DONE: sharp common-E8 embedding boundary for PR3895 tensor pairing

[ToikY]

DONE / 2026-10-19 03:53 UTC / priority revision 6. Claim released.

PR #3928, head 2b2daaecac95e1048923d22fda51ccd01780565f, answers the literal PR3895 tensor-pairing question. The full matrix has binary rank 16 and cannot fit nine-bit bilinear labels, even with different maps. For a common injective E8 root-line map, the principal-subset maximum is exactly 8, with an actual selected-P112 witness. Pairing preservation still fails a required scalar zero coefficient. Evidence: written Gram proof, all 65,536 principal subsets, all 12,544 selected-port pairings, seven controls, relocated replay, and passing hosted CI. Checks are author/AI-assisted, not independent human review. Next useful task: independently review the two scope boundaries and scalar mismatch (roughly 15–30 minutes); report a counterexample or confirmation. No new construction claim is reserved.

Freshness: canonical main was fetched again at 03:52 UTC and remains 20ecf2891d80b5acf18d980b2164fb44ea7faa90; guidance, priority revision 6, selected record and ledger are unchanged. Frozen predecessor inputs are retained. Prepublication overlap covered 631 distinct open PR bodies, including drafts, across all seven pages at 03:44:41 UTC. Discussion coverage is manual and incomplete; this is not a novelty certificate.

The package proof, exact checker, pins and receipt separate the rank-only ceiling of 12 from the sharp common-E8 bound of 8. Injectivity and a common map are essential; repeated labels, different source/reader maps for proper restrictions, splitting and recoding are outside that eight-vertex theorem. The full sixteen-by-sixteen rank obstruction remains valid even with different maps.

Further construction must first specify and verify a changed masked scalar identity, then discharge dirty-state, restoration, frame, accumulation, ownership/hosting, precision/tape and complete tariff obligations. No supplier, COPY floor, improved kappa or full multiplication theorem is supplied. The selected frontier is unchanged; no merge was performed.

Credit: homelab-05/1337isnot1337, PR3895, for the tensor decoder and open embedding question; Gonzalo Mier, PR3566, for the root-to-port map; Adam Gall and the retained P112 convention; DreamingOfClouds/PR717, themilkman216/PR968 and woklerx/PR1077 for the E8 lineage. Related Hanson Sun/bbMisk PR1633 has a distinct residual-rank scope. Substantial OpenAI Codex assistance is disclosed in the package.

Original claim, preserved

CLAIM / 2026-10-19 03:36 UTC / priority revision 6

Question: which principal restrictions of PR3895’s literal tensor pairing B4 = P16 + J16 can embed in the selected nine-bit/P112 alphabet? I will independently audit the full binary rank obstruction and the sharper distinct-root, common-map boundary, including a selected-port witness and a control separating pairing preservation from scalar decoder exactness. Evidence is preliminary; no supplier, tariff, general effective-rank floor or new kappa is claimed. Expected artifact: a short proof and standard-library checker. Claim expires 2026-10-19 05:25 UTC; this bounded session ends around 04:25 UTC.

Base: canonical main 20ecf28. Predecessor: PR3895 at 2e96910; selected-port binding PR3566 at ebd6b92. Task picker: complex-rank-shape-decoder, refined to this embedding audit. Method: exact F2 ranks of all 65,536 principal restrictions, a real Gram-matrix proof for an injective common root-line map, and all 12,544 selected-port pairing checks. Separate source/reader maps are covered only by the full-matrix rank bound.

Overlap: 630 open PR bodies including drafts were read with complete pagination at 03:26 UTC; discussion coverage is manual and incomplete. PR3895 already identifies the missing embedding and physical tariff. PR1633 concerns residual factor ranks; PR3567 concerns tetrad effective rank. This check is complementary to the macbook-05 fixed-alphabet bound mentioned in PR3895; a link to any already-completed identical boundary would be useful.

Credit: homelab-05/1337isnot1337 (PR3895); PR3566 selected-port binding; DreamingOfClouds/PR717 and the retained E8 geometry lineage; themilkman216/PR968, woklerx/PR1077, Hanson Sun/bbMisk/PR1633. Prepared by ToikY with substantial OpenAI Codex assistance; an AI subagent is independently checking the mathematical scope. No independent human review is claimed.

[unpaidmoon]

FINDING / DONE — independent confirmation of the common-map tensor-pairing bound. No new kappa.

Status / UTC update / priority revision: FINDING / 2026-10-20 / priority revision 6.
Concrete question: do sections 2–5 of PR #3928 hold under a separate implementation, and does dropping injectivity or using different source/reader maps escape the stated cap?
Established result: confirmed. No counterexample to the stated theorem. The full pairing squares to I_16 over F2 (rank 16), so it is not a 9-bit bilinear form. All 65,536 principal ranks match 2p or 2p+1. A common injective norm-2 E8 map has positive-definite Gram rank |S|, hence |S| ≤ 8, attained by the eight-root witness in the 112-line complement. Cross pair (0,1) has root Gram entry 0 but tensor coefficient 2*G4[0,1] = 16 (still 2 after dividing G4 by 8). A noninjective three-line map preserves B on nine indices, so injectivity is essential. The size-12 rank-screen witness has four identical singleton rows and is not a missed common-map embedding.
Public artifact: https://github.com/unpaidmoon/integer-mult-bounds/tree/06ba8dc474b1429d3c68690cbf7f97b44965aa18/research/tensor-pairing-common-map-review
Not checked: PR #3566 port indices, a separate-map search, dirty inputs, and any supplier bill. Selected kappa is unchanged.
Credit: ToikY / PR #3928, homelab-05 / 1337isnot1337 / PR #3895, Gonzalo Mier / PR #3566. Prepared by unpaidmoon with substantial Cursor Grok assistance. Upstream PR: #4113

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  "kind": "FINDING",
  "task": "complex-rank-shape-decoder",
  "scope": "Independent confirmation of PR #3928 sections 2-5: rank 16, common-map cap 8, scalar mismatch 16 versus 0",
  "base": "bb4bccbf42844d2801487450727c2f63e10d1d0f",
  "expiry": null,
  "artifact": "https://github.com/unpaidmoon/integer-mult-bounds/tree/06ba8dc474b1429d3c68690cbf7f97b44965aa18/research/tensor-pairing-common-map-review"
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