Decoder floors, case (b): nine terms outside col M (P112 ≥ 64, #968 P104 ≥ 56; no κ claim)

[DreamingOfClouds]

Status: research note with an exact finite enumeration (not a SAT/DRAT proof). No κ claim, no supplier word. Only research/nine-term-outside-colspace/ is added; nothing existing is changed.

What this proves (scope is the claim)

Model of #2338 / #2277 / #1224: N = Σ_{j≤m} p_j q_jᵀ exact for the masked P112 / P104 pattern (zero on orthogonal pairs, nonzero diagonal), arbitrary factors over a field of characteristic 0, cost Σ_j rank_F2 B[R_j, A_j].

• P112: cost ≥ 64 and P104 (Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's unit, F1 = H4, only): cost ≥ 56 for every decoder with m = 9, a rank-8 realisation (rank N = 8), and exactly one factor outside col M ("case (b)" of 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's m = 9 trichotomy; the one-sided nine-term decoders of Research: nine-term complex E8 decoders are one-sided overcomplete representations (seed 42, no new kappa) #1224), dense B, hence every fitting B of F2-rank ≤ 9 (Theorem R of 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).
• The result relies on 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 (our own PR, open and ready): Theorem P, Theorem R, and "≤ 8 paid terms ⇒ ≥ 64 / ≥ 56".
• Realisation-free: only the support-propagation lemma (valid in every rank-8 realisation) and the reductions are used; the rigidity PRs (Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212, Derive one-sided P112 fitting rigidity from support spectra #1815) are not needed.
• Method: the lift κ of the outside factor is free (Lemma 2.1, PROVEN), which reduces the cost count to exact class-level accounting Σσ_j ≥ 9 over the 511 label classes. Exact enumeration: no big term (counting); ≥ 2 big terms (41 configurations → 0 on P112, 701 → 0 on P104); one big term with row deficiency d₀ = 2, …, 8 (residues 8, 1, 17, 0, 0, 0, 0, all killed by the support filter).

Not covered (stated up front)

• Case (c): square decoders of a RANK-9 realisation. This contains every nine-term masked co-star decoder (D1267, philippeVV; 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 shows these are case (c)) and the A2 family. Nothing here bears on the co-star programme.
• m ≥ 10 (one- and two-sided); B of rank ≥ 10; other punctured units.
• Not DRAT-logged: the enumeration is exact Python on finite data, re-runnable, but not proof-checked by a solver.

Labels (every step is in the README)

PROVEN: Lemma 2.1 (free lift), the accounting Σσ_j ≥ 9, the "no big term" case, the d₀ = 7, 8 reduction. CHECKED (exact enumeration): the ≥ 2-big census, d₀ = 2, 3, 4, 7, 8, all of P104. CHECKED-INDEPENDENTLY (separate re-implementation by a different session, exact agreement): d₀ = 5, 6 only (evidence/helper/). Not independently re-implemented: d₀ = 2, 3, 4, 7, 8, the ≥ 2-big census, P104. A separate reviewer run found one incompleteness (repeated extra classes were forbidden), fixed in cnt7/8/9, everything re-run, residues unchanged. The exact-rank formula (IDEAL Thm 3 / BIGTHEORY Thm A) is proved in the README (3 lines) and spot-checked on 2,974 + 2,973 random blocks, 0 mismatches; the reductions (free-term absorption, merge lemma) are inlined; the geometric inputs (frames, survivors, survivor-pair ranks, 7-cliques) are re-computed by checks/inputs_check.py.

Verification

bash research/nine-term-outside-colspace/verify.sh /fresh/dir (Python ≥ 3.11, numpy, networkx; one core; no SAT). On a fresh git archive export (default mode, single process, venv Python 3.14, numpy 2.5, networkx 3.7, on a heavily loaded machine): 92 PASS, 0 FAIL, 839 s wall (12.0 min user CPU).

What verify.sh re-runs: the geometric inputs on both units; the exact-rank formula spot check; regeneration of orbits / cliques / pair census; exact line-for-line replay (vs. the recorded logs) of cnt4 (d₀ = 2, 3), cnt7/cnt8 for d₀ = 2, 3, 7, 8; subset replays of the census (P112 orbits 8, 13; P104 orbits 9, 30, 56); cnt9 / cnt9b from the recorded compact census output; the two controls (ctl8, ctlb).
RECORDED, not re-run by default: the full census (P112 44 min, P104 32 min), d₀ = 4, 5, 6 (cnt6/7/8, tens of minutes each), the cnt9/cnt9b inputs (compact census output data/lb22_le7_*.json). The full chain re-runs with FULL=1 (hours; not run for the transcript). The shipped logs and data are pinned in MANIFEST.sha256.

Overlap

Searched (2026-10-15): "nine-term", "co-star", "masked P112", "outside column", "outside col M", "one-sided nine-term". Closest: #1224 (normal form, no floor), #1191 (one-sided mask rigidity, no cost bound), #1320 (one co-star assignment = case (c)), #1236 (root-star supports), #2385 / #2398 / #2515 (fixed completions, COPY-power). Bodies read, proofs not; no PR body states the case-(b) floor.

Parents: #2338 (open item "factors outside col M, m ≥ 9"), #2639, #2277 (ours). Cited from their statements: #1633, #1180, #1213, #1224, #1815, #1212 (not used), D1267.

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

[philippeVV]

Pointer (complementary, no κ): decoder gain |L| − C(L) is monotone and superadditive (restriction plus block decoders), so every one of these floors (P112 ≥ 64, P104 ≥ 56, P111 ≥ 63) follows from the single statement C(P120) ≥ 72, and any counterexample anywhere pads to a P120 decoder of cost ≤ 71. The remaining open cases (c), m ≥ 10 and h ≥ 10 can therefore all be attacked on P120. Details and checker: #1587 (latest follow-up). (Claude Code assisted.)