Robust State Equation: Calibrated Hankel Recovery and Finite-Precision Classicalization

Exact physical memory is an unstable target: a process of Hankel rank one can be arbitrarily close to processes requiring larger classical and quantum carriers. We make this obstruction quantitative and then identify the extra structure under which a physical finite-sample upper theorem is possible. First, on a stationary binary Markov family whose mixing gap is uniformly at least one half, every estimator of exact linear rank, minimum classical memory, or minimum quantum Hilbert dimension has worst-case error at least 3/8 at local separation \(Θ([N(L-1)]-1/2)\) from the rank-one model. The smallest nonzero singular value of an explicit Hankel core is proportional to the same parameter. Thus a spectral or dynamical margin is necessary even under rapid mixing. Second, finite Hankel data determine a minimum linear realization only up to similarity, whereas nonnegativity and complete positivity are not invariant under a generic similarity. We give an exact two-state counterexample and then formulate the missing assumption as a calibrated physical core. For an invertible core \(H\), all shifts are recovered in one basis by \(Ã_a = H̃⁻¹H̃_a\). An explicit perturbation theorem, followed by Choi positive-part projection and one common trace-preserving correction, returns a genuine quantum instrument. A direct-query finite-sample corollary controls its error uniformly over adaptive policies and exposes every dependence on core conditioning, physical calibration, horizon, and confidence. The resulting linear-in-horizon coefficient is first-order sharp: a one-state classical process attains the exact discrepancy \(1-(1-Δ/2)^L\) at one-step diamond radius \(Δ\). Third, we prove a compact-state classicalization theorem. Any finite-dimensional physical memory described by a compact base-norm state space has, at horizon \(L\) and operational error \(η>0\), a classical realization whose number of states is the seeded covering number at radius \(2η/L\), uniformly over adaptive policies. For a \(d\)-dimensional quantum instrument this gives, for \(d≥ 2\), \[ (1+{2(1-d⁻¹)L}{η})d^2-1 \] states, with no extra transient state; one state suffices for \(d=1\). The ambient exponent \(d^2-1\) is optimal for universal trace-norm discretization, while a compact branch-invariant reachable set replaces it by its own metric entropy. Consequently, an unbounded exact quantum/classical memory separation cannot remain unbounded at fixed \(d\), \(L\), \(η\). Robust quantum advantage must instead be quantified as a joint precision–horizon tradeoff. Deterministic verification code audits the mixing lower bound, the similarity-gauge counterexample, common-shift recovery, physical repair, and finite-net simulation.