On the Non-existence of the Rational Perfect Parallelepiped: Algebraic Reduction and Global Parity Breakdown

The Rational Perfect Parallelepiped problem, a natural generalization of the classic Perfect Cuboid problem (Guy's Unsolved Problems in Number Theory, Problem D18), asks for the existence of a parallelepiped where all three edges, six face diagonals, and four space diagonals are strictly rational. In this paper, we resolve this question in the negative. By explicitly eliminating the inner-product cross-terms via the generalized parallelogram law (the Ichida Method), we transform the geometric conditions into an overdetermined network of simultaneous Pellian equations. Projecting the system into the ring of integers and evaluating modulo 8, we demonstrate that the highly constrained network demands mutually exclusive parity assignments. This inescapable parity breakdown proves the local insolubility in the 2-adic field Q_2, rigorously establishing by the Hasse principle that no such rational parallelepiped can exist.