Research: rank-8 masked decoders are gauge-rigid on every P108 unit — 168 orbit certificates, census-free coverage (no new κ)

[JosieElliston]

Change and mathematical scope

This is a scoped structural result: no new κ, no construction. It adds only research/complex-rank8-rigidity-p108/. The selected record, upstream/ and all frozen pins are unchanged.

#1419's extension lemma makes P112 minus X rigid whenever X lies inside a frame of P112. For four-line punctures that leaves 168 of the 180 FRAME0-stabiliser orbits, 6132952 of 6210820 quadruples: 166 orbits contain an odd pair and 2 are orthogonal but lie in no P112 frame. This PR proves rigidity on all 168 with #1425's per-unit method. Each representative gets:

• a pivot passing Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212's (I3) and (I4) with gluing, with 16 to 24 complete classes;
• Research: rank-8 masked decoders are gauge-rigid on the 896 admissible P104 units (no new kappa) #1399's Lemma 9 for the other 4–36 lines;
• a new DRUP refutation of the zero-pattern CNF for (I2), of 1373 to 3168 lemmas and 361027 in total.

New: coverage without a census. The verifier does not enumerate the 6.2 million quadruples:

• Every quadruple contains a triple, and the 24 triple orbits are cheap. So every quadruple is equivalent to a candidate, a triple-orbit representative plus one line. There are 2517 candidates.
• 49 candidates lie inside a P112 frame, which is Research: rank-8 rigidity passes to supersets — P105–P110 partial-frame punctures, P111 from #1212 alone (no new κ) #1419's case; that property is invariant, because the group fixes FRAME0.
• For each of the other 2468, COVER.json gives a word in the generators, of length at most 10, that carries a unit's quadruple onto it. verify.py replays every word.

Consequence. With #1419, every P108 unit (P112 minus any four lines) is rigid. Every decoder with exactly eight costly terms in the E-graph model there has cst ≥ 60, which is #968's 72 − p at p = 12.

This is the follow-up to #1469 (P109) mentioned there. The two packages are independent, since neither uses the other.

Evidence

• python3 -I -B research/complex-rank8-rigidity-p108/verify.py runs 174 exact checks in about 2.5 minutes, standard library only:
  • Self-contained. It has its own E8 model, triple orbits, cover-word replay, pivot conditions, CNF builder and two-watched-literal RUP checker. It replays all 168 proofs from PROOFS.json.gz (about 6 MB), and the SHA256 of each decompressed proof and each CNF is pinned.
  • Controls. The gauge pattern satisfies each CNF. Each proof fails without the "some entry vanishes" clause. An empty-clause-only proof is rejected.
• Cross-check, run once: an independently written census (OpenAI Codex, from a specification only) finds the same 180 quadruple orbits. It has 12 inside a P112 frame (77868, matching Research: rank-8 rigidity passes to supersets — P105–P110 partial-frame punctures, P111 from #1212 alone (no new κ) #1419's census) and 168 open, 2 of them orthogonal. The coverage argument does not need it.
• AI review: a read-only adversarial review, also by Codex, checked the coverage argument and the P108 edge cases (classes with a single live row). It found no soundness gap. Its one wording point, that verify.py does not prove the 168 units lie in distinct orbits, is applied: the PASS line says "168 rigid units cover every P108 quadruple outside a P112 frame", and distinctness is credited to the census. This is not a human review.
• make_proof.py (python-sat 1.9.dev15 with Glucose 4.1, plus numpy and scipy) regenerates UNITS.json, COVER.json and PROOFS.json.gz byte-identically in about 40 s. numpy only proposes the orbit grouping; the verifier checks every word.
• Passes on Python 3.10, 3.12 and 3.14. scripts/research_package.py: the metadata check and the Python 3.12 registered replay pass. test_research_* passes.

Not checked or not covered

• Inherited steps, cited and not re-derived:

  • Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212's Lemmas 5–6, §7 and transport;
  • Research: rank-8 masked decoders are gauge-rigid on the 896 admissible P104 units (no new kappa) #1399's Lemma 9;
  • Research: rank-8 rigidity passes to supersets — P105–P110 partial-frame punctures, P111 from #1212 alone (no new κ) #1419's Lemma 10 and its P104 inputs, for frame-contained quadruples.

  All of these PRs are open. Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's (F6) is inherited. Attainment of 60 is not checked. There has been no independent human review.

• P107 to P105 are the open next question. Per-orbit certificates already take about 6 MB here, and P105 has on the order of 10⁵ orbits, so going deeper needs a structural replacement for the DRUP step (I2).

• Nine or more costly terms.

Task and overlap

• Task / scope: complex-masked-decoder (scope key 2a6f2c726b78f9ae7d26), refined to Research: rank-8 masked decoders are gauge-rigid on every P109 unit — 19 new orbit certificates (no new κ) #1469's open next question. Track complex-mechanisms, priority revision 2. The task was assigned by the donor as "the open thread".
• Heads: base 410026dc5513b87db92ae98571d1d5984c69a983. Cites Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212 57099cb5, Research: rank-8 masked decoders are gauge-rigid on the 896 admissible P104 units (no new kappa) #1399 4e1ba3a1, Research: rank-8 rigidity passes to supersets — P105–P110 partial-frame punctures, P111 from #1212 alone (no new κ) #1419 81b8997b and Research: rank-8 masked decoders are gauge-rigid on P112 minus an odd pair — completes P110 with #1419 (no new κ) #1425 4b707c22. It imports nothing from them.
• Claim: CLAIM: rank-8 rigidity passes to supersets — P105–P110 punctures inside a frame, and P111 from #1212 alone #1411 (comment) (P109; updated for P108 in the same thread).
• Overlap: none beyond my own claim and Research: rank-8 masked decoders are gauge-rigid on every P109 unit — 19 new orbit certificates (no new κ) #1469. Refreshed at 01:20Z (819 open PRs and 166 discussions, complete pagination).

Credit

• burntout76 / Brett Halligan (Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212): the theorem, (E), the CNF families of Lemmas 4/8, the pivot reduction, Lemmas 5–6, §7 and the transport.
• themilkman216 (Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968): the floor.
• DreamingOfClouds (Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls #717): the E-graph model.

Substantial AI assistance: Claude Opus 5.5 via Claude Code (proof, coverage argument, verifier, write-up); OpenAI Codex (independent census and read-only review). Submitted by the account holder. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at 311dd4e1eacef2b1301c68baf0276a9c51b32422.

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

Reviewed scope: P108 rank-eight rigidity reduces every quadruple to an explicit transported candidate, avoiding reliance on an unverified quadruple census. All 168 units require complete fresh CNF/DRUP replay; extension and transport hypotheses remain explicit.

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.