A Complete Proof of the Non-Existence of Perfect Cuboids Based on Differential Algebraic Closure and Finite Representation Theory of Transcendental Functions

This paper applies the framework of differential algebraic closure to transform the perfect cuboid problem into a differential-algebraic representation of a polynomial system. By analyzing the consistency conditions and contradictions in the branching indices, we rigorously prove that there does not exist a rectangular parallelepiped (perfect cuboid) whose edge lengths, face diagonals, and space diagonal are all integers. The core steps include: (1) expressing the perfect cuboid conditions as a system of multivariate polynomial equations; (2) rigorously reducing it to a single-variable octic equation P(U) = 0 in terms of the square of the edge ratio via symmetric elimination and resultant computation; (3) constructing the differential algebraic closure for this specific polynomial and obtaining an explicit parametric representation of its solutions according to the universal formula; (4) substituting this solution representation into the auxiliary quadratic condition derived from the original integer constraints, analytically deriving a system of linear congruences modulo 8 concerning the branching indices; (5) analyzing the arithmetic properties of the polynomials using 2-adic valuations to prove that the resulting system has no solution, leading to a contradiction and thus no perfect cuboid exists. This paper not only resolves this classic number theory problem but also demonstrates the systematic power of differential algebraic finite representation theory in solving complex Diophantine equations. Building upon and correcting the framework initiated in previous work, we provide a complete and rigorous solution to this centuries-old conjecture.