Research: rank-8 rigidity on every P105 unit via #1560's criterion, and (G) does not imply (D) for frame sub-families (no new κ)

[JosieElliston]

Scope

research/complex-rank8-rigidity-p105: a finite verification of a conditional theorem. No new κ, no
construction, and the selected frontier is unchanged. It answers the first open question of #1616 (P105) and
extends #1616 by one level.

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, exactly and in the standard library, for all 242802
orbits of 7-subsets of P112 (P105) under the 8-generator FRAME0-stabiliser subgroup H. So with #1560, every
P105 unit is rigid, and through #1212 §8 PR968's floor for masked decoders with exactly eight costly terms
reaches 57 = 72 − 15 at P105, the bottom of its 72 − p range.

How the replay became affordable

#1616 left P105 open because a stdlib replay cost about 25 ms per unit plus 1.65 million coverage words. Here:

• Per unit, about 5 ms instead of 17 ms (one Apple-silicon core). The tests are Research: rank-8 rigidity on every P107 and P106 unit via #1560's frame-incidence criterion (no new κ) #1616's. Rows are int
  bitmasks over deduplicated frame traces, (G) uses union-find with early exit, and (D)'s modular elimination
  first takes a greedy chain of rows that each add a new column. Those rows are independent over every field;
  only the row order changes. A GF(2) rank never reaches the target, so there is no binary shortcut.
• Coverage transporters are element indices into H, which verify.py rebuilds by breadth-first closure (8 s,
  about 700 MB), instead of words. COVER.bin.gz is 5.1 MB; UNITS.json.gz is 0.7 MB.

Evidence

• verify.py --cover: 13 exact checks. Model and the frame identity (6) on all 2025 frames; generators; H (172032
  elements fixing FRAME0); triple orbits; four coverage levels (2517 → 180, 18782 → 1614, 156627 → 19295,
  1651528 → 242802), each candidate the image of its unit under an indexed element of H; prove a frame-incidence zero exclusion criterion with a P107 fixture #1560's fixture and
  control; (G) and (D) fail on deficient frame families; a first-frame control.
• verify.py --part i/3: every P105 pivot is a live frame passing (D), (G), (W) and nonempty blocks, ranks
  modulo 1000003 as lower bounds with the seven kernel vectors as the matching upper bound. All 242802 pass.
• The registered replay passes on head d960f8a2 (PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed, about 4 min wall on 10
  cores; timeouts 120 + 3 × 340 s assume a 4-core runner). Also: Python 3.10 and 3.14 on --cover and a 1/50
  part, and the 20 tests/test_research_* tests.
• Mutation controls: a non-live pivot, a (W)-failing pivot and one changed level-7 element index each fail.
• In scratch comparisons the new criterion matched Research: rank-8 rigidity on every P107 and P106 unit via #1560's frame-incidence criterion (no new κ) #1616's exactly (verdict and statistics) on 300 P105 pivots,
  1200 random (unit, frame) pairs and 800 sub-family controls with (W), (G) and (D) failures.
• make_units.py (numpy) regenerates the data from scratch with an exact pivot search (about 9 min on 10 processes) and reproduces both committed data files byte for byte. Its census sums orbit sizes to exactly
  C(112,k), k = 4..7, and reproduces Research: rank-8 rigidity on every P107 and P106 unit via #1560's frame-incidence criterion (no new κ) #1616's unregistered P105 census and float-rank pivots exactly.
• A read-only adversarial review by OpenAI Codex found no soundness defect in the criterion, rank bound,
  coverage induction or part split. Its one low-severity point is addressed (see below). This is not a human
  review.

#1616's question "does (G) imply (D)?": no, for arbitrary frame families. New control.g-without-d: on
P105 unit (2,3,7,12,13,30,45) at pivot frame 543, an 80-frame sub-family of the live frames passes (W), nonempty
blocks and (G), but its incidence matrix on I_3 has rank 39, not 40. The certificate is exact: the integer vector
−1, −1, +1 on lines 42, 66, 93 kills every row and is independent of the seven h_3l. With all live frames the pivot
passes. So a structural (G) ⇒ (D) lemma would have to use that the family is all live frames. (Found by a random
scratch search: one hit in 16416 sub-families of 20–300 frames at 608 P105 pivots.)

Observation for the full-family question. tally_earlier.py (unregistered, stdlib, seconds) reruns the exact
criterion on all 17429 live frames that precede a pivot: 17210 fail (W) and 219 fail nonempty blocks. So in every
P105 unit the first frame passing (W) and nonempty blocks also passes (G) and (D).

Not covered

• prove a frame-incidence zero exclusion criterion with a P107 fixture #1560's Theorem 1 and Corollary 2 are cited, not re-derived; 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).
• Nine or more costly terms, and any decoder or priced profile.

Task and overlap

• Task: complex-rank-shape-decoder (revision 3, chooser nonce e29ae3210329806ff56c7142ba041cc6), scope
  changed: revision 3 retired complex-masked-decoder into this card, whose gate asks masked supports to
  change an assumption of the eight-term bound. This is a side result on that: at P105, masking does not. See
  TASK.json.
• Base: d040f15f. prove a frame-incidence zero exclusion criterion with a P107 fixture #1560 at eb9f6f15, Research: rank-8 rigidity on every P107 and P106 unit via #1560's frame-incidence criterion (no new κ) #1616 at 7b3e4624; this package reads neither's files.
• Overlap: claim DONE: structural zero-exclusion criterion with a P107 fixture (PR1560) #1502 (comment).
  Research: rank-8 rigidity passes to supersets — P105–P110 partial-frame punctures, P111 from #1212 alone (no new κ) #1419 (open) covers the P105 deletions that lie inside a frame through its superset lemma; this package checks every 7-subset directly and does not use it. No other P105 claim or PR as of 2026-10-14T05:11Z (refreshed overlap: 593 open PRs, 292 discussions).

Credit

an664 / Andrey Mas (#1560), burntout76 / Brett Halligan (#1212), themilkman216 (PR968), DreamingOfClouds (PR717),
and #1616/#1568/#1474 for the census method and the P105 search. Prepared with substantial assistance from Claude
Opus 5.5 via Claude Code.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at d960f8a276f752934ff37617da99e489ad5e9d5e.

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

Reviewed scope: Exact coverage and pivot-incidence verification for all 242,802 P105 units, with complete four-level transporter coverage and modular lower bounds matched by exact kernels. Inherits PR1560 rigidity and PR1212/F6 floor implications; selected-family shortcut counterexample is separately scoped.

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.