prove a frame-incidence zero exclusion criterion with a P107 fixture

[an664]

This supplies a structural route to zero exclusion in the rank-eight E-graph problem: three finite frame-incidence conditions force every live support product a_ik b_ik=4 over any characteristic-zero field. The proof uses full-frame projector identities and a complete four-line class plus one opposite-class witness. It assumes neither symmetry nor nonzero support entries, and uses no Boolean solver certificate.

The self-contained checker establishes the conditions for a specific P107 deletion X=(32,45,73,78,88) outside the frame-contained family. Its 877 live frames provide eight exact rank certificates, 28 connected graphs and twelve qualifying 4+1 blocks. Seven fully live blocks alone miss three pairs. All fourteen blocks are nonempty for this fixture, supplying the separate premise for its gauge corollary.

Local normal and optimized Python 3.12.14 runs reproduce the certificate. Eleven recorded outcomes include an alternate positive fixture, a coverage-failure control, deleted-pivot rejection and certificate-tamper rejection. The registered optimized hook passes in under one second here. A separate GPT-6 Pro review found no flaw in the transcribed algebra and supplied small exact checks that also replay locally; it did not receive the package verifier or recompute its incidence ranks/connectivity. The universal implication remains a written proof, not a kernel theorem. The added exact 13-line boundary control checks all 13 diagonal and 108 ordered orthogonal-zero equations, rank eight and 15 support zeros per factor; it demonstrates why the global shared-formula hypotheses cannot be omitted. Fourteen further local outcomes cover the replacement certificate and mutation rejection, including rejection before certificate regeneration can overwrite a bad fixture. Neither restricted nongauge control is a P107 counterexample.

This addresses the structural I2 handoff under #1502. Josie Elliston's complementary I3/I4 pivot investigation remains separate. Prior rigidity, propagation and census results retain credit to Brett Halligan / @burntout76 / Ninefold (#1212), Josie Elliston (#1399, #1419, #1474), @DreamingOfClouds (#717) and @themilkman216 (#968). Prepared by Andrey Mas with substantial GPT-6 Pro mathematics/checker assistance through ChatGPT and Codex coordination, packaging and local replay.

No all-five-deletion theorem, new cost floor, supplier improvement or selected-frontier change is claimed. Failure of the sufficient criterion at the recorded bad fixture is not a nongauge P107 counterexample. The alternate positive fixture lacks the extra nonempty-block premise, so its gauge corollary is not claimed. Changes are confined to the new package.

Validation: python3.12 -B scripts/research_package.py --package research/frame-incidence-zero-exclusion.

[Mister-Meeseeks]

Batch review at eb9f6f15f15704949cbd25ca8fbf58063084dd8e.

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

Reviewed scope: Written projector/incidence theorem proves structural zero exclusion under D/G/W, with exact modular lower and integer-kernel upper rank certificates. The P107 fixture and extra nonempty-block gauge premise pass; no all-five-deletion coverage is claimed.

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.