Research audit: exact prime-power kernel root valuations (no new kappa)

[FanPUYen]

This mathematical audit / research note corroborates the nonunit-root lemma in Ali Romman's incorporated #621 and determines the complete p-adic root-valuation multiset of its prime-power kernel. No defect was found in that lemma, and no new κ is claimed.

For r=p^a and v(1-u)=1, the written proof gives root valuation 1+(k-1)(p-1)/p with multiplicity (p-1)p^(k-1), for every k=1,...,a. It supplies all Newton-polygon vertices and collinear coefficient contacts, handles radix two and every primitive root choice, and distinguishes this normalization from v_p(p)=1. These are nonarchimedean valuations, not complex root moduli or a sharper complex conditioning bound.

Only research/prime-power-newton-audit/ is added. One implementation counts factor valuations; a separate implementation reconstructs the exact cyclotomic coefficients, changes to the uniformizer basis, and extracts their valuations without using the counting formula.

python3 -I -O -B research/prime-power-newton-audit/verify.py
python3 -I -B research/prime-power-newton-audit/test_verify.py

Expected: PASS_NEWTON_NONUNIT_AUDIT: 28 radices; 697 primitive embeddings; 30224 coefficient obligations; 9 negative controls, then eight tests pass. Python 3.9+; standard library only.

Validation passed on macOS arm64 / Python 3.9.6: normal and isolated optimized replay; dense-polynomial reference tests; exact Eisenstein checks; malformed-input, source-drift and receipt-corruption rejection; a complete minimal relocated replay; unchanged #621 replay matching its published receipt; selected-record integrity; and staged whitespace. A separate Codex reviewer found no correctness issue. Finite replay supports the written proof; neither is a formal proof of integer multiplication. Full repository/cold supplier suites, optional SymPy, Linux and remote CI were not run or claimed.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Audited #621 head: 520774462b19b29af41554b74def31f9d01e997e. Source hashes, proof references, deterministic receipt, completed local claim and validation scope are included. OS assignment seed 68, adversarial-audit track. No pinned sprint issue existed. The final overlap check through #757 distinguishes #668's CRT resultants, #666's composite lifting, #628/#692's inverse execution, and #736's replay optimization.

Credit Ali Romman, OpenAI's pinned exact-DFT kernel, the classical q-Pochhammer/Newton lineage, and the standard cyclotomic/Newton-polygon references. Prepared for Hao Yan (@FanPUYen) with substantial OpenAI Codex assistance, under Apache-2.0. No literature-priority claim. The selected #626/#543 frontier and upstream/ are unchanged; paid preparation, restoration, compiler/hosting, precision/tape and analytic multiplication obligations remain.

[Mister-Meeseeks]

100-PR batch review at 22902155c7875139e43e4fdf15723142d6709bdd.

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

Reviewed scope: Exact prime-power Newton polygon and valuation census; strengthens a nonunit lemma, not complex conditioning or a multiplication bound.

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.