Research: Eisenstein E8 decoder copy floor 23 via submatrix-minrank certificates (past the clique-method limit; no κ claim)

[JosieElliston]

Result

Scoped lower bound; no supplier, no κ change, selected record untouched.

In the abstract Eisenstein E8 scalar binary decoder model of research/eisenstein-clique-copy-bound (Jonathan Lam, PR798), every fitting decomposition costs at least 23 pairing-rank units. PR798 proved 22 and showed that no weighting of cliques can prove more. Its PROOF.md says a stronger bound needs a new ingredient. This package supplies one: rank bounds on non-clique principal submatrices.

• Lemma 1. Zero-cost totals touch only free orthogonal entries, so they can be dropped.
• Lemma 2. Let m(S) be the least F2 rank of B[S,S] over fitting B. Then m(S) is at most the number of positive-cost totals meeting S on both sides. Weights satisfying sum_S λ_S [F meets S][G meets S] <= c(F,G) on every flat pair with c >= 1 prove C >= sum λ_S m(S). Cliques are PR798's special case. Asymmetric row and column sets add nothing.
• New sets. Take S_uw = N[u] ∩ N[w], where u,w are nonorthogonal; there are 540 such sets of 20 lines each. Each splits as I_2 ⊕ core, with an 18-line core of clique number 5. An exhaustive normalized search (U[Q]=V[Q]=[I_5|0]) excludes every fitting completion of rank ≤ 7, and a rank-8 completion exists, so m(S_uw) = 10. Under PR798's optimal clique dual these sets are priced at 9.5. Five pinned automorphisms generate a group transitive on all 540 nonorthogonal pairs, so the search runs once.
• Certificate. Use weights 184/6835 (singletons), 13/13670 (2,880 seven-cliques) and 7/20505 (S_uw). They satisfy all 296,321 positive-cost flat-pair inequalities exactly, with 601 tight. The objective is 30200/1367 ≈ 22.092, so C ≥ 23.

Consequences. The abstract optimum moves from [22,40] to [23,40]. PR1794's necessary window for an Eisenstein supplier shrinks from C ≤ 31 to 23..31. PR798's conditional three-stage Dfc window shrinks from 22..26 to 23..26. Neither route is closed.

How far the method goes (exploration, not replayed). I enumerated all 858,299 hit-maximal sets ⋂_{a∈A} N[a]; every set has the same price as one of them and no larger minrank. They fall into 78 (|A|, |S|, price) classes. SAT minranks on class representatives and the class-uniform LP give exactly 30200/1367 again, and only the S_uw class beats the clique dual. If those classes are automorphism orbits (not checked), principal-submatrix certificates top out at 23.

Verification

python3 -B research/eisenstein-submatrix-copy-floor/verify.py
python3.12 scripts/research_package.py --package research/eisenstein-submatrix-copy-floor

• verify.py is standard library only. It passed on CPython 3.10, 3.12 and 3.14 in 35–58 s. The registered 3.12 replay, --check-only and the 42 research tests also pass.
• There are five rejection controls: the clique rank 6/7 search sanity check, a non-automorphism, a non-transitive generator subset, an inflated S_uw weight, and cliques-only weights falling below 22.
• Independent cross-check (SAT, not shipped): the normalized rank-7 instance is UNSAT for every one of the core's 112 five-cliques.
• One OpenAI Codex read-only review found no soundness issue (Lemmas 1–3, search completeness, automorphism reduction, i ≤ j symmetry, rounding). Its two wording fixes are applied: the positive-cost pair count, and an explicit m(P,Q) definition with the method-limit statement kept conditional.
• Not done: proof assistant, independent human review, any physical or supplier reading beyond PR798's premises.

Scope and credit

The model, premises and geometry are PR798's; the pinned geometry.py is imported unchanged. Whole K-lines, scalar binary totals and the fitting pattern are essential. Nonlinear messages and sub-line operations are not covered.

Chooser seed 8bcd09ea00ac64eec3911b4b0c15e635 (complex-mechanisms / complex-rank-shape-decoder), scope changed; the reason is in TASK.json. Base 0b1cccf86e2f5132ed286a26ef0a681a495740ea. Claim: #1167 (comment)

Credit: Jonathan Lam (@jonathanlam, PR798: model, geometry, clique certificate, method-limit theorem); Joel Pulikkan / GamingPuzzled (PR620 geometry); Douglas R. Colkitt (PR703) and Sagar Thakrar (PR738) for the weighted-clique method; Maximooch (PR1794) for the window screen; DreamingOfClouds for naming the target. New material is by Josie Elliston, prepared with Claude Opus 5.5 (Anthropic) via Claude Code, with substantial AI assistance in search, proof and verifier. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at ea021c002e61fea7f2f9cd1b613f65b88f7eb7e9.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Non-clique minrank certificate improves the abstract Eisenstein scalar pairing floor from22 to23. Exhaustive normalized core search, rank-eight witness, automorphism orbit and every positive-cost flat-pair inequality replay. The stronger floor40 in #2017 supersedes its numerical bound while retaining credit for this distinct method; exploration beyond the certified sets remains conditional.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.