The Structural Incompatibility of Spatial Closures: A Geometric Proof of the Non-Existence of the Perfect Cuboid

The paper provides a rigorous, deterministic proof of the non-existence of the Perfect Cuboid (Euler brick). By defining the rectangular parallelepiped as a monolithic three-dimensional block governed by strict geometric invariants rather than amorphous algebraic variables, the author demonstrates the fundamental spatial constraint on edge parities. Through a systematic analysis of planar constraints and volumetric closures, it is shown that the generation of a primitive Pythagorean triple for the base face automatically locks the dimension of the major even edge. The subsequent volumetric extension forces a secondary Pythagorean structure that deterministically requires the minor even edge to exceed the major even edge. This irreconcilable geometric contradiction proves that a Perfect Cuboid cannot exist in full integers.