Research: exact tri-plane factorization for selected P112 (no new kappa)

[1337isnot1337]

This adds an exact algebraic decoder factorization for the selected 112 PR1094 complex labels using 37 mutually orthogonal 3-planes and one residual line. The stored F2 block solution reconstructs the full 112-by-112 pairing matrix and has B14 copy-rank profile {7:3, 8:8}, total 85, compared with 126 for the matched 2-plane factorization.

The proposed three-unpaid-coordinate local screen still misses a = 1/99: the exact relaxation at the target is 1.0002651999055365, with a fixed-word ceiling a = 0.010012347901. This is before any physical tri-plane route, endpoint-translation multiplicity, hosting, frame/restoration or precision costs. No supplier, new kappa or selected-frontier change is claimed.

Evidence

• 24,416 pairwise-orthogonal independent triples in the pinned selected labels.
• Deterministic MRV exact cover with residual port 0; 10,000 covers checked. The F2 system has rank 81 over 334 variables. Its best result under the fixed zero-free-variable solver has block-rank histogram {0:32, 1:2, 2:3, 3:1} and 11 active terms.
• Exact 9-by-9 identity and full 112-by-112 decoder checks; 3,072 Gaussian-dyadic correction-operator basis cases; four negative controls.
• The 10,000-cover domain does not vary residual ports or nonzero free-variable solutions and is not a universal lower bound.

Reproduce

python3 -B research/tri-plane-port-p112-seeddb093026/verify.py
python3 -B research/tri-plane-port-p112-seeddb093026/search.py --cover-limit 10000
python3 scripts/research_package.py --package research/tri-plane-port-p112-seeddb093026 --check-only
python3.12 scripts/research_package.py --package research/tri-plane-port-p112-seeddb093026

The registered package replay validates the saved exact certificate and controls; it does not repeat the 10,000-cover search.

Provenance

Research base: ec94fd3d2fcad7ade8e72d6bcd8e223b7fdb3f30. Selected-record SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32. This refines the recursive-interfaces / coverage-law-change assignment. It extends the 2-plane endpoint from Discussion #1587 and the matched P112 factorization in merged PR #1825 (head d16fb8936795ff101e4d2f63cfdc6a25066ebabf); PR #1900 is the distinct P111 bit-side plane result. Prepared with substantial OpenAI Codex assistance; no independent human review is claimed.

[thakrarsagar]

Scalar-endpoint audit in #2532 (Sagar Thakrar, with AI assistance), also covering #2297. The tri-plane F2 identity reproduces B0 = q_i·q_j, but that is not the masked scalar decoder the E → FULL / 0 → E^⊥ contract needs. #1889 found the same issue for #1825's paired planes.

• Free entries. Only 102 ordered pairs are free: 72 between four orthogonal three-frame pairs and 30 between the residual line and five three-frames.
• Literal lift. It has 11,755 forbidden nonzero entries, 6,608 of them parity-forced. All 112 dirty columns fail; target 0 ← source 7 has coefficient 3.
• Rank. Any valid decoder needs rank ≥ 99, from an exact clique of 33 non-orthogonal three-frames, against 11 terms.
• Cost. philippeVV's 2-integral floor gives a charge of at least Σ dim E_i = 334, above 3w = 298.7 in port-space pricing. So sharing the free pairs cannot save without even-denominator gain above about 35.

This concerns the validity of the charge-85 decoder; your README already reports that the screen misses a=1/99.