The State Equation: Causal Cone Rank and Minimal Physical Memory

We ask which finite classical or quantum memory is sufficient to generate the predictions of a stochastic process. Starting from its Hankel kernel \(H_p(u,v)=p(uv)\), we define causal cone rank by globally shift-compatible cone lifts of the canonical predictive realization. Within a specified realization class, the physical carrier minimum equals the corresponding minimum compatible lift dimension. The framework separates unconstrained Hankel rank, classical nonnegative realizations, and quantum completely positive instruments, and extends to action--outcome processes with interventionwise normalization. A declared linear coarse state has exactly \(\dim\ker(C|_{\operatorname{im} H_p})\) missing predictive coordinates. Finite temporal cuts give nonnegative- and positive-semidefinite-rank witnesses; an explicit two-time family exhibits a growing exact classical--quantum dimension gap, while its fixed-tolerance classical cost remains bounded. For an exact finite-dimensional time-homogeneous Markovian dilation, reset-conditioned closure rank obeys \(r_{\rm cl}\le d_S^2(d_E^2-1)\), with a saturating qubit--qubit example. We give nonasymptotic singular-value tests, while emphasizing that numerical rank and unconstrained truncated singular expansions do not themselves certify a physical realization.