Stable stochastic-dynamics imports

This declaration-free umbrella re-exports deterministic-start trajectory semantics for both finite and arbitrary measurable state spaces. The finite layer supports arbitrary prefix-dependent probability kernels and bounded prefix/next-state scores; the measurable-state layer uses a jointly measurable bounded score contract. It also re-exports Markov anytime-valid and posterior-uniform PAC-Bayes certificates, including finite-catalog and countable-tilt finite-state trajectory endpoints and arbitrary-measurable- hypothesis empirical-Bernstein trajectory endpoints. The arbitrary measurable state-and-hypothesis layer also exposes a countable geometric-tilt full-prefix oracle with path-selected posteriors, ordinary monitored conditional-risk semantics, an observable LIL-order envelope, and conditional vanishing width under the stated posterior KL-rate assumption. It also exposes a two-sided singular-fraction theorem: for every n >= 2, one event controls the absolute gap between posterior-averaged encountered conditional risk and posterior- averaged observed prequential risk, uniformly over eligible path-selected posteriors, with an observable LIL-order radius. Under the same jointly measurable trajectory contract, the corresponding all-admissible-fractions event supports explicit rational scalar fractions, while a separate corollary attains the exact scalar-fraction minimum noncomputably. Neither endpoint is a coin-betting or universal predictable-strategy result. Under the same jointly measurable score contract, the layer supports wake-selected suffixes and the exact finite-prefix observable suffix-variance selector. The suffix theorem has a finite-time dyadic square-root envelope but does not assert unconditional vanishing width. For arbitrary measurable state spaces, it also exposes the filtered and ambient joint measurability and deterministic finite-time envelope of the exact sleeping master process for a predeclared countable catalog of measurable prefix-predictable tilts. One common event then supports path-selected posteriors over both an arbitrary measurable hypothesis space and the active strategy prefix, with separate model and strategy KL costs. Its ordinary encountered-prefix endpoint additionally pays twice the realized total-variation discrepancy of the selected time weights from uniform ones. This is confidence allocation over a fixed catalog, not coin betting. The finite-state, arbitrary-measurable-hypothesis layer additionally exposes an exact finite-prefix polynomial sleeping mixture over a countable catalog of prefix-predictable strategies. Its normalized endpoint retains time-varying encountered conditional-risk semantics and the observable forward-predictor quadratic penalty, with logarithmic rather than linear wake-time selection cost. Constant post-wake tilts additionally give ordinary conditional-risk certificates on wake-selected trajectory suffixes. The reporting time, suffix start, and eligible hypothesis posterior may be selected from the observed path; the fixed catalog pays the polynomial wake-selection cost. The suffix average restarts, but the forward predictor may retain earlier history. General time-varying nonnegative predictable tilts also yield ordinary suffix risk after paying twice the realized total-variation discrepancy between their normalized time weights and uniform suffix weights. Constant post-wake tilts make this discrepancy exactly zero. An exact finite-prefix geometric selector minimizes the observable suffix excess width and is bounded by a finite-time variance-sensitive dyadic square-root envelope. The envelope has iterated-logarithm order only under the stated posterior-complexity conditions. An explicit finite-PMF specialization rewrites posterior integrals, relative entropy, prefix-kernel conditional risks, observable suffix variation, and the exact selected suffix-variance boundary as finite sums. The finite-state layer additionally supports fixed-before-data tilt rules that read each complete available prefix, with one event uniform over time and finite posterior PMFs. Its geometric empirical-Bernstein trajectory oracle allows path- and time-dependent posterior selection and exact post-data minimization over a growing tilt prefix, while controlling monitored conditional trajectory risk by empirical prequential risk plus an observable variance-adaptive envelope. For the finite homogeneous Markov tilt-catalog endpoint, it also re-exports the extension from a deterministic start to an arbitrary supplied finite-state initial PMF.

The stationary finite-state layer includes the supplied-Poisson endpoint, its supplied-Poisson stationary-risk specialization, and the finite-depth automatic Poisson construction under oscillation contraction. It also exposes the robust fixed-candidate Poisson bridge under an explicit row-wise total-variation misspecification budget, together with the induced Dobrushin perturbation certificate and uniqueness of supplied invariant laws. It additionally exposes time-uniform empirical transition-coordinate and row-total-variation confidence certificates for unknown finite kernels, including countably allocated geometric tilt selection with vanishing statistical radii under positive limiting row-visit frequencies. Finite-simplex Cesaro compactness constructs an invariant PMF for every nonempty finite kernel; strict Dobrushin or candidate row-TV certificates upgrade existence to uniqueness.

It additionally exports finite state--action behavior-law semantics and normalized one-step importance-weighting interfaces. For finite state-based behavior policies, the controlled prefix kernel and path law are identified exactly with the ordinary Markov law on action--state pairs. Under a positive behavior-probability floor, augmented-kernel row-TV control also yields action-conditioned environment-row control with the explicit inverse-floor factor. For finite state-based Markov target policies, it exports a stationary target-policy OPE endpoint under a known environment and behavior policy, supplied invariant target laws and exact Poisson potentials, and declared overlap and span bounds. It also exports the deterministic robust-candidate bridge from action-conditioned environment-row total variation to induced target-policy drift and stationary-residual envelopes, together with the non-variance-adaptive fixed-range approximate-Poisson OPE event and the fixed-candidate, fixed-depth robust OPE event under supplied contraction and physical action-row total-variation certificates. It also exports encountered-prefix dynamic comparators for finite catalogs of history-dependent target policies, including a known prefix/time-dependent environment kernel. For a supplied target policy, it also exports an exact finite-horizon target-path change-of-measure identity, target state occupancy identity, and the explicit C ^ n likelihood-weight range inflation. These controlled endpoints do not themselves estimate the environment, invariant law, or nuisance functions, and no endpoint gives an anytime cumulative-weight importance-sampling guarantee.