Rank-9 realisations of the E8 pattern with extra zeros (≤ 3 per row and column) are B-form or A2: philippeVV's R1 in that regime (no κ claim)

[DreamingOfClouds]

Package: research/e8-rank9-extra-zeros/. Pure matrix classification on the E8 unit. No κ claim, no window, no supplier, no decoder cost. It extends #2819 (rank-9 realisations with the full iff pattern are B-form or A2) to realisations that may have zeros on non-orthogonal pairs, and compares the result with philippeVV's normal forms and open item R1 (Discussion #1587, follow-ups 7, 8 and 9). R1 for an arbitrary number of extra zeros is not claimed.

Labels in the README: PROVEN / CHECKED (recomputed by verify.sh; SAT steps are kissat 4.0.4 UNSAT with a DRAT proof accepted by the standard drat-trim) / RECORDED / CONDITIONAL.

Results

1. Bridge to philippeVV's R1 / R2 (PROVEN; 12 exact members CHECKED). Every B-form realisation (any supp c, supp β, singular D allowed) is a member of their κ-family (expSM Proposition 0) and is a lift with R2 in the per-N form; every A2-family member equals their a2lib normal form; conversely {B-form} = {κ-family} after N_aa = 1. The README states how this relates to the definition of lift in follow-up 7 (agreement with a torus orbit of G/2 except on odd pairs touching a pairwise-odd S), under which R1 is not vacuous; that their R2 is universal rigidity on T while the per-N root-realisation form is what the Lift Lemma uses; and that when A and B both contain orthogonal pairs, S = B is a lift only in the §4 / R3 sense (the κ-family form is unaffected). Non-concurrency occurs only on A2 triangles.
2. Theorem Z3′ (PROVEN from CHECKED). If every row and every column of a rank-9 realisation of the E8 pattern (characteristic 0) contains at most 3 extra zeros, with no bound on the total, then it is B-form (D singular at most on a clique inside supp c ∩ supp β) or A2. With the bridge: R1 with R2 in the per-N form holds in this regime (universal R2 is proved only for clique complements, item 3). Its hypotheses cover those of the earlier results for one exceptional row or column with at most 3 zeros and for a total of at most 3 extra zeros (not their conclusions: Z3's is sharper), and it uses Theorem G with singular scalings (G⁻, G^K); the earlier results Z1 (any number of zeros in one row or column) and Z3 (at most one singular scaling for a total of at most 3) are kept with their inputs.
3. Theorem RK (PROVEN from CHECKED). Rank-8 completions on E8 ∖ C, C any clique, with arbitrary extra zeros, have no extra zero and are root realisations (D₁GD₂): universal R2 for clique complements. Two zero-pattern CNFs (8-clique and maximal 7-clique complements) are kissat UNSAT with drat-trim VERIFIED proofs shipped; both regenerate byte-identically; the negative control (goal clause dropped) is SAT; W(E8) is transitive on maximal cliques of each size; A2 triangles by rigidity on E8 ∖ triangle.

What remains (stated exactly in README §5)

R1 with arbitrarily many extra zeros. The proof uses the per-line budget in three places only (a frame row or column with ≥ 8 extra zeros; the step Π(+) ≠ Π(−); "no pure off-frame column"), and the SAT residue step is certified for at most 3 extra zeros per row and column. The first uncovered configuration is a pure line, which Lemma PL (PROVEN) shows is a clique.

Verification

bash research/e8-rank9-extra-zeros/verify.sh from a fresh git archive: manifest; exact finite facts (spans, P-compatible frames, clique spans and orbits, bridge members, rigidity on E8 ∖ triangle, frame choice, P-compatibility on E8 ∖ clique); byte-identical regeneration of the five CNFs; the four shipped DRAT proofs (.drat.xz, 36.6 MB in total) checked by drat-trim. FULL=1 adds fresh kissat solves with fresh drat-trim checks and the negative control. kissat only (never CaDiCaL); one heavy process at a time under nice -n 10, every external call under timeout 1200.

Credit and scope notes

The κ-family normal form, lifts, the a2lib normal form, the R1 to R4 formulation and the census are philippeVV's (artifact integer-mult-bounds @ 93766d7, decoder-law/lift-family-2 and expSM; follow-up 9's Theorem R3, lifts on dependent or non-clique supports of at most 26 lines are κ-family members, @ b5ce9d2 closures-3/decoder-law-subsets). Our R3 remark is a normal-form statement for Gram T-rows only and does not extend their range; this package compares normal forms and does not run their code. The audit is by the same author's tooling (independent read-only reviews), not a human review.

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

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