Decoder floors, case (b): the six remaining d = 8 multisets of #2816 are dead (≥ 64 with ≤ 3 non-full corrections; E8 lifts ≥ 67; no κ claim)

[DreamingOfClouds]

Follow-up to #2816 (ours), which left exactly six d = 8 weight multisets of the weight-≤ 7 basis open: [7^5,6^5,1], [7^6,6^4], [7^8,5,4,1], [7^8,5,5], [7^8,6,3,1], [7^8,6,4]. All six are impossible. Adds only research/one-sided-six-multisets/; nothing existing changes.

Results (setting, normalisation and n_nf as in #2816; P112, case (b), every m, char 0)

• Basis Lemma (PROVEN). At d = 8 the fundamental circuits of any λ-basis give 8 linearly independent codeword directions h_y with Off(h_y) ⊆ U_B ∪ Λ(F_y).
• Lemma S66 / S66+t (CHECKED, two independent implementations). For any two 6-dimensional label flats (plus one free line, chosen per direction, for S66+t) the qualifying directions span at most 3 dimensions (at most 6 directions). s66.c (C) and indep.py (Python, other coordinates and labels, direction list rebuilt from orthogonal root sums and checked against the theta counts) agree on all 159 orbit representatives of V_1 against all 776,805 line sets of V_2; indep.py runs in 6 chunks and check_logs.py asserts exact coverage. Control: two 7-dimensional flats reach rank 8 (152 pairs).
• Theorem 6 (PROVEN from the above). The six multisets cannot occur; 8 independent qualifying directions are needed and at most 3 exist.
• Consequences (inherit Decoder floors, case (b) for every number of terms: one-sided P112 decoders cost ≥ 64 without corrections (unique optimum) and with ≤ 2 non-full corrections (no κ claim) #2816's reduction at cost 66; second, independent route via a d = 0 Carrier Lemma for κ ≤ 2, proved in the README): Δ ≤ 2 ⇒ n_nf ≤ Δ (decoder level) and ρ_0 ≤ Δ (lift level).
  • One-sided P112 decoders with n_nf ≤ 3 non-full corrections (any number of full ones) cost ≥ 64; in general ≥ 67 − n_nf for n_nf ≥ 3, i.e. ≥ 64 − max(0, n_nf − 3) ≥ 59 (was ≥ 64 − max(0, n_nf − 2) ≥ 58).
  • Lift level: cost_w(S) ≥ 64 + min(|I|, 3) for every S ⊇ 𝒞_{X_I}; every lift of the E8 root code (E8/P120 decoder) costs ≥ 67, and ≥ 72 unless the P112 restriction has Δ ≥ 3; gain-law form |P120| − C(P120) ≤ 53.

Caveats

• No κ claim, no supplier word. One-sided (case (b)) decoders of P112 only; not two-sided, not case (c), not other units, characteristic 0.
• Δ ≥ n_nf for general n_nf stays OPEN; the root-swap family of Decoder floors, case (b) for every number of terms: one-sided P112 decoders cost ≥ 64 without corrections (unique optimum) and with ≤ 2 non-full corrections (no κ claim) #2816 is tight at n_nf ≤ 4. An extension to Δ = 3 is in progress and is neither claimed nor used.
• Inherited from Decoder floors, case (b) for every number of terms: one-sided P112 decoders cost ≥ 64 without corrections (unique optimum) and with ≤ 2 non-full corrections (no κ claim) #2816, cited and not recomputed: the DP/LP/Pencil/triple-rule survivor list (exactly these six at slack ≤ 2), the flat caps g(1..8), |Off| ≥ 53, the circuit lemma.
• verify.sh (fresh export, nice -n 10, at most 6 processes, each under timeout 1200, no wait loops): data regeneration, Parity-lemma search (rank-4 quadruples), s66.c (modes 0, t and the 7+7 control), indep.py in both modes, then one PASS/FAIL line per assertion. 1690 s wall (about 28 min under load), 14 PASS, 0 FAIL

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

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