Decoder floors, case (c): the B-locus of the A2 family (completes #2784's m ≥ 10 claim: P112 ≥ 64, #968 P104 ≥ 56, E8 ≥ 72; no κ claim)

[DreamingOfClouds]

Status: research note with exact finite computations and a written proof (not a SAT/DRAT proof). No κ claim, no supplier word. Only research/a2-family-b-locus/ is added; nothing existing is changed.

Results

#2784 left open its Claim M (m ≥ 10, decoders M = P Qᵀ with factors in the realisation's two 9-dimensional spaces, dense B, iff pattern, characteristic 0) on the B-locus γ₂ = γ₃ of the A2 family. This closes it.

• PROVEN (§1): B-locus members are of B-form, M = D₁G + 1_X βᵀ, supp β ⊆ X: a rank-one change of the 3×3 triangle block only.
• PROVEN from CHECKED facts (a)–(e) (§2): on the B-locus every such decoder with m ≥ 10 costs ≥ 64 (P112), ≥ 72 (E8); for P104 (all 28 triangle representatives) ≥ 58 when the decoder still has ≥ 10 terms after merging parallel terms, otherwise Theorem F's 56 (so the combined P104 bound stays 56). The key checked input is Q_XX = 62 / 56 / 70, the square lower bound with the triangle deleted on both sides (opt3u duality branch-and-bound, complete); the rest is a per-term bound c′ ≥ 7, a class-level inside relaxation (66 / 63 / 75 at 9 terms) and a rank fact on the three triangle normals. The transposed family follows by symmetry.
• With 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 (off the locus 68 / 64 / 77; Theorem F at m = 9): Claim M holds on the whole A2 family: P112 ≥ 64, Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968 P104 ≥ 56, E8 ≥ 72, every m ≥ 9, factors in the spaces.
• The relaxation values 62 / 70 (P112 / E8) that made the locus look open are not decoders: the optimum needs three cost-7 terms with row normals ∝ r₁, r₂, r₃ inside an 8-term inside decoder, impossible since they span 2 dimensions. For P104 the value 58 equals the relaxation value.

Scope and caveats

• Floors for decoders whose realisation N is a member of the A2 family and whose factors lie in that realisation's spaces; nothing is claimed for other rank-9 realisations, non-iff patterns or B of rank ≥ 10.
• Out-of-space m ≥ 10 (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 §7.3) stays open.
• Tightness of 64 / 72 on the B-locus is not claimed and was not checked (the tie decoders of 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 are off the locus).
• No κ claim.

Verification

bash research/a2-family-b-locus/verify.sh /fresh/dir (Python ≥ 3.11, numpy, networkx). On a fresh git archive export: 50 PASS, 0 FAIL, 172 s wall (nice 10; the 28 P104 representatives and the Q_XX partial punctures run 6 at a time, each under timeout 1200). It re-runs facts (a)–(e) including Q_XX on P112, E8 and all 28 P104 representatives (outputs compared byte-for-byte with the recorded logs), the relaxation with the e₉ row and column, and the inside relaxation. The needed files of #2784 (a2lib, opt3u and their imports) are vendored unchanged so the package is self-contained (NOTICE.md, SOURCES.json, MANIFEST.sha256).

Overlap

Follow-up to #2784 (ours); parents #2338, #2724, #2277 (ours). Cited: #968. No other overlap expected.

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