Research: positive half-integral Gram decoder barrier and exact paired-plane C5 gate

[tekuuu]

A positive half-integral Gram decoder looks like a possible route to more unpaid coordinates. This proves that, after normalizing the scalar diagonal to one, every positive half-integral weighting with integral Gram factors costs at least the sum of the complete port-space dimensions. The written proof allows arbitrary term counts, supports and nondegenerate port dimensions. Under the retained favorable coverage law, this class cannot pay for a positive saving. Exact known-E8 controls show that quarter weights (cost 72 on 120 line ports) and signed half weights (cost 81) escape the hypotheses.

The attempted quarter-weight construction was also tested on all 23 induced C5s of the pinned PR1825 matching, extending PR2484's one-core motivation. In the explicitly specified 120-root E8 row menu, each minimum is exactly 20, the direct baseline. Python integer backtracking and a separately written C++ search agree; the latter covers 3,705,605 nodes. This is a scoped proof and completed finite negative result, not a new κ, a universal rational-decoder floor, or supplier admission. The initial 120-second signed-half MILP was inconclusive and is labelled separately.

Reproduce with Python 3.12, standard library only:

python3.12 -I -B research/c5-dyadic-decoder-gate/verify.py

Expected: PASS_C5_DYADIC_DECODER_GATE, 23 cycles, minimum 20, controls 72/81, no_new_kappa.

Validation passed: registered optimized replay; independent C++ replay; six behavioral/mathematical control tests; Git-free read-only optimized replay; metadata/source pins; whitespace checks. The add-only package was rebased onto canonical main 6ca6f217708b075814de409ad693fea9a2aaa760, preserving identical tested bytes. Scientific base is 31452ecd309f32a97e2e84214719b21297a59018, priority revision 5, OS seed24. Selected bindings and the baseline ceiling check passed; the full selected supplier replay is not claimed or needed for this additive note.

Credit is retained for Cinna Davis, Adam Gall, ywz65, Sagar Thakrar, Jeremy Diviney, Philippe Van Velzen, flyingviper and the selected E8/native lineage. This differs from the prior 2-integral theorem by allowing half-integral weights while requiring positive Gram factors; it differs from PR2757's triad-mask rational floors and PR2484's unpriced C5 minrank result. New proof/search work by Tekalign with substantial OpenAI Codex assistance. No human review or formal-kernel verification is claimed. Apache-2.0.

Canonical claim and handoff. Final overlap refresh covered 717 open PR bodies through #2779 and fresh Discussion1587 comments; broader discussion cache was older after a network timeout. No duplicate exact scope found; this is bounded coverage, not a novelty certificate. Native realization, arbitrary signed/asymmetric decoders, other matchings and a complete two-supplier improvement remain open. No selected-frontier or frozen-source change.