Resolution of Richard Guy's Problem D18: Complete Non-existence of Perfect Cuboids via Pell Equations, Linear Forms in Logarithms, and Human-AI Collaborative Methodology

This paper investigates Richard Guy's Problem D18 concerning the existence of a perfect cuboid. By reducing the fundamental Pythagorean constraints into a coupled system of Pell's equations and applying Matveev's lower bound theorem for linear forms in logarithms, we derive an absolute upper bound for all hypothetical solutions and establish a rigorous non-existence proof across the infinite domain. Furthermore, this work reflects upon the methodological and epistemological significance of human-AI collaboration in advancing independent mathematical research.