Research: bit E8 decoder witness and 9/17 pairing-cost floor

research/bit-orthogonal-pair-decoders/CLAIM.md

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+# Local sprint claim
+
+No pinned coordination issue existed (checked 2026-10-13 12:43 -04:00; open issues were #913 and #650), so
+the claim was recorded locally after a 13-second preliminary existence check,
+before the threshold census, lower-bound argument and supplier analysis.
+
+- **Track and seed.** 40-49, structural decoder research. Seed 40 from the OS CSPRNG
+  (`System.Security.Cryptography.RandomNumberGenerator`, uint32 mod 100).
+- **Base commit.** `2921874911956718fcaaaf9396fe8267b3cfa9f0` (selected PR942, κ = 0.001661960547713947).
+- **Object.** Bit label decoders I_9 = Σ_j p_j q_jᵀ on the 120 E8 lines minus a puncture, priced by
+  rational pairing rank (PR446 Theorem C), on punctures other than the fixed P111 of PR757/PR761.
+- **Hypothesis.** Terms pairing two orthogonal punctured lines cost 6; nine such pairs with F2-dual labels
+  give cost 54 on 102 ports.
+- **Deliverable.** This directory: proof, exact verifier, independent checker, witness, controls,
+  favorable pricing, no κ claim.
+- **Expected finish.** 2026-10-13 about 16:30 -04:00.
+
+## Candidates considered
+
+1. Complex exact 8-term decoders for 25 ≤ p ≤ 56, extending PR968. Not chosen: complex-only gains are
+   capped by the PR589 bit supplier, and PR968's P104 budget already reaches that cap.
+2. Complex decoders with n ≥ 9 terms on P112 beyond the single-added-total class of PR845 and #998. Not
+   chosen: #998 (also seed 40) covers the nearest class, and general n-term bounds were not bounded work.
+3. **Chosen:** bit label decoders under Theorem C pricing on other punctures. PR757/PR761 fix P111;
+   PR939/PR954/PR631 use star pricing; PR620's deeper search used star pricing; PR595 lists x = 54/102 as
+   unrealised.
+
+## Stopping rule and outcome
+
+Stop the experiment if no nine-pair dual system exists after an exhaustive backtracking pass with one
+pair fixed, or once the floor argument and the witness are both checked. Outcome: a system was found in
+13 s; the floor was proved for p ≤ 25 with the thresholds 1, 14 and 26; the full count of systems was
+stopped after 15 minutes and is not claimed.
+
+The exploratory backtracking domain consisted of ordered pairs of distinct
+orthogonal root lines, with one fixed pair (0,1) in sorted doubled coordinates;
+extensions had to satisfy B0[a_j,b_k] = delta_jk. It returned three systems and
+the first was retained. This was a deterministic witness search after the OS
+assignment draw, not a randomly sampled experiment. No complete count or
+impossibility conclusion follows from the timed-out counting attempt.
+
+## Submission refresh
+
+The 2026-10-13 refresh covered all 757 open PRs through #1062, rather than the
+initial capped 400-item inventory. No issue was pinned. PR1020 treats complex
+F2 label-pairing ranks at 26 punctures; PR1008 treats tensor shears on fixed
+P111. Neither is a rational pairing-rank result for bit-side punctures through
+25. PR757, PR761, PR717 and PR939 retained the source heads listed in SOURCES.json.
+An immediate pre-submission refresh through #1068 also inspected #1060's
+complex P111 COPY construction: it prices a different physical complex word,
+not these bit rational-pairing decoder witnesses. Main remained at the base.
+
+## Result and handoff
+
+Submitted as [PR #1084](https://github.com/CrocSwap/integer-mult-bounds/pull/1084):
+research note / exact algebraic decoder and scoped lower bound, not a hosted
+supplier or new kappa. The retained witness has cost 54 on 102 ports, and the
+written argument bounds the label-decoder cost ratio by 9/17 through 25
+punctures. All eight package checks, the staged-export replay and selected
+source integrity pass. The checked changed-reader example costs 68/67 and
+guards the corrected fixed-reader scope of the PR589-to-P110 diagnostic.
+
+The immediate submission refresh reached #1081. PR1077's arbitrary-complex
+eight-term result is also distinct from the binary label-factor/rational
+pairing-rank model here. Upstream advanced to
+`abdc6a7002abdbcd9e21bfc5e1a795784deacfdf` while this branch retained its exact
+base; all seven pinned executable/data inputs remained byte-identical, and
+PR942's selected kappa remained 0.001661960547713947. No coordination issue
+was pinned, so this is the single local result update.

research/bit-orthogonal-pair-decoders/NOTICE.md

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+# Attribution and scope
+
+Contribution prepared for @navinor, under Apache-2.0.
+
+Substantial AI assistance through OpenCode covered the research, written argument,
+both implementations, verification and documentation. The resumed review and
+packaging used GPT-6.1 Sol (OpenAI). Neither two code implementations nor an
+AI-assisted review constitute independent human review.
+
+Predecessor credit:
+
+- Joel Pulikkan / GamingPuzzled (PR620): the bit label-decoder model and the
+  orthogonal-punctured-pair mechanism.
+- DreamingOfClouds (PR446, PR595, PR717 and PR589): the inherited pairing-rank
+  copy premise, accounting, recorded invoice diagnostics and selected bit word.
+- Russell Seligmann (PR757) and Cody Harris (PR761): solver-free cost-63
+  certificates for their fixed P111 fixtures. That value is not claimed as new.
+- themilkman216 (PR968): the complex frame-pair analogue, hyperplane census and
+  favorable P104 budget. The pricing code here follows that scenario.
+- Chi Zhao (PR631), Ross Richardson (PR939 and PR954), and Philipp (@mightyOil,
+  PR875): the distinct star-cost results and root-subspace coverage argument.
+- The selected supplier lineage: Jonathan Capalbo, Chafik Boukhalfa, Jeremy
+  Kothe, Hao Yan, DreamingOfClouds and eumemic, with their predecessors credited
+  in the retained notices.
+
+Only this research directory is added. No supplier word, selected certificate,
+upstream source, existing notice or existing verifier is modified. Pricing
+engines are imported from their immutable, SHA-256-pinned repository paths;
+predecessor research documents are referenced, not copied into this package.
+
+No physical realization of the new decoder, improved multiplication bound,
+formal multiplication theorem, or hosted CI pass is claimed.