Sharp Affine Dynamical Determinants and Finite-State Spectralization

We study finite graph-directed transfer operators whose branches are axis-aligned affine hyperbolic maps with characteristic source and target rectangles in renewal-compatible cross form. These discontinuous multipliers are bounded on a fixed tensor anisotropic Sobolev scale. For the resulting sharp operators we prove an exact finite-state expansion of every flat trace. The expansion resums the stable and unstable fixed-point denominators into a normally convergent product of finite matrix determinants. The central result identifies the zeros of the flat dynamical determinant with the external spectrum of the sharp transfer operator, including algebraic multiplicities and holomorphic parameter dependence. The proof combines a two-sided symbolic renewal formula, pole detection on the sharp space, a finite-dimensional detuning argument, and an exponential estimate for the remaining affine jets. In particular, the difference between the flat trace and the trace of the external Riesz restriction decays exponentially. The theorem supplies a finite-state spectral carrier for a class in which the usual smooth-weight determinant theorems do not apply directly.