Congruence Jumping in Polynomial Divisibility Systems: Arithmetic Structure and Solution Sets

Let $A$ and $B$ be polynomials with integer coefficients. We study nonzero integer solutions of $$y\mid A(x),\qquad x\mid B(y),$$ using quotient transformations that may change the polynomial pair. We call this process \emph{congruence jumping}. We establish a general reciprocal-polynomial rule gove