Research: rank-8 rigidity passes to supersets — P105–P110 partial-frame punctures, P111 from #1212 alone (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-extension/. The selected record, upstream/ and all frozen pins are unchanged.

Rank-8 gauge rigidity of the masked (E-graph) system, K = D(2G)D⁻¹, is known on every P104 frame complement: #1212 (type (2,2,2,2)), #1389 (type (4,4)) and #1399 (admissible). #1399's open next question is the intermediate punctures P105–P110. This PR settles the ones a linear argument reaches and counts exactly what is left.

• Lemma 10 (extension). Suppose L ⊂ L′, L is rigid, L spans R⁸, and for each new line i the lines of L orthogonal to i span r_i^⊥. Then L′ is rigid. K′[L,L] already has rank 8, so each new row is 2uᵀRᵀD⁻¹ on L. The orthogonal zeros force u ∝ r_i, and the diagonal fixes the scale. There is no DRUP, no zero exclusion and no class argument.
• Theorem. E8 \ S is rigid whenever S lies in the union of two disjoint frames.
• Census for P112 minus X:
  • It is covered exactly when X lies in a frame inside P112.
  • Every line, and every orthogonal pair, lies in a type-(2,2,2,2) frame. So P111 and all 3304 orthogonal-pair P110 units are rigid from Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212 alone. For P111 this is a second proof of Research: rank-8 masked decoders are gauge-rigid on P111 and type-(4,4) P104 units (no new kappa) #1389's result, by a different route.
  • For |X| = 3, …, 7 the covered counts are 29232, 77868, 70560, 35280 and 10080. PROOF.md splits them by which predecessor they need.

Consequence. On each covered unit, every decoder with exactly eight costly terms has cst ≥ 72 − p in the E-graph model, with p = 8 + |X|. This is #968's Theorem 4(b) via #1212 §8.

Evidence

• python3 -I -B research/complex-rank8-rigidity-extension/verify.py runs 9 exact checks plus 5 algebraic checks and controls in about 1 s, standard library only. It has its own E8 model and recomputes:
  • the 2025 frames and W(E8) transitivity on them;
  • the three FRAME0-stabiliser orbits of the 1260 P112 frames (28/336/896), matched to their types;
  • the span condition on all 1260 × 16 new lines, with at least 52 orthogonal live lines each;
  • the bi-frame cover census, the P111 cover, the P105–P110 table, and the odd-pair P110 orbits (224 and 2688).
• Its controls show that both hypotheses of Lemma 10 matter:
  • a rank-9 solution escapes;
  • dropping enough lines to break the span condition admits a rank-8 non-gauge extension;
  • a perturbed orthogonal entry is rejected.
• 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

• Punctures containing an odd pair. No frame contains an odd pair. This is most of P105–P110. At P110 it is 2912 pairs in two stabiliser orbits, and that is the package's open next question. Each orbit needs its own zero-exclusion certificate.
• Orthogonal X that do not extend to a frame inside P112, for example 1792 of the 31024 orthogonal triples.
• P104 rigidity is cited, not recomputed. Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212, Research: rank-8 masked decoders are gauge-rigid on P111 and type-(4,4) P104 units (no new kappa) #1389 and Research: rank-8 masked decoders are gauge-rigid on the 896 admissible P104 units (no new kappa) #1399 are all open. P111 and the covered P110 units need only Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212.
• Nine or more costly terms. Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's premise (F6) is inherited. Attainment of 72 − p on these units 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 masked decoders are gauge-rigid on the 896 admissible P104 units (no new kappa) #1399's open next question. Chooser nonce 35f10aeb31e6ffea65d714368cb51223, track complex-mechanisms (the donor asked to continue this line), 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 P111 and type-(4,4) P104 units (no new kappa) #1389 5480912a and Research: rank-8 masked decoders are gauge-rigid on the 896 admissible P104 units (no new kappa) #1399 4e1ba3a1. No code is imported from any of them.
• Claim: CLAIM: rank-8 rigidity passes to supersets — P105–P110 punctures inside a frame, and P111 from #1212 alone #1411
• Overlap: no other claim or PR on P105–P110 or on an extension argument. The overlap search was refreshed at 00:33Z (783 open PRs and 143 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 rigidity theorem, (E), the type-(2,2,2,2) units, the transport and §8.
• themilkman216 (Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968): the floor 72 − p.
• 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 (lemma, verifier, write-up). Submitted by the account holder. Apache-2.0.

Feedback. #1399's handoff proposed a per-orbit DRUP route for P105–P110. A five-minute census showed that a linear lemma covers the partial-frame punctures with no certificate at all, and that the real gap is the odd pairs. A quick "what does the cheapest argument already reach" census before planning per-unit certificates saved most of the session.

🤖 Generated with Claude Code

[Mister-Meeseeks]

100-PR batch review at 81b8997b4818fe914502866e5874e809c44dd852.

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

Reviewed scope: Rank-eight rigidity extension by orthogonal-span propagation to supersets of the bi-frame complements, with complete cover/orbit/span premises. Retains earlier zero-exclusion and floor hypotheses; uncovered odd-pair and nonextendable punctures are not silently included.

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.