Independent verification: algebraic radix-9 coefficient and factor audit

.github/workflows/radix9-algebraic-audit.yml

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+name: Radix-9 algebraic coefficient audit
+
+on:
+  push:
+    paths:
+      - 'research/radix9-algebraic-audit/**'
+      - '.github/workflows/radix9-algebraic-audit.yml'
+  pull_request:
+    paths:
+      - 'research/radix9-algebraic-audit/**'
+      - '.github/workflows/radix9-algebraic-audit.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  audit:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 5
+    strategy:
+      matrix:
+        python-version: ['3.11', '3.14']
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97 # v7
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Check independent coefficient and factor evidence
+        run: python -B -I -O research/radix9-algebraic-audit/verify.py
+      - name: Check failure detection and offline relocation
+        run: python -B -I -O research/radix9-algebraic-audit/test_verify.py
+      - name: Require unchanged tracked inputs and receipts
+        run: git diff --exit-code

research/radix9-algebraic-audit/CLAIM.md

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+# Sprint claim
+
+- Contributor: William Lawyer, with substantial OpenAI Codex assistance.
+- Track: adversarial mathematical audit; operating-system assignment seed 73
+  (one Python `secrets.randbelow(100)` draw).
+- Base commit: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+- Scope: PR628's radix-9 kernel coefficient enclosures and the coefficient-error
+  interface of its submitted approximate factor certificate. Pin:
+  `0768f97d06c623b8211e651543c66135796b62bf`.
+- Method: isolate the real algebraic root of `x^3-3x+1` in `(3/2,8/5)`,
+  use a rational square-root enclosure for its positive imaginary component,
+  and derive kernel coefficients by division in `Q[X]/(X^6+X^3+1)`.
+  This is separate from PR628's Machin-pi/Taylor-exponential enclosure engine.
+- Domain: all six primitive embeddings of radix 9, plus every radix-9 row in
+  the pinned factor certificate. Seed 73 selects index `73 mod 3 = 1` from
+  the candidate radices `[7,9,16]` and will select coefficient mutation sites.
+- Deliverable: scoped audit note, independent exact checker, deterministic
+  evidence and mathematical negative controls; no new kappa claim.
+- Start: 2026-10-13 05:27:13 UTC (2026-10-12 23:27:13 America/Denver).
+- Finish deadline: 2026-10-13 06:27:13 UTC. Verification and handoff receive
+  at least the final 15 minutes of the 60-minute budget.
+
+## Selection and overlap
+
+Three narrow candidates were inspected: (1) PR628's coefficient enclosure /
+factorization interface; (2) PR666's excluded-prime modular lifting interface;
+(3) selected cold-helper lifetime / no-later-touch accounting. Candidate (1)
+was chosen because the independent closed-coefficient checks in PR628 cover
+radices 4 and 8, while radix 9 can use a different exact algebraic method.
+PR666 already supplies its own modular witnesses, and PR729 specifically
+audits helper lifetimes. Neither rejected candidate is reopened here.
+
+The required contribution guide, selected certificate, status, review ledger,
+selected construction/review, retained recurrence note and research index
+were inspected. At the recorded base, main selected #626 with #543, conditional
+`kappa=0.001635527380375392`; this audit does not modify those sources.
+
+The public inventory contained 597 distinct open PRs across six pages, with
+a created-order refresh through #786 immediately before this local claim.
+Related #663/#673/#689/#730/#744/#758/#784 concern dirty echoes;
+#692 concerns weighted-stride inverses; #764 sharpens scan rounding;
+#666/#668/#736 concern cyclic nonvanishing or arithmetic replay.
+Their work is distinct from the radix-9 coefficient audit. Pending work is
+not treated as accepted. A final overlap refresh is required before handoff.
+
+The open issue search returned only #650 (Discussions), with no pinned sprint
+coordination issue visible. The claim is therefore recorded locally, as
+permitted by the sprint instructions; no reservation issue is created.
+
+## Result update
+
+Completed a scoped independent audit: all 54 coefficient rectangles across
+six primitive radix-9 embeddings contain the independent algebraic enclosures.
+Both submitted radix-9 factor rows pass the inherited target budget, and all
+15 mathematical controls are rejected. No defect or new kappa is claimed.
+See [PROOF.md](PROOF.md), [EXPECTED.json](EXPECTED.json) and
+[VALIDATION.md](VALIDATION.md). Attribution is to William Lawyer with the
+assistance disclosure and predecessor credits in [NOTICE.md](NOTICE.md).
+
+The initial handoff refresh through #799 found the related #796 p-adic
+valuation audit; it is credited and distinguished from this complex-enclosure
+check. Main and the pinned PR628 head remained unchanged. Publishing access
+was unavailable at that handoff, so a focused clean patch and prepared PR
+description were produced. The subsequent publication refresh at
+2026-10-13 06:07:46 UTC covered arrivals through #810. Related #803 and #804
+were inspected and credited in NOTICE.md; neither checks these complex
+coefficient rectangles and submitted factor residuals. Main and PR628
+remained at the recorded pins. No remote coordination claim was posted.
+
+The final submission check at 2026-10-13 06:14:25 UTC covered arrivals
+through #811 (an unrelated JP8 circuit contribution). Main had advanced to
+`7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261`; its selected certificate and
+current-status note now identify #741 with #589 on #555, conditional
+`kappa=0.001661202643272239`. The research base and audited PR628 head
+remain pinned above. This audit neither changes nor verifies that new frontier.