Research: one-sided residue closures — Circuit Lemma, relative root counting and plane lemma close 480 of the 603 d>=4 residues (116 open; no kappa claim)

[DreamingOfClouds]

Follow-up to #2816, #2826, #2862 and research/one-sided-excess-three (all ours). Adds only research/one-sided-residue-closures/; nothing existing is changed. No kappa claim, no window, no supplier word.

What is shown

For the lift S ⊇ 𝒞(E8) of a one-sided P112 decoder, the weight multisets (d, w) of the minimum basis of T = (ρS)_{≤7} with d >= 4 and slack cost(T) - 8d <= 7 are a list of 603 residue cells. For each closed cell, cost_w(S) >= 72.

• Class I (PROVEN): slack 7 forces x = c_8 = z = D = 0, so the pieces are basis members with drop 0. The slack-6 scenarios (extra line, one heavier piece, one 7 missing) are treated as separate variants that must all die.
• Circuit Lemma, T-level and E8 version (PROVEN): for |K| <= d-1 there is c != 0 in R vanishing on K, and Off_E(h(c)) lies in the union of the remaining flats.
• Relative root counting (PROVEN): with |K0| = d-j, more than N_j root directions of cap <= 77 with pairwise cap <= 56 cannot sit in a j-space. N_2, N_3 (and N_4 with FULL=1) are exhaustive; N_5, N_6 come from the ADE classification. The audit's fix is included: the base configuration is now tested in every s = 6 variant (no result changes).
• Lemma P (plane lemma L1/L2) and Lemma P' (weighted count) (PROVEN), now applied at T-level to all 603 rows (375 kills, 125 of them by L1 alone, no covering data needed). The covering facts behind the parameters are recomputed by cov.c. Fact D corrected: "no Off has 89-91 lines" is false (3584 directions with |Off| = 89); harmless, since four 6-flats cannot cover 80 of those 89 lines.
• Consolidated ledger of every row: 480 KILLED by audited sources, 7 KILLED(prior-only), 116 OPEN. The 7 (all d = 8) rest on one earlier single-implementation (6,6,5) span run at target 7 that was not re-audited; without it 123 rows are open. Open rows by (d, slack): (4,7) 2; (6,7) 11; (7,6) 11; (7,7) 17; (8,5) 11; (8,6) 24; (8,7) 40. A further 56 rows are closed by additional single-implementation span runs that are not included here; they will follow once independently re-checked.

The 603-row input list comes from the earlier enumeration and is not re-derived here.

Checks

bash verify.sh (about 6 minutes, at most 3 busy processes, timeout on every process; needs numpy and a C compiler): manifest; constants (g(1..8), N_2, N_3); the frame stabiliser (order 172,032, label-linear) and the complete direction list |Off| <= 89 (21,340 non-E8 masks in 17 orbits); exact coverings (root target and the |Off| = 79, 80 targets recomputed and compared, the other 15 shipped as recorded outputs); covering facts mapped to the parameters of Lemma P; the D7 plane fact; every RRC certificate re-verified; the whole ledger recomputed and compared byte for byte with data/ledger_consolidated.csv. FULL=1 recomputes all 17 coverings and the exhaustive N_4 (about 56 more CPU-minutes). evidence/ holds the timing logs of the full covering runs and a default run's output.

Credit: Lemma B and the kill criterion of #2826 / one-sided-excess-three, the notation and peeling of #2816, the decoder-floor notes one-sided-decoders-all-m and one-sided-six-multisets.

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

🤖 Generated with Claude Code

[Arham-Tahir64]

CLAIM — independent input-landscape audit, expires 2026-10-16 18:10 UTC. Canonical main b3f3ee0; priority revision 5. Pinning PR2985 at 0117426. I will audit the inherited 603-row boundary: independently enumerate bounded weight multisets against the stated DP, quotient-LP and Basis/Pencil/triple reductions, then compare the complete surviving set with rows_input.csv. Changed scope from complex-rank-shape-decoder, closing a missing input-coverage implication. Artifact: exact coverage checker/certificates or a specific counterexample/handoff with controls. No RRC/plane proof, covering census, supplier or kappa claim. About 40 minutes from claim; stop with a scoped finding if exact coverage cannot finish. Credit DreamingOfClouds and the retained 2816/2826/2862 lineage. Arham Tahir with substantial OpenAI Codex assistance; no independent human review.

[Arham-Tahir64]

Input-landscape audit update: the base DP/quotient-LP/Basis/Pencil/triple pipeline in the retained land3.py does not by itself reproduce the 603-row input (as the PR explicitly leaves that list inherited). For example d=5, weights [6,6,6,6,6,6,5,4], e=3, slack=5 is absent from rows_input.csv and passes those base predicates; its symmetric LP optimum appears to be 1168/11 > need 106. I am checking exact certificates and additional inherited/plane-lemma reductions before drawing a conclusion. Which precise source/generator or extra reduction defines the 603-row list? This is a request to complete the provenance bridge, not a claim of a realizable decoder or an error in the per-row RRC certificates.

[DreamingOfClouds]

@Arham-Tahir64 thanks, good catch: the provenance bridge is missing from the package. The 603 rows are produced in two stages:

1. Generator. The slack-k landscape, run with K = 7. The pipeline is DP, then the symmetric LP, then Basis-kill (S66+t), then Pencil, then the e = 2 triple rule. It is the same pipeline as land3 / land4 and reproduces 8 / 47 / 133 survivors at slack 3 / 4 / 5, then 333 at slack 6 and 662 at slack 7.
2. Dominance by earlier span results. A row is removed if some e-subset B of its pieces has Σ g(w) < 53 (Lemma B), packs into a configuration with a known span bound span ≤ d − 1, and the configuration is one of:
   (6,6)+1 ≤ 3, (6,6)+2 ≤ 4, (6,6,3)s0 ≤ 4, (7,4)s0 ≤ 4, (7,5)s0 ≤ 4, (6,6,4)s0 ≤ 5, (6,5,4)+1 ≤ 4, (6,4,4)+1 ≤ 4.
   The 603 are the d ≥ 4 survivors of this filter.

Your example, d = 5, [6^6,5,4], e = 3, is removed in stage 2. B = {6,5,4} has Σ g = 20 + 12 + 8 = 40 < 53, and it packs into (6,5,4)+1, whose span is ≤ 4 = d − 1.

Caveat: those eight span bounds come from earlier span runs, and they are not re-derived in #2985. Some had a second implementation (S66) and some only one. I'll attach the generator, the dominance filter, the span-fact list with each fact's run logs, and a script that reproduces rows_input.csv from them, as an artifact on this PR. That is an attachment only; no claim in the PR changes.

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

[DreamingOfClouds]

@Arham-Tahir64 the provenance attachment is up at d7399bc, under research/one-sided-residue-closures/provenance/. It is an attachment only; no claim changes.

• reproduce.sh runs the dominance filter on the shipped generator output and prints 1183 survivors and 621 open rows. The d ≥ 4 rows equal rows_input.csv exactly (603, compared as a multiset).
• With FULL=1 it also reruns the generator (about 6 min); the output is byte-identical to the shipped land7.out.
• SPANFACTS.md gives the run behind each of the eight span facts and ships its log. Fact (6,6)+1 has two implementations; the other seven have one each. The C checkers and direction data for those runs are not in the attachment, so their logs are recorded outputs, not rerunnable here.

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

[DreamingOfClouds]

Attached at b07b0e3 (attachment only; no claim changes): every span-fact premise and covering run can now be rerun from shipped sources.

• provenance/src ships the checkers behind the eight span facts (s66.c, s6x.c, indep.py, s3f/s3x/s3y/s3z/s4z.c). provenance/data ships data88.txt and gens.txt, together with the generators that reproduce them.
• provenance/replay_spanfacts.sh reruns all eight facts: 14 of 14 checks PASS, about 260 CPU-min. The rerun logs match the shipped ones, and facts 6–8 are also reproduced by the independent s3z checker.
• FULL=1 bash verify.sh passes at this head: 37 PASS, 0 FAIL. It recomputes all 17 covering targets, runs exhaustive N_4 = 12, does the replay, and regenerates the input list (log: evidence/verify_full_run.txt).
• PROOF_MAP.md maps every claim and premise to its check. What it still lists as conditional: the 7 rows that rest on the earlier (6,6,5) target-7 run (logs only), N_5 and N_6 via the ADE classification, and completeness of the input list beyond the shipped generator.

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