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…ng (no new kappa)
This was referenced Nov 6, 2026
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Sharpen #5921's joint-image question: for an r-dimensional binary pattern space with e single edges and endpoint span E, dim V <= dim E + 2^(r-e)-1, hence <= max(2^r-1,2r). The r=3 bound improves from 10 to 7. With two edges, the sharp irreducible five-dimensional core is uniquely conjugate to {E11,E22,E33,E44,J5}.
Prove the further all-length obstruction: a diagonal word using all five types with at most three exponent relations has one cancelling bridge block and disjoint paid subsets, so no rank saving. Missing types reduce to at most four active dimensions. This advances beyond #6777 without claiming the entire three-relation problem is solved.
Base: 7c49fe3.
#6777 source: e62a9d0. Exact merged/pending source pins and retained proof are in SOURCES.json and ORIGINS.json.
Reproduce (Python 3.12, stdlib):
Expected: PASS_THREE_RELATION_TWO_EDGE_CORE, PASS_FIVE_CORE_WORD_QUADRATIC and PASS_REGISTERED_RESEARCH_PACKAGE.
Executed: all eight registered commands; normal/optimized checks; all 496 width-five projectors and 372 proper subspaces; 11,064 bounded quadratic word patterns; independent full Laurent replay of 720 orders (240 pass, 480 fail). Missing/reducible bridges, interleavings and a noncancelling same-linear-kernel control are rejected. Two separately assigned proof reviews, committed-byte/mode checks and all eight Git-free replays pass. The complete remote tree matches tested files.
No new κ. The universal claims rest on the written proofs. The exponent group must be paid-generated; literal deletion applies to the restricted core, not arbitrary dirty spectators. Four-dimensional, zero-edge and one-edge word cases remain open. Native realization, complete invoice and all five framework assumptions remain; no selected cold replay or formal-kernel acceptance is claimed.
By @CubeJerry with substantial OpenAI Codex assistance. Credit Nick Petalas (#5921), David Govea (#720), Ali Romman (#468), woklerx (#1456/#1846), eumemic (#288) and retained lineage. Apache-2.0.
Proof, exact evidence and open questions:
https://github.com/CubeJerry/integer-mult-bounds/tree/685457252b51b9c7932ece363365bdf0b12c9c31/research/bit-three-relation-core