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Ceilings for linear rank-counting decoder certificates (360/17 on P120) and a one-sided lift bound (cost >= 72 for dim S >= 23) - #2962
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Mister-Meeseeks merged 1 commit intoOct 16, 2026
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…-sided lift bound Theorem C: the orthogonal-set (Hall-charging) certificate family F gives at most 360/17 on P120, 15968/757 on P112, 945/71 on E7, 12 on E6 (explicit fractional witness, exact enumeration). Corrected scope: the larger family F' has proven ceilings 24 (P120) and 23.94 (P112); E7/E6 values are MILP-computed. Lemma S: cost_w(S) >= 72 for every S containing C(E8) with dim S >= 23, tight at e = 15. No kappa claim. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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research/decoder-certificate-ceilings/. Pure finite combinatorics on the E8 root lines and the units P112, E7, E6. No kappa claim.Theorem C (ceiling). For the family F of dual-feasible rectangle (Hall-charging / orthogonal-set) certificates,
rank N0[C,C'] <= #terms meeting (C,C'), every certificate gives at most 360/17 = 21.18 on P120 (target 72), 15968/757 = 21.09 on P112 (64), 945/71 = 13.31 on E7 (49), 12 on E6 (36). Proof: an explicit fractional "fake decoder" (weight 8/765 on each of the 2025 frames, weight 8 on every cost-0 single-entry term) is feasible for every rectangle, plus weak duality; the meet counts come from enumerating all 352,875 orthogonal sets, with exact fractions.Scope correction. An independent audit showed that F, as defined, does not contain #1977 Lemma 2 / #798 as originally stated (it forces zero weight on rectangles with a non-orthogonal pair). The larger family F' (zero-cost terms dropped, set value at most min(T, |D|)) has proven ceilings P120 <= 24 and P112 <= 23.94 (Bonferroni bound on the meet counts); E7 <= 13.90 and E6 <= 13.06 are MILP-computed (HiGHS), not proven by hand. Whether 360/17 also holds for F' is open. All values are far below the targets. Mixed char-0 / F2 certificates and nonlinear counting are not covered.
Lemma S (one-sided). For every subspace S containing the 8-dimensional root span with dim S >= 23, cost_w(S) >= 72, tight in the counting relaxation at e = dim S - 8 = 15. Proof: Rado-type independence count, root-subsystem sizes in subspaces, flat sizes over all 417,199 subspaces of F2^8, and an exact dynamic programme for e in [15, 112]. It says nothing for e <= 14 or about two-sided decoders.
Credit. The certificate shape (clique / orthogonal-set rank counting) is that of #1977 (Lemma 2) and #798; #968 for the P104 setting (not computed here). The family definitions here are our formalisation.
Verification.
bash verify.sh(about 9 minutes, numpy and scipy): manifest, exact recomputation of all four Theorem C ceilings, the F' ceilings, and Lemma S; prints PASS/FAIL, all PASS on the committed state.MANIFEST.sha256lists every file.Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.
🤖 Generated with Claude Code