Research: rank-8 masked decoders are gauge-rigid on P112 minus an odd pair — completes P110 with #1419 (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-p110-odd/. The selected record, upstream/ and all frozen pins are unchanged.

#1419 shows that rank-8 rigidity of #1212's system (E) passes to supersets. That makes P112 minus any orthogonal pair rigid. It cannot reach an odd pair, because no frame contains one. The 2912 odd pairs of P112 form two FRAME0-stabiliser orbits, of sizes 224 and 2688. This PR proves rigidity on both, by #1212's method plus #1399's Lemma 9. For each orbit representative it ships a pivot and a new zero-exclusion certificate.

• Inputs, per representative:
  • (I2) zero exclusion: a DRUP refutation, of 1893 and 2320 lemmas, of Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212's zero-pattern CNF families (two/four/diag/trans). The CNF builder and the RUP checker are reimplemented here.
  • (I3) and (I4), plus gluing, on the complete classes: 24 and 22 of them.
  • Lemma 9 witness connectivity for every line of an incomplete class: 6 and 14 lines, all six column pairs witnessed.
• Proof: Research: rank-8 masked decoders are gauge-rigid on the 896 admissible P104 units (no new kappa) #1399's Step 1 (Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212 Lemmas 5–6 on the complete classes), then Lemma 9 on the rest, then Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212 §7.

Consequence. With #1419, every P110 unit (P112 minus any two lines) is rigid. Every decoder with exactly eight costly terms in the E-graph model there has cst ≥ 62, which is #968's 72 − p at p = 10.

Evidence

• python3 -I -B research/complex-rank8-rigidity-p110-odd/verify.py runs 26 exact checks in about 2 s, standard library only. It is independent of Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212's code: its own E8 model, orbits, classes, witnesses, gluing, Lemma 9 checks, CNF and two-watched-literal RUP checker. It replays the shipped proofs against CNFs it builds itself, and the SHA256 of both the proofs and the CNFs is pinned. Its controls:
  • the gauge zero pattern satisfies the CNF without the "some entry vanishes" clause;
  • the proofs fail without that clause;
  • a bare empty-clause proof is rejected.
• Cross-check, 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 both proofs on this package's CNFs, with identical lemma counts, and rejects them without the zero clause.
• make_proof.py (python-sat 1.9.dev15, Glucose 4.1) regenerates the pivots and proofs in about 2 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

• P109 and deeper punctures with an odd pair; this is the open next question. Orthogonal punctures outside a frame (see Research: rank-8 rigidity passes to supersets — P105–P110 partial-frame punctures, P111 from #1212 alone (no new κ) #1419's census). Nine or more costly terms.
• Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212's Lemmas 5–6, §7 and transport, and Research: rank-8 masked decoders are gauge-rigid on the 896 admissible P104 units (no new kappa) #1399's Lemma 9, are cited, not re-derived. Both 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 62 is not checked. There has been no independent human review.

Task and overlap

• Task / scope: complex-masked-decoder (scope key 2a6f2c726b78f9ae7d26), refined to Research: rank-8 rigidity passes to supersets — P105–P110 partial-frame punctures, P111 from #1212 alone (no new κ) #1419's open next question. Chooser nonce 35f10aeb31e6ffea65d714368cb51223, track complex-mechanisms, priority revision 2.
• Heads: base bce54fb3f895cbe5cd4c0eb477c3afeed100190f. 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, and Research: rank-8 rigidity passes to supersets — P105–P110 partial-frame punctures, P111 from #1212 alone (no new κ) #1419 (orthogonal pairs only). 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 (same session as Research: rank-8 rigidity passes to supersets — P105–P110 partial-frame punctures, P111 from #1212 alone (no new κ) #1419).
• Overlap: none on odd-pair P110 or on P109. Refreshed at 00:41Z (788 open PRs and 146 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). Submitted by the account holder. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

100-PR batch review at 4b707c22e3caad28172f3c80479684f4e402fd93.

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

Reviewed scope: Scoped rank-eight rigidity for both odd-pair P110 orbits, with complete finite premises and freshly checked DRUP zero exclusions. Combined with retained orthogonal-pair results this covers P110 in the eight-term model, not larger decoder classes.

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.