An Elementary Obstruction to the Existence of a Perfect Cuboid

Abstract

We study arithmetic constraints arising from the three faces meeting along the space diagonal of a rectangular cuboid. Using a propagation mechanism along this diagonal, based on the appearance of a minimal odd prime in certain triangular remainders, we derive strong structural restrictions on possible configurations. These constraints induce an infinite descent along the space diagonal, preventing the existence of a compatible integral structure. This approach provides an elementary obstruction to the existence of a perfect cuboid, relying only on divisibility and congruence arguments, and avoiding the use of Gaussian integers or classical quadratic factorizations.