Research: rank-8 masked decoders are gauge-rigid on P111 and type-(4,4) P104 units (no new kappa)

[JosieElliston]

Change and mathematical scope

This is a scoped structural result: no new κ, no construction. It adds only research/complex-rank8-rigidity-p111-p104-44/. The selected record, upstream/ and all frozen pins are unchanged.

PR #1212 (burntout76) proves rank-8 gauge rigidity, K = D(2G)D⁻¹, of the masked (E-graph) system on P112, P120 and the type-(2,2,2,2) P104 units. Its PROOF.md §10 leaves two gaps, and this PR closes both:

• P111 (all 112 units): Lemma 5′ covers a class with three live rows. Case (ii) has rank 1 < 3, as Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212 noted. The new part is the b side: the three pairwise sign products span the non-trivial characters, so γγ′ is constant.
• Type-(4,4) P104 (28 units): Lemma 6′ is a gluing lemma. It needs only connectivity of the usable classes through each pivot pair, plus an explicit three-triple path. The argument is multiplicative, so it holds over any field of characteristic 0. It works with 12/14 usable blocks.

Consequence. In the E-graph model, decoders with exactly eight costly terms have cst ≥ 63 on every P111 unit (attained by PR717's P111 decoders) and cst ≥ 56 on every type-(4,4) unit. Sharpness of 56 on the type-(4,4) units is not checked.

Evidence

• python3 -I -B research/complex-rank8-rigidity-p111-p104-44/verify.py runs 26 exact checks and 4 controls in about 1 s. It is an independent implementation with its own E8 model, classes, witnesses, gluing and assembly, and imports nothing from Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212. The controls show:
  • two-row classes are not forced;
  • P111 needs Lemma 5′;
  • the path condition rejects a rank-17 block set;
  • a perturbed entry breaks (2).
• --pr1212-dir <checkout of 57099cb5> checks the git blobs of Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212's verify.py and its two .drup files. It then replays their G.P111.zero-exclusion, G.P104-44.zero-exclusion, G.line-orbits and G.frame-orbits, and matches the pivots by root vectors. All pass locally on Python 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

• The 896 admissible P104 units, which have two-row classes. This is the open next question. PR Research: masked rank-8 complex decoders are locally rigid on all P104 frame-pair punctures (no new kappa) #1292 gives only local rigidity there.
• Nine or more costly terms.
• Zero exclusion is Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212's DRUP result, replayed but not re-derived. PR968's F6 premise is inherited.
• There has been no independent human review of the written proof.
• All inherited conditions of the selected record remain.

Task and overlap

• Task / scope: complex-masked-decoder (scope key 2a6f2c726b78f9ae7d26), refined to the Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212 §10 obligations on P111 and type (4,4). Chooser nonce 118c1c4d3856655a6b7cb222d8d50aef, track complex-mechanisms, priority revision 2.
• Heads: base bce54fb3f895cbe5cd4c0eb477c3afeed100190f; depends on Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212 head 57099cb5684f38877f31d149d598801671a633db.
• Claim: CLAIM: global rank-8 rigidity on P111 and type-(4,4) P104 units (completes #1212 §10 two gaps) #1384. This is the next step named in Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212's handoff in Handoffs and requests for independent checks #1166.
• Overlap: no other claim or PR on P111 or type-(4,4) global rigidity. The overlap search was refreshed before submission. Research: masked rank-8 complex decoders are locally rigid on all P104 frame-pair punctures (no new kappa) #1292 and Research: audit P104 tangent-to-orbit local regularity (no new kappa) #1367 address local (tangent) rigidity on P104.

Credit

• burntout76 / Brett Halligan (Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212): the theorem, the proof skeleton, zero exclusion and the transport.
• themilkman216 (Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968): the eight-term 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 and the P111 decoders.
• Navinor (Research: masked rank-8 complex decoders are locally rigid on all P104 frame-pair punctures (no new kappa) #1292): local P104 rigidity.

Substantial AI assistance: Claude Opus 5.5 via Claude Code (proof, verifier, write-up). Submitted by the account holder. Apache-2.0.

Feedback. The biggest friction was crowding: the overlap search returned 38 matches for this card, and finding an unclaimed gap meant reading handoffs, not the card. A structured "open obligations" index built from package open_questions would have saved most of the orientation time.

🤖 Generated with Claude Code

[Mister-Meeseeks]

100-PR batch review at 5480912a95c3edd0a9ef7532dcc7fc7060d12380.

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

Reviewed scope: Scoped rank-eight masked-decoder rigidity extension to P111 and type-(4,4) P104, using a three-row class lemma and explicit multiplicative gluing paths. Exact finite premises pass; zero exclusion and orbit transport remain inherited from the reviewed PR1212 package.

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.