Research: connected E8 frame subfamilies need not saturate diagonal incidence

[navinor]

Result and exact boundary

347 genuine E8 frames on the fixed P107 live set have all 28 two-point
frame-intersection graphs connected, but their coordinate-2 diagonal-incidence
matrix has rational rank 43 instead of the required 44. An explicit +/-1
vector supplies an eighth independent kernel direction. This refutes
(G) implies (D) for selected frame subfamilies, not for the family of all
live frames.

The complete 877-frame family satisfies all eight required ranks. Adding one
explicit omitted frame, whose extra-vector sum is 2, already repairs all eight
ranks. No nongauge decoder, supplier, copied-information floor, physical
adoption, new kappa or selected-frontier change is claimed. In particular this
does not refute the predecessor theorem, which assumes (D) independently.

Task card and sources

• Assigned track: complex-mechanisms; OS nonce
  0d6468a107c6f96e18bdef01e3e8f622; original card
  complex-rank-shape-decoder, scope key 02b31ebd160542c86299.
• Changed scope to the connectivity-versus-rank handoff:
  Handoffs and requests for independent checks #1166 (reply in thread).
  This is prerequisite-verification side work, not satisfaction of the card's
  physical rank-shape acceptance gate. No proposed supplier means no replacement
  moment or a<D/E invoice; selected baseline bindings and the card's ceiling
  positive control were replayed.
• Base: 858771bcca3b8dd5d4ef4aa682a13cc9b7e23920.
• Concrete object: FRAME0=(0,1,6,11,20,29,106,119),
  X=(32,45,73,78,88), F=(2,5,8,9,30,41,108,117); coordinate 2;
  select all live frames annihilating the recorded signed vector.
• Criterion/fixture: prove a frame-incidence zero exclusion criterion with a P107 fixture #1560
  at eb9f6f15f15704949cbd25ca8fbf58063084dd8e.
• Open implication question: Research: rank-8 rigidity on every P107 and P106 unit via #1560's frame-incidence criterion (no new κ) #1616
  at 7b3e4624e31647fc2a8b777a181823739b12b36e.
• Entry and immediate refined-object overlap checks were run before computation.
  Final refresh at 2026-10-14T05:10Z covered 593 open PRs and 292 discussions,
  complete pagination, plus local notes and ledger. Relevant heads were unchanged.
  No competing selected-subfamily investigation was found; unsearched diffs and
  closed work outside local notes remain coverage limitations.
• Claim: Handoffs and requests for independent checks #1166 (comment).

Files and verification

Only research/e8-connectivity-incidence-boundary/ is added:
README.md, TASK.json, SOURCES.json, VERIFY.json, CERTIFICATE.json,
verify.py, test_verify.py. Frozen inputs and selected record are unchanged.

python3.12 -B scripts/research_package.py --package research/e8-connectivity-incidence-boundary --check-only
python3.12 -B scripts/research_package.py --package research/e8-connectivity-incidence-boundary

Observed locally on Windows, Python 3.12: metadata/pins PASS; registered replay
PASS; 2 unittest groups PASS, including five semantic/receipt rejection cases.
Verifier output:

PASS_E8_CONNECTIVITY_INCIDENCE_BOUNDARY live=877 selected=347 graphs=28 rank=43 required=44 canonical_kernel=7 extra_kernel=1 full_family_control=PASS one_frame_repair=PASS

Integer kernel identities supply exact upper rank bounds, modular nonzero minors
supply lower bounds, and the bounds meet. No floating point or predecessor
checker imports. A separate read-only AI review independently regenerated the
geometry/graphs and used Fraction elimination to reproduce selected/full ranks
43/44, canonical/augmented kernel ranks 7/8 and repaired rank 44. Its two minor
test-feedback items were addressed and the registered replay rerun.

Remaining obligations and credit

The all-live-frame-family implication remains open and is the next question
in the README/TASK. No downstream rigidity, cost-floor or full multiplication
proof obligations are discharged. No full selected cold replay, independent
human review or hosted-CI success is claimed.

Credit Andrey Mas / an664 for the criterion and fixture, Josie Elliston for the
handoff and modular-rank methodology, and Brett Halligan / burntout76 /
Ninefold (#1212),
DreamingOfClouds (#717),
themilkman216 (#968)
for the retained geometry/decoder lineage.

Prepared by Aaron Wang / navinor, with help from chatgpt. Substantial AI
assistance includes research, implementation, writing and read-only review.
Apache-2.0.

Workflow feedback: the concrete-term recheck found the precise open handoff;
the annihilator-selected first vector falsified the broader implication without
a larger search. The supplied Python-3.12 path lacked a backslash: the working
path here is C:\Users\warha\.local\bin\python3.12.exe. PowerShell 5's default
redirection writes UTF-16; avoid it for JSON intended for UTF-8 tools. No packages
were installed.

[Mister-Meeseeks]

Batch review at 66711b58077a660341359e9baacedc304adc791f.

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

Reviewed scope: Exact chosen-family E8 counterexample to connectivity implying diagonal-incidence saturation: 347 genuine frames, 28 connected graphs but one deficient rank, with exact kernel/modular bounds and one-frame repair. Does not refute the all-live-frame claim or rigidity theorem.

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.