[petalas]

Summary

Scoped written analytic proof with exact finite corroboration: for every integer r >= 3, the cyclotomic coefficient heights of #1025/#1428 satisfy

sum_(k unit mod r) log H_k <= r log r (1+log r) + phi(r) log r.

With n=r-1, D=phi(r), ell=ceil(log2 r), and the same nonzero-resultant hypothesis, the sufficient uniform circle-gap exponent changes as follows:

- d = 1+2ell+min(nDr, 3nr(ell+1)^2/2 + 2nD ell)
+ d = 1+2ell+min(nDr, ceil(n[3r ell^2/4+(r+D)ell]))

At the certified composite radix 8978=2*67^2, the new exponent is 13,531,624,611, compared with 28,312,479,536 from the cited old formula. These are sufficient analytic bounds. No inverse at either precision was executed, and no new kappa or selected-frontier change is claimed.

The new step combines Hanson Sun / bbMisk's #1428 arbitrary-radix unit average with Nick Petalas's older #5930 all-radix zero-mean Fourier identity. The latter's height application was restricted to prime powers. Credit Chafik Boukhalfa #1025 for averaged heights, Ali Romman #621 for the resultant/cofactor argument, Jason Willems #666 for the certified composite nonvanishing family, and the Rohan Guliani #846 / Charles Bethin #764 solver lineage. Their results are cited premises; no predecessor verifier or census is replayed. This contribution was produced by Nick Petalas. Apache-2.0.

Only research/composite-zero-mean-gap/ is added. The uniform-transfer-audit card changes scope to an analytic precision-interface lemma. Base and refreshed canonical main: e54523517d0f03e58352e647c403a5af3e525a57, priority revision 8, selected #7000. The first proved improvement is the complete delivered scope.

Evidence

• Before: Research audit: averaged-height bounds for certified composite radices (no new kappa) #1428 supplies the older mixed-radix exponent. After: the written all-radix proof gives the new height estimate and a never-larger sufficient exponent; the coarse branch can make the full exponents equal.
• Exact rational interval arithmetic encloses actual coefficient-height sums at 100 primitive embeddings over thirteen mixed radices, using Machin's identity, Taylor remainders, integer square roots and an atanh logarithm series. This is new finite corroboration of the delivered bound.
• The checker passes 254 natural-to-binary conversion cases and 2,062 exponent cases, with 1,964 strict improvements. Cases count executions and can repeat a radix. Eight invalid inputs reject; a half-integer case checks the necessary ceiling.
• Both registered warning-strict normal/optimized commands pass, at about 0.45 seconds each locally, below their 60-second limits. The generated JSON receipts bind all six files and agree between modes.
• Metadata/source pins, AST compilation, whitespace inspection, committed-byte and declared source-history checks pass at 238fccbde08d7a630477b541dcbb61d3021e3836. The official six-file export and declared-input integrity check pass. Both exported modes pass with read-only package files and Git absent from PATH, with identical receipts. The working tree is clean.

Local environment: CPython 3.12.15, macOS arm64, standard library only. From the repository root:

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

Another CPython 3.12 path may replace the local interpreter. Markers are PASS_COMPOSITE_ZERO_MEAN_GAP, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES. The standalone checker writes no files and needs no Git, network or external dependencies.

Limitations

The universal result is the written argument with cited premises, not proof-kernel acceptance. The height estimate is unconditional; gap/operator consequences retain the nonzero-resultant hypothesis. The certified family r=2p^a, p>=5 prime, comes from #666/#1428 and is not re-audited. Exponent arithmetic does not certify other radices.

No paid composite solver, coefficient-preparation or rounding policy, dirty-restoring physical word/caller, tape/stock invoice, compiler closure, supplier admission or multiplication theorem is supplied. All five selected framework assumptions remain. Selected scientific cold replay, native suites, Lean, Docker and external static analysis are unrun. Independent human and formal proof-kernel review are unclaimed.

Open next questions

Can one certified composite inverse execute at reduced precision with all costs retained? Start with r=10 or r=134 and one explicit preparation/inverse error recurrence. Paid coefficient work, rounding propagation, restoration and tape/caller costs remain blockers.

Related open work

• #7199, head 54b76434dd07ecfdeeaa5958268693f176f37e65, studies the same cyclotomic coefficient-height and precision quantity. It proves a new primitive-unit harmonic envelope within prime powers, building on Research: averaged cyclotomic heights give O(r² log² r) precision (no new kappa) #1025 and Sharpen averaged cyclotomic gap from 8ell^2 to 3ell^2 (no new kappa) #5930; this PR uses Research audit: averaged-height bounds for certified composite radices (no new kappa) #1428's existing unit envelope and derives a mixed-radix height estimate from the zero-mean Fourier input. Its displayed radix-4096 example is not one of this package's computed cases, while this PR's displayed radix-8978 example is outside its prime-power domain. No shared numerical claim is compared, and no extra computation checks that work.

The own-open-PR refresh since the CLAIM read nine other contributions: #7189, #7191, #7192, #7193, #7194, #7196, #7197, #7198 and #7199. Only #7199 studies this cyclotomic height/precision quantity; the others concern a Gaussian reader, polynomial insertions, rounded owner residues, address twists, three-shear cycles, Jordan maps, compact switches and query-cache stock.

The advance CLAIM preceded substantial work. Live open/account PR bodies, active Discussions, reservations, all-main distinctive-term and merged searches were checked. The task helper covered 488 open PRs and 7,152 Discussion records with complete remote pagination; only 202 of 10,178 local records were materialized, so main-tree searches supplied separate coverage. The dedicated #1428 source query found no recent account target. Arbitrary diffs and unpublished work remain outside the search.

CI notes

CI did not certify this package. At unchanged head 238fccbde08d7a630477b541dcbb61d3021e3836, run 43001319803, plan job 130457874416, 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 document still describes #6030 and make frontier6030-verify. Direct source inspection confirms that mismatch. The selected record, document and test are unchanged by this PR.

I also read the failed plan-job logs for every run below and confirmed the same exact sole failing test, missing command, 760-test total, one failure and 24 skipped tests. Each metadata record identifies Onboarding helper tests as the failed step:

• Other contributor: run 42905094314, job 130140894651.
• Other contributor: run 42903313790, job 130134720799.
• Save eight further ADDs in the translated Gaussian reader (no κ) #7189: run 43000211746, job 130454190330.
• Classify polynomial two-gap insertions with exact dimension 64d+50 (no κ) #7191: run 43000676071, job 130455727586.
• Prove rounded inverse injectivity from eight owner residues (no κ) #7192: run 43000790178, job 130456104520.
• Prove the sharp rank-two bound for five address-twist contributions (no κ) #7193: run 43000937833, job 130456595356.
• Prove the paired-root floor for three-shear cycles off tsu = ±1 (no κ) #7194: run 43001222481, job 130457550993.

The package replay job 130459820128 was skipped because plan failed. The package evidence therefore remains local: the registered normal/optimized checks, exact interval/exponent checks, metadata/pins, committed-byte/source-history checks and read-only Git-free export replay described above. No fix, retrigger commit, push or job rerun was made for this repository-level failure. Maintainers will repair main. Mathematical acceptance remains separate from these bounded checks.

Merge Danger

Door: two-way.

Blast Radius: package. Six new files; shared code, predecessors and selected-result records keep their bytes.