Research audit: averaged-height bounds for certified composite radices (no new kappa)

[bbMisk]

PR1025's prime-power Fourier average uses q/phi(q) <= 2. That intermediate step fails outside its stated domain, for example 134/phi(134) = 67/33. This package replaces two by r/phi(r) and carries the factor through the resultant estimate, where it cancels the field degree. This is an extension of the predecessor's domain, not a reported defect in its prime-power theorem.

Status: mathematical audit / conditional precision bound; no new kappa.

For n=r-1, D=phi(r) and ell=ceil(log2 r), the written proof gives

T = min(n D r, 3 n r (ell+1)^2 / 2 + 2 n D ell),
d_r = 1 + 2 ell + T,
|P_k(z)| >= 2^(-d_r) on |z|=1,
||P_k(tau Z_L^w)^(-1)||_infinity <= 2^(2 d_r + r + 2).

The resultant must be nonzero. PR666's seed-2 lifting argument supplies that condition for every r=2*p^a, prime p>=5, a>=1; its specialization is written out and credited here. Both exponents remain O(r^2 log^2 r). At radix 8,978 the sufficient gap exponent drops from the common coarse dyadic envelope 356,393,327,561 to 28,312,479,536. These are analytic bound values, not executed working precision or runtime savings.

Reproduce from the repository root, with Python 3.12 and no third-party dependencies:

python3.12 -B scripts/research_package.py --package research/composite-height-audit-seed67

Expected: PASS_COMPOSITE_HEIGHT_AUDIT_S67, eight passing tests, then PASS_REGISTERED_RESEARCH_PACKAGE.

Local validation on macOS arm64:

• All 8,910 character maps at radices 3 through 134, with 487,412 unit-image occurrences and exact rational envelopes.
• All 19 admitted modular lifting instances through radix 134, checked by polynomial Bezout identities and separately implemented dense Sylvester ranks.
• 136 composite-family budget cases, nine internal rejection controls and eight isolated package tests, including source/receipt corruption, a dropped totient factor and relocation.
• Python 3.12 registered replay, Python 3.14 tests, selected-record integrity, package/source pins and staged whitespace checks pass.

The finite checks support specific identities and failure modes. The universal analytic claim is a written proof, not proof-assistant verification. No paid composite solver, all-size compiler, hosting admission, full cold replay or multiplication improvement is claimed. Only the new research package is added.

Exact base: bce54fb3f895cbe5cd4c0eb477c3afeed100190f. Predecessors: Chafik's #1025 at 6ef2168ac90d42926bceb7ac9b2e029463800c76, Jason Willems's #666 at 8e5a8d4da5bf6d83dc0b39ec90fea15e3cf627d6, and Ali Romman's retained #621 proof. SOURCES.json records file hashes and Git blobs. The final overlap refresh covered 788 open PRs and 146 discussion records with complete pagination; both dependency heads were unchanged. Nearby #668/#796/#803/#832 concern different obligations. No novelty claim follows merely from that search.

OS assignment seed 67 selected the adversarial-audit track and the 2*67^a example family. The local claim, three candidates, stopping rule and open next question are recorded in TASK.json; no pinned coordination issue existed.

Prepared for Hanson Sun (@bbMisk) with substantial OpenAI Codex assistance in derivation, implementation, verification and writing. Credit the predecessor authors, the retained OpenAI exact-DFT kernel and classical cyclotomic/Fourier arguments. Apache-2.0; no independent human review or literature-priority claim.

[Mister-Meeseeks]

100-PR batch review at 65ba60e3074f7b3968858f95eed514e8fe87e787.

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

Reviewed scope: General-radix averaged-height proof retains the r/phi(r) factor and cancels it in the norm budget, with exact character/lifting checks. Conditional on nonzero resultant except for the proved 2p^a family; analytic bound only, not paid inverse implementation 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.