[aliromman]

Every characteristic-zero E7 frame relation with minimum support twelve belongs to one of two quadratic types: 630 rank-six supports and 1,008 rank-two supports. Every support has exactly one scalar family of relations. This completes the equality question in Nick Petalas's PR #5238. It gives a complete list of sparse obstructions at the decoder-incidence boundary. No new complete kappa is claimed.

The proof, exact receipt, verifier, source pins and validation record are pinned to 700efb538a3d919409c9506d117846b7895b02ea. The result covers all fields of characteristic zero and every E8 root slice. It classifies all frame-sum relations of support twelve, including inputs not supplied as quadratics. The separate arbitrary-deletion extension in the released handoff remains open.

The checker enumerates all 134,217,727 nonzero binary frame-code words and finds 1,638 minimum supports. Exact root reflections split these into the two stated orbits. Both representatives have six positive and six negative values. Exact rational rank eleven proves a unique lift on each twelve-line support. The written proof uses rational minors to extend the result to arbitrary characteristic zero. A diagonal ansatz alone misses the second type.

python3.12 -B scripts/research_package.py --package research/e7-minimum-support-classification
python3.12 -B scripts/check_research_commit.py --package research/e7-minimum-support-classification

Expected: PASS_E7_MINIMUM_SUPPORT_CLASSIFICATION, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES. Python 3.12 and the standard library suffice. Registered normal and optimized checks pass, including all 63 root reflections on all 135 frames, exact quadratic ranks, ambient transport, 24 coordinate-omission controls and four corruption controls. Committed-byte and export checks pass. The scientific files also pass a read-only Git-free optimized replay with empty PATH; every exported byte is unchanged. Actual corruption of the retained source is rejected before execution.

Canonical base and final refresh: df0137efd35e11b4445734bf612bdd4c7ebc330c, priority revision 7. The authenticated selected metadata remains at conditional kappa 0.00310045, limited by the main-transform row. Only this package is added. The advance claim and final complete metadata search found no competing equality claim. Full-history preflight is unavailable in this shallow checkout; immediate ancestry and exact source bytes were checked.

Credit Nick Petalas for the support bound, frame-code geometry and injectivity proof, retained in the source note. Credit JoyboyBrian for the handoff and the source note's Josie Elliston, Andrey Mas, Aaron Wang, Brett Halligan, themilkman216 and DreamingOfClouds lineage. New equality census, orbit classification, characteristic-zero argument, proof and checks: Ali Romman with substantial OpenAI Codex assistance. Apache-2.0.

No arbitrary-deletion extension, new decoder floor, physical supplier, native/outer admission, selected cold replay, full historical suite, Lean, Docker, independent human review or formal proof-kernel acceptance is claimed. CI is pending at publication. The next useful step is the missing extension lemma or a literal comparison of the two types with the next deletion boundary; neither is reserved here.