[petalas]

Summary

Scoped written analytic proof with exact finite corroboration. For every prime-power radix r >= 3, a new primitive-unit harmonic estimate sharpens the averaged coefficient heights in Chafik Boukhalfa / chafreaky's #1025:

- sum_(k unit mod r) log H_k <= phi(r) [2(log r)^2+3 log r]
+ sum_(k unit mod r) log H_k <= phi(r) [(log r)^2+3 log r]

The previous line is the cited older #5930 consequence. Its zero-mean Fourier identity remains a premise. The new step retains the p-divisible harmonic subtraction and proves the primitive-unit geometric-sum envelope log q + 2, replacing 2(1+log r).

With ell=ceil(log2 r), n=r-1, D=phi(r), and B_ell=ceil(3ell^2/4+3ell), the cited resultant/cofactor implication gives

|P_k(z)| >= 2^(-d_new),  |z|=1,
d_new = 1+2ell+nD min(r,B_ell).

At r=4096 the sufficient gap exponent is 1,207,664,665, versus the cited 3,622,993,945. These are analytic bound values. No reduced-precision inverse was executed and no kappa or selected-frontier change is claimed. The first verified improvement is the complete delivered scope.

Only research/prime-power-harmonic-gap/ is added. Canonical base and refreshed main are e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8, selected #7000. The formal-interface-lemma card explicitly changes scope to this analytic side result. It does not claim proof-assistant formalization or an actual selected-caller binding.

The new proof and checker were produced by Nick Petalas. Credit Chafik Boukhalfa #1025 for averaged heights and normalization, Nick Petalas #5930 for the older Fourier identity, Ali Romman #621 for prime-power resultant nonvanishing and the norm/cofactor implication, and Rohan Guliani #846, Chafik Boukhalfa #862/#977 and Charles Bethin #764 for the retained solver lineage. Harmonic integral estimates, sine chords, Fourier inversion and Machin/Taylor identities are classical. Existing source files and notices retain their bytes. Apache-2.0.

Evidence

• Before: the cited primitive-unit estimate is 2(1+log r). After: PROOF.md derives log q+2 by exact prime-power harmonic cancellation and carries it through the height and gap formulas.
• The standalone checker passes 604 exact harmonic bounds through 4096, 117 directly enumerated identities through 512, and 44 rational cosecant-average bounds through 128. Its 1,083 pairing classes cover all 2,165 primitive residues in that domain, with the q=2 midpoint explicit.
• 968 exact exponent cases pass, with 948 strict improvements. Large dyadic and ternary cases check exponent arithmetic only. Thirteen controls cover eleven rejected inputs, the missing harmonic subtraction and the required ceiling.
• Warning-strict normal and optimized replays each took about 0.47 seconds locally, below their separate 120-second registered limits. Both VERIFY.json commands and the package runner pass. Generated six-file SHA256 receipts agree between modes.
• Metadata/source pins, in-memory AST compilation, function-size inspection and staged whitespace checks pass. Committed-byte, executable-mode and declared source-history checks pass at 54b76434dd07ecfdeeaa5958268693f176f37e65. The official six-file export and declared-input check pass. Both exported replays pass with read-only package files and Git absent from PATH, producing identical receipts. The worktree is clean.

Local environment: CPython 3.12.15, macOS arm64, standard library only. Reproduce from the worktree root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/prime-power-harmonic-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/prime-power-harmonic-gap/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/prime-power-harmonic-gap
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/prime-power-harmonic-gap --check-source-history --check-modes

Other machines may substitute their CPython 3.12 path. Expected markers are PASS_PRIME_POWER_HARMONIC_GAP, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The direct checker writes no files and uses no Git, network or predecessor executable. The source proofs are pinned bibliographic provenance.

Limitations

The universal result is the written proof with cited premises, supported by bounded exact corroboration. Independent human and formal proof-kernel review are unclaimed. The Fourier identity, prime-power nonzero resultant and norm implication are credited inputs, not re-audited deliverables.

Composite radices, optimal constants, actual inverse execution, coefficient preparation, rounding, dirty restoration, physical tapes/hosting, complete costs, compiler closure, supplier admission and a complete multiplication theorem remain outside scope. All five selected framework assumptions remain. No selected scientific cold replay, native suite, Lean, Docker or external static-analysis tool was run.

Open next questions

• Improve the explicit log q+2 envelope. Start with the exact subtracted harmonic expression or a sharper sine bound for one prime-power family. No mathematical blocker is identified; it was not pursued after the first gain.
• Execute one actual small prime-power inverse at the improved gap. Begin with the cited arbitrary-stride solver and a complete preparation, rounding, restoration and tape error/cost recurrence. That physical implementation and ledger are absent.

Related open work

• #7195 extends the zero-mean Fourier estimate to arbitrary radices using Research audit: averaged-height bounds for certified composite radices (no new kappa) #1428's existing unit envelope. This PR improves the primitive-unit envelope within prime powers. Its displayed composite example is outside this PR's domain; no extra computation checked its numbers. Neither contribution audits the other.

The closeout own-open-PR check read the eight other account PRs created since the CLAIM: #7191 through #7198. Only #7195 studies this cyclotomic coefficient-height and precision quantity. The others concern polynomial insertions, rounded owner residues, address twists, three-shear root cycles, Jordan maps, compact switches and persistent query stock. #7195 now also lists this PR. Its displayed radix-8978 example is outside this prime-power domain, and this PR's displayed radix-4096 example is not among its computed cases. No extra scientific computation compared either contribution's numbers.

The advance CLAIM preceded substantial work. Live open/account metadata and relevant bodies, recent handoffs, active Discussions, local reservations, all-main distinctive-term and merged searches were checked before claiming and refreshed before committing. Source #1025/chafreaky differs from the concurrent #1428/bbMisk question. The delayed task-helper result covered 488 open PRs and 7,157 Discussion records, with sparse local coverage of 232 out of 10,178 HEAD notes; its output arrived after CLAIM and was not counted as completed preclaim evidence. Arbitrary diffs and unpublished work remain outside that coverage.

CI notes

CI did not certify this package. At unchanged head 54b76434dd07ecfdeeaa5958268693f176f37e65, run 43001851217, plan job 130459654862, failed in Onboarding helper tests. I read the checks once and confirmed the cause directly in the job log:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record
AssertionError: '`make frontier7000-verify`' not found
Ran 760 tests
FAILED (failures=1, skipped=24)

This was the sole failing test. At tests/test_research_current_status.py:58, it expects the selected reproduction command in the current section of docs/ci-verification.md. Canonical main e54523517d0f03e58352e647c403a5af3e525a57 names make frontier7000-verify in certificates/selected-result.json, while the CI guide still describes #6030 and make frontier6030-verify. Direct inspection confirms that mismatch. The selected record, guide and test are unchanged by this PR.

I read the failed logs for every comparison run below and confirmed the same exact sole failing test, missing command, 760-test total, one failure and 24 skipped tests, each in Onboarding helper tests:

• Other contributor: 42905094314.
• Other contributor: 42903313790.
• Prove the paired-root floor for three-shear cycles off tsu = ±1 (no κ) #7194: 43001222481.
• Sharpen the composite cyclotomic precision exponent (no κ) #7195: 43001319803.
• Prove q² coupled optima for two resonant Jordan blocks (no κ) #7196: 43001498725.
• Prove the sharp five-translation minimum for a two-target switch (no κ) #7197: 43001666262.
• Prove exact minimum persistent stock for query streams (no κ) #7198: 43001765438.

The package replay job 130461520358 was skipped because plan failed. Automated package evidence therefore remains local: the normal/optimized registered checks, exact arithmetic and controls, metadata/source pins, committed bytes/modes/source history and read-only Git-free exported replay described above. No fix, retrigger commit, push or job rerun was made for this repository-level failure. Maintainers will repair main. Independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way.

Blast Radius: package. Six new files in one isolated research directory.