Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa)

[burntout76]

Change and mathematical scope

Status: research note / scoped structural result (a proof). No κ claim, no new supplier, no scalar saving. Only research/complex-rank8-decoder-rigidity/ is added; upstream/, the selected record (κ = 0.001672660197076918, PR1094 at the base) and every existing file are unchanged.

The complex supplier's copied-information cost is the decoder constant cst = 64 on P112 (one cst unit ≈ 2.8% of b in the PR942 lineage on which this work started; the constant is the same in the current selection). PR717 (DreamingOfClouds) reduced exactness to the E-graph (odd entries are fixed by cost-0 terms) and showed that decoders with exactly eight costly terms cost at least 64, but recorded its support step (that the costly supports are hyperplane-flat complements, the W(E8)-orbits of the fundamental weights) as ARGUED: it rested on the tangent-space rigidity D1(d) plus an orbit census, with global rigidity only sampled (10 of 12 random starts, RECORDED). PR968 proved the floor rigorously only under all-pairs exactness, where the normal form D = RA, T = RB is immediate, and listed E-graph-only exactness as not covered (so does #1071).

This package proves the missing global statement:

• Theorem 1. On P112 and P120, every real matrix K of rank ≤ 8 with K_ii = 4 and K_ij = 0 on orthogonal root pairs equals D (2G) D⁻¹ for a diagonal D with nonzero entries. The rank-8 solutions of the E-graph system form one orbit of the diagonal torus; no odd entry vanishes. "P112" is the complement of any frame: W(E8) is transitive on the 2025 frames (checked) and the system is invariant up to a sign conjugation that lies in the torus, so the result covers the Hamming coordinate frame punctured by the shipped word as well as PR717's FRAME0.
• Corollary 2. In PR717's E-graph-only model every decoder with exactly eight costly terms has PR968's normal-form supports C(a_k), C(b_k) for a dual basis ABᵀ = 2I; (RA, RB) is then literally an all-pairs-exact decoder with the same supports and costs, so PR968's floor theorem gives cst ≥ 64 on P112 and ≥ 72 on P120. This reaches PR717 D1(b)'s conclusion by a proved route — the flat-support step itself does not hold for a general dual basis (checked on a sampled basis), so PR968's Lemma 1 is what carries the floor — and extends PR968's eight-term floor to E-graph exactness at p = 0 and for any single-frame puncture.

Changed hypotheses: none. Remaining obligations / not covered: nine or more costly terms (PR717's mixed-β problem stays open on P112 in this model — its locus is the rank ≥ 9 solutions together with ≥ 9-term factorisations of the rank-8 ones), rigidity on P111 and on the P104 units other than type (2,2,2,2) (see the update below; numerical evidence only), Corollary 2 over complex coefficients (Theorem 1 itself is algebraic and holds over any field of characteristic zero), any physical word. Corollary 2 inherits PR968's one cited premise (the classification of E8 hyperplane flats, PR968 (F6)) and PR968's written argument.

Proof shape: pivot on a live frame F chosen so that the punctured lines form whole classes (28 of the 1260 live frames of P112 qualify): K[F,F] = 4I₈ gives K = ¼ K[:,F] K[F,:], so the problem is bilinear in rows a_i = K[i,F] and columns b_j = K[F,j]; relative to F the lines fall into 14 D4-blocks forming the Steiner system S(3,4,8), each with two classes of four mutually orthogonal lines whose sign matrices are Hadamard; orthogonal lines in different blocks share exactly two pivot coordinates, so their equations have two terms. (i) Zero exclusion: the necessary conditions on the zero pattern form a CNF whose only model is the trivial pattern; DRUP refutations of "some entry vanishes" (for the full CNF and for the reduced one with only the two-term and diagonal equations) are shipped and replayed by the verifier's own unit-propagation checker. (ii) Within a class the parallelism relations forced by live witnesses leave two solution families; one has a rank-1 class matrix and is excluded, the other is A_C = diag(δ) H_C diag(γ). (iii) The two-term equations glue the column scalings across blocks and the 3-design turns the ratios into a global potential on F. Then K_if = 2G_if d_i/d_f and K = D(2G)D⁻¹.

Update (second commit, same package): beyond one frame

Section G of verify.py and PROOF.md §10 extend the analysis to the other units PR968/PR717 study:

• The stabiliser of FRAME0 generated by single and double root reflections (204 line permutations) is transitive on the 112 lines of P112 and has exactly three orbits on the 1260 frames inside P112, of sizes 28 / 336 / 896 — PR968's 364 inadmissible frames and its 896 admissible ones (PR717's P104 clique lies in the 896-orbit). So P111 is one unit up to symmetry and the P104 units are three.
• Theorem 1 transfers verbatim to the 336 P104 units of type (2,2,2,2): with the pivot {12,13,22,23,30,31,36,37} every live class is complete, zero exclusion holds (shipped refutation), all live group witnesses exist and every block has a complete live class — exactly the finite inputs the written proof uses. Hence cst ≥ 56 for eight costly terms in the E-graph model on those units (PR968's floor at p = 16).
• Zero exclusion (Lemma 4) holds on P111 and on all 1260 P104 units (four shipped refutations, transported by the stabiliser). For the type-(4,4) units the two-term and diagonal conditions alone admit a zero pattern (576 zeros; the verifier records this) and the refutation needs all four clause families.
• What is still missing, named per unit (verdict pinned in code): type (4,4) — the all-complete pivot leaves two blocks without a live class, so Lemma 6's cocycle step has no witness; type (1,…,1), incl. PR717's P104 — four classes keep two live rows, so Lemma 5 does not apply; P111 — one three-row class (the three-row variant of Lemma 5 is sketched in §10 as the smallest next step). Rigidity on these units rests on numerical evidence only (type (4,4): 8/8 random starts gauge-equivalent, seed 5; type (1,…,1): 5/6).

Validation now: 61 PASS on P112 (41 + 20 in section G), 41 on P120, eight shipped DRUP refutations, all replayed by the verifier's own checker; every count, class profile and CNF digest in section G is pinned. Expected final lines unchanged.

Task and overlap

Base commit: 1b59b624d21924f8e3537cdc1ac52d3be6eda2fe (current main). The work started at 2921874 (selected κ 0.001661960547713947, PR942) and the additive package was rebased before submission; its statements do not depend on the selected record. Dependency heads: PR717 at 4a97676 (model and definitions, pinned by blob and SHA-256), PR968 as merged (normal form and puncture floor), both pinned in SOURCES.json. Concrete object and bounded domain: real matrices of rank ≤ 8 satisfying the E-graph system on the P112 and P120 line sets; eight-costly-term decoders in PR717's D1(a) model on those units.

Task origin: the external sprint prompt (track 40–49 structural decoder research, OS seed 47), not an autoresearch.md task card; the local claim in CLAIM.md records the three candidates considered. No pinned coordination issue existed at the start; the discussion board's claims were checked before submission (the only adjacent one, #1210 "masked root-star copy-cost floor on P112", concerns star decoders, a different class).

Nearest overlapping PRs and added value: PR717 D1(b)/(d) (the ARGUED step this proves) and PR968 (the all-pairs floor this extends to E-graph exactness). Concurrent, opened while this was being prepared: #1191 proves that E-graph ("mask") exactness already yields the all-pairs identity when one factor is root-linear, for any term count (P112, all P111, all P104); its hypothesis and Theorem 1's are incomparable — Theorem 1 fixes the rank at 8 and restricts neither factor (its solutions D(2G)D⁻¹ have non-root-linear factors in general), #1191 restricts one factor but neither rank nor term count — so together they close, on P112, the E-graph decoders of rank ≤ 8 and those with a root-linear factor, leaving rank ≥ 9 with both factors off the root code. #1180 extends the all-pairs floor to any term count with both factors root-linear (explicitly excluding masked identities); #1197 (ten-term decoders with retained coordinate readers), #1187 (P104 COPY-drop thresholds), #1077 (complex coefficients), #1071 (delivery bounds), #1020, #998/#845 (single lifts) stay in the all-pairs or single-lift models. None proves rank-8 rigidity of the E-graph system. The sprint seed (47) coincides with PR968's; the object is distinct.

Validation

Standard library only, ~10 s CPU (up to ~30 s wall) per unit, from the repository root:

python3 -I -B research/complex-rank8-decoder-rigidity/verify.py               # P112
python3 -I -B research/complex-rank8-decoder-rigidity/verify.py --live P120   # control

Expected: 61 PASS lines on P112 (41 on P120), no FAIL, exit 0, final lines

PASS_COMPLEX_RANK8_DECODER_RIGIDITY: live P112; pivot [2, 3, 16, 17, 46, 47, 52, 53]; blocks 14; classes 26 complete; frames 2025; zero-pattern CNF 3744 vars / 29820 clauses; proof lemmas 1722 (+3001 reduced); coordinate cst 64
PASS_COMPLEX_RANK8_DECODER_RIGIDITY: live P120; pivot [0, 1, 26, 27, 44, 45, 54, 55]; blocks 14; classes 28 complete; frames 2025; zero-pattern CNF 4032 vars / 33768 clauses; proof lemmas 2045 (+2439 reduced); coordinate cst 72

Finite tests versus proof: every finite fact the written proof (PROOF.md) uses carries a check id that verify.py recomputes exactly — Steiner/Hadamard/orthogonality structure, class matchings, live witnesses (all counts asserted, nothing passes vacuously), W(E8)-transitivity on the 2025 frames, the two RUP replays (deletions ignored; proofs produced by Glucose via python-sat, never trusted), exact rational corollary mechanics, and an exact recomputation of the gluing/assembly step. Failure controls: relaxed CNFs accept the all-zero pattern, a proof with an inserted non-RUP lemma or truncated before its empty clause is rejected, the proof does not close without the "some zero" clause, a missing proof file fails loudly, a zero-free non-Hadamard block matrix shows a single D4 block is not rigid, class-dependent column scalings break the assembly, and a rank-9 solution is not of gauge form. The within-class algebra (Lemma 5), the gluing (Lemma 6) and the assembly are written arguments; the zero-exclusion step is a checked propositional refutation whose minimal core still spans 380 equations over 73 lines, so no hand argument is offered for it. probe_numeric.py (numpy) records non-proof evidence: tangent nullity equals the gauge dimension on P120/P112/P104 (183/175/167) and 8/8, 5/6, 5/6 converged random starts are gauge-equivalent to 2G (the rest stalled; none non-gauge).

Checks not run: hosted CI (no workflow or Makefile group added — the token lacks the workflow scope to push one; the existing CI groups do not touch this directory), repository-wide suites (nothing outside the new directory changes), Python versions other than 3.14. Environment: Linux x86-64, CPython 3.14.4, 61/61 checks on P112 and 41/41 on P120, also from a fresh git archive of the head. Review: two rounds of AI-assisted adversarial review (22 refuters across mathematician/code-auditor lenses) and a reviewing agent's from-scratch re-implementation of the finite facts (own model, own CNF — UNSAT for four different pivots, Gröbner-basis derivation of the two class-matrix families, own RUP checker, clause-by-clause audit of the CNF; not shipped); no independent human review.

Credit

DreamingOfClouds, PR717 (open, head 4a97676): the model, the E-graph reduction, the eight-term floor with its ARGUED step and the local rigidity this completes. themilkman216, PR968 (merged): the normal form and puncture floor that Corollary 2 invokes. DreamingOfClouds PR446/PR595 (Theorem C, format ceiling); GamingPuzzled PR620 (cost 64); Mihir Srivastava PR845 and woklerx PR998 (single lifts); mightyOil #1191 and woklerx #1180 as concurrent complementary results. Source licenses retained (Apache-2.0 throughout).

Prepared by the agent Ninefold (Claude Fable 5.1 via Claude Code) for Brett Halligan (@burntout76), with substantial AI assistance in research, proof, implementation, review and documentation. Submitted by Brett Halligan. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

100-PR batch review at 57099cb5684f38877f31d149d598801671a633db.

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

Reviewed scope: Rank-eight E-graph rigidity via live D4 classes, exact zero-pattern refutations and cocycle gluing; real eight-term floors 64/72 inherit PR968 classification. P104 transfer is restricted to the verified 336-unit orbit; no overcomplete-decoder exclusion.

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.