The Deterministic Statistical Feedback Law in Finite Dimensions: Residuals, Channels, and Nonlocal Correlations

We develop the Deterministic Statistical Feedback Law (DSFL) as a concrete finite-dimensional framework for quantum information. DSFL works in a single calibrated Hilbert “room”, compares statistical blueprints to physical responses through one residual of sameness, and declares as lawful exactly those updates that contract this residual in a fixed instrument norm. We prove that admissibility of a channel is equivalent to a spectral data–processing inequality and that iterated admissible dynamics admit a Lyapunov-style envelope, with a canonical DSFL clock making the decay of the residual a straight line in semilog coordinates. On bipartite systems we reinterpret standard entanglement proxies, such as negativity, sandwiched Rényi divergences, and trace distance to product form, as correlation residuals that cannot increase under local completely positive maps. Embedding CHSH and GHZ or Mermin tests into the same room, we recover the usual quantum violations while all local hidden-variable models remain confined to their classical bounds.