Independent verification: integral-divisibility kernel replay and radix 81

[navinor]

Result and scope

Independent verification / research note; no new kappa. This corroborates

Hao Yan's concurrent PR 796 and Ali Romman's incorporated PR 621 using a different

valuation-extraction method: repeated exact integral division by pi=1-u.

It also adds every primitive embedding at radix 81, beyond PR 796's cutoff of 68.

The shared Newton-polygon formula is credited to the concurrent contribution;

no mathematical priority or new archimedean conditioning bound is claimed.

The standalone integer engine reconstructs actual tail-product coefficients,

checks the shifted cyclotomic polynomial is Eisenstein, and compares three

obligations: weighted-basis minima, exact integral-divisibility counts, and

the factor-valuation sum. An integer convex hull is checked against the rational

segment formula. No predecessor field/SymPy/determinant code is imported.

• Complete fixed domain: 13 radices, **214 primitive embeddings / 10,268 exact

  coefficients**, with eight rejection controls and per-embedding coefficient hashes.

• Pinned PR 796 receipt comparison: all twelve shared radices, **160 embeddings /

  5,894 coefficient obligations**, hulls and root multiplicities agree.

• Additional radix 81: 54 embeddings / 4,374 coefficients, checked by both

  extraction methods. The ideal-residue-preserving control doubles a coefficient while

  retaining all modulo-ideal residues and both kernel endpoints; its stronger

  valuation obligation correctly fails.

Reproduce

Python 3.11+; standard library only, from the repository root:

python -B -I -O research/prime-power-newton-valuation/verify.py --check research/prime-power-newton-valuation/EXPECTED.json

Expected: PASS 214 primitive embeddings; 10268 exact coefficients; 8 negative controls.

Nine focused tests and the exact PR 796 comparison snippet are in the package.

Pins and verification

• Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60.

• PR 621 head: 520774462b19b29af41554b74def31f9d01e997e.

• PR 796 comparison head: 22902155c7875139e43e4fdf15723142d6709bdd.

• Kernel source: openai/math@adc7f1241b42e322a6451854ab7e4b4c146bf78a.

Normal/isolated optimized replay passes on Windows x86-64, CPython 3.14.7 and

3.12.14; nine tests pass, including optimized changed-receipt and source-drift

rejection. The unchanged PR 621 primary replay equals its complete published JSON

receipt. Selected-record/source-inventory integrity passes. A scoped AI proof

worker and separate read-only AI code reviewer found no correctness issue;

the reviewer also checked every coefficient using a descending long-division

reference. No independent human review is claimed.

Only research/prime-power-newton-valuation/ is added. upstream/, the selected

PR 626/PR 543 frontier and every existing generator are unchanged. Full make verify,

the selected cold supplier replay, Linux and remote CI were not run or claimed;

GNU make is absent on this session's PATH. The scoped checks are complete.

OS-random seed 80, independent-verification track. No pinned sprint issue

exists; the local claim, three candidates, stopping rule and final overlap are

retained. PR 796 was discovered in the submission refresh, and the handoff was

narrowed to distinct divisibility evidence and radix-81 coverage. The final queue

refresh also found PR 800's unrelated Clifford path valuations, not a kernel overlap.

Prepared for @navinor with substantial Codex assistance.

Apache-2.0; predecessor and concurrent credits are in NOTICE.md. Written uniform

valuation reasoning and finite replay are not proof-assistant formalization, a

paid inversion/preparation/restoration/tape program, or a full multiplication

theorem. No merge or selected-frontier edit is requested by this contribution.