Six-Channel Conservation–Duality, the Conservation Move Algebra and Two-Turn Square-Preserving Repair

Normalising by g² gives six cells A…F with A + B + C = 1 and A + F = B + E = C + D = 1, so a perfect cuboid exists iff A, B, C, 1 − A, 1 − B, 1 − C are all rational squares on the open 2-simplex; the conservation moves form a 2-dimensional abelian group, any repair of the defective cell needs at least 4 cells, and the two one-axis orbits (k = 7/32 and 11/45) both expose the same defect 233/1913; equivalently a perfect cuboid exists iff the dual-square set P = {X : X and 1 − X are rational squares} contains X + Y = Z.