An AI Proof of 18-Variable Undecidability for Diophantine Equations over $\mathbb{Z}[i]$

This paper presents an AI proof that there is no algorithm deciding whether a polynomial equation over the Gaussian integers in $18$ unknowns has a solution. The proof improves the $20$-unknown theorem of Matiyasevich and Sun. It follows their rationality criterion and integer test, but saves two variables: the clearing-denominator variable is avoided by imposing two integer conditions, and the remaining nonzero condition is absorbed into the relation-combining lemma by the one-variable gadget $(2R+1)(3R+1)$.