Deterministic Projection, Diffusive Limits, and Knee–Like Threshold Transitions

A finite-dimensional deterministic system can produce a stochastic effective law when only a non-injective observable is retained and hidden initial conditions are sampled from an invariant ensemble. We give an explicit invertible example. Its fast subsystem is the hyperbolic toral automorphism generated by the integer matrix with rows (2,1) and (1,1), its slow coordinate is an affine contraction driven by the centered sine observable of the first torus coordinate, and the observed map forgets the torus coordinate. Standard deterministic homogenization then gives an affine Ornstein-Uhlenbeck limit with exact asymptotic variance one half. At any fixed readout time, the limiting probability of a positive outcome is an exact Gaussian sigmoid. We compute its midpoint, maximal slope, and every quantile width, and prove uniform convergence of the finite deterministic readout curves on compact protocol intervals. The construction is an existence theorem for deterministic microscopic realizations of diffusive threshold laws. It does not derive intrinsic randomness, a first-passage law, quantum collapse, the Born rule, or an MTT-selected physical protocol.