the nonzero resultant and cofactor argument, Rohan Guliani / #846 and Charles Bethin / #764 for the retained solver/suffix lineage, and the classical logarithm and finite Fourier identities. Exact proof and notice provenance is pinned in SOURCES.json. No historical priority is claimed for the classical ingredients.

Evidence

Local CPython 3.12.15 checks at head 84262199a8ec626ad89b04d2d2b12222b08eef24 pass:

• Warning-strict normal and optimized commands, each about 0.16 seconds, produce identical generated package-hash-bound receipts.
• PASS_ZERO_MEAN_CYCLOTOMIC_HEIGHT: 4,560 exact polynomial kernel identities, 78 exact harmonic checks, three formal logarithmic Fourier fixtures, 235 integer budget cases, 179 strict budget improvements and eleven rejected controls.
• Registered package replay, metadata/source pins, committed bytes and source-history preflight pass. Only the six new package files are changed.
• Official six-file export and declared-input integrity pass. The exported verifier passes with a read-only package, Git absent from PATH, isolated optimized Python, and an identical receipt.
• AST compilation, function-size inspection and staged whitespace checks pass. The verifier's managed-allocation and bounded rejection-callback deviations are documented in README.md.

From the worktree root, reproduce with the exact local interpreter path:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/zero-mean-cyclotomic-height
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/zero-mean-cyclotomic-height/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/zero-mean-cyclotomic-height/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/zero-mean-cyclotomic-height --check-source-history

Other machines may substitute their CPython 3.12 path. Only the standard library is required.

Limitations

The universal result is the written argument in PROOF.md. The finite checks corroborate its algebra and arithmetic; they do not prove every radix. The primitive-unit estimate and prime-power nonzero resultant are credited premises. No optimal diagonal choice or optimal constant 3 is claimed.

No actual inverse at the reduced precision, rounded preparation, common-label compiler, dirty restoration, physical tapes/hosting, complete transform, supplier admission, new kappa or complete multiplication theorem is supplied. No native suite, Lean build or selected cold replay was run. Independent human and formal proof-kernel review are unclaimed. The selected frontier remains unchanged.

Open next questions

• Optimize the complete interval estimate over the unused zero-residue value, retaining its nonzero mean. Begin with the piecewise-linear Fourier mass; mass alone is insufficient.
• Execute one actual small prime-power inverse at reduced precision, retaining all coefficient-preparation, rounding and restoration costs. The analytic bound alone does not establish that execution.

Structured first steps and blockers are in TASK.json.

Related open work

No open PR studying the averaged cyclotomic-height or uniform circle-gap quantity was found in the refreshed live searches. Merged #1025 is the direct predecessor; merged #1428 extends its older estimate to a separate composite-radix domain. Merged #5080 improves an independent inverse-envelope factor for a given circle gap and does not prove this height estimate. Its open follow-ups were not taken. Sparse local notes and unpublished work remain coverage limits.

Related-work refresh covered #5931 through #5940, the ten other open account PRs published since the advance CLAIM. None computes the same cyclotomic averaged-height or circle-gap quantity.

• Prove exact tensor annihilator coefficient heights (no new κ) #5934 determines Gaussian-rational tensor-annihilator coefficient denominators and primitive heights. It concerns a different polynomial family and coefficient quantity; this PR computes no shared numerical value. No extra mathematical computations were run to compare it.

CI notes

Both hosted checks passed at exact head 84262199a8ec626ad89b04d2d2b12222b08eef24 in run 40523487818: plan in 4m11s and replay (research/zero-mean-cyclotomic-height) in 1m17s. The registered replay runs warning-strict normal and optimized checks. Local mathematical checks, metadata/source pins, committed bytes/source history and portable export replay also pass. Hosted execution certifies these bounded checks; independent human and formal proof-kernel review of the analytic theorem remain unclaimed.

Merge Danger

Two-way. The change adds one isolated six-file research package. Blast radius: package. No shared implementation, historical package, pinned source or selected-frontier file is changed.