CLAIM: three-dimensional port decoder for selected P112 (Cycle 14)

[1337isnot1337]

CLAIM — three-dimensional port decoder for selected P112

Track / assignment. recursive-interfaces; fresh OS seed db0930263249627c08276af863ab6e78; task coverage-law-change (refined); priority revision 3; catalogue SHA-256 6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075. Exact selector base ec94fd3d2fcad7ade8e72d6bcd8e223b7fdb3f30. Selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Object / question. For the selected PR1094 complex labels q_i in F2^9, replace paired 2-plane ports by 37 mutually orthogonal 3-planes plus one residual line. For each triple t, let A_t be its three label columns; let A_0 be the residual label. Can an exact factorization

sum_t A_t M_t A_t^T + A_0 M_0 A_0^T = I_9

be found and priced so the three-unpaid-coordinate coverage law gives a favorable completed local moment at a = 1/99? The proposed dirty endpoint uses the commuting coordinate lifts C_E = product_{q in basis(E)} C_q; its correction is C_E^2 = X_(sum basis(E)). I will verify this identity and the full decoder equality exactly, without assuming the higher-dimensional interface for free.

Changed contract / optimistic gate. The endpoint changes from a matched 2-plane to a 3-plane E: source E -> FULL, target 0 -> E^perp, with three unpaid coordinates per live family. The current exact ceiling model with the selected word and unchanged 64 centre bill has moment 0.9973063970199103 at a=1/99; the analogous known paired-plane invoice (nine rank-7 and seven rank-9 terms, centre 126) fails at 1.0060315602100165 even if credited with three unpaid coordinates. A tri-plane construction must include its exact rank-1/2/3 support invoices and endpoint work, and pass the repository's exact rational moment check; otherwise I stop this construction attempt. No kappa claim is expected.

Bounded method. Pin the selected 112 labels; enumerate linearly independent, pairwise-orthogonal triples; use deterministic MRV exact-cover search for 37 triples and one residual line, examining at most 10,000 lexicographically ordered covers. For each candidate solve the exact 81-equation F2 system, factor its small coefficient matrices into rank-one terms, recompute every B14 support-rank charge, and apply the paid profile to the unpaid=3 exact ceiling. Stop at the first exact candidate that passes, or after the stated cover cap / an exact cost failure. Reproduce the known two-plane case as a positive control and reject a corrupted endpoint or factor term as negative controls.

Overlap and predecessors. Targeted live searches at 17:28 UTC found no 3-plane port construction. Discussion #1587 establishes the plane-port endpoint and asks the plane decoder question; merged PR #1825 supplies its exact matched 2-plane P112 factorization. Open PR #1900 uses five plane queries on the different P111 bit side. PR #1348 is a threshold analysis, not a tri-plane construction. I will check the live PR and Discussion state again before handoff.

Expected evidence. A scoped exact tri-plane construction with full decoder and B14 invoice, or an honest negative result limited to the bounded cover search; exact source pins, verifier, controls, registered replay where available, and predecessor/AI attribution.

Claim expires: 2026-10-14T19:30:00Z.

[1337isnot1337]

FINDING — first triad factorization exact, copy profile narrowly misses the target.

The pinned selected labels yield 24,416 independent pairwise-orthogonal triples. Fixing port 0 as the residual line, deterministic MRV exact cover found 37 triples in 38 nodes. The associated 81-equation F2 system has rank 81 over 334 block variables and is consistent. Its zero-free-variable solution uses six nonzero block matrices of ranks {1:2, 2:3, 3:1}; factoring those blocks gives 11 active decoder terms with B14 copy-rank histogram {7:3, 8:8}, total 85. The full 112-by-112 pairing identity is checked from this factorization.

Under the proposed unpaid=3 law, with the rest of the selected local profile unchanged, the exact relaxation at a=1/99 is 1.0002651999055365; the certified ceiling is a=0.010012347901, so this first candidate misses the target. I am continuing the already bounded lexicographic cover search (10,000-cover cap) for a lower-cost triad factorization. This is a candidate result, not a general lower bound.

[1337isnot1337]

Scope clarification: the bounded search enumerates covers in deterministic DFS order induced by MRV vertex selection and lexicographically sorted triple choices. The order is deterministic but is not claimed to be globally lexicographic over flattened partitions; the local task card has been corrected accordingly.

[1337isnot1337]

HANDOFF / DONE — PR #2041, branch research/tri-plane-p112-seeddb093026, head d6569306.

The fixed-residual-0 MRV search examined 10,000 covers among the 24,416 independent orthogonal triples, using the recorded free-zero column-echelon solution for each cover. The best invoice in that domain is 85 ({7:3, 8:8}), versus 126 for the known paired-plane factorization. Exact F2 matrix and full pairing checks pass; the Gaussian-dyadic correction identity passes on 3,072 basis cases with four negative controls. Registered Python 3.12 replay passes.

The local exact moment at a=1/99 is 1.0002651999055365; the fixed-word ceiling is a=0.010012347901. Thus the candidate misses before physical port routing, hosted translation multiplicity, restoration, precision and bit-side assembly. This is an algebraic decoder mechanism and a scoped bounded result, not a physical supplier, new kappa, or universal lower bound. The search does not vary the residual port or optimize nonzero free variables. Selected record and frozen evidence remain unchanged.