Finite bit construction: exact R439 four-helper fusion below the pair floor

research/r439-helper-quartets-seed07/CLAIM.md

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+# Cycle 3 claim
+
+Claimed 2026-10-13 at 16:48 UTC. Assignment seed 7 was drawn once with
+`secrets.randbelow(100)`, using OS randomness. Track: bit supplier.
+Base: `2921874911956718fcaaaf9396fe8267b3cfa9f0`, clean fetched main.
+Branch: `research/cycle03-seed07-helper-quartets`.
+Finish by 17:23:24 UTC; reserve 17:13:24 onward for verification and handoff.
+
+Hypothesis: a general GL(4,2) basis on the union of two disjoint helper pairs
+can reduce the complete ADD count of the precision-admitted PR941 R439
+candidate while retaining its 104-bit local and 192-bit nine-stage caps.
+Freeze the PR915 target basis and every operation outside D. No new kappa,
+selected-supplier adoption, global regeneration or changes to upstream/.
+
+Method and stopping rule: choose exactly 64 distinct pair-of-pair indices
+by sorting SHA256 of `7:i:j` over all 19,701 unordered pairs of the 199
+PR941 pairs. Enumerate all 20,160 GL(4,2) matrices by breadth-first search
+using lexicographically ordered XOR generators, retaining one shortest
+word per matrix. Price all 1,290,240 combinations with preparation and
+literal reverse cleanup included. For each quartet reconstruct its cheapest
+representative, breaking ties by matrix tuple, and check the complete word's
+positive precision bounds. Pick one admitted improvement by complete ADDs,
+then F, B and quartet indices. No adaptive restarts or combination of winners.
+If none passes, report the finite failed search, not a general impossibility.
+
+Optimistic complete price before implementation:
+`6220 + 612 + 2*(197 + length(H)) + weight(E*K*H_inverse)`.
+The 6,220 outside-D ADDs, 306 target preparations and 306 restorations,
+197 untouched helper pairs, complete injection and both quartet passes
+are all charged. Paid recursive ranks, helpers, copies and frame paths stay
+fixed; a smaller local word alone does not establish a better supplier bound.
+
+Three candidates considered: this four-helper fusion; a simultaneous
+helper-and-target basis change; further phase-A retiming. Choose the first
+because its exact finite class and complete price fit this cycle. The second
+would change two interacting contracts; the third overlaps existing prefix
+and moment work.
+
+Coordination checked immediately before claiming: no pinned sprint and only
+open issues #913 and #650, neither coordination. No reservation issue opened.
+Live PR metadata checked through #1021; all-state searches for GL(4,2),
+four-helper, quartet and helper inspected. PR941 disjoint matching, PR965
+pair-plus-singleton GL(3,2), PR967 three-helper stars and PR978 historical
+R420 sharing are predecessors or adjacent scopes. PR1015 is phase B and
+PR1016 is a fixed-frame moment obstruction. This search is a four-helper
+fusion around the frozen PR941 matching, with all other pairs retained.
+
+Deliverable: exact witness and bounded-search census with source pins,
+independent finite replay, negative controls, cost and scope proof; one PR
+only if useful evidence and relevant checks pass, otherwise a clean local
+patch and handoff. Sagar Thakrar / @thakrarsagar, with substantial OpenAI
+Codex assistance. Apache-2.0; predecessor attribution retained.
+
+## Completed result
+
+Submitted 2026-10-13 at 17:10:45 UTC:
+https://github.com/CrocSwap/integer-mult-bounds/pull/1045
+Implementation head: `51871a0aa67132565acba1de38bb97db14a187fb`.
+This completion note is the one result update to the local claim.
+
+All 64 quartets and all 1,290,240 basis prices completed. The chosen quartet
+`[269,88,288,153]` uses three preparations in place of two, saves five
+injection ADDs and costs two additional preparation/restoration ADDs.
+Complete cost: 11,949 to 11,946; this is the exact minimum in the declared
+64-quartet class. The independently replayed all-disjoint-pair floor is
+11,948. Sixty-two quartets give no improvement and one other saves one ADD.
+Every cheapest representative passes the two inherited precision caps.
+F becomes 70,355; B remains 10,548,832. No new kappa or best-circuit claim.
+
+Verification includes independent group/domain/distance and full-price
+censuses; the 96,141-edge pair-bound dual; 661 input columns in both D
+directions; 11 local and four external adverse cases plus a charged neutral
+case; unchanged owning checks; native 5/9/11-stage replay on both words and
+96 rejected bridge controls; 43 source pins; 25 selected-record and nine
+finite-bit-contract tests. The native receipt records 186 dependency hashes.
+Verification and handoff occupied the final part of the cycle, beginning
+before 17:00 UTC, ahead of the reserved 17:13:24 start.
+
+Metadata and claims were refreshed through #1043 immediately before
+submission. PR1040 changes the target basis after PR967, whereas this
+construction freezes E and changes four PR941 helpers; it is credited.
+PR1023 is a hosting-width result, PR1027 is an R480 six-wire circuit and
+PR1043 changes source access and helper stock. No same-object claim was
+found. Refreshed main is still the recorded base; all 43 source blobs and
+the selected kappa are unchanged. No pinned sprint exists. The repository
+has neither a codex nor a codex-automation label available.
+
+PR1045 is open and attached to the originating chat. GitHub CI was queued
+at the 17:11 UTC submission check; success is not asserted. Whole-hosted
+adoption, the complete supplier invoice, outer assembly and inherited
+uniform/realization/tape/analytic assumptions remain. No full repository
+suite, selected supplier cold replay, Linux run or formal proof was done.