Non-dense fitting B: every E8 term with x·y ≠ 0 has rank ≥ 9; square root-code floors E8 ≥ 72, P112 ≥ 64, P104 ≥ 56, D4 ≥ 12 for any fitting B (no κ claim)

[DreamingOfClouds]

What this adds

Only research/nondense-pairing-square-floors/. Extends the dense-pairing floor of research/ideal-decoder-floors/ (Theorem 4) to any fitting B (B_tt = 1, B_ts = 0 on non-orthogonal pairs, free on orthogonal pairs) for square root-code decoders (m = T, XYᵀ = I). Related: #968, #1744 (rank ≥ 9 on P111/P112 in its scope), #2176, #2338 (Theorems P, R).

• Lemmas 6.1-6.3 (PROVEN, written proofs): per-term fitting minrank = mr₂ of the orthogonality graph on R∩A; normal-pair reduction; orthogonal-set deletion lowers mr₂ by ≤ 1.
• Theorem 6 (CHECKED): all 4022 non-orthogonal root-hyperplane pair types of E8 (440 880 hyperplanes, W_J = Stab(n₁)-orbits, min |D| = 30) are UNSAT at fitting rank ≤ 8, each with a drat-trim VERIFIED proof; two independent orbit implementations and two independent CNF writers; brute force over all fitting B agrees with both writers on 60 instances.
• Corollary 6.4: square root-code decoders cost ≥ 72 (E8), ≥ 64 (P112), ≥ 56 (P104), ≥ 12 (D4) for any fitting B, all sharp (exact decoders re-built and checked).
• Counterexamples to the per-term claim (CHECKED): E8 orthogonal normals |D| = 24, mr₂ = 6; D6 non-orthogonal types with mr₂ 3-4 (scan |D| ≤ 19 only); one E7 type |D| = 11, mr₂ = 5. No decoder below any floor is shown (B is shared across terms).
• Open / not claimed: overcomplete decoders with x_j ⊥ y_j terms, E7/D6 square (global-B coupling), factors outside the root code. No κ claim.

Verification

bash verify.sh (a few minutes, ≤ 2 processes): manifest, vendored drat-trim build, both orbit recounts and their agreement, fixed 32-type sample re-solved by both writers with DRAT VERIFIED, brute-force agreement, the counterexamples, the sharp decoders. FULL=1 bash verify.sh re-solves all 4022 types with both writers, every proof drat-trim VERIFIED; it was run in full at this head and passes (log in evidence/verify_FULL.log). PROOF_MAP.md lists claim → check; anything unchecked is CONDITIONAL in the README.

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

🤖 Generated with Claude Code