Decoder floors, case (c): the A2 family of non-B-form rank-9 realisations (P112 ≥ 64, #968 P104 ≥ 56, E8 ≥ 72; 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/a2-family-decoder-floors/ is added; nothing existing is changed.

What this proves (scope is the claim)

Model of #2338 / #2277 / #2724: exact masked decoders N = Σ_{j≤m} p_j q_jᵀ of a pattern realisation (zero on orthogonal pairs, nonzero elsewhere on the diagonal and off it: the "iff pattern"), arbitrary factors over a field of characteristic 0, dense pairing table B, cost Σ_j rank_F2 B[R_j, A_j].

• A1/A2: the A2 family. For P112, P104 and E8 there are rank-9 iff-pattern realisations that equal the Gram matrix except on the three rows of one A2 triangle, where each row is the Gram row rescaled by one factor on N̄(u_i) ∩ O_{u_j} and another on N̄(u_i) ∩ O_{u_k} (7 parameters plus diagonal scalings). They are not of B-form D₁GD₂ + cβᵀ (with c_uβ_s = 0 on orthogonal pairs): a counterexample to our working "Conjecture B" for rank-9 realisations. Exact certificate (three random exact members per unit, plus the two tie members below).
• Theorem G (PROVEN). If the rows outside a set X of a rank-9 pattern realisation are Gram rows, it is B-form with supp c ⊆ X, or in the A2 family on an A2 triangle inside X (columns: transposed). Lemma E (PROVEN): rank ≤ 8 on V∖X plus a rigid sub-unit (Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212 and extensions) gives Gram rows. The existence step ("every rank-9 realisation has such an X") is OPEN and not claimed.
• Floors (F). Every 9-term (square) decoder of every A2-family member, at all parameters and on all A2 triangles, costs P112 ≥ 64, P104 (Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's unit) ≥ 56, E8 ≥ 72. Triangle coverage: P112 one orbit of 896, E8 one orbit of 1120, P104 ≤ 28 orbits (all 28 representatives run). Case I (some row normal ∥ e_9) and Case II (none; II(b) by an exact class-level min-cost relaxation, 65 / 62 / 74 off the B-locus, 66 / 63 / 75 on it, using the checked fact that two merged column classes are co-killable only on γ₂ = γ₃); includes the E8 near-miss (the three punctured normals r_1, r_2, r_3 are dependent).
• Sharpness (S). Exact tie decoders of cost 64 (P112) and 72 (E8) at the stated special parameter points (exact over Q, independently re-evaluated; the tie members are certified non-B-form); the generic members checked reach only 65 / 73. P104 sharpness is not claimed.
• m ≥ 10, both factors in the realisation's two 9-dimensional spaces (M), off the B-locus γ₂ ≠ γ₃. Cost ≥ 68 (P112), 64 (P104), 77 (E8) (merge lemma + exact class-level min-cost relaxation). On the B-locus the relaxation is lower (62 / 58 / 70) and that case is open.

Not covered (stated up front)

• Decoders with factors outside col M / row M on the A2 family with m ≥ 10 are NOT covered (for m = 10 they are exactly the one-sided frame-expansion decoders; only a heuristic search was done, which found ties and nothing proven).
• Other rank-9 realisations: this is one explicit family, not all of case (c). In particular nothing here bears on the co-star decoders (Discussion 1267, Prove a nine-root obstruction for one masked P112 co-star decoder #1320) unless their realisation lies in the family.
• m ≥ 10 on the B-locus γ₂ = γ₃: open (not claimed).
• Non-iff patterns, B of rank ≥ 10, other P104 units. Not claimed: that the A2 family is an irreducible component; the variant of the non-B-form statement without the orthogonal-pair condition.

Labels (every step is in the README)

PROVEN: square-decoder normal form, merge lemma, A2 facts, the construction, Theorem G, Lemma E, the non-B-form argument, Case I and II(a). PROVEN from CHECKED: Case II(b) (row census parameter-free, column census treats γ symbolically; merged-kill fact; class-level relaxation) and m ≥ 10 off the B-locus. CHECKED (exact computation): members with the iff pattern and rank 9; rank certificates (mod p, safe direction); span facts S1/S2; the unit census on all three units and all 28 P104 representatives; survivor-pair ranks; the row-puncture floors P_k (exact branch-and-bound on P112/P104, structural on E8); the tie decoders; exact optima at fixed members (65 / 64 / 73 / 72); the merged-kill fact (P112, E8, all 28 P104 representatives); the class-level relaxations; pair tables at the members checked (corroboration). RECORDED (logs shipped, hash-pinned, headline lines asserted, not re-run): the robust CP-SAT / CEGAR runs over all parameters (no square decoder ≤ 63 on P112, ≤ 71 on E8) and the all-parameter special-locus census behind the pair tables.

Verification

bash research/a2-family-decoder-floors/verify.sh /fresh/dir (Python ≥ 3.11, numpy, networkx; one process at a time; no SAT solver). On a fresh git archive export: 102 PASS, 0 FAIL, 941 s wall (15.4 min user CPU, one process at a time, nice 10; venv Python 3.14, numpy 2.5, networkx 3.7).

What verify.sh re-runs: units; survivor classes and survivor-pair ranks; S1/S2; exact A2 members and the non-B certificate on all three units; the unit census (replayed and compared line for line with the recorded logs) on the first triangle of each unit and all 28 P104 representatives; triangle-orbit counts; P_k and the case arithmetic; the merged-kill fact and the Case II(b) / m ≥ 10 class-level relaxations (P112, E8, all 28 P104 representatives); the tie decoders and the tie members' certificates; exact optima at fixed members; pair tables at members (corroboration). RECORDED, not re-run: the CP-SAT / CEGAR logs and special-locus census (evidence/logs/hunt-a2/, pinned in MANIFEST.sha256).

Overlap

Searched (2026-10-15): "A2 family / triangle / subsystem", "rank 9 / rank-9 / rank nine realisation", "non-B-form", "B-form", "Conjecture B", "square decoder", "decoder floor", "masked P112 decoder", "co-star", "rank-8 rigidity", "non-Gram rank" and the unit sizes. No PR body found that states an A2-family (non-B-form) rank-9 realisation, Theorem G / Lemma E, or these floors. Nearest: #2338 / #2724 (ours; case (c) open item and sibling), #1212 and extensions (rank 8 only), #2321 / #2385 / #2398 / #2515 (one fixed rank-9 P112 completion under a COPY-power cost). Bodies read, proofs not; discussions not searched.

Parents: #2338 (open item "factors outside col M, m ≥ 9", case (c)), #2724, #2639, #2277 (ours). Cited from their statements: #1212, #1389, #1399, #1419, #1425, #1616 (rank-8 rigidity, used only to state Lemma E), #968 (the P104 unit), #1224, #1180, #1320, D1267 (context, not covered).

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