Research: rank-8 masked decoders are gauge-rigid on the 896 admissible 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-p104-admissible/. 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 336 type-(2,2,2,2) P104 units. Its PROOF.md §10 leaves the 896 admissible P104 units open (PR717's P104 among them). On #1212's pivot there, four classes keep only two live rows. For those classes Lemma 5 leaves a one-parameter family. This PR closes that orbit.

• What already holds. On Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212's admissible pivot, all 14 blocks carry a complete class, so Lemmas 5–6 fix the global scaling g on the 22 complete classes unchanged. Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212 §10 records that (I2), (I3) and (I4) hold; only (I1) fails.
• New: Lemma 9 (propagation). A two-row line i needs no in-class argument. Take a witness j in a complete class with T_i ∩ T_j = {k, k′}. Then (E2) gives a_ik ε_ik / g_k = a_ik′ ε_ik′ / g_k′, and the reverse order gives the same for b. If the witnessed pairs connect T_i, then a_i ∝ ε_i ∘ g and b_i ∝ ε_i / g, and (E0) fixes the product. On this unit, every two-row line has all six pairs witnessed, with 40 witnesses each. The lemma is linear, needs no zero exclusion for i, and uses no equation inside its class.

Consequence. In the E-graph model, every decoder with exactly eight costly terms has cst ≥ 56 on every admissible P104 unit, and PR717's P104 decoders attain it. With #1212 (type (2,2,2,2)) and #1389 (type (4,4), unmerged), rigidity then covers all 1260 P104 frame complements.

Evidence

• python3 -I -B research/complex-rank8-rigidity-p104-admissible/verify.py runs 11 exact checks and 4 controls in about 0.1 s. It is an independent implementation (own E8 model, classes, witnesses, gluing, propagation) and imports nothing from Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212. It recomputes the orbit type of the punctured frame. It cross-checks Lemma 9's algebra on the actual (E2) equations: rank 3, with the gauge vector as kernel. The controls show:
  • in-class conditions admit a non-gauge two-row solution;
  • witnessed cross-block equations reject it (28/40);
  • removing the witnesses at one column leaves a 2-dimensional family;
  • 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 proof-P104-11111111.drup. It replays G.P104-11111111.zero-exclusion, G.P104-11111111.class-witnesses, G.frame-orbits and G.pr717-p104-orbit, and matches the unit and pivot by root vectors. That is 22 checks, all passing locally.
• Passes on Python 3.10.21, 3.12.14 and 3.14.7. scripts/research_package.py: the metadata check and the Python 3.12 registered replay pass. test_research_* passes.

Not checked or not covered

• Nine or more costly terms (PR717's mixed-β problem).
• The intermediate punctures P105–P110. This is the package's open next question. Lemma 9 makes incomplete classes free once g is fixed, but each orbit needs its own zero-exclusion certificate.
• 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 obligation on the admissible orbit. Chooser nonce 6dba39f7133593e3228f8833f5d6594a, 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. Does not depend on Research: rank-8 masked decoders are gauge-rigid on P111 and type-(4,4) P104 units (no new kappa) #1389.
• Claim: CLAIM: global rank-8 rigidity on the 896 admissible P104 units (closes the last P104 orbit after #1212/#1389) #1395. This is the open next question of Research: rank-8 masked decoders are gauge-rigid on P111 and type-(4,4) P104 units (no new kappa) #1389.
• Overlap: no other claim or PR on global rigidity for the admissible units. 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 only.

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, the pivot, 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 P104 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. This session was quick because #1212's §10 had recorded exactly which finite input failed on each unit, and #1389's handoff named the next step. A short "which inputs hold" table in each package is cheap, and it was the most useful thing in the predecessors.

🤖 Generated with Claude Code

[Mister-Meeseeks]

100-PR batch review at 4e1ba3a1ded94772cce5f8c0ad667a92afd6c493.

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

Reviewed scope: Rank-eight rigidity on admissible P104 units via connected cross-class propagation, with exact finite witnesses and controls. Uses the retained zero-exclusion/orbit framework; closes the stated eight-term class, not arbitrary larger masked factorizations.

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.