39. Integrated controlled stochastic realization
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Proposition 37: Conditional observation-based stochastic completion
Suppose a causal finite-alphabet signal process has behavior rank at most \(r\), and a selected \(r\times r\) history/test block containing the empty test is invertible. Exact values of that block, every one-symbol shifted block, and the initial core-test vector determine a minimal realization of every future experiment probability. Its normalized state is recursively accessible after each positive-probability history, without reconstructing the hidden physical state.
The exact realization has the canonical probability cone of Proposition 29. Under any invertible atlas, transform the state, symbol operators, mass row and cone together:
Every likelihood and posterior prediction is then unchanged. A complete packet decomposition preserves these predictions after full synthesis. Packetwise probability interpretation requires the extra positivity condition of Proposition 30.
For estimates obtained under Proposition 35's reset protocol, the simultaneous event there bounds the finite table perturbations. A separated finite-table spectrum yields Proposition 32's subspace certificate. A well-conditioned fixed core yields Proposition 33's operator certificate. On branches satisfying Proposition 34's likelihood margins, the resulting normalized state errors obey its recursive bound. These perturbation statements alone do not certify a learned cone or global positivity of an estimated model.
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Proof
Proposition 28 reconstructs the unique minimal behavior realization, and Proposition 27 gives its conditional interpretation. Proposition 29 supplies probability validity for the exact realization. Substituting the transformed quantities cancels every \(T^{-1}T\), including in the scalar normalization. Full packet synthesis is the identity, whereas Proposition 30 characterizes the additional probability-mixture condition. The perturbation conclusions are the stated chain of propositions with their separate assumptions retained. No missing gap, coverage or positivity assumption follows from matrix inversion alone. \(\square\)
This theorem closes a finite controlled and stochastic sector of the foundational model. It does not claim unrestricted discovery of a nonlinear environment from a single trajectory. It does identify what a learned predictive state represents, why its operations compose, where its evidence comes from, and which defects change likelihoods or posteriors.