Boundary-Aligned Ulam Approximation and Grid Leakage in a Cyclic Postcritically Finite Quadratic Map

Finite Markov approximations can turn an exact cyclic decomposition into an apparently mixing matrix when a partition cell crosses the decomposition boundary. We isolate this mechanism for the postcritically finite quadratic map f_u(x) = 1 - u * x^2 at the parameter satisfying u_c^3 - 2 * u_c^2 + 2 * u_c - 2 = 0. The map has an invariant interval J = [-(u_c - 1), 1] and two bands exchanged modulo their shared endpoint, so its Perron-Frobenius operator has an L^1 sign mode at -1. We prove that an exact cell-overlap Ulam matrix whose mesh contains the band boundary is block anti-diagonal and inherits this mode to machine precision. A phase scan over four resolutions then shows that one crossing cell creates positive projected same-band mass and a reproducible displacement of the tracked finite-dimensional branch. We separate this discretization effect from an independently defined reflecting-noise operator using direct and split constructions, Gaussian, uniform, and triangular kernels, and a resolution condition sigma/h >= 10.6. The retained data support a square-root-like numerical trend over one decade, but not a continuum perturbation theorem. The LRL quantity used for validation is a dynamical cylinder mass; it is not assigned an interpretation in terms of primes or Riemann zeros.