Shared formal perturbation for declared determinant tie-breaking

Input: a finite set of exact dyadic points in dimension $d=2$ or $d=3$, with unique stable integer IDs, an ordered tuple with pairwise distinct IDs of the required size, the coefficient order of Dedekind-complete ordered real field, and an explicit caller declaration that this algorithm uses symbolic tie-breaking. If an ID is repeated, return the ordinary exact zero for the repeated-row determinant. Otherwise order all coordinate slots by point ID and then coordinate index; let $k$ be the zero-based rank of a slot. Form the requested orientation or lifted determinant polynomial after replacing that coordinate $x$ formally by $x+{\epsilon }^{3^k}$. Expand with exact dyadic coefficients, combine equal powers of the formal positive infinitesimal $\epsilon$, discard zero coefficients, and return the sign of the coefficient with the least remaining exponent.

This is a symbolic procedure: no numeric $\epsilon$ is chosen or evaluated. It supplies a tie sign only to algorithms that declare this same shared point-index perturbation. The ordinary exact predicate continues to return zero on an exact zero determinant.