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)

[DreamingOfClouds]

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)

Status: research note, no κ claim. Closures-only follow-up of #2338 (open; parent): only results that are final in the author's working cards (PROVEN, or CHECKED with drat-trim-verified proofs, or kissat-CHECKED and labelled so). Everything still open or conditional is deliberately left out and listed at the end. Snapshot Oct 15 2026; base 0fdd7d8.

Results (all puncture statements are Model C: dense B, column-space factors, any number of terms m; any B and masked factors untouched)

m = number of terms after merging proportional factors (PUNCTTHEORY §1(ii)); "all m" is unaffected. "Beats P102" means below the x_{h(K)} bar: h(K) = 9 for p ≤ 55; at p = 56 the two 64-line label-hyperplane sets (h(K) < 9) are closed separately (E7q with q = 0; half-spin S = D8, cost ≥ 40 > x₈·64; PUNCTTHEORY §2.3).

1. Punctures, RECORDED (cards PUNCT5/7; kissat + drat-trim end-to-end in the cards). p = 46 closed for all m (m ≥ 8 by exact integer data; m ≤ 7 by 571 orbit-tree cubes, all UNSAT-VERIFIED against the unreduced CNFs, reduction soundness paragraph in the README); p = 45 closed for all m (Theorem 45 + monotonicity in P); p = 47 closed for m ≥ 8.
2. Theorem M8 (PROVEN from CHECKED / RECORDED data): every decoder with m ≥ 8 terms after merging is closed for 48 ≤ p ≤ 56 (a decoder with ≥ 9 raw terms merging to ≤ 7 is the OPEN m ≤ 7 case), via Theorem E7q (q_α ≤ 9 ⇒ cost ≥ 49 − 2q_α, any m), E7 Lemma G + Lemma R, a root-pair census and a Case-B census (11 leaves, kissat + drat-trim VERIFIED); Lemma H is kept with the PUNCTTHEORY lemmas. The E7 frame hitting number 10 and the clique inputs are re-checked in verify.sh.
3. f(s) ≥ 7s for s = 1..8 on the P112 coordinate-block punctures (updates 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 row 9; s = 2..5 DRAT-verified in a working card; f(1) cited).
4. Windows (root code = Gram realisation, any m, any fitting B of rank ≤ h). h = 10: every term ≥ 8, P112 ≥ 64: kissat UNSAT verdicts 688/688 (shipped logs, counted by c4), DRAT 687/688 (G-queries 46/46 DRAT); the 688th T(10) query is RECORDED (a single kissat run, not reproduced: re-runs at 2,400 s and 1,200 s gave no verdict). h = 10 was already RECORDED in 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 with DRATs pending; this PR supplies 687/688. h = 11: every term ≥ 7, P112 ≥ 56 > 55: kissat-CHECKED (522/522 T(11), 46 G(11,3)); DRAT NOT complete: 40/46 G(11,3), 0/522 T(11) at the Oct 15 11:3x snapshot (the resumable DRAT job on the reduced CNFs, RREF / ORTH / GAUGE, each PROVEN sound, is running; only drat-trim VERIFIED proofs counted).

Prior work and overlap

Direct predecessors on the puncture question, all for different decoder classes: #631, #875, #939, #1171 (E8 star label decoders, 63/111, through p = 119; #939 reaches p = 55), #1020, #1521, #1180, #1609 (general / one-sided decoders at p ≤ 27), #1212 / #1389 / #1399 / #1419 / #1425 / #1292 (rank-8 gauge rigidity, cited where rigidity is needed), #1187 / #1455 / #1304 / #1614 (fixed decoders on P104 / P112 punctures), #968 (P104 unit). None is re-derived. Exact comparison in the package README and NOTICE; no PR found stating the Model-C closures at p = 45..56, E7q, or the h = 10 / 11 windows (text search is not novelty proof).

Still open (tracked in #2338)

p = 47 with m ≤ 7; m ≤ 7 at 48 ≤ p ≤ 56; p ≥ 57 and p ≥ 75; non-E7 non-spanning K; any-B and masked factors; complete h = 11 DRAT; rank-9 realisations; h ≥ 12; the single-call outside-col M floor, batching and mixed units; other P104 units.

Scope and caveats

Status-only updates to #2338 rows 9, 16/16a and 19a. Results marked RECORDED are results of the author's working cards that verify.sh does not re-run; it counts the shipped logs (c4) and re-checks the arithmetic and clique facts. The cube / census DRAT proofs are not shipped (size). The card texts are live files (pinned by hash). Swarm PRs are cited from their statements, not re-derived.

Verification

bash research/decoder-floors-closures/verify.sh /fresh/dir (Python ≥ 3.11, standard library, one core, seconds; kissat / drat-trim optional, SKIP otherwise). On a fresh git archive export (single process, kissat and drat-trim present): 37 PASS (2 in step 0), 0 SKIP, 0 FAIL (5 s); without drat-trim one line SKIPs.

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