[RossRichardson]

Research note / scoped negative result; no new kappa or physical supplier.

PR231 leaves the geometry and pricing of its five-port XOR decoder open. This contribution checks the natural quadratic pairing G(S,T)=2(|S∩T|-1)(|S∩T|-3) and shows that it cannot reach the selected bound under explicit separate-star, drop-free-correction, compressed-window accounting.

• The old incidence-linear geometry is impossible: one cube forces 16 mutually orthogonal nonzero labels into a span of dimension at most six.
• The quadratic lift repairs the geometry, but its copy and active-frame costs alone force supplier saving < 125/76433 ≈ 0.00163541925608 for every p ≥ 5 and s ≥ 3. All other expenses are optimistically granted zero. This lies below the base's selected conditional κ = 0.001661957774647029.
• The proof uses 17 explicit integer minors, including conservative copy-rank witnesses at the degenerate p=11/12 cases. An embedding of the p=13 witness covers every larger p; the stage count is eliminated algebraically. This is not an extrapolation from a failed finite search.

Only research/five-port-quadratic-decoder/ is added. Start with its README and PROOF.md. The scope excludes other Gram pairings, coded totals, punctures, paid non-drop-free gathers/corrections, and different stage accounting. The selected certificate and upstream/ are unchanged. All-size/compiler, common-label/mutable-source, hosting, precision/tape, bit H1–H5 and analytic obligations remain inherited.

Reproduce and validation

Offline, Python 3.10+ and the standard library:

python3 -I -B research/five-port-quadratic-decoder/verify.py
python3 -I -B research/five-port-quadratic-decoder/independent.py
python3 -I -B research/five-port-quadratic-decoder/test_package.py

Expected primary marker: PASS_FIVE_PORT_QUADRATIC_DECODER witnesses=17 ceiling=125/76433 tail=250/170739 controls=9 no_new_kappa.

• Primary checker passes on CPython 3.12.12, including isolated optimized execution: all 17 minors, 37,888 source/target coefficient checks, rational pricing, source pins and nine adverse controls.
• Separately written set-feature/Bareiss implementation passes on CPython 3.14.6 at a second prime, with two exact integer determinants. This is implementation independence by the same AI-assisted contributor, not independent human review.
• All six package tests pass, including 123 determinant comparisons against the permutation definition, singular valid-label minors, source corruption, and omitted obligations.
• Canonical staged Git-blob export to a path containing spaces passes both checkers and regenerates byte-identical receipts. Existing output preservation, make selected-record-check, and whitespace checks pass.
• The final commit differs from the tested export only in the claim completion, validation note and execution receipts. Full unchanged repository/supplier suites and hosted CI are not claimed.

Provenance and coordination

Base 47481c8; seed 40, structural decoder track. Main refreshed to 007664306fff84fbb1c9a1b4053b69f439764034 with unchanged selected PR890 κ. No pinned coordination issue exists; the completed local claim links the result. A full overlap refresh covered 707 open PRs through #954, followed by a submission check through #955; neither adds this quadratic geometry/cost result.

Credit Joel Pulikkan / GamingPuzzled (PR231), icekylinx and eumemic for the paired-cube/star-decoder lineage, and DreamingOfClouds (PR446 and format-limits) for the frame, pairing-rank and explicit accounting premises. PR446 is pinned to 309482849b6ddf34db998f53612d34c19f6e3233; four retained sources have exact SHA256/Git-blob/base pins. PR669's proper-star puncture results and PR868's Gaussian clique ceiling are distinct adjacent work and are credited. No predecessor implementation code is imported.

Prepared by Ross Richardson (@RossRichardson) with substantial OpenAI Codex assistance in mathematics, implementation, verification and writing. Apache-2.0. The written proof needs external review; no formal multiplication theorem or new kappa is claimed.