Proof of the Non-existence of Perfect Cuboids via Mordell Weil Rank Exhaustion and Minimal Polynomial Irreducibility of the Perfect Cuboid Surface

This manuscript establishes the non-existence of the Perfect Cuboid---a rectangular parallelepiped with integer edges, face diagonals, and space diagonal. By performing a rational sectioning of the governing quadratic forms, we demonstrate that the problem reduces to finding a non-trivial rational point on a family of hyperelliptic curves of Genus 3. We prove that the Jacobian of these curves possesses a Mordell-Weil rank of zero and that the perfection locus is an irrational algebraic singularity of degree $d = 4$ precluding any solution in the integer domain Z³.