Part II Extended mathematical development
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Abstract
A bounded generator and a finite packet system turn a semigroup into a matrix of transfer blocks. We prove convolution, differential, path, Laplace, resolvent, and defect formulas. Exact intertwining yields an inverse limit semigroup. Weighted summable adjacent defects give convergent reconstructed evolutions, while range leakage determines whether the limit remains a semigroup. For finite Markov chains we identify a canonical lumping obstruction and quantify its failure. Conditional expectations give orthogonal martingale packets, and the Ornstein Uhlenbeck semigroup gives a countable Hermite model. Bayesian updating is treated as normalized transfer. We characterize exact coarse sufficiency, determine the coarsest nontrivial fixed sufficient partition for a likelihood family when the common level sets are nontrivial, and split relative information into coarse and local terms. Positive fractional powers are handled through spectral, Balakrishnan, and subordination formulas. Green operators, finite differences, and finite state Feynman Kac formulas provide analytic realizations. Schur Feshbach elimination produces a rational self energy and an exact memory kernel. Every stable finite hidden system yields a polynomial exponential expansion, including Jordan blocks. A certified optimal primary block selection and a Volterra estimate control reduced trajectories. State dependent killing, survival laws, moments, and competing risks complete the finite theory.