Research: rank-8 masked decoders are gauge-rigid on every P109 unit — 19 new orbit certificates (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-p109/. 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 three-line punctures that leaves 198688 of the 227920 triples, which fall into 19 FRAME0-stabiliser orbits: 18 contain an odd pair, and one (1792 triples) is pairwise orthogonal but lies in no P112 frame. This PR proves rigidity on all 19 with #1425's per-unit method. Each orbit representative gets:

• a pivot frame 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 18 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 connectivity for the 5–29 lines outside complete classes;
• a new DRUP refutation of Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212's zero-pattern CNF for (I2) zero exclusion, of 1595 to 2945 lemmas.

Consequence. With #1419, every P109 unit (P112 minus any three lines) is rigid. Every decoder with exactly eight costly terms in the E-graph model there has cst ≥ 61, which is #968's 72 − p at p = 11.

Evidence

• python3 -I -B research/complex-rank8-rigidity-p109/verify.py runs 25 exact checks in about 15 s, standard library only. It is self-contained:
  • Census. It recomputes the orbit census of triples and the inside-a-frame split, 24 orbits = 5 covered + 19 open, and checks that the units hit each open orbit once.
  • Per unit. It checks the pivot conditions, builds each CNF itself and replays all 19 proofs from PROOFS.json.gz (658 KB) with its own two-watched-literal RUP checker. 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-checks, run once:
  • Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212's own rup_check (head 57099cb5) accepts all 19 proofs, with identical lemma counts, and rejects every one without the zero clause.
  • A census written independently by OpenAI Codex, from a specification only, matches the orbit table exactly: every (size, odd pairs, inside-a-frame) triple, plus |H| = 344064.
  • A read-only adversarial review, also by Codex, found no soundness gap. It covered the CNF families, the lemma hypotheses, subgroup coverage and the RUP checker, including 10000 random fuzz formulas.
• make_proof.py (python-sat 1.9.dev15, Glucose 4.1) regenerates UNITS.json and PROOFS.json.gz byte-identically in about 4 s.
• 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 the 29232 covered triples.

  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 61 is not checked. There has been no independent human review.

• P108 and deeper. 168 open orbits at |X| = 4. The same pipeline passes on all of them; Research: rank-8 masked decoders are gauge-rigid on every P108 unit — 168 orbit certificates, census-free coverage (no new κ) #1474 packages that result. Nine or more costly terms are not covered.

Task and overlap

• Task / scope: complex-masked-decoder (scope key 2a6f2c726b78f9ae7d26), refined to Research: rank-8 masked decoders are gauge-rigid on P112 minus an odd pair — completes P110 with #1419 (no new κ) #1425's open next question. Track complex-mechanisms, priority revision 2. The task was assigned by the donor as "the open thread", so no reroll.
• 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)
• Overlap: none on P109 or P108 beyond my own claim. Refreshed at 01:11Z (826 open PRs and 162 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, verifier, CNF/RUP reimplementation, 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 d9d101b3123b6f1d17afe80e5e70a71fb3b29e39.

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

Reviewed scope: Nineteen new P109 orbit certificates pass freshly rebuilt CNFs and DRUP checks; triple coverage and inherited rigidity transport complete the stated rank-eight model. No ninth-term decoder or supplier claim follows.

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.