Finite-Memory Noise and the White-Noise/Markov Limit Exact Colored-OU Convergence and Its MTT Scope

A delta-correlated covariance is the distributional limit of many finite-memory kernels, but covariance convergence alone does not imply convergence of stochastic processes, Gaussianity, or the Markov property. We separate these statements and prove an exact process-level model. Nonnegative approximate-identity kernels with total mass 2D converge to a Dirac distribution of mass 2D. A valid stationary covariance must additionally be of positive type. For an explicit stationary colored Ornstein-Uhlenbeck source with memory time tau, the integrated disturbance converges weakly on every finite continuous-path interval to Brownian motion with diffusion scale 2D. The damped response therefore converges to the white-noise Ornstein-Uhlenbeck equation. Its exact finite-memory stationary variance is D divided by gamma times one plus gamma tau, and tends to D divided by gamma. This supplies a rigorous effective Markov limit, not a derivation of fundamental randomness. A general MTT source theorem would still need a selected preparation measure or invariant state, positivity of the covariance, mixing or a functional limit theorem, closure of the resolved variables, and a derivation of the finite-memory coefficients from the physical carrier.