Decoder floors, closures: punctured E8 units (Model C) for 48 ≤ p ≤ 56, all m (no κ claim)

[DreamingOfClouds]

Closures-only follow-up of #2639 (open; this PR is based on it and uses its tools and Theorem M8 as stated there). No κ claim.

#2639 §4 lists m ≤ 7 at 48 ≤ p ≤ 56 as open. This note closes it, in the same setting (punctured E8 units, Model C: dense B, column-space factors), so every p with 48 ≤ p ≤ 56 is closed for every m (m ≥ 8: #2639 Theorem M8 with its Case-B census; m ≤ 7: this PR).

• Reduction (PROVEN from 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's E7q, Lemma G/H/R, monotonicity in P). 48 ≤ p ≤ 53: a root with q ≤ Q(p) is closed by E7q, otherwise all q_γ ≥ 6; 54 ≤ p ≤ 56: q ≤ 16 is closed by Lemma R (m = 7: cost ≥ 35 ≥ need(p)), otherwise all q_γ ≥ 17. m ≤ 7 then needs a Lemma-G failure, i.e. SAT of one CNF per rank-7 flat type: (I-a) P = 53, all q_γ ≥ 6 (covers 48–53); (I-b) P = 56, all q_γ ≥ 17 (covers 54–56).
• Result (RECORDED from the author's working card; sample re-checked). All 8 flat types are UNSAT-VERIFIED for both families: every leaf of the orbit trees is kissat + drat-trim VERIFIED against its own CNF, or a counting leaf (|T| > P); no SAT, no TIMEOUT. Per-leaf tables with the md5 of every leaf CNF, state files and run summaries are shipped.
• verify.sh (≈ 15 min, one core, nice 10): manifest/pins; tallies and arithmetic; rebuilds the option sets for 4 of the 8 types from scratch (E8 lines, 440,880 rank-7 flats); encoding checks (leaf CNF vs. direct numpy test, 880 sets, kissat); tree coverage (c3: each orbit tree re-walked with an oracle leaf, every recorded leaf visited, none missing; CHECKED); then regenerates 60 leaf CNFs per (type, family), compares md5 with the recorded runs and sha256 with pins, and re-solves 15 per (type, family) with kissat + drat-trim. FULL=1 does every leaf and the coverage of all 8 types (hours).
• Proofs shipped: 12 small leaf DRAT proofs (xz) with sha256 of proof and CNF, drat-trimmed by verify.sh against regenerated CNFs; the other leaf proofs are RECORDED (md5-pinned CNFs, regenerable).
• Imported from 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 (open), labels as there: E7q PROVEN from CHECKED data; Lemma H PROVEN; Lemma R PROVEN from card-verified data; Theorem M8 PROVEN from RECORDED data; Case-B census RECORDED. They are not re-derived here; Q(p) and need(p) are recomputed by c1.
• Not claimed / not re-run: p = 47 with m ≤ 7 stays open. The other ≈10⁴ leaf proofs are not shipped (tens of GB); their verdicts are RECORDED, regenerable on demand.
• Overlap: a search of PR bodies found no other closure of these punctures; Scoped negative result: E8 star-copy ratio cannot improve through 55 punctures (no new kappa) #939 / Research: all-puncture E8 star decoder exclusion and p=56 classification (seed 49, no new kappa) #1171 treat star label decoders, a different class.

Package: research/punctured-e8-48-56/ (README with PROVEN / CHECKED / RECORDED labels, verify.sh, checks/, sat/, data/, evidence/, MANIFEST.sha256, SOURCES.json).

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

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