15 One common operator language
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The preceding constructions are not identical, but they use the same packet operations. The domain dictionary is summarized in Table 1.
| Realization | Seed | Transfer | Packet/reconstruction law |
|---|---|---|---|
| Markov | point mass \(\delta_i\) | \(K^n\) or \(e^{tQ}\) | state lumping and \(CK^n=\bar K^nC\) |
| Conditional | random variable | \(P_{\mathcal Q}\) | nested partitions and tower law |
| Bayesian | prior \(p\) | \({\mathcal B}_\lambda\) | packet evidences and normalized synthesis |
| Fractional | spectral source | \(e^{-tL^\alpha}\) | spectral projectors \(P_j\) |
| PDE | source \(f\) or \(\delta_y\) | resolvent/Green operator | eigenmode or spatial packet synthesis |
| Feynman–Kac | terminal data \(f\) | \(({\operatorname{diag}}(e^{-V})K^T)^n\) | weighted path reconstruction |
| Latent system | retained state | Feshbach/self-energy | hidden packets return as memory |
| Survival | initial transient law | \(e^{tA}\) | sum over transient packets |
For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For \(A\) acting on \(\mathbb C^d\), its generalized eigenspaces \(E_\mu=\ker(A-\mu I)^d\) form a direct-sum decomposition over the eigenvalues. Define the unique linear operator \(F(A)\) by
The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of \(F\) at the spectrum. If \(F\) is holomorphic on an open neighborhood \(U\) of the spectrum, each eigenvalue admits a radius \(r_\mu>0\) with \(\overline{D(\mu,r_\mu)}\subset U\), and Cauchy's formula gives
Consequently the same canonical operator is obtained from any such local Cauchy circles, even when \(U\) is disconnected. This is the standard finite Taylor-jet definition of the holomorphic matrix function.
Theorem 15.1: Naturality under exact intertwining
For finite complex matrices, let \(F\) be holomorphic on a neighborhood of the union of the spectra of \(A_f\) and \(A_c\). Real matrices are treated by complexification. If
then
In particular, exact generator refinement propagates simultaneously to semigroups, resolvents, integer powers, and analytic transfer functions.
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Proof of Theorem 15.1
The intertwining identity implies \(C(A_f-\mu I)^k=(A_c-\mu I)^kC\) for every \(k\ge0\). It therefore sends the generalized \(\mu\)-eigenspace of \(A_f\) into that of \(A_c\). Use a common truncation index at least as large as both dimensions in the canonical formula above. Multiplying its finite sum by \(C\) now gives the desired identity on every generalized eigenspace. The direct-sum decomposition gives it on the full space. The argument covers nilpotent Jordan parts and uses only local spectral Taylor jets; it does not require one Taylor series to converge on both spectra. The zero-dimensional case is immediate.
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Remark 15.2: Where nonlinearity enters
Bayesian conditioning does not fit the preceding linear functional calculus because normalization depends on the current state. It nevertheless obeys an exact composition law and has a closed packet reconstruction. Paper III supplies the general nonlinear coordinate theorem; the Bayesian map is a particularly transparent probabilistic realization.