Conclusion

A bounded semigroup becomes a matrix of packet transfers once a finite packet system is fixed. The matrix obeys convolution in time, a closed differential system, a Laplace resolvent formula, and a directed path expansion. A generator defect propagates through Duhamel's formula and through the resolvent calculus. Exact compatibility gives a semigroup on the bounded inverse limit. Summable adjacent defects give convergent reconstructed evolutions. Range leakage, and in particular an off diagonal generator block, determines whether the limit retains the semigroup law.

In probability, the same structure appears in several forms. Markov transfer is linear and exact lumpability is the vanishing of a canonical kernel obstruction. Conditional expectation is an orthogonal stochastic projector whose differences form martingale packets. The Ornstein Uhlenbeck semigroup resolves exactly into Hermite packets. Bayesian transfer is nonlinear because of evidence normalization. Its exact two stage decomposition separates coarse packet evidence from local conditional change. Relative entropy then records exactly how much information lies at each level. A fixed sufficient coarse description exists for every prior precisely when the likelihood is constant inside each packet. The common level sets of a likelihood family give its unique coarsest fixed sufficient partition.

Positive fractional powers preserve exact intertwining. Under a common spectral gap, the Balakrishnan formula turns an ordinary generator defect into an explicit fractional defect. Green operators and finite differences show how source reconstruction fits the same transfer language. Feynman Kac weights Markov paths by a potential without leaving the semigroup framework.

Hidden packets provide the final structural lesson. Exact elimination does not erase them. It returns them as a rational self energy in the resolvent and a causal memory kernel in time. The primary decomposition of the hidden generator produces every finite memory kernel as a finite sum of polynomial exponential modes. Stable blocks can be truncated with an integrated error certificate, and the Volterra estimate transfers that certificate to the retained trajectory. State dependent killing identifies the Feynman Kac potential with a survival mechanism. Survival and competing risks are then scalar reconstructions of a transient semigroup.

The article therefore separates four operations that are often compressed into one informal idea of coarse dynamics:

Each arrow has its own hypotheses and its own obstruction.

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