Rank-9 realisations of the E8 pattern are B-form or the A2 family (classification; no κ claim)

[DreamingOfClouds]

Adds only research/e8-rank9-realisations/. No κ claim, no decoder cost, no window or supplier. E8 only.

Result. Over a field of characteristic 0, every 120 × 120 matrix M indexed by the E8 lines with rank M = 9, M_tt ≠ 0 and M_ts = 0 ⇔ t ⊥ s (t ≠ s) is B-form (D₁GD₂ + cβᵀ, rank-one biclique with c_uβ_s = 0 on orthogonal pairs) or in the A2 family of #2784. Both families occur (exact members in checks/a2e8.py). This is the existence step ("Conjecture B*") that #2784 left open.

• PROVEN from: Theorem G (CONJB / Decoder floors, case (c): the A2 family of non-B-form rank-9 realisations (P112 ≥ 64, #968 P104 ≥ 56, E8 ≥ 72; no κ claim) #2784 §3; span facts S1, S2 CHECKED), Theorem P (pinning) and Theorem A (an orthogonal pair in Σ_a or Σ_b; all 2025 E8 frames P-compatible, all 3780 orthogonal pairs in a frame, CHECKED), Theorem S, Theorem Σ, Theorem R.
• Theorem S (Σ_a ∩ Σ_b ≠ ∅): reduction PROVEN; finite step = a 56-line point/line incidence problem, CHECKED: kissat 4.0.4 UNSAT, standard drat-trim VERIFIED, regenerated by verify.sh for f₁ = 0..15 (all 120 under FULL=1).
• Theorem Σ (Σ_a ≠ ∅ or Σ_b ≠ ∅): PROVEN from S, A, G and CHECKED (S3′), (W2).
• Theorem R (Σ_a = Σ_b = ∅ is empty): reduction PROVEN; finite step CHECKED: the pinning-model CNF (56 784 variables, 3 513 012 clauses) is regenerated byte-identically (sha256 pinned), kissat UNSAT with --factor=false --preprocessfactor=false, DRAT proof shipped xz-compressed (11.7 MB; 163 MB raw, sha256 pinned), standard drat-trim VERIFIED (shipped proof: 137 s). FULL=1 also re-solves (≤ 20 min) and drat-trims a fresh proof. The auxiliary C checker rupc.c (RUP-only, deletions honoured, RAT rejected) agrees; drat-trim is the reference.
• Rule soundness of the model: one-line proofs in the card text; soundtest.py finds 0 violated clauses on true data (A2 members, 5 parameter draws on 5 triangles; B-form), and a mutation control (one flipped sign) is caught.
• Independent read-only audit: re-verified both certificates with drat-trim, regenerated the CNFs byte-identically, confirmed the soundness test catches a flipped rule (RECORDED; the two fixes it asked for are applied: characteristic 0 is stated, the A2 parameter is varied).

Downstream (case (c) on E8). With #2784 (A2-family floors, E8 ≥ 72), case (c) reduces to decoders on B-form realisations. Closed there: A2 family (#2784). For B-form, the working cards (not shipped) have: a term with p ∝ c and q ∝ β costs ≥ 64; |supp c| = 1 (the #2321 / #2385 K_λ family) ≥ 64 modulo a CHECKED puncture census; the generic 8-clique type ≥ 65 for sampled values. Not closed: B-form with all x_j, y_j ≠ 0 and |supp c| ≥ 2, non-generic c, β, singular D, m ≥ 10. So the E8 floor 72 for all rank-9 square decoders is not proven.

Relation to philippeVV's open item (R1) (Discussion #1587, follow-up 8, decoder-law/lift-family-2 @ 93766d7: "every rank-9 completion is a lift"). The lift normal form, credit machinery and census are theirs; nothing here uses them. Their completions (decoder-law/README.md) have N_aa = 1, N_ab = 0 for a ⊥ b and non-orthogonal entries free (zeros allowed), a larger class than our iff pattern; our theorem covers only completions with all non-orthogonal entries nonzero. It would give R1 for that subclass only if every B-form and A2 realisation is a κ-lift, which we have not compared (their R3 covers dependent supports). So R1 does not follow. Remaining for R1: (i) show B-form ∪ A2 ⊆ lifts; (ii) drop the iff assumption: Theorem G survives (CONJB §5), but pinning (Theorem P), the NON rule and the S-step use nonzero entries, so both SAT models must be rebuilt and re-certified. Not attempted.

Not claimed. P112 / P104 (P112 Theorem R certification is in progress and nothing is claimed; P104 has few P-compatible frames); positive characteristic; formal-kernel check.

Overlap. A search of PR bodies for rank-9 / rank-nine realisation classifications found none besides #2784 (A2 family, same author) and #1212 (rank 8 only).

Verify: bash verify.sh [outdir] (numpy, sympy, kissat 4.0.4, drat-trim, cc, xz); default run re-solves and drat-trims the Theorem S step, regenerates and hash-checks both CNFs, checks the shipped proof's hash, and runs the soundness test; FULL=1 adds the Theorem R re-solve and drat-trim.

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

🤖 Generated with Claude Code