Positive Realization and Hidden-State Complexity: Spectral Completion, Exact Minima, and the Necessity of Hidden Dynamics

How many nonnegative internal states are needed to reproduce a rational response exactly? This sequel to Diagonal Transport and Spectral Fibers studies minimum state number, transition rank, and the distinct roles of hidden reachable recurrences, unobserved spectral multiplicities, and positive polyhedral lifts. A sharp duality bound controls excess reachable dimension. Explicit rational families show that one compulsory hidden mode can reduce positive order by an unbounded factor. Positive-circuit elimination bounds the corresponding ordinary-dimensional cone cost, with optimal quadratic growth for one hidden coordinate. A short trace argument identifies strictly positive responses approaching a constant for which every minimum-order realization requires arbitrarily many stationary directions, even though no additional reachable recurrence is needed. Equality classifies every order optimum. Nonnegative companion constructions determine exact portions of the damping staircase, including a complete classification for two conjugate pairs and a divisibility criterion at trace thresholds. The least stationary dimension among minimum-order realizations can be nonmonotone along damping. Eventually periodic responses admit exact order and rank formulas through their transient length and an optimal nonnegative cycle decomposition. Fourier interval interpolation identifies the hidden modes required by overlapping cycles and supplies sparse separation certificates. Cyclotomic orbit budgets bound periodic complexity at fixed ordinary and coefficient-field degrees; Ramanujan sums give the exact least rational degree at any prescribed period. Digit arithmetic certifies state costs that fixed-window winding observations cannot control. Puiseux valuations provide a separate transfer of lower bounds from tropical weighted automata. Spectral completion is exact near constant responses with fixed simple poles, while cubic residue examples show why poles alone do not determine minimality. Compression, resolvent normalization, and shift-compatible factorizations give finite certificates and closed semialgebraic strata. All theorem proofs are analytical, with explanatory vector figures and supplementary exact identity checks. The unrestricted structural minimum problem remains open, together with the general ordinary-degree bound for hidden complexity, the proposed five-state cubic separation from six cone facets, and general intermediate damping and residue boundaries.