Independent verification: algebraic radix-9 coefficient and factor audit

[WilliamLawyer]

PR628's numerical factor certificate depends on pi/Taylor enclosures of the
prime-power kernel coefficients. This independent finite audit corroborates
that input for radix 9 using an isolated cubic root, integer square-root
bounds, and forward division in Q[X]/(X^6+X^3+1). Status: mathematical audit /
independent verification; no new kappa and no defect alleged.

All 54 complex coefficient enclosures across the six primitive embeddings
contain the separately computed algebraic rectangles. Both submitted
radix-9 factor rows have coefficient residual at most 2^-1020; with their
existing 768-bit settings, the inherited error formulas stay below the
2^-128 target. Exact field coefficients and the root rectangle are included.
Fifteen mathematical mutations are rejected; valid conjugation and root
reordering are accepted.

Reproduce with Python 3.10+ and the standard library:

python -B -I -O research/radix9-algebraic-audit/verify.py
python -B -I -O research/radix9-algebraic-audit/test_verify.py

Expected: PASS_RADIX9_ALGEBRAIC_AUDIT, then eight passing tests. Local
Windows / Python 3.10.9 validation includes optimized isolated replay,
offline relocation, source/receipt corruption, output protection, the selected
record integrity check and staged whitespace. The scoped Linux workflow is
included but has not been run. Full repository/supplier and PR628 execution
suites were not rerun for this isolated package.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Audited PR628 head:
0768f97d06c623b8211e651543c66135796b62bf. Original subject/input bytes are
pinned by SHA256 and Git blob identity. PR628 supplies the operator and scan
analysis, which remains inherited; this is not a full multiplication proof,
an all-radix audit or a new execution algorithm. upstream/, selected sources
and the frontier are unchanged.

Credit Chafik Boukhalfa for PR628, Ali Romman for PR621, and the retained
OpenAI/q-Pochhammer/Newton lineage. Related concurrent PR796 concerns p-adic
root valuations; it does not audit these complex rectangles or factor
residuals. PR764's scan-bound improvement and other related work are credited
without being imported. The publication overlap refresh covered arrivals through
#811, with PR628 unchanged. Rafael Guevara's #803 improves an
inverse-norm bound, and navinor's #804 verifies p-adic valuations by integral
division; neither checks these complex coefficient rectangles or submitted
factor residuals. Both are credited as related context without code dependencies.

Upstream advanced during connection setup to
7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261, selecting #741 with #589 on
#555 at conditional kappa 0.001661202643272239. This audit retains its
original research base; its selected-record integrity check refers to that
base. The subsequent integration is not a dependency or an audited result.

Contribution attributed to William Lawyer, with substantial OpenAI
Codex assistance in derivation, implementation, testing and writing.
Apache-2.0; no independent human review or literature-priority claim.

[Mister-Meeseeks]

100-PR batch review at 00f326845c0614366bef11ee6901ae6295af4999.

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

Reviewed scope: Independent algebraic radix-nine coefficient enclosures and two factor certificates, with corruption controls; no all-radix claim.

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.