Exact discrete logarithms from a flagged polygonal transform

Assume the flagged polygonal-transform and exact-factoring interfaces specified in the paper. We construct a polynomial-time uniform quantum circuit family over one fixed finite gate set that computes discrete logarithms in prime fields with probability one and polynomial worst-case resources. For prime-order inputs, each verified Fourier-bin label has probability at least 1/(81q). Given its verified logarithm, that probability is computable exactly by a bounded number of lattice-point counts in dimension at most thirteen. Dyadic completion produces success probability exactly one quarter, followed by one exact amplification step. All sampled probabilities are dyadic; no input-dependent rotation or postselection is used.