The Rational-Sphere Baseline, Fixed-Hypotenuse Synchronization and the Primitive Gate: Bounded Exact Searches That Find No Perfect Cuboid

An exact scan of 478,765 rational-sphere shapes (stereographic denominators ≤ 60) finds no perfect cuboid, and its unique two-face closure is the seed (448, 495, 840; 1073) with missing face 233·1913; an exhaustive search for g ≤ 300,000 (three legs of right triangles with hypotenuse g) finds no cuboid and 3,508 two-channel near-hits; a primitive cuboid has an odd space diagonal g with all prime factors ≡ 1 (mod 4), needs at least 3 representations and edge parities (odd, 0 mod 4, 0 mod 4), and an exhaustive search for g ≤ 5,000,000 finds no primitive cuboid.