Research theorem: common-flag rational-frame obstruction (no new kappa)

research/common-flag-frame-obstruction/CLAIM.md

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+# Sprint claim
+
+- Track: structural decoder research; OS-random assignment seed **42**.
+- Started: 2026-10-13 06:03:27 UTC. Expected finish: 06:48 UTC;
+  verification and handoff reserved from 06:37 UTC.
+- Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60` (fresh `origin/main`).
+- Branch: `research/seed42-structural-decoder`, isolated worktree.
+- Scope: the all-role rational-frame XOR interface, with every assigned
+  matrix preserving one common complete flag after a common additive gauge.
+  Test whether nonprojection, possibly noncommuting upper-triangular labels
+  can yield a positive rank deficit. Include the commuting, nonsplit case
+  by passing to a characteristic-zero splitting field in the written proof.
+- Method: one-dimensional characters of the additive frame group, a
+  telescoping rank bound over a characteristic-two Laurent fraction field,
+  and exact finite circuit/flag controls. No optimization search is planned.
+- Deliverable: a scoped theorem or counterexample, independently checkable
+  finite evidence, source pins, and one reproduction command. No kappa claim.
+
+## Selection and overlap
+
+The three candidates were (1) mixed-beta overcomplete P112 decoders,
+(2) Gaussian E8 asymmetric support costs, and (3) nonprojection rational
+frame labels. Candidate 1 overlaps #717's unfinished mixed-beta question;
+candidate 2 overlaps #738/#767. Seed 42 selects the third mechanism and
+the finite controls' deterministic generation seed. Its common-flag
+restriction changes the search class; this does not reopen a closed
+puncture or star-decoder search with more computation.
+
+Read CONTRIBUTING, the selected record, current status, PR ledger, selected
+#626 assembly, its source inventory and maintainer review, and the actual
+K492 construction proof. Accepted conditional kappa is
+`51110230636731/31250000000000000`, with #626 head
+`660de893d27bb7abd1d5b01b1640207528a5c851` and #543 head
+`2065fe4f72a914d907f1046603f5a36f17018535`.
+
+The initial complete open-PR metadata scan reached #808; the immediate
+pre-claim refresh reached #810. Relevant predecessors: the historical
+commuting-projection core obstruction, the general finite-bit contract and
+routing obstruction, #717's explicitly open nonprojection-label question,
+#791's distinct idempotent endpoint penalty, and #705/#800's Clifford
+path-valuation criteria. None is treated as an accepted frontier claim.
+GraphQL `pinnedIssues` returned an empty list. The only open issue was #650,
+a request for Discussions, so this claim is local.
+
+All workers are restricted to this common-flag question. The selected
+record, existing generators and `upstream/` are outside the edit scope.
+
+Prepared for @navinor with substantial OpenAI GPT-6 Astra assistance.
+New material is under Apache-2.0.
+
+## Result update
+
+Submitted one focused PR:
+[#825](https://github.com/CrocSwap/integer-mult-bounds/pull/825).
+The common affine complete flag forces **s >= Wm** and hence a normalized
+rank-child moment of at least one. This is a scoped negative theorem, with
+no new kappa and no claim about unrestricted nontriangular frames.
+
+Completed finite evidence: 14 circuits, 13 tests and all 729 ordered
+triangular-matrix differences. The existing scorer agrees on every edge
+rank and all-role endpoint contract. Optimized/relocated replay, source and
+receipt corruption rejection, output protection and regeneration passed;
+see `VALIDATION.md` for commands, review responses and limitations.
+
+Final overlap snapshot: 637 open PRs through #824, no matching theorem
+identified; pinned issues still empty. Upstream main was refreshed to
+`7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261`, selecting #741/#589/#555.
+The original research base is retained; all four mathematical source pins
+are unchanged. The submission awaits human review of the written proof
+and its interface scope.