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)

[DreamingOfClouds]

Status: research note with exact finite computations and written proofs (not a SAT/DRAT proof). No κ claim, no supplier word. Only research/one-sided-decoders-all-m/ is added; nothing existing is changed.

What this proves (scope is the claim)

Model of #2338 / #2277 / #2724 / #1224: exact masked decoders N = Σ_{j≤m} p_j q_jᵀ of the Gram realisation of the P112 pattern, field of characteristic 0, dense pairing table B, cost Σ_j rank_F2 B[R_j, A_j]. Case (b): one factor outside col M (the other in col M), every number of terms m. Reduced to the (y, U) form; the decoder is a lift S ⊇ 𝒞 of the root code, cost ≥ cost_w(S) (min-weight basis).

• R1 (CHECKED + PROVEN). ε-free one-sided P112 decoders cost ≥ 64 for every d = dim(𝒞 ∩ S_{≤7}): exact DP for d ≤ 6, an LP with exact integer dual certificates for d = 7, 8 (lp3.py), re-derived by an independent symmetrised LP with exact rational duals (symlp.py); the multiset enumeration is complete (enum.py: none for d ≤ 6, one at d = 7, seven at d = 8, all killed).
• R2 (Theorem U′). The cost-64 ε-free decoder is unique: the coordinate decoder (8 terms (e_j, W_j)). Cost-65 decoders have at most one non-full correction; the cost-66 case is reduced to six OPEN d = 8 weight multisets.
• R3 (corrections). Lemma P: a full corrected term (U = F2⁹) is absorbed at no cost. Corrected cost ≥ 64 whenever at most 2 corrected terms of the normalised decoder are non-full (normalise = merge proportional corrected terms and absorb full ones, repeated until stable; terminates since (#terms, #full corrected) drops lexicographically; the version for raw decoders is the lift-level form R4 with I = all corrected terms); in general ≥ 64 − max(0, n_nf − 2) ≥ 58. The root-swap family attains 64 exactly with n_nf = 1, 2, 3, 4 (exact P·Y = C), so Δ ≥ n_nf is tight.
• R4 (lift level). Every S ⊇ 𝒞_{X_I} has cost_w(S) ≥ 64 + min(|I|, 2); every lift of the E8 root code has cost_w ≥ 66.
• R5. Lemma N (≤ 2 directions per 7-flat after removing ≤ 4 lines; exact counts for |L| = 1, 2, an upper bound for |L| ≤ 4, and an independent C implementation) and the Pencil Lemma (d ≤ 2).
• Gain-law reading (philippeVV, Discussion CLAIM: where does the one unpaid coordinate per live family come from, and is it forced? (ctoth, expires 07:30Z) #1587 follow-up 6): in this class the gain on P112 is at most 48 (112 − 64).

Open residue (stated exactly, not claimed)

Δ ≥ n_nf for n_nf ≥ 3: at n_nf = 3 exactly six d = 8 multisets of the weight-≤7 basis remain ([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]). Not claimed here.

Not covered (stated up front)

• Case (c) (rank-9 realisations, see Decoder floors, case (c): the A2 family of non-B-form rank-9 realisations (P112 ≥ 64, #968 P104 ≥ 56, E8 ≥ 72; no κ claim) #2784), two-sided decoders, P104 and other units, B of rank ≥ 10, positive characteristic.
• Imported (merged results, not re-derived): Gram realisation (Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212 Theorem 1, plus a diagonal normalisation); dense table (Decoder floors are tight in the frontier format: one F2^9 label table, factors in col M of a rank-8 realisation (P112 64, E8 72, P104 (#968's unit) 56; two-call batches 128) — no κ claim #2338 Theorem R, DRAT-checked there). The parabolic-flat fact is proved in the README (C3).
• Not DRAT / Lean: written proofs plus exact Python/C computations, re-runnable.

Labels and verification

PROVEN / CHECKED / RECORDED / IMPORTED / OPEN per claim in the README. bash verify.sh (about 12–15 minutes, two processes, no SAT) re-runs every exact check (DP, enumeration, LP certificates, Lemma N in two implementations, the root-swap family) and asserts them in checks/check_logs.py; fresh git archive export: 34 PASS / 0 FAIL in 714 s (logs in evidence/logs).

Overlap

Scanned the 1,000 most recent PRs (titles), 119 decoder-related bodies and 100 recent Discussions: no ε-free one-sided floor for all m, LP/charging certificates, uniqueness statement, root-swap family or Lemma N. Nearest: #1224 (model), #1180 (both factors in the root code), #1633, #1815, #1609, and our #2338 / #2277 / #2724 (this is the every-m extension of #2724's case (b) at m = 9, by a different method).

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