← Method Lines of the Perfect Cuboid: FORC, GERES, SIO-X4, X4-DMSD, IGPC and TIFV / X4-DMSD-P1 and P2 · X4 shared-edge coupling

Method Lines of the Perfect Cuboid: FORC, GERES, SIO-X4, X4-DMSD, IGPC and TIFV X4-DMSD-P1 and P2 · X4 shared-edge coupling unsigned in the source

Exact X4 Joint Algebra and the Shared-Edge Scale-Coupling Engine: Three Face Joints Share Edges iff A·C·B′ = B·C′·A′

P1: the triple-joint coupling theorem — three oriented primitive face joints share edges iff A·C·B′ = B·C′·A′ (an integer scale witness is given); an Euler brick corresponds to a compatible triple; a working prototype, with no novelty claimed. P2: a ratio-indexed join: at m ≤ 80 there are 2,628 oriented joints, 6,906,384 pair emissions, 288 compatible triples, 48 Euler bricks and 0 perfect cuboids (cross-checked against N³ at m ≤ 15).

An elementary and correct shared-edge coupling algebra plus a real engine: 48 Euler bricks and 0 perfect cuboids at m ≤ 80. The engine was re-run and checked with an independent count in the pre-publication read-through; m ≤ 80 is a bounded range. 'EXACT' in X4-DMSD means an exact face coupling (an Euler brick), not a solution.

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This is a full-text translation of the Chinese source, which remains authoritative for its research claims. Translation is not independent mathematical verification and does not upgrade the original evidence level.

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