Deterministic Microdynamics, Projected Laws, and the Superdeterminism Boundary Factorization, Conditional Kernels, Markov Closure, and Stabilization in MTT

A deterministic microscopic evolution need not define a closed deterministic law for a reduced state. The exact obstruction is factorization: a reduced map exists precisely when the next reduced state is constant on every fiber of the retained map. Probability does not follow from failure of this condition. After a preparation measure is supplied, regular conditional measures on the fibers define a one-step kernel. That kernel is Dirac exactly when the next reduced state is conditionally almost surely constant, a weaker statement than global fiber constancy. We give finite counterexamples proving that the same projection and microdynamics can induce different kernels under different preparations and that a stationary projection of deterministic dynamics need not be Markov. Thus a stochastic or Markov effective theory requires a measure, a closure theorem, and often a scaling limit. Existing deterministic homogenization results provide such limits under declared assumptions; projection alone does not. We also prove a coupled-basin contraction theorem: once a record basin is selected, a nonnegative influence matrix with spectral radius below one gives geometric joint stabilization. This does not select the basin. Finally, determinism and superdeterminism are separated. In a Bell experiment, measurement independence, Bell factorization, and operational no-signaling are distinct conditions. The stated MTT route retains measurement independence and upper or base locality while allowing a nonseparable state to violate Bell factorization. Its selected state, instruments, and probability source remain completion obligations.