Decoder floors: every label window h ≥ 12 is closed in the root-code scope (clique bound + DRAT-verified h = 12–14; no κ claim)

[DreamingOfClouds]

Follows #2639 (windows h = 10, 11; parent #2338). Research note and verifier; no κ claim, no window, no supplier, no word-side cost. Package: research/label-windows-h12-plus/.

Result

In the root-code scope of #2639 §3, for any number of terms m ≥ 8 and any fitting B of F2-rank ≤ h, every label window h ≥ 12 is closed (no decoder pays), under the pricing model inherited from #2639 / #2338. Labels: PROVEN / CHECKED / RECORDED / CONDITIONAL as in the README.

• Lemma C (PROVEN). rank B[R,A] ≥ ω(R ∩ A), the clique number of the non-orthogonality graph on R ∩ A: a clique gives an identity submatrix of any fitting B (any h, any gauge, any factorisation).
• Census (CHECKED). min ω = 5 over all 47 × 440,880 = 20,721,360 section pairs, by two independently written exact C++ programs (+ networkx on the exceptional pairs, pynauty for the orbits): 3 pairs with ω = 5, 141 with ω = 6, all others ≥ 7; 6 exceptional orbits under Aut (order 172,032) and transposition. Hence every term has rank ≥ 5, cost ≥ 40, which exceeds the break-even for every h ≥ 15 (X(112,40,h)/base is strictly decreasing for h = 15..200, computed, not extrapolated).
• h = 12, 13, 14 (CHECKED, DRAT VERIFIED). 61 orbit queries UNSAT (kissat 4.0.4, drat-trim): terms ≥ 7 / 6 / 6, cost ≥ 56 > 51, 48 > 46, 48 > 42.
• The 696 T(12, l+k = 7) queries that Decoder floors, closures: punctured E8 units (Model C) to p = 46 (all m) and p ≤ 56 (merged m ≥ 8), windows h = 10–11 (no κ claim) #2639 lists as not run: 696/696 UNSAT, DRAT VERIFIED (a side result, implied by the above).
• Break-even h = 11..18: 55 / 51 / 46 / 42 / 38 / 33 / 29 / 25 is RECORDED from the inherited pricing model (reproduced by verify.sh, not derived or audited here); the closures are CONDITIONAL on it.
• Scope. Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212 Theorem 1 (merged) makes every realisation of rank ≤ 8 of the P112 pattern (rank exactly 8) a diagonal rescaling D₁ G D₂ of the Gram matrix with the Gram zero pattern, so the root-code scope covers non-Gram rank-8 realisations (the card H11 §3 gap). Not claimed: realisations of rank ≥ 9, and decoders whose factors lie outside col M.

Evidence and reproduction

The SAT encoder sat/hk3.py is shared with #2639 (two marked path/IO edits). CNFs are regenerated deterministically; their sha256 are recorded with the kissat times and drat-trim verdicts in evidence/drat_61.log and evidence/drat_696.log (DRAT files are not shipped: the largest is about 1.6 GB). bash research/label-windows-h12-plus/verify.sh /fresh/dir (inside the repository tree: it needs the sibling directories format-limits-corrected and complex-local-word-ceiling) runs the lemma check on all exceptional pairs, the full cen.cpp census and a partial mc.cpp census, the orbit checks, the break-even / X(40,h) computation, the log accounting, the 4 SAT controls' accounting, and 14 SAT queries (CNF hash + kissat + drat-trim); FULL=1 runs everything (all pairs, all 757 queries, about 1.5 hours, at most two processes).

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.

🤖 Generated with Claude Code