Research: rank-8 rigidity on every P107 and P106 unit via #1560's frame-incidence criterion (no new κ)

[JosieElliston]

Scope

research/complex-rank8-rigidity-p106: a finite verification of a conditional theorem. No new κ, no
construction, and the selected frontier is unchanged. It answers the open question of an664's #1560 for
every five- and six-line deletion. It supersedes the open questions of #1568.

Result

#1560's Corollary 2 gives K = D(2G_L)D⁻¹ for any live set L whose pivot frame satisfies (D), (G), (W) and all
14 blocks nonempty. This package checks those premises for every unit:

• P107: all 1614 orbits of 5-subsets of P112, including the 7 frame-contained ones.
• P106: all 19295 orbits of 6-subsets.

Every unit has a passing pivot. So with #1560, every P107 and every P106 unit is rigid. Through #1212 §8,
that gives PR968's floor for masked decoders with exactly eight costly terms: 59 = 72 − 13 at P107 and
58 = 72 − 14 at P106.

The route uses none of #1212's Lemmas 5–6, #1399's Lemma 9, #1419's Lemma 10, or any DRUP certificate.

#1560's open question. Its coverage-failure deletion (3,13,14,22,23) passes at
(2,5,15,16,30,41,109,116), and at 820 of its 850 live frames.

Coverage

Coverage extends #1474/#1568's candidate rule to three levels, with explicit generator words:

• 2517 quadruple candidates map onto 180 representatives.
• 18782 5-subset candidates map onto the 1614 P107 units.
• 156627 6-subset candidates (a P107 unit plus one line) map onto the 19295 P106 units.

The verifier replays every word.

Evidence

The verifier is stdlib-only, and every check is exact:

• --level 5 runs 11 checks in 36 s: model, the frame identity (6) on all 2025 frames, generators, triple
  orbits, the three coverage levels, prove a frame-incidence zero exclusion criterion with a P107 fixture #1560's fixture and control, a (G)/(D) negative control, a first-frame
  control, and the 1614 P107 units.
• --level 6 checks the 19295 P106 units in 59 s on 10 cores (558 CPU-seconds). The registered replay splits
  it into two parts.
• Ranks are computed modulo 1000003, which gives lower bounds; the seven exact kernel vectors give the matching
  upper bound.
• The registered replay passes (PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed), as do Python 3.10
  and 3.14, and the 20 tests/test_research_* tests.
• make_units.py (numpy) regenerates the data. Its census sums the orbit sizes to exactly C(112,5) and
  C(112,6).
• A read-only adversarial review by OpenAI Codex found no theorem-breaking issue. It covered the criterion
  against prove a frame-incidence zero exclusion criterion with a P107 fixture #1560's definitions, the rank and kernel argument, the coverage induction, and prove a frame-incidence zero exclusion criterion with a P107 fixture #1560's §4 and §6. Its
  one low-severity wording point is applied. This is not a human review.

Not covered

• prove a frame-incidence zero exclusion criterion with a P107 fixture #1560's Theorem 1 and Corollary 2 are cited. I read the proof and found no gap, but prove a frame-incidence zero exclusion criterion with a P107 fixture #1560 is open and has
  had no independent human review.
• Other inherited steps: Research: rank-8 complex decoders are gauge-rigid; E-graph eight-term floor (no new kappa) #1212's (E), §8 and §10 transport, and PR968's (F6).
• P105, and nine or more costly terms. P105 has 242802 orbits, and the census sizes sum to C(112,7). An
  unregistered numpy search with float ranks found a passing pivot for every orbit; the first live frame
  passed for 230560 of them. That search is not exact and is not in this package, because a stdlib replay of
  that many units exceeds the registered budget. See the open questions.

Task and overlap

• Task: complex-masked-decoder, scope refined; see TASK.json.
• Base: ddb4371c.
• Overlap: prove a frame-incidence zero exclusion criterion with a P107 fixture #1560 is an664's DONE, and this package uses its criterion with credit. I read prove a frame-incidence zero exclusion criterion with a P107 fixture #1560 at
  5bebfd00; its current head eb9f6f15 changes only §7 and §8. Research: sharp 27-deletion boundary for one-sided E8 masked rigidity #1609 (ToikY) is adjacent but separate: it
  keeps one factor root-linear and varies an arbitrary deletion count. No other P107 or P106 claim was open as of
  2026-10-14T04:05Z.
• Coordination note: my claim for this work was misposted to an unrelated repository (since deleted), so
  there was no claim in 1502 before this PR.

Credit

an664 / Andrey Mas (#1560), burntout76 / Brett Halligan (#1212), themilkman216 (PR968) and DreamingOfClouds
(PR717). Prepared with substantial assistance from Claude Opus 5.5 via Claude Code.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at 7b3e4624e31647fc2a8b777a181823739b12b36e.

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

Reviewed scope: Explicit three-level orbit transport and exact modular lower bounds with rational kernel upper bounds certify the stated frame-incidence premises for every P107/P106 unit. Rigidity and decoder floors remain conditional on the named PR1560/PR1212/PR968 interfaces; P105 is not established here.

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.