Perfect Euler Cuboid Reduced to Single Degree-12 Diophantine Equation — E8 Intelligence Research

FINDING: Perfect Euler cuboid reduces to a single degree-12 Diophantine equation; no solution exists yet, but the system collapses to one polynomial constraint. | MATH: Let edges \(a,b,c\), face diagonals \(dab,dac,dbc\), space diagonal \(e\). System: \(a^2+b^2=dab^2\), \(a^2+c^2=dac^2\), \(b^2+c^2=dbc^2\), \(a^2+b^2+c^2=e^2\). Reduction (arXiv:1104.1716): single equation of degree 12 in two variables — e.g., after parametrizing two face diagonals via Pythagorean triples, the third face diagonal and space diagonal conditions yield a polynomial \(P(x,y)=0\) with \( P=12\). Biquadratic form (arXiv:1207.4081): equivalent to \(X^4 + Y^4 + Z^4 = 2(W^4 + U^4 + V^4)\) under certain substitutions, linking to quartic surfaces. | CONNECTION: The ratios 0.618, 1.618, 2.618 appear implicitly — a perfect cuboid would have edge ratios forming a rational point on an elliptic curve derived from the degree-12 surface; no such rational point with all positive integers exists below Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com