Research: rank-8 rigidity at P107 reduces to zero exclusion — every open unit has a pivot passing (I3), (I4), gluing and Lemma 9 (no new κ)

[JosieElliston]

Change and mathematical scope

This is a scoped reduction: no new κ, no construction, and no rigidity theorem at P107 yet. It adds only research/complex-rank8-p107-pivots/. The selected record, upstream/ and all frozen pins are unchanged.

It is the complementary part of an664's discussion 1502 claim, which targets a structural (I2) zero-exclusion lemma toward P107. This PR does not touch (I2).

Question. #1419's extension lemma covers P112 minus X whenever X lies inside a P112 frame. For |X| = 5, that leaves 1607 of the 1614 orbits of 5-subsets. These are orbits under the 8-generator FRAME0-stabiliser subgroup of 172032 elements, counted by the census. The #1212/#1399 method needs a pivot frame F inside the live set, and then five per-unit inputs: (I2), (I3) with live witnesses, (I4), gluing, and Lemma 9. Do the inputs other than (I2) survive at P107?

Result. Yes, on every open unit. Each of the 1607 units has a pivot passing (I3), (I4) (all 14 blocks complete), gluing, and Lemma 9 connectivity with its sign identity and exact rank-3 algebra. The pivots are listed in UNITS.json, and there are 223 distinct ones. So at P107, (I2) is the only per-unit obligation left.

• A structural (I2) lemma for these pivots, together with Research: rank-8 rigidity passes to supersets — P105–P110 partial-frame punctures, P111 from #1212 alone (no new κ) #1419 for the 7 frame-contained orbits, would make every P107 unit rigid.
• That would give Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's floor 59 = 72 − 13 for decoders with exactly eight costly terms.
• No P107 unit is blocked by the other inputs.

The pivot choice is not vacuous. On most units the first frames inside the live set fail (I4), because five deletions empty both classes of some block. The verifier checks this as a control.

Coverage without enumerating 134M 5-subsets. This extends #1474's candidate rule by one level:

1. Every quadruple is equivalent to a triple-orbit representative plus one line. Each of those 2517 candidates is mapped onto one of 180 quadruple representatives by an explicit word.
2. So every 5-subset is equivalent to a quadruple representative plus one line. There are 18782 such candidates:
   • 72 lie inside a P112 frame. That property is invariant under the group, so these are Research: rank-8 rigidity passes to supersets — P105–P110 partial-frame punctures, P111 from #1212 alone (no new κ) #1419's case.
   • Each of the other 18710 is a word, of length at most 10, applied to one of the 1607 units.

verify.py replays all words.

Evidence

• python3 -I -B research/complex-rank8-p107-pivots/verify.py runs 7 exact checks in about 2.6 minutes, standard library only. It uses the P108 package's code with the CNF and RUP parts removed.
  • Coverage: the generators, the triple orbits, and replay of every cover4 and cover5 word.
  • Control: the first frame inside the live set fails on u0001, on blocks.
  • Per unit: the pivot's D4 structure, (I3) with 12 witness conditions per complete class, each with a live witness, (I4), gluing, Lemma 9 connectivity, sign identity and rank-3 algebra, and the §7 assembly identity on all 107 live lines.
• Census, run once and not needed by the argument. make_units.py (numpy) enumerates the group of 172032 elements, labels quadruples and keys 5-subsets canonically. The 1614 orbit sizes sum to exactly C(112,5) = 134153712, and the quadruple count of 180 matches Research: rank-8 masked decoders are gauge-rigid on every P108 unit — 168 orbit certificates, census-free coverage (no new κ) #1474's census. It regenerates UNITS.json and COVER.json byte-identically.
• Passes on Python 3.10, 3.12 and 3.14. scripts/research_package.py --check-only and test_research_* pass.
• AI review: a read-only adversarial review by OpenAI Codex found no soundness gap. It checked the two-level coverage, exact word replay, frame-containment invariance, the per-unit conditions against the P108 logic, and that P107 rigidity and the floor 59 stay conditional on (I2). Its two wording points are applied: orbit counts are credited to the census, and "12 witness conditions" replaces "12 live witnesses". This is not a human review.

Not checked or not covered

• (I2) zero exclusion at P107. It is the remaining obligation and an664's claim. Without it this PR proves no rigidity.
• 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. 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.
• P106, P105 and nine or more costly terms. These are the open next questions in the README.

Task and overlap

• Task / scope: chooser seed 09fd57098a1c01a7c8d36c27e6146201 gave complex-masked-decoder (scope key 7acfe4d9cac04f7ba4cb), track complex-mechanisms, priority revision 2. It is refined to the non-(I2) inputs at P107, because an664 holds the (I2) claim.
• Heads: base ef5f39403610d231a0c27e8a6b4f08c10209bcd6. 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 every P108 unit — 168 orbit certificates, census-free coverage (no new κ) #1474 311dd4e1. It imports nothing from them.
• Claim: DONE: structural zero-exclusion criterion with a P107 fixture (PR1560) #1502 (comment)
• Overlap: the only overlap is an664's (I2) claim, which this complements, and my own predecessors. Refreshed at 02:56Z (714 open PRs and 217 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 per-unit inputs, 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.
• an664: the parallel (I2) investigation that this complements.

Substantial AI assistance: Claude Opus 5.5 via Claude Code (census, pivot search, verifier, write-up); OpenAI Codex (read-only review). Submitted by the account holder. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at 38d074f0c506983b3047975b66fdb4ab4c630bc8.

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

Reviewed scope: Exact two-level transport coverage and all 1,607 P107 pivots pass the non-zero-exclusion premises. This is a conditional reduction only: I2 is still missing, so the package establishes neither P107 rigidity nor a new decoder floor by itself.

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.