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Rank-8 rigidity of the E8 pattern with arbitrary extra zeros after deleting any at most 8 lines (hypothesis (1) of Lemma RIGD; no kappa claim) - #3040
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…rches; R1 hypothesis (1) in that range; no kappa claim) Ten DRAT-verified UNSAT zero-pattern cores, their W(E6)-images, and two independent exhaustive hitting searches (cube.cpp and s.c) show no D with |D| <= 8 avoids every core; two independent frame searches show no |D| <= 8 kills all 2025 H-frames. verify.sh reruns everything (FULL=1 for the complete s.c HIT run). Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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Scope: pure matrix classification on the E8 unit; no kappa claim, no decoder cost. Extends #2864 Theorem RK (deleted set a clique) to every deleted set D with |D| <= 8: a rank <= 8 matrix on E8 \ D with the forced orthogonality zeros, nonzero diagonal and all other entries free has no extra zero and equals D1 G D2 with D1, D2 invertible. This is hypothesis (1) of Lemma RIGD for |D| <= 8.
Method: restriction to ten local cores (27 to 38 lines) whose zero-pattern models are UNSAT (kissat, drat-trim VERIFIED); W(E8) is transitive on ordered non-orthogonal pairs (6720) so the pair is fixed and W(E6) (order 51840) acts on the cores; an exhaustive search shows no D with |D| <= 8 meets every image. A second exhaustive search shows no D with |D| <= 8 spoils all 2025 frames (needed for the iff-pattern-implies-Gram step). Both searches exist in two independent implementations, with controls.
Verification:
bash research/e8-rank8-rigidity-small-deletions/verify.sh(default about 15 CPU-minutes; FULL=1 adds the complete independent searches in chunks under 20 minutes).PROOF_MAP.mdmaps every claim to its check.Not claimed: R1 for an arbitrary number of extra zeros stays open (the P/A/Sigma/S steps without the per-line budget of #2864). The zero-pattern model's soundness is inherited from #2864 and not re-proved here.
Credit: philippeVV (R1/R2/R3, Discussion #1587); #1212 (rank-8 rigidity, iff pattern); #2864 and #2819 (model and frame calculus).
Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.
🤖 Generated with Claude Code