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

research/prime-power-newton-audit/CLAIM.json

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+{
+  "track": "Adversarial mathematical audit",
+  "assignment_seed": 68,
+  "randomness": "secrets.randbelow(100), operating-system randomness",
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "source_pr": 621,
+  "source_head": "520774462b19b29af41554b74def31f9d01e997e",
+  "scope": "Audit the prime-ideal nonunit lemma by deriving the complete p-adic Newton polygon of the integral q-Pochhammer kernel; no complex conditioning or kappa improvement claim.",
+  "hypothesis": "Vertices (r-p^k, k*(p-1)*p^(k-1)) give root valuations 1+(k-1)*(p-1)/p with multiplicities (p-1)*p^(k-1), normalized by v(1-u)=1.",
+  "candidates": [
+    {
+      "task": "Cold-helper lifetime assumption",
+      "decision": "Not selected: overlaps PR702/712/729."
+    },
+    {
+      "task": "Fourier inverse transfer for weighted strides",
+      "decision": "Not selected: overlaps PR692 and rounded PR628."
+    },
+    {
+      "task": "Prime-power root nonunit lemma via exact Newton polygon",
+      "decision": "Selected: different proof obligation and valuation method from PR668 CRT resultant and PR666 composite lifting."
+    }
+  ],
+  "method": "Written uniform proof plus independent integer valuation / convex-hull implementations; bounded complete domain prime powers r<=68, with higher-power and primitive-root controls as appropriate. Seed sets upper radix 68.",
+  "stopping_rule": "One fixed finite domain; reject or narrow the formula if a counterexample appears; no adaptive record search.",
+  "expected_deliverable": "Research audit note, exact deterministic receipt, standard-library verifier and negative controls.",
+  "expected_finish": "2026-10-13T13:38:00+08:00",
+  "coordination": "Open-issue search returned only issue650, no pinned sprint issue; local claim.",
+  "status": "completed",
+  "completed_at": "2026-10-13T13:11:33+08:00",
+  "result": "Uniform exact Newton polygon and p-adic root-valuation formula; no defect found in PR621 nonunit lemma; 28 radices, 697 primitive embeddings, 30224 coefficient obligations, 9 rejecting controls, 8 focused tests. No new kappa.",
+  "result_url": "https://github.com/FanPUYen/integer-mult-bounds/tree/audit/seed68-cyclic-inverse/research/prime-power-newton-audit",
+  "overlap_refresh": "Newest 100 open PRs through #757, earlier 200 entries, targeted Newton-polygon/p-adic/valuation searches, and open issues. #736 affects old replay performance only. No overlapping root-valuation audit or pinned sprint issue found.",
+  "delegation": "Two scoped Codex workers: one proof derivation/review, one independent cyclotomic implementation; no repository-wide delegated investigation."
+}