← Method Lines of the Perfect Cuboid: FORC, GERES, SIO-X4, X4-DMSD, IGPC and TIFV / GERES-P5 and P7 · body-cuboid near-misses

Method Lines of the Perfect Cuboid: FORC, GERES, SIO-X4, X4-DMSD, IGPC and TIFV GERES-P5 and P7 · body-cuboid near-misses unsigned in the source

World Encoding and Candidate Ingestion for the Perfect Cuboid, and Rational-Point Fibre Search: 25 'Body Cuboid' Records That Are Not Perfect Cuboids

P5: a canonical world schema and ingestion; it ships 25 exact-integer 'body cuboids' (a² + b² + c² = g² with exactly two integer face diagonals), every record tagged claim_status NOT_PERFECT_CUBOID, led by G2 = (448, 495, 840; 1073) with the open channel √445729. P7: exact bounded scans: 1,931,595 parameter pairs (denominator ≤ 80) give 113 structural points and 0 perfect points; the two G2 coordinate fibres (denominator ≤ 500, 76,115 rationals each) hold only 2 structural points each.

'Candidate' means an integer triple with exactly two integer face diagonals and an integer space diagonal (a body cuboid): every record is tagged NOT_PERFECT_CUBOID; the bounded scan finds no perfect point. 'Candidate' is misleading: 5 of the 25 records are ×5 multiples (only 20 distinct shapes) against the paper's own de-duplication rule, and the selection rule is unstated (25 of the 76 body cuboids with g ≤ 9061, omitting two primitive ones with g = 9061). P7 says '20 of 25 regenerated', which is really 20/20 distinct shapes. All 25 records were re-derived independently in the pre-publication read-through.

Connections · Connections

PreviousGERES-P8 and P9 · the G2 fibres · Arithmetic Geometry and Birational Analysis of the G2 Fibre, and the Twin-Elliptic Perfect Gate: Each Coordinate Fibre Is a Genus-1 Curve and With the Perfect Gate It Is a Genus-5 Curve
NextX4-DMSD-P1 and P2 · X4 shared-edge coupling · Exact X4 Joint Algebra and the Shared-Edge Scale-Coupling Engine: Three Face Joints Share Edges iff A·C·B′ = B·C′·A′

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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