Research: (G) does not imply (D) for full live families at eight deletions (no new kappa)

[JosieElliston]

Scope

research/full-family-g-without-d: an exact counterexample to a proposed verification shortcut, plus scoped
search evidence. No new κ, no construction, and the selected frontier is unchanged. It answers open question 1
of #1650: does a pivot that passes (W), nonempty blocks and (G) always pass (D) when the frame family is all
live frames? The earlier counterexamples (#1650's control.g-without-d, navinor's #1649) use sub-families.

Result

No, once eight lines are deleted from P112. Fixture 1: X = (3, 4, 8, 9, 107, 108, 117, 118), pivot
F = (19, 25, 37, 47, 67, 82, 94, 115), all 784 live frames of P112 \ X.

• Nonempty blocks, (W) and (G) hold.
• (D) fails at pivot k = 2: the live-frame incidence on I_2 (52 lines) has exact rank 44 over Q, not 45.
• Certificate: x = +1 on lines {30, 31, 40, 41} and −1 on {2, 5, 7, 10}. Every live frame is orthogonal to x, and
  x is outside the span of the seven h_2l, which also lie in the kernel. So the kernel has dimension at least 8.
• Mechanism. Every one of the 2025 E8 frames meets the coordinate-half D4 subsystems (lines on coordinates
  0–3, and on 4–7) in equally many lines. FRAME0 removes 4 lines from each and X removes 4 more. The 4 lines left
  in each are all odd to F[2], and x is the difference of their indicators.
• Restoring any one of the eight deleted lines makes the same pivot pass all four conditions.
• The unit is still rigid through other pivots: 674 of its live frames pass all four conditions.

Fixture 2 has the same pattern at another pivot, using a non-coordinate pair of D4 subsystems. Its
certificate is checked the same way.

Evidence

• verify.py (standard library, about 3 s): both fixtures. It checks the premises on the full family, exact
  Fraction ranks for every k (only the stated k fails), the ±1 certificates, and the D4 frame identity on all
  2025 frames. Controls: 8 restore checks per fixture, a mutated certificate, and the pivot alone failing (G). Its
  conditions are re-typed from 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) #1650's criterion, so it does not depend on that unmerged package.
• Unregistered search (search/, imports 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) #1650's verify.py through PR1650_PACKAGE):
  • Deletion walks: fix a pivot and delete lines while (W) and blocks hold. 176 of 5000 walks reach a
    (G)-pass/(D)-fail step. Greedy restoration shrinks them to locally minimal sets of size 8 to 28, none below 8.
  • D4-mechanism census: all 7 D4 pairs that contain FRAME0 as 4 + 4, all 1260 pivots disjoint from FRAME0, all k.
    The deletion this mechanism needs has size 4, 6, 8 or 10, and (D) fails only at size 8 (32256 cases).
  • Random 8-, 9- and 10-line deletions, every live frame: no (G) or (D) failure.
  • Every live frame of all 1614 P107 units (1393377 frames): 1386438 pass all four, 6936 fail (W), 3 fail
    blocks, and none fails (G) or (D).
• The registered replay and --check-only pass. verify.py also passes on Python 3.10 and 3.14 and refuses -O; the 20 tests/test_research_* tests pass.
• A read-only OpenAI Codex review found that the four conditions match 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) #1650's criterion and that the rank-44 proof
  is exact. It independently reproduced both (D) verdicts and all 16 restores with 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) #1650's code. Its three
  low/medium points (a placeholder, a search-script label, and two README counts not in verify.py) are fixed. This is not a human review.

Not covered

• |X| ≤ 7 (P105–P111) is still open. The searches found no counterexample there, and the D4 mechanism needs
  exactly eight deletions, but neither is a proof. An every-frame census of the 19295 P106 units would take about
  40 CPU-hours (9 CPU-s per unit at P107).
• Nothing in Research: rank-8 rigidity on every P107 and P106 unit via #1560's frame-incidence criterion (no new κ) #1616 or 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) #1650 changes, since both check (D) explicitly.
• prove a frame-incidence zero exclusion criterion with a P107 fixture #1560's Theorem 1 and Corollary 2 are cited, not re-derived.

Credit

an664 / Andrey Mas (#1560, criterion); navinor / Aaron Wang (#1649, the sub-family case and the full-family
question); #1650/#1616 (implementation and question); burntout76 (#1212), themilkman216 (#968) and
DreamingOfClouds (#717) for the model. Chooser nonce 083eb6ba0dd52fa857761981ccb9156c drew recursive-interfaces.
Its three cards were crowded tonight (the plane-port lever is claimed in #1587), so this session took a recorded
open question instead. TASK.json records the scope as changed.

Prepared by JosieElliston with substantial assistance from Claude Opus 5.5 (Anthropic) via Claude Code. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at 5c959793a63332f54e7436cd8ded0e979774117e.

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

Reviewed scope: Two exact full-live-family counterexamples to dropping diagonal-incidence rank condition at eight deletions, with integer extra kernels and one-line restoration controls. Does not refute established rigidity or prove absence at smaller deletions; broader search tables remain author-reported.

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.