Independent verification: integral-divisibility kernel replay and radix 81

research/prime-power-newton-valuation/CLAIM.md

@@ -0,0 +1,80 @@
+# Sprint claim and implementation plan
+
+- Track: independent verification; OS-random assignment seed **80**.
+- Exact base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+- Branch: `research/seed80-independent-verification`.
+- Claimed at 2026-10-13 05:40 UTC; expected finish by 06:15 UTC.
+- Object: the coefficient valuations and lower Newton polygon of PR621's
+  integral prime-power kernel, not a proposed multiplication construction.
+- Method: build tail products in the integer cyclic group ring, reduce in the
+  cyclotomic power basis, substitute `u=1-pi`, and extract the ramified valuation.
+  Compare with the independent tail-sum formula and a convex-hull computation.
+- Domain: every primitive embedding at orders 3, 5, 7 and every repeated
+  prime power through `seed+1=81`: 4, 8, 9, 16, 25, 27, 32, 49, 64, 81.
+- Deliverable: one standard-library verifier, exact deterministic receipt,
+  tests, scoped mathematical note, source pins and handoff. No new kappa.
+- Stop after the complete fixed domain, or report the first discrepant exact
+  coefficient. No adaptive search or extra computation on a closed search.
+
+## Selection and coordination
+
+Three candidates were inspected: (1) R420 compensation algebra, already covered
+by open #731/#737/#760/#762; (2) prime-power resultant replay, already covered by
+#668 and composite extensions #666; (3) ramified coefficient valuations and
+Newton polygons for #621. Candidate 3 changes the proof obligation and arithmetic
+method, rather than merely using a new seed. The open-PR inventory and primary
+package contain no Newton-polygon or p-adic-valuation contribution at selection.
+The pinned-issue GraphQL query returned no issues, so this is a local claim.
+No separate reservation issue was opened.
+
+Selected status was followed from `certificates/selected-result.json` to
+`research/community-k492-audit/pr626/replay/ASSEMBLY.json`, its source manifest,
+the #626 construction README and maintainer review. The accepted conditional
+value is `51110230636731/31250000000000000`, not the pending PR-title values.
+
+## Implementation plan
+
+Goal: independently check exact valuations before deriving any conclusion.
+Architecture: a standalone integer engine and rational convex hull; no imports
+from the predecessor's field, determinant, inverse or SymPy implementations.
+Tech stack: Python 3.11+ standard library. Scope and design are the claim above.
+
+1. Write tests with hand-calculated radix-4 coefficients and radix-8/27 hulls,
+   plus nonprimitive/composite and altered-coefficient controls; observe failure.
+2. Implement `tail_coefficients(r, power)`, `coefficient_valuation(c, p, a)` and
+   `lower_hull(values)`; compare exact expansion against the tail-sum formula.
+3. Replay the fixed domain, recording every primitive power and a digest of
+   every reconstructed coefficient. Add controls that preserve the old mod-ideal
+   zero/nonzero pattern while changing the stronger valuation obligation.
+4. A single scoped background worker checks the uniform mathematical argument
+   and writes only `PROOF.md`; it does not change the implementation or scope.
+5. Reserve the final ten minutes for fresh normal/optimized replay, predecessor
+   checks, receipt-mutation failure, source integrity and overlap refresh.
+6. Submit at most one independent-verification/research-note PR, with exact
+   attribution, AI disclosure and unresolved obligations. Do not alter upstream
+   or the selected record.
+
+## Final result and handoff (one claim update)
+
+Completed 2026-10-13 at approximately 06:00 UTC, with the final verification
+and handoff phase beginning at 05:51 UTC. Submitted one focused contribution:
+[PR804](https://github.com/CrocSwap/integer-mult-bounds/pull/804).
+
+The fixed domain completed: 214 primitive embeddings, 10,268 exact coefficient
+valuations, two valuation extractors and eight rejecting controls. Nine focused
+tests pass on installed Python 3.12.14 and 3.14.7. The unchanged #621 replay
+matches its full published receipt; selected-record/source integrity passes.
+
+The submission refresh discovered Hao Yan's concurrent #796, posted at
+05:47:56 UTC, with the same formula and weighted-minimum extraction. The
+handoff was therefore narrowed to distinct repeated integral-divisibility
+evidence and radix-81 coverage, not priority for the shared theorem. Its pinned
+receipt agrees on 160 shared embeddings and 5,894 coefficients, with all shared
+hulls and root multiplicities identical. Radix 81 adds 54 embeddings and 4,374
+coefficients. The final metadata refresh retained #796 head
+`22902155c7875139e43e4fdf15723142d6709bdd`; #800's Clifford path-valuation work
+is a different object. Exact comparison pins are in `CORROBORATION.json`.
+
+No kappa change, full repository-suite pass, Linux/CI result, paid inverse,
+proof-assistant formalization or independent human review is claimed.
+`VALIDATION.md` supplies the commands, scope and remaining obligations.

research/prime-power-newton-valuation/CORROBORATION.json

@@ -0,0 +1,22 @@
+{
+  "status": "PASS_EXACT_SHARED_RECEIPT_COMPARISON",
+  "source_pr": 796,
+  "author": "Hao Yan (@FanPUYen)",
+  "source_head": "22902155c7875139e43e4fdf15723142d6709bdd",
+  "source_receipt": "research/prime-power-newton-audit/EXPECTED.json",
+  "source_receipt_sha256": "d074f1c2f87324e81cf8aa94784077694230cf35d95f8b0c173755c89f876333",
+  "shared_radices": [3, 4, 5, 7, 8, 9, 16, 25, 27, 32, 49, 64],
+  "shared_primitive_embeddings": 160,
+  "shared_coefficient_obligations": 5894,
+  "equal_fields": [
+    "primitive powers",
+    "every coefficient valuation",
+    "lower hull vertices",
+    "every root valuation and algebraic multiplicity"
+  ],
+  "additional_radices": [81],
+  "additional_primitive_embeddings": 54,
+  "additional_coefficient_obligations": 4374,
+  "distinct_method": "Repeated exact integral division by pi=1-u, independent of the weighted-minimum valuation extractor shared with PR796.",
+  "scope": "Pinned receipt corroboration, not execution or review of the full PR796 package and not independent human review."
+}