Exact Vanishing Orders for Exponentially Deformed Toeplitz Minors

Let H(z) be a finite product of positive linear numerator factors divided by positive geometric denominator factors, and let T_t be the lower triangular Toeplitz matrix generated by exp(tz)H(z). For every structurally admissible minor, we give an explicit index-sum formula for its order of vanishing at t=0. Every coefficient from that order to the total index displacement is strictly positive. A unique maximal intermediate index tuple gives the leading coefficient as a product of two positive determinants. The proof uses elementary factor supports, interlacing and the Cauchy-Binet identity. The classical zero-pattern classification and qualitative exponential strictification are explicitly credited. This is a quantitative refinement, not a claim to newly solve the already classified AIM source question. The note is unrefereed and makes no absolute-priority claim.