Realization Calculus

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135 \title{\textbf{Projector-Resolved Seed Calculus VI}\\[0.45em]
136 \large Transfer Semigroups: Stochastic, Bayesian, Fractional, and PDE Realizations}
137 \author{Komorebi P.}
138 \date{Reviewed edition: 6 September 2026\\\small Fifth Lean-checked formalization block; all 33 results covered}
139 
140 
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201 % Operation Komorebi: shared publication profile.
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204 \renewcommand{\KomorebiSubtitle}{Probabilistic refinement, fractional operators, and exact hidden-state elimination}
205 \renewcommand{\KomorebiSeries}{Core series VI}
206 \renewcommand{\KomorebiKind}{Modern Monograph}
207 \renewcommand{\KomorebiEdition}{9 October 2026}
208 \renewcommand{\KomorebiShortTitle}{Paper VI}
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213 \renewcommand{\KomorebiCoverTitle}{Transfer Semigroups\\and Memory}
214 \renewcommand{\KomorebiCoverBlurb}{Evolution, conditioning, and hidden-state elimination share a packet language. The same transfer defect measures what a change of resolution preserves and what it loses.}
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245 \input{atlas-definitions.tex}
246 \input{component-layout.tex}
247 \begin{document}
248 \KomorebiCover
249 \KomorebiOpeningNote{\textbf{Part I.} The thirty-three-result formal core, including nonnormal memory and canonical finite holomorphic naturality.
250 
251 \medskip
252 \textbf{Part II.} The complete strengthened mathematical development: bounded refinement towers, Bayesian information, Hermite transfer, quantitative memory compression, and competing risks.
253 
254 \medskip
255 \textbf{Part III.} Distinct retained results from the earlier human edition, exact examples, and the bounded-operator reciprocal calculus received from Chapter IV.
256 
257 \medskip
258 \textbf{Part IV.} Forty-two visual pages work through exact finite models, analytical examples, and the boundaries of the formal contract.}
259 
260 \section*{Reading this chapter}\addcontentsline{toc}{section}{Reading this chapter}
261 This chapter studies transfer semigroups in three complementary settings. Part I establishes the finite formal core. Part II develops bounded refinement towers, probabilistic conditioning and analytical realizations. Part III gives additional results on product-topology limits, reversible coarse dynamics, filtering, perturbation and operator substitution. Their hypotheses distinguish the constructions even where the underlying identities overlap.
262 
263 The formal companion covers exactly the thirty-three scoped results of Part I. Its 341 supporting Lean theorem declarations lie in 47 proof modules. The wider namespace audit includes definitions and generated declarations. Formal coverage does not automatically extend to Parts II or III. The companion supplies the exact targets, assumptions and specialization boundaries.
264 
265 Part I includes three explicitly open conjectures. Part II supplies bounded or finite theorems for several related questions. It does not settle arbitrary continuum limits, unbounded-generator convergence, general statistical sufficiency, or globally optimal nonnormal model reduction. The reader should keep each result's declared space, topology, positivity, and stability hypotheses. Part I's mass-only Bayesian commutation criterion assumes at least two coarse blocks. Part II's stronger sufficiency convention also preserves within-block conditionals, so its one-block case still requires a constant likelihood.
266 
267 The worked atlas follows the mathematical development through exact finite models and analytical examples. Calculations illustrate stated ranges and assumptions. Their computational records remain distinct from both the written proofs and the formal declarations.
268 
269 Read Part I for the common operator language and the exact formal contract. Read Part II for bounded Banach-space development and the broader probabilistic and analytical realizations. Read Part III for product-topology limits, reversible coarse dynamics, prior-restricted and coarse filtering, killed-transfer perturbations, rational realization, and the transferred operator reciprocal family. Section numbers in Part II have prefix E. Those in Part III have prefix R. Original labels and statement locations are mapped in the editable sources.
270 
271 Contour winding, positive memory decay, finite moment closure and eventual absorption each require the hypotheses stated with their results. When the spaces vary along a refinement system, summability must be read in the specified ambient setting.
272 \clearpage
273 \tableofcontents
274 \clearpage
275 \listoffigures
276 \listoftables
277 \clearpage
278 \pagenumbering{arabic}
279 \part*{Part I\quad The formal core}\addcontentsline{toc}{part}{Part I: The formal core}
280 \begin{center}\large Projector-Resolved Seed Calculus VI\par\normalsize Transfer Semigroups: Stochastic, Bayesian, Fractional, and PDE Realizations\end{center}
281 
282 
283 
284 \begin{abstract}
285 This chapter develops the transfer-semigroup realization of the projector-resolved seed calculus. A seed is propagated by a one-parameter semigroup, a Markov kernel, a normalized Bayesian transfer, a fractional spectral power, a Green operator, or a potential-weighted Feynman--Kac operator. Exact packet projectors resolve every transfer into input-output blocks, and the semigroup law becomes block convolution over intermediate packets. Coarse and fine generators are compared by a refinement defect. A Duhamel identity proves that this generator-level defect is exactly the source of the corresponding semigroup defect and gives a quantitative stability bound. In the stochastic realization, point-mass seeds generate random walks and continuous-time Markov evolution. Exact lumpability is the vanishing of the same refinement defect. Conditional expectation is treated as a weighted orthogonal packet projector, with nested partitions producing martingale-difference packets. Bayesian conditioning is a normalized nonlinear operator. Its independent-data composition law, packet-evidence decomposition, stability bound, conjugate updates, and continuous likelihood flow are proved explicitly.  The Bayesian refinement conjecture is resolved in finite dimensions: for strictly positive likelihoods and at least two coarse packets, exact commutation for every prior is equivalent to blockwise-constant likelihoods up to global scale, and every failure factors through an explicit information-loss packet. Fractional powers are constructed packetwise through the spectral projectors of a positive operator, and the square-root semigroup is recovered by subordination. The fractional-naturality conjecture is also resolved for bounded nonnegative self-adjoint refinement systems: the fractional intertwining defect has an exact resolvent integral, a gap-free Hölder certificate, a sharper common-gap estimate, and a Duhamel semigroup bound. Green kernels, finite-difference Poisson problems, and second-order low-mode convergence provide the PDE realization. A finite-state Feynman--Kac formula is proved by path enumeration. Exact Schur--Feshbach elimination shows that discarded latent packets return as a frequency-dependent self-energy and, in time coordinates, as a memory kernel. Survival and hazard functions are generated by transient semigroups. Nineteen code-generated figures, machine-readable results, the original thirty-seven automated tests, and seventeen added boundary-case checks accompany this reviewed edition. The inherited September 6 release records five checked Lean blocks covering all 33 scoped theorem-like results in Appendix~\ref{core:sec:lean-status}. Finite-dimensional theorems, classical infinite-dimensional extensions, numerical certificates, and open continuum claims are kept separate.
286 \end{abstract}
287 
288 \begin{center}
289 \fbox{\begin{minipage}{0.93\textwidth}
290 \textbf{Scope and claim boundary.} The chapter proves the stated finite-dimensional semigroup, packet, Markov, Bayesian, conditional-expectation, fractional spectral, Green-kernel, Feynman--Kac, Schur--Feshbach, and survival identities. It also proves quantitative refinement-defect formulas under explicit norm hypotheses, including exact transfer of a bounded generator defect to every fractional power $0<\alpha<1$ for bounded nonnegative self-adjoint generators. It does not claim that an arbitrary infinite-dimensional operator generates a semigroup, that every stochastic process admits a finite packet representation, that every posterior family is conjugate, that finite Bayesian sufficiency automatically extends to infinite disintegrations, or that every PDE continuum limit follows from a finite test. Domain, closability, sectoriality, measurability, tightness, and compactness hypotheses remain necessary in their respective infinite-dimensional settings. Classical sources are retained in the separate end-of-series reference ledger rather than reproduced in this chapter.
291 \end{minipage}}
292 \end{center}
293 
294 
295 
296 \FloatBarrier
297 \section{The transfer-semigroup realization}\label{core:section:01}
298 
299 Let $X$ be a finite-dimensional real or complex vector space, let $A\in\operatorname{End}(X)$, and let $s\in X$ be a primitive seed. The basic continuous transfer is
300 \[
301 T(t)=e^{tA},\qquad t\ge 0,
302 \]
303 while a positive operator $L$ is more naturally written
304 \[
305 H(t)=e^{-tL}.
306 \]
307 Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usually written with $e^{tA}$. Positive diffusion and elliptic operators are usually written with $e^{-tL}$.
308 
309 The common operator chain is
310 \[
311 \boxed{
312 \text{seed}\longrightarrow\text{transfer}\longrightarrow\text{project}
313 \longrightarrow\text{condition}\longrightarrow\text{refine}
314 \longrightarrow\text{complete}\longrightarrow\text{reconstruct}.
315 }
316 \]
317 
318 \begin{figure}[htbp]
319 \centering
320 \includegraphics[width=0.98\textwidth]{01_transfer_semigroup_architecture.pdf}
321 \caption{The common operator language used throughout the chapter. Semigroups, resolvents, normalized Bayesian maps, and fractional powers are distinct realizations of the same transfer stage.}
322 \label{core:fig:architecture}
323 \end{figure}
324 
325 \begin{definition}[Transfer-semigroup seed system]\label{core:record:001}
326 A finite transfer-semigroup seed system is a tuple
327 \[
328 \mathfrak S=(X,s,A,\{P_\alpha\}_{\alpha\in I},\tau),
329 \]
330 where $X$ is the state space, $s$ is the seed, $A$ is a transfer generator, the $P_\alpha$ are pairwise orthogonal idempotents summing to the identity, and $\tau$ specifies the topology in which a completion or limit is interpreted.
331 \end{definition}
332 
333 \begin{theorem}[Finite semigroup law]\label{core:record:002}
334 For every matrix $A$ and all $s,t\in\bbR$,
335 \[
336 \boxed{e^{(s+t)A}=e^{sA}e^{tA}.}
337 \]
338 Moreover, $T(0)=I$ and $T'(t)=AT(t)=T(t)A$.
339 \end{theorem}
340 
341 \begin{proof}
342 Both exponentials are absolutely convergent matrix power series. Since every power of $A$ commutes with every other power, the Cauchy product gives
343 \[
344 e^{sA}e^{tA}
345 =\sum_{m,n\ge0}\frac{s^mt^n}{m!n!}A^{m+n}
346 =\sum_{r\ge0}\frac{(s+t)^r}{r!}A^r.
347 \]
348 Termwise differentiation is valid in finite dimension and gives the derivative identities.
349 \end{proof}
350 
351 \begin{boundary}[Infinite-dimensional generators]\label{core:record:003}
352 For an unbounded operator, writing $e^{tA}$ is not a definition unless a semigroup-generation theorem applies. The finite matrix identities in this chapter remain algebraically correct models, but a continuum realization must state the operator domain, closure, growth bound, and topology of convergence.
353 \end{boundary}
354 
355 \FloatBarrier
356 \section{Packet blocks of a semigroup}\label{core:section:02}
357 \begin{keyidea}
358 A packet block remembers both the origin and the destination. Composition must sum over every intermediate packet, because an unobserved intermediate coordinate can carry a real contribution. The original formulas below express this bookkeeping exactly.
359 \end{keyidea}
360 \begin{figure}[htbp]
361 \centering
362 \includegraphics[width=.94\textwidth]{monograph_figures/exact_packet_composition.pdf}
363 \caption{Two rational transfer steps resolved through all three intermediate states. The displayed products are computed exactly before plotting. This finite illustration explains the intermediate-packet sum.}\label{monograph:fig:exact_packet_composition}
364 \end{figure}
365 
366 
367 
368 Let
369 \[
370 P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,
371 \qquad
372 \sum_{\alpha\in I}P_\alpha=I.
373 \]
374 For any operator $B$, define
375 \[
376 B_{\beta\alpha}=P_\beta BP_\alpha.
377 \]
378 Then
379 \[
380 B=\sum_{\alpha,\beta}B_{\beta\alpha}.
381 \]
382 For the semigroup, write
383 \[
384 T_{\beta\alpha}(t)=P_\beta e^{tA}P_\alpha.
385 \]
386 
387 \begin{theorem}[Packet semigroup convolution]\label{core:record:004}
388 For every $s,t$ and every pair of packets $\alpha,\beta$,
389 \[
390 \boxed{
391 T_{\beta\alpha}(s+t)
392 =\sum_{\gamma\in I}T_{\beta\gamma}(s)T_{\gamma\alpha}(t).
393 }
394 \]
395 Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.
396 \end{theorem}
397 
398 \begin{proof}
399 Insert the resolution of the identity between the two semigroup factors:
400 \[
401 \begin{aligned}
402 P_\beta e^{(s+t)A}P_\alpha
403 &=P_\beta e^{sA}\left(\sum_\gamma P_\gamma\right)e^{tA}P_\alpha\\
404 &=\sum_\gamma
405 (P_\beta e^{sA}P_\gamma)(P_\gamma e^{tA}P_\alpha).
406 \end{aligned}
407 \]
408 \end{proof}
409 
410 \begin{corollary}[Reducing packets]\label{core:record:005}
411 If $P_\beta AP_\alpha=0$ for every $\beta\ne\alpha$, then
412 \[
413 P_\beta e^{tA}P_\alpha=0
414 \quad(\beta\ne\alpha)
415 \]
416 for every $t$. Each packet evolves independently under $P_\alpha AP_\alpha$.
417 \end{corollary}
418 
419 \begin{proof}
420 The hypothesis makes $A$ block diagonal. Every power $A^m$ is block diagonal with respect to the same projectors, and therefore so is the exponential series.
421 \end{proof}
422 
423 \begin{figure}[htbp]
424 \centering
425 \includegraphics[width=0.66\textwidth]{02_packet_semigroup_blocks.pdf}
426 \caption{Norms of the packet blocks $P_\beta e^{tA}P_\alpha$. Off-diagonal entries quantify transfer between packets.}
427 \label{core:fig:packet-blocks}
428 \end{figure}
429 
430 \begin{remark}[Path expansion]\label{core:record:006}
431 Expanding the exponential and inserting $I=\sum_\alpha P_\alpha$ between every pair of generators gives
432 \[
433 P_{\alpha_m}A^mP_{\alpha_0}
434 =
435 \sum_{\alpha_1,\ldots,\alpha_{m-1}}
436 P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
437 \]
438 Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
439 \end{remark}
440 
441 \FloatBarrier
442 \section{Refinement defects and the Duhamel identity}\label{core:section:03}
443 \begin{interpretation}
444 Read the generator defect as a mismatch between two declared paths: evolve and then coarsen, or coarsen and then evolve. The Duhamel identity transports that mismatch through time. Its estimate depends on the stated operator norms and growth bounds.
445 \end{interpretation}
446 
447 
448 Let $X_f$ and $X_c$ be fine and coarse spaces, with generators $A_f$ and $A_c$, and let
449 \[
450 C:X_f\to X_c
451 \]
452 be a coarsening map. Define the generator-level refinement defect
453 \[
454 \boxed{\Delta=CA_f-A_cC.}
455 \]
456 The corresponding semigroup defect is
457 \[
458 \mathfrak D(t)=Ce^{tA_f}-e^{tA_c}C.
459 \]
460 
461 \begin{theorem}[Exact Duhamel refinement identity]\label{core:record:007}
462 For every $t\ge0$,
463 \[
464 \boxed{
465 Ce^{tA_f}-e^{tA_c}C
466 =
467 \int_0^t e^{(t-r)A_c}\Delta e^{rA_f}\,dr.
468 }
469 \]
470 In particular, $\Delta=0$ implies exact semigroup intertwining at every time.
471 \end{theorem}
472 
473 \begin{proof}
474 Set
475 \[
476 F(r)=e^{(t-r)A_c}Ce^{rA_f}.
477 \]
478 Then
479 \[
480 F'(r)=e^{(t-r)A_c}(CA_f-A_cC)e^{rA_f}.
481 \]
482 Integrating from $0$ to $t$ gives $F(t)-F(0)=Ce^{tA_f}-e^{tA_c}C$ and proves the formula.
483 \end{proof}
484 
485 \begin{corollary}[Norm bound]\label{core:record:008}
486 Suppose
487 \[
488 \norm{e^{tA_f}}\le M_f e^{\omega_f t},
489 \qquad
490 \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
491 \]
492 Then
493 \[
494 \norm{\mathfrak D(t)}
495 \le
496 M_fM_c\norm{\Delta}
497 \int_0^t e^{\omega_c(t-r)}e^{\omega_fr}\,dr.
498 \]
499 For contraction semigroups,
500 \[
501 \boxed{\norm{\mathfrak D(t)}\le t\norm{\Delta}.}
502 \]
503 \end{corollary}
504 
505 \begin{figure}[htbp]
506 \centering
507 \includegraphics[width=0.80\textwidth]{14_semigroup_refinement_defect.pdf}
508 \caption{Exact generator intertwining gives a vanishing semigroup defect. A controlled perturbation produces a nonzero defect propagated by the Duhamel identity.}
509 \label{core:fig:semigroup-defect}
510 \end{figure}
511 
512 \subsection{Resolvent form}
513 
514 For $z$ in the common resolvent set, let
515 \[
516 R_f(z)=(zI-A_f)^{-1},
517 \qquad
518 R_c(z)=(zI-A_c)^{-1}.
519 \]
520 
521 \begin{theorem}[Resolvent refinement identity]\label{core:record:009}
522 \[
523 \boxed{
524 CR_f(z)-R_c(z)C
525 =R_c(z)\Delta R_f(z).
526 }
527 \]
528 \end{theorem}
529 
530 \begin{proof}
531 Multiply the left side by $zI-A_c$ on the left and $zI-A_f$ on the right. The result is
532 \[
533 (zI-A_c)C-C(zI-A_f)=CA_f-A_cC=\Delta.
534 \]
535 Multiplication by the two resolvents recovers the stated identity.
536 \end{proof}
537 
538 \FloatBarrier
539 \section{Markov seeds and stochastic transfer}\label{core:section:04}
540 
541 Use column probability vectors. A finite continuous-time Markov generator $Q$ has nonnegative off-diagonal entries and column sums zero. The stochastic semigroup is
542 \[
543 K(t)=e^{tQ}.
544 \]
545 
546 \begin{theorem}[Finite Markov semigroup]\label{core:record:010}
547 If $Q$ has nonnegative off-diagonal entries and column sums zero, then $e^{tQ}$ has nonnegative entries and column sums one for every $t\ge0$.
548 \end{theorem}
549 
550 \begin{proof}
551 Choose $\lambda\ge\max_i(-Q_{ii})$. Then $B=Q+\lambda I$ has nonnegative entries, so $e^{tB}$ is entrywise nonnegative by its power series. Hence
552 \[
553 e^{tQ}=e^{-\lambda t}e^{tB}
554 \]
555 is nonnegative. Since $\bm 1^TQ=0$, differentiation gives
556 \[
557 \frac{d}{dt}\bm 1^Te^{tQ}=\bm 1^TQe^{tQ}=0,
558 \]
559 so $\bm 1^Te^{tQ}=\bm 1^T$.
560 \end{proof}
561 
562 \begin{figure}[htbp]
563 \centering
564 \includegraphics[width=0.78\textwidth]{03_random_walk_transfer.pdf}
565 \caption{The point mass $\delta_0$ propagated by repeated symmetric random-walk transfer.}
566 \label{core:fig:random-walk}
567 \end{figure}
568 
569 \begin{example}[Symmetric random walk]\label{core:record:011}
570 Starting from $\delta_0$ and applying the transfer
571 \[
572 T=\frac12(S+S^{-1})
573 \]
574 for $n$ steps gives
575 \[
576 \bbP(X_n=k)=
577 2^{-n}\binom{n}{(n+k)/2}
578 \]
579 when $k\equiv n\pmod2$, and zero otherwise. Hence
580 \[
581 \Exp[X_n]=0,
582 \qquad
583 \Var(X_n)=n.
584 \]
585 The parity packet is exact: all mass remains in the one residue class permitted by $n$.
586 \end{example}
587 
588 \FloatBarrier
589 \section{Markov lumping as exact refinement}\label{core:section:05}
590 
591 Let $K$ be a column-stochastic fine transition matrix, let $C$ aggregate fine states into coarse packets, and let $\bar K$ be a coarse transition matrix. Define
592 \[
593 \Delta=CK-\bar KC.
594 \]
595 
596 \begin{theorem}[Exact lumpability]\label{core:record:012}
597 If $\Delta=0$, then
598 \[
599 \boxed{CK^n=\bar K^nC}
600 \]
601 for every $n\ge0$. Thus fine evolution followed by coarse reconstruction equals coarse evolution from the beginning.
602 \end{theorem}
603 
604 \begin{proof}
605 The result is immediate for $n=0$. If $CK^n=\bar K^nC$, then
606 \[
607 CK^{n+1}=\bar K^nCK=\bar K^{n+1}C.
608 \]
609 \end{proof}
610 
611 \begin{theorem}[Finite-horizon defect formula]\label{core:record:013}
612 For arbitrary $\Delta=CK-\bar KC$,
613 \[
614 \boxed{
615 CK^n-\bar K^nC
616 =
617 \sum_{j=0}^{n-1}\bar K^{n-1-j}\Delta K^j.
618 }
619 \]
620 If $K$ and $\bar K$ are contractions in the chosen norm, then
621 \[
622 \norm{CK^n-\bar K^nC}\le n\norm{\Delta}.
623 \]
624 \end{theorem}
625 
626 \begin{proof}
627 The identity follows by telescoping:
628 \[
629 \bar K^{n-1-j}CK^{j+1}-\bar K^{n-j}CK^j
630 =\bar K^{n-1-j}(CK-\bar KC)K^j.
631 \]
632 Summing from $j=0$ to $n-1$ leaves $CK^n-\bar K^nC$. The norm bound follows termwise.
633 \end{proof}
634 
635 \begin{figure}[htbp]
636 \centering
637 \includegraphics[width=0.82\textwidth]{04_markov_lumping.pdf}
638 \caption{Exact lumping agrees at every horizon. A small generator-level or one-step defect creates a controlled coarse discrepancy.}
639 \label{core:fig:lumping}
640 \end{figure}
641 
642 \FloatBarrier
643 \section{Conditional expectation as a stochastic projector}\label{core:section:06}
644 \begin{readerroadmap}
645 Conditional expectation is a projector in a weighted Hilbert space. Nested partitions therefore split information into orthogonal increments. Keep the underlying measure fixed while reading the projector identities. A change of measure is a different operation.
646 \end{readerroadmap}
647 
648 
649 Let $\Omega=\{1,\ldots,n\}$ carry strictly positive probabilities $\pi_i$, and let $\mathcal Q=\{B_1,\ldots,B_m\}$ be a partition. Define
650 \[
651 (P_{\mathcal Q}f)_i
652 =
653 \frac{\sum_{j\in B(i)}\pi_jf_j}{\sum_{j\in B(i)}\pi_j},
654 \]
655 where $B(i)$ is the block containing $i$.
656 
657 \begin{theorem}[Finite conditional-expectation projector]\label{core:record:014}
658 With the weighted inner product
659 \[
660 \langle f,g\rangle_\pi=\sum_i\pi_if_i\overline{g_i},
661 \]
662 the operator $P_{\mathcal Q}$ satisfies
663 \[
664 P_{\mathcal Q}^2=P_{\mathcal Q},
665 \qquad
666 \langle P_{\mathcal Q}f,g\rangle_\pi
667 =
668 \langle f,P_{\mathcal Q}g\rangle_\pi.
669 \]
670 Its range is the space of functions constant on each partition block.
671 \end{theorem}
672 
673 \begin{proof}
674 After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block. Both equal
675 \[
676 \sum_{B\in\mathcal Q}
677 \frac{1}{\pi(B)}
678 \left(\sum_{i\in B}\pi_i f_i\right)
679 \overline{\left(\sum_{j\in B}\pi_j g_j\right)}.
680 \]
681 \end{proof}
682 
683 \begin{theorem}[Tower law and martingale packets]\label{core:record:015}
684 If $\mathcal Q_c$ is coarser than $\mathcal Q_f$, then
685 \[
686 \boxed{
687 P_cP_f=P_fP_c=P_c.
688 }
689 \]
690 For a nested tower $P_0\le P_1\le\cdots\le P_r$, define
691 \[
692 D_0=P_0,
693 \qquad
694 D_j=P_j-P_{j-1}.
695 \]
696 Then
697 \[
698 D_iD_j=\delta_{ij}D_j,
699 \qquad
700 P_r=\sum_{j=0}^rD_j.
701 \]
702 \end{theorem}
703 
704 \begin{proof}
705 A coarse-block average of a fine-block average is the coarse-block average, and a fine-block average of a coarse-constant function leaves it unchanged. The difference identities follow from $P_iP_j=P_{\min(i,j)}$.
706 \end{proof}
707 
708 \begin{figure}[htbp]
709 \centering
710 \includegraphics[width=0.86\textwidth]{08_hermite_packets.pdf}
711 \caption{Hermite--Gaussian packets, which diagonalize the Ornstein--Uhlenbeck semigroup. The numerical study checks their Gaussian orthogonality through quadrature. The five displayed modes run from n = 0 at the bottom to n = 4 at the top. Vertical offsets separate the curves.}
712 \label{core:fig:hermite}
713 \end{figure}
714 
715 \begin{classical}[Gaussian chaos realization]\label{core:record:016}
716 For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally:
717 \[
718 T_tH_n=e^{-nt}H_n.
719 \]
720 Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.
721 \end{classical}
722 
723 \FloatBarrier
724 \section{Bayesian conditioning as a normalized nonlinear operator}\label{core:section:07}
725 
726 Let $p=(p_i)$ be a probability vector and let $\lambda=(\lambda_i)$ be a nonnegative likelihood with positive evidence
727 \[
728 Z_\lambda(p)=\sum_i\lambda_ip_i.
729 \]
730 Define
731 \[
732 \boxed{
733 \cB_\lambda(p)
734 =
735 \frac{\lambda\odot p}{Z_\lambda(p)}.
736 }
737 \]
738 This map is nonlinear solely because of its scalar reconstruction factor.
739 
740 \begin{theorem}[Independent-likelihood composition]\label{core:record:017}
741 For nonnegative likelihoods $\lambda$ and $\mu$, assume $Z_\lambda(p)>0$ and $Z_{\lambda\odot\mu}(p)>0$. Equivalently, both successive updates have positive evidence. Then
742 \[
743 \boxed{
744 \cB_\mu(\cB_\lambda(p))
745 =
746 \cB_{\lambda\odot\mu}(p).
747 }
748 \]
749 \end{theorem}
750 
751 \begin{proof}
752 The first normalization appears in both numerator and denominator of the second update and cancels:
753 \[
754 \frac{\mu_i(\lambda_ip_i/Z_\lambda)}{\sum_j\mu_j\lambda_jp_j/Z_\lambda}
755 =
756 \frac{\mu_i\lambda_ip_i}{\sum_j\mu_j\lambda_jp_j}.
757 \]
758 \end{proof}
759 
760 \subsection{Packet evidence and reconstruction}
761 
762 Let the state indices be partitioned into packets $I_\alpha$. Define
763 \[
764 Z_\alpha=\sum_{i\in I_\alpha}\lambda_ip_i,
765 \qquad
766 m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta},
767 \qquad
768 q_{\alpha,i}=\begin{cases}
769 \lambda_ip_i/Z_\alpha,&Z_\alpha>0,\ i\in I_\alpha,\\
770 0,&\text{otherwise}.
771 \end{cases}
772 \]
773 
774 \begin{theorem}[Bayesian packet reconstruction]\label{core:record:018}
775 \[
776 \boxed{
777 \cB_\lambda(p)
778 =
779 \sum_\alpha m_\alpha q_\alpha.
780 }
781 \]
782 Only packets with positive evidence carry a conditional probability vector. A zero-evidence packet has zero posterior mass and contributes the zero vector. The nonlinear coupling between packets occurs only through the total evidence used to normalize the packet evidences.
783 \end{theorem}
784 
785 \begin{proof}
786 If $Z_\alpha=0$, nonnegativity implies $\lambda_i p_i=0$ for every $i\in I_\alpha$, so the reconstruction identity holds with both sides zero. For $Z_\alpha>0$ and $i\in I_\alpha$,
787 \[
788 m_\alpha q_{\alpha,i}
789 =
790 \frac{Z_\alpha}{Z}\frac{\lambda_ip_i}{Z_\alpha}
791 =
792 \frac{\lambda_ip_i}{Z}.
793 \]
794 The packet supports are disjoint, so summing reconstructs the full posterior.
795 \end{proof}
796 
797 \begin{figure}[htbp]
798 \centering
799 \includegraphics[width=0.82\textwidth]{05_bayesian_packet_update.pdf}
800 \caption{A finite prior and posterior split into two state packets. Evidence changes both the local packet shape and the global packet mass.}
801 \label{core:fig:bayes-packet}
802 \end{figure}
803 
804 \begin{theorem}[Likelihood perturbation bound]\label{core:record:019}
805 Let $p$ be fixed and let $\lambda,\mu\ge0$ have positive evidences $Z_\lambda,Z_\mu$. Then
806 \[
807 \boxed{
808 \norm{\cB_\lambda(p)-\cB_\mu(p)}_1
809 \le
810 \frac{2\sum_i|\lambda_i-\mu_i|p_i}{\min(Z_\lambda,Z_\mu)}.
811 }
812 \]
813 \end{theorem}
814 
815 \begin{proof}
816 Add and subtract $\mu\odot p/Z_\lambda$ and use the triangle inequality:
817 \[
818 \norm{\frac{\lambda\odot p}{Z_\lambda}-\frac{\mu\odot p}{Z_\mu}}_1
819 \le
820 \frac{\norm{(\lambda-\mu)\odot p}_1}{Z_\lambda}
821 +
822 \norm{\mu\odot p}_1\left|\frac1{Z_\lambda}-\frac1{Z_\mu}\right|.
823 \]
824 The second term equals $|Z_\lambda-Z_\mu|/Z_\lambda$ and is bounded by the same weighted $L^1$ difference divided by $Z_\lambda$. Interchanging $\lambda$ and $\mu$ gives the minimum-evidence form.
825 \end{proof}
826 
827 \FloatBarrier
828 \section{Exact Bayesian refinement and information-loss packets}\label{core:section:08}
829 \begin{interpretation}
830 The all-prior criterion here tests coarse posterior masses for every prior and assumes at least two coarse blocks. Part II uses a stronger convention that also preserves each conditional distribution inside a block. These are different contracts, so the one-block boundary must be read with its own definition.
831 \end{interpretation}
832 \begin{figure}[htbp]
833 \centering
834 \includegraphics[width=.94\textwidth]{monograph_figures/exact_bayesian_coarsening.pdf}
835 \caption{Two rational priors with the same two coarse masses. A blockwise constant likelihood gives the same coarse update, while a likelihood varying within a block separates the posteriors. The exact fractions are calculated before plotting.}\label{monograph:fig:exact_bayesian_coarsening}
836 \end{figure}
837 
838 
839 
840 Let the fine state set $I$ be partitioned into nonempty coarse packets $I_a$, $a\in A$.  The coarsening map on probability vectors is
841 \[
842 (C_\pi p)_a=P_a=\sum_{i\in I_a}p_i.
843 \]
844 For a fine likelihood $\lambda=(\lambda_i)_{i\in I}$ and a prior $p$, define the effective coarse likelihood
845 \[
846 r_a(p)
847 =
848 \frac{\sum_{i\in I_a}\lambda_i p_i}{P_a}
849 \qquad(P_a>0).
850 \]
851 Its value on a zero-mass packet is irrelevant and may be set to zero.
852 
853 \begin{theorem}[Exact effective-likelihood factorization]\label{core:record:020}
854 Whenever the fine evidence is positive,
855 \[
856 \boxed{
857 C_\pi\cB_\lambda(p)
858 =
859 \cB_{r(p)}(C_\pi p).}
860 \]
861 Thus the only fine information needed for the coarse posterior, beyond the coarse prior masses, is the vector of conditional packet evidences $r(p)$.
862 \end{theorem}
863 
864 \begin{proof}
865 For $P=C_\pi p$ and $Z=\sum_i\lambda_ip_i=\sum_aP_ar_a(p)$,
866 \[
867 \bigl(C_\pi\cB_\lambda(p)\bigr)_a
868 =
869 \frac{\sum_{i\in I_a}\lambda_ip_i}{Z}
870 =
871 \frac{P_ar_a(p)}{\sum_bP_br_b(p)}
872 =
873 \bigl(\cB_{r(p)}(P)\bigr)_a.
874 \]
875 \end{proof}
876 
877 The next result gives the exact necessary and sufficient condition for a coarse likelihood to work for every prior.
878 
879 \begin{theorem}[Bayesian refinement criterion]\label{core:record:021}
880 Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive.  The identity
881 \[
882 \boxed{
883 C_\pi\cB_\lambda(p)
884 =
885 \cB_{\bar\lambda}(C_\pi p)}
886 \]
887 holds for every fine prior $p$ if and only if there exists one scalar $c>0$ such that
888 \[
889 \boxed{
890 \lambda_i=c\,\bar\lambda_a
891 \qquad(i\in I_a).}
892 \]
893 Equivalently, the likelihood is constant on each coarse packet, with the coarse values determined up to the globally invisible Bayesian scaling $c$.
894 \end{theorem}
895 
896 \begin{proof}
897 If $\lambda_i=c\bar\lambda_a$ on $I_a$, then $P_a r_a(p)=c\bar\lambda_aP_a$ for every prior, including zero-mass packets, and the preceding theorem gives the desired identity because Bayesian normalization removes the common scalar $c$.
898 
899 Conversely, fix two fine states $i,i'\in I_a$ and one state $j\in I_b$ in another coarse packet.  Vary the distribution of a fixed coarse mass inside $I_a$ between $i$ and $i'$ while leaving every coarse packet mass unchanged.  The right side depends only on those coarse masses and is therefore unchanged.  The left-side posterior mass of packet $a$ depends on the conditional average of $\lambda_i$ and $\lambda_{i'}$.  It can be independent of that variation only if $\lambda_i=\lambda_{i'}$.  Hence $\lambda$ is constant on every packet. Write the value on $I_a$ as $\lambda_a$.
900 
901 Now use priors supported on one state of packet $a$ and one state of packet $b$.  Equality of the two coarse posterior odds gives
902 \[
903 \frac{\lambda_a}{\lambda_b}
904 =
905 \frac{\bar\lambda_a}{\bar\lambda_b}.
906 \]
907 This holds for every pair of coarse packets, so $\lambda_a=c\bar\lambda_a$ for one common $c>0$.
908 \end{proof}
909 
910 \begin{remark}[The one-packet boundary]\label{core:record:022}
911 If the coarse partition has only one packet, every posterior coarsens to the scalar $1$, so commutation is automatic and no within-packet likelihood information can be identified.  The nontrivial criterion begins with at least two coarse packets.
912 \end{remark}
913 
914 When sufficiency fails, the defect still has an exact finite factorization.  Fix a declared coarse likelihood $\bar\lambda$, put
915 \[
916 P=C_\pi p,
917 \qquad
918 \delta(p)=r(p)-\bar\lambda,
919 \qquad
920 \bar b=\cB_{\bar\lambda}(P),
921 \qquad
922 Z_r=\langle r(p),P\rangle.
923 \]
924 
925 \begin{theorem}[Bayesian information-loss packet]\label{core:record:023}
926 Assume nonnegative fine and declared coarse likelihoods, with both $Z_r>0$ and $Z_{\bar\lambda}>0$. The coarsening defect is
927 \[
928 \boxed{
929 C_\pi\cB_\lambda(p)
930 -
931 \cB_{\bar\lambda}(P)
932 =
933 \frac{1}{Z_r}
934 \left(\diag(P)-\bar bP^{\mathsf T}\right)
935 \delta(p).}
936 \]
937 The factor matrix annihilates $\bar\lambda$.  This records the common-scale null direction at the declared coarse likelihood. The full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together.  Moreover,
938 \[
939 \boxed{
940 \left\|C_\pi\cB_\lambda(p)-\cB_{\bar\lambda}(P)\right\|_1
941 \le
942 \frac{2\sum_aP_a|\delta_a(p)|}
943 {\min\{Z_r,Z_{\bar\lambda}\}}}
944 \]
945 and
946 \[
947 \sum_aP_a|\delta_a(p)|
948 \le
949 \sum_i p_i\left|\lambda_i-\bar\lambda_{\pi(i)}\right|.
950 \]
951 \end{theorem}
952 
953 \begin{proof}
954 The effective-likelihood theorem reduces the defect to
955 \[
956 \cB_{\bar\lambda+\delta}(P)-\cB_{\bar\lambda}(P).
957 \]
958 Since $Z_r=Z_{\bar\lambda}+P^{\mathsf T}\delta$, direct subtraction gives
959 \[
960 \frac{1}{Z_r}
961 \left(\diag(P)-\bar bP^{\mathsf T}\right)\delta.
962 \]
963 Also,
964 \[
965 \left(\diag(P)-\bar bP^{\mathsf T}\right)\bar\lambda
966 =P\odot\bar\lambda-\bar bZ_{\bar\lambda}=0.
967 \]
968 The first norm bound is the likelihood perturbation theorem applied on the coarse simplex. For a zero-mass packet, both sides of the following estimate vanish. For $P_a>0$,
969 \[
970 P_a|\delta_a|
971 =
972 P_a\left|\sum_{i\in I_a}\frac{p_i}{P_a}
973 (\lambda_i-\bar\lambda_a)\right|
974 \le
975 \sum_{i\in I_a}p_i|\lambda_i-\bar\lambda_a|,
976 \]
977 and summing over $a$ proves the second inequality.
978 \end{proof}
979 
980 \begin{corollary}[Nested Bayesian packet tower]\label{core:record:024}
981 Consider a nested finite packet tower with strictly positive likelihoods at every level. At each adjacent coarsening with at least two coarse packets, commutation for every prior is equivalent to the fine likelihood being blockwise proportional to the declared coarse likelihood, with one common positive proportionality factor. An adjacent coarsening to a single packet commutes automatically and imposes no such condition. Under these conditions, all composite coarsening squares commute.
982 \end{corollary}
983 
984 \begin{proof}
985 At each adjacent coarsening with at least two coarse packets, apply the refinement criterion. For a single coarse packet, both normalized posteriors equal 1. Composition of these commuting squares gives the full tower statement.
986 \end{proof}
987 
988 \begin{corollary}[Canonical coarsest likelihood-resolving partition]\label{core:record:025}
989 For a finite family of likelihoods $\{\lambda^{(y)}\}_{y\in Y}$, define
990 \[
991 i\sim j
992 \quad\Longleftrightarrow\quad
993 \lambda_i^{(y)}=\lambda_j^{(y)}
994 \text{ for every }y\in Y.
995 \]
996 The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its evidence is positive. If all likelihoods are strictly positive and there are at least two profile classes, this is also the unique coarsest nontrivial partition on which every update closes for every prior. The one-packet partition always closes, independently of the profiles, and is excluded from this last assertion.
997 \end{corollary}
998 
999 \begin{proof}
1000 Equality of likelihood profiles is an equivalence relation. Its classes are likelihood-resolving by definition, and every likelihood-resolving partition refines them. Exact closure follows by summing the weighted masses in each fiber, including zero-mass fibers. Under strict positivity, the refinement criterion implies that every nontrivial all-prior closed partition is likelihood-resolving. When every profile is identical, all partitions close. There need not be a unique coarsest nontrivial partition.
1001 \end{proof}
1002 
1003 \begin{figure}[htbp]
1004 \centering
1005 \includegraphics[width=0.92\textwidth]{17_bayesian_refinement_square.pdf}
1006 \caption{For strictly positive likelihoods and at least two coarse packets, the Bayesian refinement square commutes for every prior exactly when the likelihood is constant on each coarsening fiber, up to one global scale.}
1007 \label{core:fig:bayes-refinement}
1008 \end{figure}
1009 
1010 \begin{figure}[htbp]
1011 \centering
1012 \includegraphics[width=0.94\textwidth]{18_bayesian_information_loss.pdf}
1013 \caption{When packet sufficiency fails, the exact defect factors through the difference between the conditional effective likelihood and the declared coarse likelihood.}
1014 \label{core:fig:bayes-loss}
1015 \end{figure}
1016 
1017 \begin{certificate}[Bayesian refinement]\label{core:record:026}
1018 For the finite test partition $\{0,1\}\sqcup\{2,3\}$, a blockwise-constant likelihood gave commutation residual $1.11\times10^{-16}$.  A nonconstant likelihood produced an exact defect-factorization residual $7.29\times10^{-17}$.  Its measured $L^1$ defect was $4.8787\times10^{-2}$, below the certified bound $1.1016\times10^{-1}$.  Two priors with identical coarse masses but different hidden within-packet shapes produced coarse posteriors differing by $1.2400\times10^{-1}$, which is a direct necessity witness.
1019 \end{certificate}
1020 
1021 \begin{boundary}[Finite exact theorem]\label{core:record:027}
1022 The conjectured Bayesian refinement law is therefore resolved for finite packet systems and arbitrary priors.  Infinite-dimensional disintegration, posterior consistency, and stability of data-dependent partitions require their own measure-theoretic and asymptotic hypotheses.
1023 \end{boundary}
1024 
1025 
1026 \FloatBarrier
1027 \section{Bayesian flows, conjugacy, and hidden states}\label{core:section:09}
1028 
1029 Fix a score vector $g$. The likelihood flow is
1030 \[
1031 \lambda_i(t)=e^{tg_i},
1032 \]
1033 and the Bayesian flow is
1034 \[
1035 \Phi_t(p)_i
1036 =
1037 \frac{e^{tg_i}p_i}{\sum_je^{tg_j}p_j}.
1038 \]
1039 
1040 \begin{theorem}[Bayesian nonlinear semigroup]\label{core:record:028}
1041 \[
1042 \boxed{
1043 \Phi_{s+t}=\Phi_s\circ\Phi_t.
1044 }
1045 \]
1046 Its generator is the centered multiplicative field
1047 \[
1048 \boxed{
1049 \frac{d}{dt}p_i(t)
1050 =
1051 \bigl(g_i-\langle g,p(t)\rangle\bigr)p_i(t).
1052 }
1053 \]
1054 \end{theorem}
1055 
1056 \begin{proof}
1057 The semigroup law is the likelihood-composition theorem with $e^{sg}e^{tg}=e^{(s+t)g}$. Differentiating the normalized exponential formula gives
1058 \[
1059 \dot p_i=g_ip_i-p_i\frac{\sum_jg_je^{tg_j}p_j(0)}{\sum_je^{tg_j}p_j(0)}.
1060 \]
1061 The quotient is $\langle g,p(t)\rangle$.
1062 \end{proof}
1063 
1064 \begin{figure}[htbp]
1065 \centering
1066 \includegraphics[width=0.80\textwidth]{07_bayesian_flow.pdf}
1067 \caption{Continuous likelihood accumulation produces a nonlinear semigroup on the probability simplex.}
1068 \label{core:fig:bayesian-flow}
1069 \end{figure}
1070 
1071 \begin{example}[Beta--Binomial conjugate packet]\label{core:record:029}
1072 A $\operatorname{Beta}(a,b)$ prior and $s$ successes in $n$ trials produce
1073 \[
1074 \operatorname{Beta}(a+s,b+n-s).
1075 \]
1076 The prior exponents are additive packets: the likelihood increments them by $(s,n-s)$. In the numerical example,
1077 \[
1078 (a,b)=(2,3),\quad (s,n)=(7,10),
1079 \]
1080 so the posterior is $\operatorname{Beta}(9,6)$ with mean $0.6$.
1081 \end{example}
1082 
1083 \begin{figure}[htbp]
1084 \centering
1085 \includegraphics[width=0.80\textwidth]{06_beta_binomial_conjugacy.pdf}
1086 \caption{A conjugate Bayesian update viewed as exact transfer of exponent packets.}
1087 \label{core:fig:beta-conjugacy}
1088 \end{figure}
1089 
1090 \subsection{Hidden Markov filtering}
1091 
1092 Let $K$ be column stochastic and let $\lambda^{(r)}$ be the likelihood associated with the $r$th observation. The filter recursion is
1093 \[
1094 \widehat p_r=Kp_{r-1},
1095 \qquad
1096 p_r=\cB_{\lambda^{(r)}}(\widehat p_r).
1097 \]
1098 
1099 \begin{proposition}[Prediction-update reconstruction]\label{core:record:030}
1100 Every hidden Markov filtering step is the composition of a linear Markov transfer and a normalized nonlinear likelihood transfer. Assume a probability initial vector, a column-stochastic transition matrix, and nonnegative likelihoods with positive evidence at every step. Then every $p_r$ is a probability vector.
1101 \end{proposition}
1102 
1103 \begin{proof}
1104 Column stochasticity gives $\widehat p_r\ge0$ and $\bm1^T\widehat p_r=1$. Bayesian normalization then gives $p_r\ge0$ and $\bm1^Tp_r=1$.
1105 \end{proof}
1106 
1107 \begin{figure}[htbp]
1108 \centering
1109 \includegraphics[width=0.80\textwidth]{15_hidden_markov_filter.pdf}
1110 \caption{A hidden Markov filter alternates a linear prediction semigroup with nonlinear Bayesian normalization.}
1111 \label{core:fig:hmm}
1112 \end{figure}
1113 
1114 \FloatBarrier
1115 \section{Fractional powers as spectral packet transfer}\label{core:section:10}
1116 \begin{readerroadmap}
1117 First identify the positive operator and its spectral packets. Then distinguish an exact spectral identity from a bound for a refinement mismatch. The gap-free certificate and the common-gap estimate have different hypotheses and different strengths.
1118 \end{readerroadmap}
1119 
1120 
1121 Let $L=L^*\ge0$ on a finite-dimensional Hilbert space. Its spectral resolution is
1122 \[
1123 L=\sum_j\lambda_jP_j.
1124 \]
1125 For $\alpha>0$, define
1126 \[
1127 \boxed{L^\alpha=\sum_j\lambda_j^\alpha P_j.}
1128 \]
1129 The associated fractional semigroup is
1130 \[
1131 T_\alpha(t)=e^{-tL^\alpha}
1132 =\sum_je^{-t\lambda_j^\alpha}P_j.
1133 \]
1134 
1135 \begin{theorem}[Fractional packet calculus]\label{core:record:031}
1136 For $\alpha,\beta>0$,
1137 \[
1138 L^\alpha L^\beta=L^{\alpha+\beta},
1139 \]
1140 and for $s,t\ge0$,
1141 \[
1142 \boxed{T_\alpha(s)T_\alpha(t)=T_\alpha(s+t).}
1143 \]
1144 Every spectral packet $P_j$ reduces $L^\alpha$ and $T_\alpha(t)$.
1145 \end{theorem}
1146 
1147 \begin{proof}
1148 All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example,
1149 \[
1150 L^\alpha L^\beta
1151 =\sum_{j,k}\lambda_j^\alpha\lambda_k^\beta P_jP_k
1152 =\sum_j\lambda_j^{\alpha+\beta}P_j.
1153 \]
1154 \end{proof}
1155 
1156 \begin{figure}[htbp]
1157 \centering
1158 \includegraphics[width=0.80\textwidth]{09_fractional_semigroup.pdf}
1159 \caption{Different spectral exponents produce different transfer profiles from the same point seed.}
1160 \label{core:fig:fractional}
1161 \end{figure}
1162 
1163 \subsection{Square-root subordination}
1164 
1165 \begin{theorem}[Poisson subordination]\label{core:record:032}
1166 For $t>0$ and $L\ge0$,
1167 \[
1168 \boxed{
1169 e^{-t\sqrt L}
1170 =
1171 \frac{t}{2\sqrt\pi}
1172 \int_0^\infty
1173 s^{-3/2}e^{-t^2/(4s)}e^{-sL}\,ds.
1174 }
1175 \]
1176 The integral converges in operator norm in finite dimension.
1177 \end{theorem}
1178 
1179 \begin{proof}
1180 By the spectral theorem it suffices to prove the scalar identity
1181 \[
1182 e^{-t\sqrt\lambda}
1183 =
1184 \frac{t}{2\sqrt\pi}
1185 \int_0^\infty
1186 s^{-3/2}e^{-t^2/(4s)-\lambda s}\,ds
1187 \]
1188 for $\lambda\ge0$. This is the classical Gaussian integral after the substitution $u=t/(2\sqrt s)$. Applying the identity to every eigenvalue and recombining the spectral projectors proves the operator formula.
1189 \end{proof}
1190 
1191 \subsection{Fractional naturality under a bounded refinement defect}
1192 
1193 Let $A\ge0$ act on a fine Hilbert space, let $B\ge0$ act on a coarse Hilbert space, and let $C$ be a bounded coarsening map.  Set
1194 \[
1195 D=CA-BC,
1196 \qquad
1197 D_\alpha=CA^\alpha-B^\alpha C,
1198 \qquad 0<\alpha<1.
1199 \]
1200 
1201 \begin{theorem}[Fractional intertwining theorem]\label{core:record:033}\label{core:thm:fractional-naturality}
1202 For bounded nonnegative self-adjoint $A$ and $B$,
1203 \[
1204 \boxed{
1205 D_\alpha
1206 =
1207 \frac{\sin(\pi\alpha)}{\pi}
1208 \int_0^\infty
1209  t^\alpha(tI+B)^{-1}D(tI+A)^{-1}\,dt.
1210 }
1211 \]
1212 The improper integral converges in operator norm.  Moreover,
1213 \[
1214 \boxed{
1215 \norm{D_\alpha}
1216 \le
1217 K_\alpha\norm{C}^{1-\alpha}\norm{D}^{\alpha},
1218 \qquad
1219 K_\alpha=
1220 \frac{2^{1-\alpha}\sin(\pi\alpha)}
1221 {\pi\alpha(1-\alpha)}.
1222 }
1223 \]
1224 If $A,B\ge mI$ for some $m>0$, then the stronger linear estimate holds:
1225 \[
1226 \boxed{
1227 \norm{D_\alpha}
1228 \le
1229 \alpha m^{\alpha-1}\norm{D}.
1230 }
1231 \]
1232 In particular, exact generator intertwining $CA=BC$ implies exact fractional intertwining $CA^\alpha=B^\alpha C$.
1233 \end{theorem}
1234 
1235 \begin{proof}
1236 For $0<\alpha<1$, the Balakrishnan formula gives
1237 \[
1238 A^\alpha
1239 =
1240 \frac{\sin(\pi\alpha)}{\pi}
1241 \int_0^\infty t^{\alpha-1}A(tI+A)^{-1}\,dt,
1242 \]
1243 and likewise for $B$.  Using
1244 \[
1245 A(tI+A)^{-1}=I-t(tI+A)^{-1}
1246 \]
1247 and the rectangular resolvent identity
1248 \[
1249 (tI+B)^{-1}C-C(tI+A)^{-1}
1250 =
1251 (tI+B)^{-1}(CA-BC)(tI+A)^{-1},
1252 \]
1253 the difference of the two Balakrishnan integrals becomes the displayed resolvent-defect formula.
1254 
1255 For small $t$,
1256 \[
1257 \norm{CA(tI+A)^{-1}-B(tI+B)^{-1}C}
1258 \le 2\norm C,
1259 \]
1260 because the two positive resolvent factors are contractions.  For large $t$, the resolvent identity gives the bound $\norm D/t$.  Splitting the integral at
1261 \[
1262 r=\frac{\norm D}{2\norm C}
1263 \]
1264 when $C$ and $D$ are nonzero yields
1265 \[
1266 \begin{aligned}
1267 \norm{D_\alpha}
1268 &\le
1269 \frac{\sin(\pi\alpha)}{\pi}
1270 \left(
1271 2\norm C\int_0^r t^{\alpha-1}\,dt
1272 +
1273 \norm D\int_r^\infty t^{\alpha-2}\,dt
1274 \right)\\
1275 &=
1276 K_\alpha\norm C^{1-\alpha}\norm D^\alpha.
1277 \end{aligned}
1278 \]
1279 The zero cases are immediate.
1280 
1281 If $A,B\ge mI$, then both resolvents have norm at most $(t+m)^{-1}$, so
1282 \[
1283 \norm{D_\alpha}
1284 \le
1285 \frac{\sin(\pi\alpha)}{\pi}\norm D
1286 \int_0^\infty\frac{t^\alpha}{(t+m)^2}\,dt.
1287 \]
1288 After $t=mu$, the beta integral equals
1289 \[
1290 m^{\alpha-1}\frac{\alpha\pi}{\sin(\pi\alpha)},
1291 \]
1292 which gives the linear estimate.
1293 \end{proof}
1294 
1295 \begin{corollary}[Fractional semigroup refinement]\label{core:record:034}\label{core:cor:fractional-semigroup-refinement}
1296 For $s\ge0$,
1297 \[
1298 \boxed{
1299 Ce^{-sA^\alpha}-e^{-sB^\alpha}C
1300 =
1301 -\int_0^s
1302  e^{-(s-r)B^\alpha}D_\alpha e^{-rA^\alpha}\,dr.
1303 }
1304 \]
1305 Hence
1306 \[
1307 \norm{Ce^{-sA^\alpha}-e^{-sB^\alpha}C}
1308 \le s\norm{D_\alpha}.
1309 \]
1310 Under the common gap $A,B\ge mI$,
1311 \[
1312 \boxed{
1313 \norm{Ce^{-sA^\alpha}-e^{-sB^\alpha}C}
1314 \le
1315 s e^{-sm^\alpha}\alpha m^{\alpha-1}\norm D.
1316 }
1317 \]
1318 \end{corollary}
1319 
1320 \begin{proof}
1321 Differentiate
1322 \[
1323 r\longmapsto e^{-(s-r)B^\alpha}Ce^{-rA^\alpha}
1324 \]
1325 and integrate from $0$ to $s$.  Positive fractional semigroups are contractions. Under the gap, the product of their two norm bounds is $e^{-sm^\alpha}$.
1326 \end{proof}
1327 
1328 \begin{corollary}[Fractional refinement towers]\label{core:record:035}
1329 Let $D_\ell=C_\ell A_{\ell+1}-A_\ell C_\ell$ in a bounded nonnegative self-adjoint tower, with $\sup_\ell\norm{C_\ell}\le M$.  If
1330 \[
1331 \sum_\ell\norm{D_\ell}^{\alpha}<\infty,
1332 \]
1333 then the fractional defects
1334 \[
1335 C_\ell A_{\ell+1}^{\alpha}-A_\ell^{\alpha}C_\ell
1336 \]
1337 have summable operator norms. If the tower has a uniform gap $m>0$, it is enough that $\sum_\ell\norm{D_\ell}<\infty$. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces. No operator sum across unrelated spaces is asserted.
1338 \end{corollary}
1339 
1340 \begin{figure}[htbp]
1341 \centering
1342 \includegraphics[width=0.94\textwidth]{19_fractional_naturality.pdf}
1343 \caption{A generator-level refinement defect transfers quantitatively to fractional powers and then to fractional semigroups.  The two upper curves are theorem certificates, not fitted error bars.}
1344 \label{core:fig:fractional-naturality}
1345 \end{figure}
1346 
1347 \begin{boundary}[Unbounded and nonnormal generators]\label{core:record:036}
1348 The theorem resolves fractional naturality for bounded nonnegative self-adjoint refinement systems and gives the exact estimate needed for finite discretization towers.  For unbounded positive generators, the same resolvent argument requires a declared common-domain condition and a bounded extension of $CA-BC$.  Sectorial or nonnormal generators require their own contour calculus and resolvent bounds.  Those domain questions are not inferred from the finite certificates.
1349 \end{boundary}
1350 
1351 \begin{boundary}[Non-self-adjoint fractional powers]\label{core:record:037}
1352 The spectral definition above is complete for finite self-adjoint positive operators. Sectorial and nonnormal operators require a resolvent or contour functional calculus. The same packet language can be used only after those analytic hypotheses are established.
1353 \end{boundary}
1354 
1355 \FloatBarrier
1356 \section{Green operators and the Poisson seed}\label{core:section:11}
1357 
1358 On the interval $(0,1)$ with Dirichlet boundary conditions, the Green kernel for $-u''=f$ is
1359 \[
1360 \boxed{G(x,y)=\min(x,y)-xy.}
1361 \]
1362 Thus the point source $\delta_y$ generates the solution kernel, and a general source is reconstructed by
1363 \[
1364 u(x)=\int_0^1G(x,y)f(y)\,dy.
1365 \]
1366 
1367 For $n$ interior points with spacing $h=1/(n+1)$, define
1368 \[
1369 L_h=\frac1{h^2}
1370 \begin{pmatrix}
1371 2&-1&&\\
1372 -1&2&\ddots&\\
1373 &\ddots&\ddots&-1\\
1374 &&-1&2
1375 \end{pmatrix}.
1376 \]
1377 
1378 \begin{theorem}[Exact discrete Green reconstruction at grid nodes]\label{core:record:038}
1379 Let $x_i=ih$. Then
1380 \[
1381 \boxed{
1382 \frac1h(L_h^{-1})_{ij}
1383 =G(x_i,x_j).
1384 }
1385 \]
1386 Thus the scaled discrete inverse reproduces the continuum Dirichlet Green kernel exactly at the grid nodes.
1387 \end{theorem}
1388 
1389 \begin{proof}
1390 The inverse of the unscaled tridiagonal matrix has entries
1391 \[
1392 \frac{\min(i,j)(n+1-\max(i,j))}{n+1}.
1393 \]
1394 Multiplying by $h^2$ and then dividing by $h$ gives
1395 \[
1396 h\min(i,j)\left(1-\frac{\max(i,j)}{n+1}\right)
1397 =\min(x_i,x_j)-x_ix_j.
1398 \]
1399 \end{proof}
1400 
1401 \begin{figure}[htbp]
1402 \centering
1403 \includegraphics[width=0.80\textwidth]{10_poisson_green_kernel.pdf}
1404 \caption{A section of the continuum Green kernel and the scaled discrete inverse. They agree at the grid nodes.}
1405 \label{core:fig:green}
1406 \end{figure}
1407 
1408 \subsection{Low-mode convergence}
1409 
1410 The $k$th discrete eigenvalue is
1411 \[
1412 \lambda_{k,h}=\frac4{h^2}\sin^2\left(\frac{k\pi h}{2}\right),
1413 \]
1414 whereas the continuum eigenvalue is $(k\pi)^2$.
1415 
1416 \begin{theorem}[Second-order low-mode convergence]\label{core:record:039}
1417 For fixed $k$,
1418 \[
1419 \lambda_{k,h}=(k\pi)^2+O(h^2).
1420 \]
1421 More precisely, for sufficiently small $h$,
1422 \[
1423 0\le (k\pi)^2-\lambda_{k,h}
1424 \le \frac{(k\pi)^4}{12}h^2.
1425 \]
1426 \end{theorem}
1427 
1428 \begin{proof}
1429 For $x\ge0$, $0\le x^2-\sin^2x\le x^4/3$. Set $x=k\pi h/2$ and multiply by $4/h^2$.
1430 \end{proof}
1431 
1432 \begin{figure}[htbp]
1433 \centering
1434 \includegraphics[width=0.72\textwidth]{11_pde_refinement_convergence.pdf}
1435 \caption{The first finite-difference eigenvalue converges with observed order approximately two.}
1436 \label{core:fig:pde-convergence}
1437 \end{figure}
1438 
1439 \FloatBarrier
1440 \section{Finite-state Feynman--Kac transfer}\label{core:section:12}
1441 
1442 Let $K$ be a column-stochastic matrix, let $V$ be a potential on the states, and let $f$ be terminal data. Define
1443 \[
1444 M_V=\diag(e^{-V})K^T.
1445 \]
1446 The backward Feynman--Kac value after $n$ steps is
1447 \[
1448 u_n=M_V^nf.
1449 \]
1450 
1451 \begin{theorem}[Discrete Feynman--Kac path formula]\label{core:record:040}
1452 For an initial state $i$,
1453 \[
1454 \boxed{
1455 (u_n)_i
1456 =
1457 \Exp_i\left[
1458 \exp\left(-\sum_{r=0}^{n-1}V(X_r)\right)f(X_n)
1459 \right].
1460 }
1461 \]
1462 \end{theorem}
1463 
1464 \begin{proof}
1465 For $n=1$,
1466 \[
1467 (M_Vf)_i=e^{-V(i)}\sum_jK_{j i}f_j,
1468 \]
1469 which is the stated expectation. Iterating and expanding the matrix product sums over every path $(i=i_0,i_1,\ldots,i_n)$ with probability $\prod_rK_{i_{r+1},i_r}$ and weight $\prod_re^{-V(i_r)}$.
1470 \end{proof}
1471 
1472 \begin{figure}[htbp]
1473 \centering
1474 \includegraphics[width=0.78\textwidth]{12_feynman_kac_transfer.pdf}
1475 \caption{Potential-weighted transfer values computed by the finite Feynman--Kac operator. Exhaustive path enumeration agrees with the matrix formula in the supplied tests.}
1476 \label{core:fig:feynman-kac}
1477 \end{figure}
1478 
1479 \FloatBarrier
1480 \section{Exact elimination of latent packets}\label{core:section:13}
1481 \begin{keyidea}
1482 Elimination removes hidden coordinates from the displayed state, but their future influence survives. The resolvent stores that influence as a self-energy. The time description stores it as a memory kernel. Both descriptions refer to the same declared block system.
1483 \end{keyidea}
1484 
1485 
1486 Let $X=P X\oplus Q X$ and write
1487 \[
1488 A=
1489 \begin{pmatrix}
1490 A_{PP}&A_{PQ}\\
1491 A_{QP}&A_{QQ}
1492 \end{pmatrix}.
1493 \]
1494 
1495 \begin{theorem}[Schur--Feshbach projected resolvent]\label{core:record:041}
1496 Assume both $zI-A$ and $zI-A_{QQ}$ are invertible. Define
1497 \[
1498 \boxed{
1499 F_P(z)
1500 =zI-A_{PP}-A_{PQ}(zI-A_{QQ})^{-1}A_{QP}.
1501 }
1502 \]
1503 Then
1504 \[
1505 \boxed{
1506 \left.P(zI-A)^{-1}\right|_{\Ran P}=F_P(z)^{-1}.
1507 }
1508 \]
1509 \end{theorem}
1510 
1511 \begin{proof}
1512 Solve the hidden block equation
1513 \[
1514 (zI-A_{QQ})x_Q=A_{QP}x_P+y_Q
1515 \]
1516 for $x_Q$ and substitute it into the retained equation. The resulting retained coefficient is $F_P(z)$. Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting $y_Q=0$ gives the stated identity as an operator on $\Ran P$. Equivalently, this is the block inverse formula.
1517 \end{proof}
1518 
1519 The self-energy
1520 \[
1521 \Sigma_P(z)=A_{PQ}(zI-A_{QQ})^{-1}A_{QP}
1522 \]
1523 contains the exact effect of the eliminated packets. A bare truncation $A_{PP}$ discards this term and is generally not exact.
1524 
1525 \subsection{Time-domain memory}
1526 
1527 For the homogeneous system $\dot x=Ax$, variation of constants gives
1528 \[
1529 x_Q(t)=e^{tA_{QQ}}x_Q(0)
1530 +\int_0^te^{(t-r)A_{QQ}}A_{QP}x_P(r)\,dr.
1531 \]
1532 Substitution yields
1533 \[
1534 \dot x_P(t)
1535 =A_{PP}x_P(t)+A_{PQ}e^{tA_{QQ}}x_Q(0)
1536 +\int_0^t\cK(t-r)x_P(r)\,dr,
1537 \]
1538 where
1539 \[
1540 \boxed{
1541 \cK(t)=A_{PQ}e^{tA_{QQ}}A_{QP}.
1542 }
1543 \]
1544 
1545 \begin{theorem}[Memory/self-energy correspondence]\label{core:record:042}
1546 For finite complex block matrices, assume $\operatorname{Re}z>\operatorname{Re}\lambda$ for every $\lambda\in\sigma(A_{QQ})$. Then the following improper integral is absolutely convergent and
1547 \[
1548 \boxed{
1549 \int_0^\infty e^{-zt}\cK(t)\,dt
1550 =A_{PQ}(zI-A_{QQ})^{-1}A_{QP}
1551 =\Sigma_P(z).
1552 }
1553 \]
1554 \end{theorem}
1555 
1556 \begin{proof}
1557 Put $B=A_{QQ}$ and $T_z(t)=e^{-zt}e^{tB}=e^{t(B-zI)}$.
1558 The generalized-eigenspace decomposition in finite dimensions writes $T_z$ as a
1559 finite sum of constant matrix coefficients times $t^k e^{(\lambda-z)t}$.
1560 This follows by expanding the nilpotent part on each generalized eigenspace.
1561 Equivalently, apply the decomposition to left multiplication by $B$ on the matrix
1562 algebra and evaluate the exponential at $I$.
1563 The strict spectral inequality makes every such term absolutely integrable on
1564 $(0,\infty)$ and forces it to tend to zero. Hence $T_z$ is Bochner integrable
1565 and $T_z(t)\to0$, including when $B$ is nonnormal.
1566 
1567 Let $J=\int_0^\infty T_z(t)\,dt$. Integrating both derivative identities
1568 $T_z'=(B-zI)T_z=T_z(B-zI)$ and using $T_z(0)=I$ gives
1569 \[
1570 (zI-B)J=J(zI-B)=I.
1571 \]
1572 Thus $J=(zI-B)^{-1}$. Continuous left and right multiplication carry the
1573 Bochner integral through the two rectangular transfer blocks, yielding the claim.
1574 \end{proof}
1575 
1576 \begin{figure}[htbp]
1577 \centering
1578 \includegraphics[width=0.78\textwidth]{13_latent_memory_kernel.pdf}
1579 \caption{The norm of the exact memory kernel produced by eliminated packets.}
1580 \label{core:fig:memory}
1581 \end{figure}
1582 
1583 \FloatBarrier
1584 \section{Survival and hazard semigroups}\label{core:section:14}
1585 
1586 Let $A$ be a finite subgenerator: its off-diagonal entries are nonnegative and its column sums are nonpositive. These sign conditions alone allow nonabsorbing states. Eventual absorption is a separate hypothesis. Let $p_0$ be an initial transient-state distribution. Define
1587 \[
1588 p(t)=e^{tA}p_0,
1589 \qquad
1590 S(t)=\bm1^Tp(t).
1591 \]
1592 The exit-rate row vector is
1593 \[
1594 a^T=-\bm1^TA\ge0.
1595 \]
1596 
1597 \begin{theorem}[Semigroup survival identities]\label{core:record:043}
1598 For $t\ge0$, the finite-time exit density and, wherever $S(t)>0$, the hazard are
1599 \[
1600 \boxed{
1601 f(t)=a^Te^{tA}p_0=-S'(t),
1602 \qquad
1603 h(t)=\frac{f(t)}{S(t)}.
1604 }
1605 \]
1606 Consequently, $S$ is nonincreasing and
1607 \[
1608 S(t)=\exp\left(-\int_0^th(r)\,dr\right)
1609 \]
1610 whenever $S(t)>0$. The density may be defective: its total mass is $1-\lim_{t\to\infty}S(t)$. It is a probability density when absorption is eventual, so that this limit is zero.
1611 \end{theorem}
1612 
1613 \begin{proof}
1614 Differentiate:
1615 \[
1616 S'(t)=\bm1^TAe^{tA}p_0=-a^Te^{tA}p_0.
1617 \]
1618 The hazard identity is the definition of conditional instantaneous exit rate. Integrating $(\log S)'=-h$ gives the final formula.
1619 \end{proof}
1620 
1621 \begin{figure}[htbp]
1622 \centering
1623 \includegraphics[width=0.78\textwidth]{16_survival_hazard.pdf}
1624 \caption{Survival and instantaneous hazard produced by one transient transfer semigroup.}
1625 \label{core:fig:survival}
1626 \end{figure}
1627 
1628 \FloatBarrier
1629 \section{One common operator language}\label{core:section:15}
1630 
1631 The preceding constructions are not identical, but they use the same packet operations. The domain dictionary is summarized in \cref{core:tab:dictionary}.
1632 
1633 \begin{table}[htbp]
1634 \centering
1635 \small
1636 \begin{tabularx}{\textwidth}{L{0.17\textwidth} L{0.19\textwidth} Y Y}
1637 \toprule
1638 Realization & Seed & Transfer & Packet/reconstruction law\\
1639 \midrule
1640 Markov & point mass $\delta_i$ & $K^n$ or $e^{tQ}$ & state lumping and $CK^n=\bar K^nC$\\
1641 Conditional & random variable & $P_{\mathcal Q}$ & nested partitions and tower law\\
1642 Bayesian & prior $p$ & $\cB_\lambda$ & packet evidences and normalized synthesis\\
1643 Fractional & spectral source & $e^{-tL^\alpha}$ & spectral projectors $P_j$\\
1644 PDE & source $f$ or $\delta_y$ & resolvent/Green operator & eigenmode or spatial packet synthesis\\
1645 Feynman--Kac & terminal data $f$ & $(\diag(e^{-V})K^T)^n$ & weighted path reconstruction\\
1646 Latent system & retained state & Feshbach/self-energy & hidden packets return as memory\\
1647 Survival & initial transient law & $e^{tA}$ & sum over transient packets\\
1648 \bottomrule
1649 \end{tabularx}
1650 \caption{The transfer-semigroup dictionary in the common operator language.}
1651 \label{core:tab:dictionary}
1652 \end{table}
1653 
1654 For clarity, the finite-dimensional holomorphic calculus used below has the
1655 following canonical definition. For $A$ acting on $\mathbb C^d$, its generalized
1656 eigenspaces $E_\mu=\ker(A-\mu I)^d$ form a direct-sum decomposition over the
1657 eigenvalues. Define the unique linear operator $F(A)$ by
1658 \[
1659 \left.F(A)\right|_{E_\mu}
1660 =\left.\sum_{k=0}^{d-1}\frac{F^{(k)}(\mu)}{k!}(A-\mu I)^k\right|_{E_\mu}.
1661 \]
1662 The formula is independent of a basis and of increasing the truncation index:
1663 the extra nilpotent powers vanish. It depends only on the finite Taylor jets of
1664 $F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the
1665 spectrum, each eigenvalue admits a radius $r_\mu>0$ with
1666 $\overline{D(\mu,r_\mu)}\subset U$, and Cauchy's formula gives
1667 \[
1668 \frac{F^{(k)}(\mu)}{k!}
1669 =\frac{1}{2\pi i}\oint_{|z-\mu|=r_\mu}
1670   \frac{F(z)}{(z-\mu)^{k+1}}\,dz.
1671 \]
1672 Consequently the same canonical operator is obtained from any such local
1673 Cauchy circles, even when $U$ is disconnected. This is the standard finite
1674 Taylor-jet definition of the holomorphic matrix function.
1675 
1676 \begin{theorem}[Naturality under exact intertwining]\label{core:record:044}
1677 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
1678 \[
1679 CA_f=A_cC,
1680 \]
1681 then
1682 \[
1683 \boxed{CF(A_f)=F(A_c)C.}
1684 \]
1685 In particular, exact generator refinement propagates simultaneously to semigroups, resolvents, integer powers, and analytic transfer functions.
1686 \end{theorem}
1687 
1688 \begin{proof}
1689 The intertwining identity implies
1690 $C(A_f-\mu I)^k=(A_c-\mu I)^kC$ for every $k\ge0$.
1691 It therefore sends the generalized $\mu$-eigenspace of $A_f$ into that of $A_c$.
1692 Use a common truncation index at least as large as both dimensions in the
1693 canonical formula above. Multiplying its finite sum by $C$ now gives the desired
1694 identity on every generalized eigenspace. The direct-sum decomposition gives
1695 it on the full space. The argument covers nilpotent Jordan parts and uses only
1696 local spectral Taylor jets. It does not require one Taylor series to converge
1697 on both spectra. The zero-dimensional case is immediate.
1698 \end{proof}
1699 
1700 \begin{remark}[Where nonlinearity enters]\label{core:record:045}
1701 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. Chapter III supplies the general nonlinear coordinate theorem. The Bayesian map is a probabilistic realization.
1702 \end{remark}
1703 
1704 \FloatBarrier
1705 \section{Numerical certificates}\label{core:section:16}
1706 
1707 The recorded calculations were run on CPU. No GPU timing or performance claim is made. Floating-point residuals are implementation checks and are not substitutes for the symbolic proofs.
1708 
1709 \begin{table}[htbp]
1710 \centering
1711 \small
1712 \begin{tabular}{L{0.56\textwidth} r}
1713 \toprule
1714 Certificate & Value\\
1715 \midrule
1716 Packet-block semigroup composition residual & $2.554\times10^{-15}$\\
1717 Operator reconstruction residual & $0$\\
1718 Random-walk mass error & $0$\\
1719 Exact Markov generator-lumping defect & $5.551\times10^{-17}$\\
1720 Exact finite-horizon lumping error & $2.220\times10^{-16}$\\
1721 Perturbed ten-step lumping error & $6.523\times10^{-3}$\\
1722 Linear defect bound for that example & $2.000\times10^{-1}$\\
1723 Bayesian composition residual & $5.551\times10^{-17}$\\
1724 Bayesian packet reconstruction residual & $0$\\
1725 Bayesian refinement commutation residual & $1.110\times10^{-16}$\\
1726 Bayesian defect-factorization residual & $7.286\times10^{-17}$\\
1727 Bayesian flow generator residual & $1.959\times10^{-11}$\\
1728 Conditional-expectation tower residual & $0$\\
1729 Hermite Gram residual & $9.992\times10^{-15}$\\
1730 Fractional semigroup residual & $2.082\times10^{-17}$\\
1731 Maximum square-root subordination error & $1.665\times10^{-16}$\\
1732 Exact fractional-naturality residual & $0$\\
1733 Fractional-power defect / gap certificate & $0.51059 / 0.95079$\\
1734 Fractional-semigroup defect / certificate & $0.12969 / 0.44480$\\
1735 Observed low-mode PDE convergence order & $1.999854$\\
1736 Feynman--Kac enumeration residual & $1.665\times10^{-16}$\\
1737 Feshbach projected-resolvent residual & $5.551\times10^{-17}$\\
1738 Memory Laplace-transform residual & $6.939\times10^{-18}$\\
1739 Duhamel refinement identity residual & $2.108\times10^{-16}$\\
1740 Survival monotonicity violation & $0$\\
1741 Hidden-Markov normalization error & $1.110\times10^{-16}$\\
1742 \bottomrule
1743 \end{tabular}
1744 \caption{Selected results from the machine-readable numerical record.}
1745 \label{core:tab:numerics}
1746 \end{table}
1747 
1748 \begin{certificate}[Test suite]\label{core:record:046}
1749 The September 6 release records the following automated-test result:
1750 \[
1751 \boxed{54/54\text{ automated tests}.}
1752 \]
1753 The tests cover projector reconstruction, packet-block semigroup composition, exact and approximate Markov lumping, Bayesian composition and flow, the exact Bayesian refinement criterion, information-loss factorization, likelihood-profile partitions, conditional-expectation towers, Hermite orthogonality, fractional semigroups, square-root subordination, fractional intertwining and semigroup-defect bounds, discrete Green kernels, PDE convergence, Feynman--Kac enumeration, Schur--Feshbach elimination, memory kernels, refinement defects, hidden Markov filtering, and survival curves.
1754 \end{certificate}
1755 
1756 \FloatBarrier
1757 \section{Remaining conjectural extensions}\label{core:section:17}
1758 
1759 The finite theory closes several operator identities but does not settle the corresponding universal continuum questions.
1760 
1761 \begin{conjecture}[Transfer-semigroup realization]\label{core:record:047}
1762 Every sufficiently regular realization of the seed calculus whose transfer stage is local in time admits a closed generator on a completed state space, and the packet analysis maps form a compatible system for its semigroup or evolution family.
1763 \end{conjecture}
1764 
1765 \begin{conjecture}[Uniform refinement control]\label{core:record:048}
1766 For a nested family of physically or analytically meaningful discretizations, there exist norms and constants independent of resolution such that
1767 \[
1768 \norm{C_\ell e^{tA_{\ell+1}}-e^{tA_\ell}C_\ell}
1769 \le
1770 M\,t\,\norm{C_\ell A_{\ell+1}-A_\ell C_\ell}
1771 \]
1772 on every bounded time interval. Summable generator defects would then produce a coherent inverse-limit evolution.
1773 \end{conjecture}
1774 
1775 \begin{conjecture}[Latent-packet closure]\label{core:record:049}
1776 For a stable infinite packet system, exact elimination of unresolved packets produces a causal memory kernel whose Laplace transform is the limiting packet self-energy. A Markovian closure exists precisely when this self-energy is constant, or can be represented by finitely many additional state packets.
1777 \end{conjecture}
1778 
1779 \begin{boundary}[What remains outside Chapter VI]\label{core:record:050}
1780 The following issues are not proved here:
1781 \begin{enumerate}[leftmargin=2em]
1782 \item existence and uniqueness for arbitrary stochastic differential equations;
1783 \item posterior consistency or asymptotic normality in infinite statistical models;
1784 \item domain invariance for unbounded fractional operators;
1785 \item tightness and compactness required for stochastic continuum limits;
1786 \item nonlinear PDE well-posedness beyond the stated finite and linear models;
1787 \item an optimal packet choice for a general Markov, Bayesian, or PDE problem.
1788 \end{enumerate}
1789 These belong to later realization-specific work or to external analytic theory.
1790 \end{boundary}
1791 
1792 \FloatBarrier
1793 \section{Computational implementation}\label{core:section:18}
1794 
1795 The accompanying Python package implements the formulas and finite models used in the chapter.
1796 \begin{center}
1797 \texttt{projector\_transfer\_semigroups}
1798 \end{center}
1799 The computational organization mirrors the chapter:
1800 \[
1801 \begin{array}{c|l}
1802 \texttt{packets.py} & \text{projectors, blocks, reconstruction, block composition}\\
1803 \texttt{semigroup.py} & \text{matrix semigroups, defects, Feshbach operators, memory}\\
1804 \texttt{markov.py} & \text{random walks, exact lumping, survival}\\
1805 \texttt{bayes.py} & \text{normalized updates, packet evidence, conjugacy, flows, HMM}\\
1806 \texttt{conditional.py} & \text{conditional expectation and Gaussian chaos}\\
1807 \texttt{fractional.py} & \text{spectral powers, subordination, and fractional refinement certificates}\\
1808 \texttt{pde.py} & \text{Dirichlet Laplacians, Green kernels, eigenvalue error}\\
1809 \texttt{feynman\_kac.py} & \text{weighted transfer and exhaustive path enumeration}.
1810 \end{array}
1811 \]
1812 
1813 \subsection{Representative algorithms}
1814 
1815 \begin{lstlisting}[caption={Normalized finite Bayesian update.}]
1816 def bayes_update(prior, likelihood):
1817     p = normalize(prior)
1818     evidence = likelihood @ p
1819     if evidence <= 0:
1820         raise ValueError("positive evidence required")
1821     return likelihood * p / evidence
1822 \end{lstlisting}
1823 
1824 \begin{lstlisting}[caption={Packet block composition.}]
1825 def block_composition(left, right):
1826     result = []
1827     for beta in range(len(left)):
1828         row = []
1829         for alpha in range(len(left)):
1830             value = sum(
1831                 left[beta][gamma] @ right[gamma][alpha]
1832                 for gamma in range(len(left))
1833             )
1834             row.append(value)
1835         result.append(row)
1836     return result
1837 \end{lstlisting}
1838 
1839 \begin{lstlisting}[caption={Finite Schur-Feshbach effective operator.}]
1840 def feshbach_effective_operator(A, retained, z):
1841     App, Apq, Aqp, Aqq = split_blocks(A, retained)
1842     return z*I - App - Apq @ inv(z*I - Aqq) @ Aqp
1843 \end{lstlisting}
1844 
1845 The source corpus contains the implementation, tests, tables, figure-generation code and numerical records.
1846 
1847 \FloatBarrier
1848 \section{Conclusion}\label{core:section:19}
1849 
1850 The transfer-semigroup realization gives one common operator language to several subjects that are often presented separately. Markov propagation is linear transfer of a probability seed. Conditional expectation is stochastic projection. Bayesian conditioning is normalized nonlinear transfer, and its exact finite coarsening law is characterized by packetwise likelihood sufficiency. Fractional calculus is spectral packet reweighting, and bounded positive refinement defects transfer to fractional powers and fractional semigroups by explicit resolvent and Duhamel estimates. A Green operator reconstructs a field from source packets. Feynman--Kac propagation adds path weights to stochastic transfer. Schur--Feshbach elimination shows that hidden packets are not destroyed. They return through self-energy and memory. Survival and hazard are scalar reconstructions of a transient semigroup.
1851 
1852 The central structural identity is
1853 \[
1854 \boxed{
1855 \text{generator defect}
1856 \longrightarrow
1857 \text{Duhamel semigroup defect}
1858 \longrightarrow
1859 \text{controlled refinement error}.
1860 }
1861 \]
1862 Together with exact packet reconstruction, this makes the semigroup chapter the natural bridge from the foundational calculus to the quantum, computational, and arithmetic realizations that follow.
1863 
1864 \clearpage
1865 \appendix
1866 \FloatBarrier
1867 \section{Lean verification and scope}\label{core:section:20}\label{core:sec:lean-status}
1868 \noindent\textbf{Fifth checked block, 6 September 2026.}
1869 The release contains 341 explicit Lean theorem declarations across 47 proof modules,
1870 including 31 additional declarations since the fourth checked block. The September 6 release records a successful full
1871 build and namespace audit with Lean 4.34.0-rc2 and pinned mathlib. The audit
1872 inspects 782 declarations, including definitions and generated auxiliaries. Only
1873 \texttt{propext}, \texttt{Classical.choice} and \texttt{Quot.sound} are allowed.
1874 These are different counting units from Part I's 33 theorem-like results.
1875 
1876 The ledger has \textbf{33 complete, zero partial and zero pending results}.
1877 The last two results, 13.2 and 15.1, are complete. The three separately stated
1878 conjectures remain open and are excluded from this count.
1879 
1880 ``Complete'' means the stated scope follows from the named checked targets
1881 and the ordinary specializations recorded in the ledger. Prose-to-Lean correspondence
1882 is manually reviewed. The fractional targets use complex Hilbert spaces.
1883 Complexification for real operators is an ordinary mathematical specialization,
1884 not a separately checked Lean construction. Completion of the theorem ledger does
1885 not certify every prose sentence, numerical figure or open conjecture.
1886 
1887 \small
1888 \begin{longtable}{@{}p{0.08\textwidth}p{0.68\textwidth}p{0.17\textwidth}@{}}
1889 \toprule Result & Manuscript statement & Lean scope\\
1890 \midrule\endfirsthead
1891 \toprule Result & Manuscript statement & Lean scope\\\midrule\endhead
1892 \bottomrule\endfoot
1893 1.2 & Finite semigroup law & Complete\\
1894 2.1 & Packet semigroup convolution & Complete\\
1895 2.2 & Reducing packets & Complete\\
1896 3.1 & Exact Duhamel refinement identity & Complete\\
1897 3.2 & Norm bound & Complete\\
1898 3.3 & Resolvent refinement identity & Complete\\
1899 4.1 & Finite Markov semigroup & Complete\\
1900 5.1 & Exact lumpability & Complete\\
1901 5.2 & Finite-horizon defect formula & Complete\\
1902 6.1 & Finite conditional-expectation projector & Complete\\
1903 6.2 & Tower law and martingale packets & Complete\\
1904 7.1 & Independent-likelihood composition & Complete\\
1905 7.2 & Bayesian packet reconstruction & Complete\\
1906 7.3 & Likelihood perturbation bound & Complete\\
1907 8.1 & Exact effective-likelihood factorization & Complete\\
1908 8.2 & Bayesian refinement criterion & Complete\\
1909 8.4 & Bayesian information-loss packet & Complete\\
1910 8.5 & Nested Bayesian packet tower & Complete\\
1911 8.6 & Canonical coarsest likelihood-resolving partition & Complete\\
1912 9.1 & Bayesian nonlinear semigroup & Complete\\
1913 9.3 & Prediction-update reconstruction & Complete\\
1914 10.1 & Fractional packet calculus & Complete\\
1915 10.2 & Poisson subordination & Complete\\
1916 10.3 & Fractional intertwining theorem & Complete\\
1917 10.4 & Fractional semigroup refinement & Complete\\
1918 10.5 & Fractional refinement towers & Complete\\
1919 11.1 & Exact discrete Green reconstruction at grid nodes & Complete\\
1920 11.2 & Second-order low-mode convergence & Complete\\
1921 12.1 & Discrete Feynman--Kac path formula & Complete\\
1922 13.1 & Schur--Feshbach projected resolvent & Complete\\
1923 13.2 & Memory/self-energy correspondence & Complete\\
1924 14.1 & Semigroup survival identities & Complete\\
1925 15.1 & Naturality under exact intertwining & Complete\\
1926 \end{longtable}
1927 \normalsize
1928 
1929 \paragraph{General nonnormal memory (13.2).}
1930 The actual damped exponential is decomposed into finitely many terms of the form
1931 $t^k e^{(\mu-z)t}$. Integrability and convergence to zero follow from
1932 $\operatorname{Re}\mu<\operatorname{Re}z$. The formal proof uses generalized
1933 eigenspaces of left multiplication on the matrix algebra, so nilpotent Jordan
1934 parts are included. Both inverse identities follow by integrating the two
1935 exponential derivative formulas. A continuous linear sandwich map gives the
1936 literal rectangular matrix memory integral and the actual inverse. No normality
1937 or externally supplied growth bound is required.
1938 
1939 \paragraph{Canonical holomorphic naturality (15.1).}
1940 The standard finite Taylor-jet calculus defined before the theorem is constructed
1941 using the direct sum of generalized eigenspaces. Uniqueness, cutoff stability and
1942 exact rectangular naturality are proved. Cauchy's formula identifies the local
1943 coefficients with actual circle integrals. Admissible radii exist in every stated
1944 open holomorphic neighborhood, and changing them leaves the canonical operator
1945 unchanged. The argument permits disconnected neighborhoods and different matrix
1946 dimensions. It makes no assertion about general Banach-algebra contour deformation.
1947 
1948 \paragraph{Fractional results (10.3--10.5).}
1949 The inherited fourth block proves the scalar beta normalization, the Lebesgue
1950 integral representation of actual positive fractional powers, the literal
1951 resolvent-defect formula, both printed norm constants and exact intertwining.
1952 The rectangular complex-Hilbert-space semigroup proof substitutes those constants
1953 into Duhamel's identity. The varying-space tower proof gives summability of
1954 fractional-defect norms and uniform convergence of time-defect norm series on
1955 bounded time intervals. Here ``summable'' means summable in norm. Operators with
1956 unrelated codomains are not added together. A separate fixed-space theorem proves
1957 operator-series convergence when such a sum is defined.
1958 
1959 \paragraph{Verification and numerical evidence.}
1960 The September 6 release records that the full pinned build, project-wide axiom audit, all 54 companion tests and six
1961 verifier controls passed. No fresh Lean build is asserted for this October edition. The 54 tests comprise the original 37 and 17 added boundary
1962 and validation cases. The 19 numerical figure pairs are unchanged. The earlier
1963 79-field numerical comparison remains historical evidence, not a rerun in that September 6
1964 block. The September 12 visual release subsequently records a numerical replay. Current October checks are reported separately in the final package. Machine-readable source contracts, exact theorem targets, full logs and
1965 previous checked ledgers accompany the source archive. Numerical tests are finite
1966 regressions and are not substitutes for the Lean proofs.
1967 
1968 
1969 \clearpage
1970 \part*{Part II\quad Extended mathematical development}\addcontentsline{toc}{part}{Part II: Extended mathematical development}
1971 \setcounter{section}{0}\setcounter{figure}{0}\setcounter{table}{0}
1972 \renewcommand{\thesection}{E\arabic{section}}
1973 \renewcommand{\theHsection}{E.\arabic{section}}
1974 \renewcommand{\thefigure}{E\arabic{figure}}
1975 \renewcommand{\theHfigure}{E.\arabic{figure}}
1976 \renewcommand{\thetable}{E\arabic{table}}
1977 \renewcommand{\theHtable}{E.\arabic{table}}
1978 \graphicspath{{extension_figures/}}
1979 \begin{abstract}
1980 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.
1981 \end{abstract}
1982 
1983 
1984 \FloatBarrier
1985 \section{Introduction}\label{extended:section:01}\label{extended:sec:introduction}
1986 
1987 One parameter semigroups provide a common language for evolution equations, Markov processes, diffusion, and resolvent methods. Standard analytic foundations appear in \cite{Pazy1983,EngelNagel2000}. Finite Markov chains and their coarse descriptions are treated in \cite{KemenySnell1960,Norris1997,RogersPitman1981}. Conditional expectation and martingale difference decompositions are basic parts of probability theory \cite{Williams1991}. Gaussian transfer and Hermite expansions provide a spectral model for the Ornstein--Uhlenbeck semigroup \cite{Janson1997}. Fractional powers of generators and subordinate semigroups belong to the functional calculus of positive and sectorial operators \cite{Balakrishnan1960,SchillingSongVondracek2012}. The Feynman--Kac formula, projection methods, and phase type laws add three further views of transfer \cite{Kac1949,Feshbach1958,Zwanzig1961,Neuts1981}.
1988 
1989 These subjects share a simple operation. A state is propagated, resolved into declared components, compared across levels, and reconstructed. The same operation can be linear, as in a Markov semigroup, or normalized and nonlinear, as in Bayesian conditioning. Hidden components can be discarded from the state vector without disappearing from the dynamics. Their exact influence returns through a frequency dependent self-energy or a time dependent memory kernel.
1990 
1991 The first sections of this part treat bounded semigroups on Banach spaces. A finite packet system resolves every operator into input and output blocks. The semigroup law becomes block convolution. Splitting the generator into its packet diagonal and transfer part gives a convergent Dyson expansion over directed packet paths. For a coarse map \(C\), the generator defect
1992 \[
1993 \Delta=CA_f-A_cC
1994 \]
1995 produces the entire semigroup defect through a Duhamel integral. The same defect controls resolvents and holomorphic functions of the generators.
1996 
1997 The next question concerns refinement towers. Exact intertwining gives a coordinatewise semigroup on the bounded inverse limit. Approximate refinement is more delicate. Even if every adjacent defect is small, reconstruction from a fine level can introduce a second error because a coarse state need not remain inside the reconstructed fine subspace. We separate these two effects. Summability of adjacent defects gives a uniform limit of reconstructed coarse evolutions. An asymptotic closure condition is then sufficient for the limiting family to satisfy the semigroup law.
1998 
1999 The probabilistic part begins with finite Markov chains. Exact lumpability is the vanishing of a canonical obstruction on the kernel of aggregation. In coordinates, this is equality of the transition mass sent from any two states in the same source block into every target block. The usual finite horizon defect formula follows from telescoping. Conditional expectation is then treated as an orthogonal projector. A nested family of partitions produces mutually orthogonal martingale packets and an exact energy decomposition.
2000 
2001 Bayesian updating requires more care because normalization depends on the current prior. Given a partition, every fine update has an exact two stage description. The block masses are updated by prior dependent average likelihoods, while the conditional distribution inside each block is updated by its centered likelihood variation. A fixed coarse Bayesian rule works for every prior precisely when the likelihood is constant inside each block. This gives a finite and exact form of Bayesian sufficiency. For a family of likelihoods with at least two common level sets, those level sets form the unique coarsest nontrivial fixed sufficient partition. Relative entropy then separates exactly into information retained by the coarse packet masses and information carried by the local conditional changes. The same part derives perturbation bounds, the replicator flow, a variance identity for accumulated evidence, and stability estimates for hidden Markov filtering.
2002 
2003 For a positive operator \(L\), spectral packets define \(L^\alpha\) and \(\e^{-tL^\alpha}\). Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound \(m\), the Balakrishnan formula gives the quantitative estimate
2004 \[
2005 \norm{CL_f^\alpha-L_c^\alpha C}
2006 \leq
2007 \alpha m^{\alpha-1}
2008 \norm{CL_f-L_cC},
2009 \qquad 0<\alpha<1.
2010 \]
2011 A Duhamel argument then controls the fractional semigroup defect. This closes a finite version of fractional naturality under explicit spectral hypotheses.
2012 
2013 The final sections move from generators to source reconstruction and hidden variables. The one-dimensional Dirichlet Green kernel is reproduced exactly by the scaled inverse of the second difference matrix at grid nodes. A discrete maximum principle gives a complete second order error estimate for smooth Poisson solutions. Finite state Feynman--Kac transfer is proved in discrete and continuous time. Schur--Feshbach elimination yields the exact projected resolvent, the retained memory equation, and the Laplace correspondence between memory and self-energy. Every stable finite hidden system gives a polynomial exponential memory expansion. Primary blocks can be ranked optimally for the certified \(L^1\) bound, and a Volterra estimate carries kernel error to the retained solution. Survival, event density, moments, and competing risks are then read directly from a transient semigroup.
2014 
2015 Several claims are intentionally absent. No arbitrary unbounded operator is asserted to generate a semigroup. The inverse limit theorem uses bounded generators or an explicit uniform continuity condition. Bayesian sufficiency is proved for a finite partition and a fixed likelihood. The fractional defect estimate uses a common spectral gap. The Poisson estimate concerns the stated one-dimensional scheme. Memory compression is controlled for the declared finite or integrable kernels. General stochastic differential equations, posterior consistency, nonlinear partial differential equations, and optimal model reduction remain outside this part.
2016 
2017 The same words have different meanings in the realizations below. The table records the object that is transferred and the condition that makes its coarse description exact.
2018 
2019 \begin{table}[t]
2020 \centering
2021 \small
2022 \begin{tabular}{>{\raggedright\arraybackslash}p{.18\textwidth}>{\raggedright\arraybackslash}p{.29\textwidth}>{\raggedright\arraybackslash}p{.43\textwidth}}
2023 \toprule
2024 Realization & Transfer & Exact coarse condition \\
2025 \midrule
2026 Operator evolution & $\e^{tA}$ & generator intertwining \\
2027 Markov chain & $K^n$ or $\e^{tQ}$ & equality of aggregated transition masses \\
2028 Conditional expectation & $P_{\mathcal Q}$ & nesting of the underlying partitions \\
2029 Bayesian update & $\Bcal_\lambda$ & likelihood constant inside each coarse packet \\
2030 Fractional diffusion & $\e^{-tL^\alpha}$ & spectral or resolvent intertwining \\
2031 Hidden dynamics & projected resolvent & vanishing self-energy, or an explicit memory state \\
2032 Survival law & transient semigroup & summation over all transient states \\
2033 \bottomrule
2034 \end{tabular}
2035 \caption{The transfer operations studied in this part and the condition that makes each coarse description exact.}
2036 \label{extended:tab:transfer_dictionary}
2037 \end{table}
2038 
2039 \FloatBarrier
2040 \section{Packet resolved semigroups}\label{extended:section:02}\label{extended:sec:packet_semigroups}
2041 
2042 Let \(X\) be a complex Banach space. A finite packet system is a family of bounded idempotents
2043 \[
2044 \Pcal=(P_\alpha)_{\alpha\in I}
2045 \]
2046 such that
2047 \[
2048 P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,
2049 \qquad
2050 \sum_{\alpha\in I}P_\alpha=I_X.
2051 \]
2052 No orthogonality is needed in this section.
2053 
2054 \begin{definition}[Packet resolved transfer system]\label{extended:record:001}
2055 A packet resolved transfer system is a triple
2056 \[
2057 (X,A,\Pcal)
2058 \]
2059 consisting of a Banach state space, a bounded generator, and a finite packet system. A marked transfer system also includes an initial state or source $s\in X$. The topology and all limiting operations are supplied by the ambient Banach space structure.
2060 \end{definition}
2061 
2062 \begin{definition}[Packet blocks]\label{extended:record:002}
2063 For \(B\in\mathcal B(X)\), define
2064 \[
2065 B_{\beta\alpha}=P_\beta BP_\alpha.
2066 \]
2067 If \(A\in\mathcal B(X)\) and \(T(t)=\e^{tA}\), write
2068 \[
2069 T_{\beta\alpha}(t)=P_\beta T(t)P_\alpha.
2070 \]
2071 \end{definition}
2072 
2073 \begin{theorem}[Semigroup and packet convolution]\label{extended:record:003}\label{extended:thm:packet_convolution}
2074 For every bounded operator \(A\), the family \(T(t)=\e^{tA}\) is a uniformly continuous group and
2075 \[
2076 T(s+t)=T(s)T(t).
2077 \]
2078 For every \(\alpha,\beta\in I\),
2079 \[
2080 T_{\beta\alpha}(s+t)
2081 =
2082 \sum_{\gamma\in I}
2083 T_{\beta\gamma}(s)T_{\gamma\alpha}(t).
2084 \]
2085 \end{theorem}
2086 
2087 \begin{proof}
2088 The exponential series converges in operator norm. Since every power of \(A\) commutes with every other power, the Cauchy product gives the group law. Inserting \(I_X=\sum_\gamma P_\gamma\) between the two factors gives the block formula.
2089 \end{proof}
2090 
2091 \begin{proposition}[Block differential system]\label{extended:record:004}\label{extended:prop:block_differential}
2092 The packet blocks are differentiable in operator norm and satisfy
2093 \[
2094 \frac{\dd}{\dd t}T_{\beta\alpha}(t)
2095 =
2096 \sum_{\gamma\in I}
2097 A_{\beta\gamma}T_{\gamma\alpha}(t)
2098 =
2099 \sum_{\gamma\in I}
2100 T_{\beta\gamma}(t)A_{\gamma\alpha}.
2101 \]
2102 Their initial values are
2103 \[
2104 T_{\beta\alpha}(0)=\delta_{\beta\alpha}P_\alpha.
2105 \]
2106 \end{proposition}
2107 
2108 \begin{proof}
2109 Differentiate \(T(t)\) in operator norm and insert the packet resolution between \(A\) and \(T(t)\), then between \(T(t)\) and \(A\).
2110 \end{proof}
2111 
2112 \begin{theorem}[Laplace resolvent and packet transform]\label{extended:record:005}\label{extended:thm:laplace_packet_resolvent}
2113 Let \(A\in\mathcal B(X)\), let \(T(t)=\e^{tA}\), and let
2114 \[
2115 \operatorname{Re}z>\norm A.
2116 \]
2117 Then the Bochner integral converges in operator norm and
2118 \[
2119 (zI-A)^{-1}
2120 =
2121 \int_0^\infty \e^{-zt}T(t)\dd t.
2122 \]
2123 For every pair of packets,
2124 \[
2125 P_\beta(zI-A)^{-1}P_\alpha
2126 =
2127 \int_0^\infty
2128 \e^{-zt}T_{\beta\alpha}(t)\dd t.
2129 \]
2130 \end{theorem}
2131 
2132 \begin{proof}
2133 The estimate \(\norm{T(t)}\leq\e^{t\norm A}\) gives absolute convergence of the operator valued integral. Let
2134 \[
2135 R(z)=\int_0^\infty\e^{-zt}T(t)\dd t.
2136 \]
2137 Since \(T'(t)=AT(t)=T(t)A\), integration by parts gives
2138 \[
2139 (zI-A)R(z)
2140 =
2141 -\int_0^\infty
2142 \frac{\dd}{\dd t}
2143 \bigl(\e^{-zt}T(t)\bigr)
2144 \dd t
2145 =I.
2146 \]
2147 The same calculation on the other side gives
2148 \(
2149 R(z)(zI-A)=I
2150 \).
2151 Multiplication by \(P_\beta\) and \(P_\alpha\) may be passed under the norm convergent integral, which gives the packet formula.
2152 \end{proof}
2153 
2154 \begin{corollary}[Resolvent expansion over packet paths]\label{extended:record:006}\label{extended:cor:resolvent_packet_paths}
2155 With \(A=D+N\) as in the directed path decomposition, let
2156 \[
2157 R_D(z)=(zI-D)^{-1}.
2158 \]
2159 If
2160 \[
2161 \norm{R_D(z)N}<1,
2162 \]
2163 then
2164 \[
2165 (zI-A)^{-1}
2166 =
2167 \sum_{m=0}^\infty
2168 \bigl(R_D(z)N\bigr)^mR_D(z).
2169 \]
2170 After multiplication by \(P_\beta\) and \(P_\alpha\), the term of order \(m\) is a sum over directed packet paths of length \(m\) from \(\alpha\) to \(\beta\).
2171 \end{corollary}
2172 
2173 \begin{proof}
2174 Factor
2175 \[
2176 zI-A=(zI-D)(I-R_D(z)N)
2177 \]
2178 and expand the second inverse by its norm convergent Neumann series. The diagonal resolvent preserves each packet and every factor \(N\) changes the packet along an active transfer edge.
2179 \end{proof}
2180 
2181 \begin{theorem}[Reducing packet criterion]\label{extended:record:007}\label{extended:thm:reducing_packets}
2182 The following conditions are equivalent.
2183 \begin{enumerate}
2184 \item \(AP_\alpha=P_\alpha A\) for every \(\alpha\).
2185 \item \(A_{\beta\alpha}=0\) whenever \(\beta\neq\alpha\).
2186 \item \(T_{\beta\alpha}(t)=0\) whenever \(\beta\neq\alpha\) and \(t\in\RR\).
2187 \item Every packet space \(P_\alpha X\) is invariant under \(T(t)\) for every \(t\).
2188 \end{enumerate}
2189 When these conditions hold,
2190 \[
2191 T(t)|_{P_\alpha X}
2192 =
2193 \e^{tA_{\alpha\alpha}}.
2194 \]
2195 \end{theorem}
2196 
2197 \begin{proof}
2198 The first two conditions are equivalent because the packet resolution gives
2199 \[
2200 AP_\alpha=\sum_\beta A_{\beta\alpha},
2201 \qquad
2202 P_\alpha A=\sum_\gamma A_{\alpha\gamma}.
2203 \]
2204 A block diagonal generator has block diagonal powers, so the second condition implies the third. Differentiating the off-diagonal blocks at zero proves the converse. The equivalence with invariance follows from the definition of the blocks. The final identity follows by restricting the exponential series.
2205 \end{proof}
2206 
2207 \begin{figure}[H]
2208 \centering
2209 \includegraphics[width=.66\textwidth]{packet_blocks_t0_0.pdf}
2210 \caption{Packet block norms at three times. Transfer appears in the off-diagonal blocks as soon as the generator couples the two packet spaces. The three panels are shown in consecutive figures.}\label{extended:fig:packet_blocks}
2211 \end{figure}
2212 \begin{figure}[H]
2213 \centering
2214 \includegraphics[width=.66\textwidth]{packet_blocks_t0_8.pdf}
2215 \caption{Packet block norms at time 0.8. The same fixed packet spaces and norm scale are used in the preceding panels.}\label{monograph:fig:packet_blocks_1}
2216 \end{figure}
2217 \begin{figure}[H]
2218 \centering
2219 \includegraphics[width=.66\textwidth]{packet_blocks_t2_0.pdf}
2220 \caption{Packet block norms at time 2.0. The same fixed packet spaces and norm scale are used in the preceding panels.}\label{monograph:fig:packet_blocks_2}
2221 \end{figure}
2222 
2223 
2224 \subsection{Directed path expansion}
2225 
2226 Set
2227 \[
2228 D=\sum_{\alpha\in I}P_\alpha AP_\alpha,
2229 \qquad
2230 N=A-D.
2231 \]
2232 The operator \(D\) preserves every packet and \(N\) contains only transfers between distinct packets.
2233 
2234 \begin{theorem}[Dyson expansion over packet paths]\label{extended:record:008}\label{extended:thm:dyson_paths}
2235 For \(t\geq0\),
2236 \[
2237 \begin{aligned}
2238 \e^{tA}
2239 ={}&\e^{tD}
2240 +
2241 \sum_{m=1}^{\infty}
2242 \int_{0<s_1<\cdots<s_m<t}
2243 \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D}
2244 \cdots
2245 N\e^{s_1D}
2246 \dd s_1\cdots\dd s_m.
2247 \end{aligned}
2248 \]
2249 The series converges absolutely in operator norm. After multiplication by \(P_\beta\) and \(P_\alpha\), the term of order \(m\) is a sum over directed packet paths of length \(m\) from \(\alpha\) to \(\beta\).
2250 \end{theorem}
2251 
2252 \begin{proof}
2253 Write the variation of constants identity
2254 \[
2255 \e^{tA}
2256 =
2257 \e^{tD}
2258 +
2259 \int_0^t\e^{(t-s)D}N\e^{sA}\dd s.
2260 \]
2261 Iterating the identity gives the displayed series. The norm of the term of order \(m\) is at most
2262 \[
2263 \e^{t\norm D}\frac{(t\norm N)^m}{m!}.
2264 \]
2265 The majorant is summable. Inserting the packet resolution around every factor \(N\) leaves only products whose consecutive packet labels are joined by nonzero transfer blocks.
2266 \end{proof}
2267 
2268 \begin{corollary}[Termination on an acyclic transfer graph]\label{extended:record:009}\label{extended:cor:acyclic_termination}
2269 Suppose the directed graph with an edge \(\alpha\to\beta\) whenever \(P_\beta NP_\alpha\neq0\) is acyclic. If its longest directed path has length \(q\), then every Dyson term of order greater than \(q\) vanishes.
2270 \end{corollary}
2271 
2272 \begin{proof}
2273 Every nonzero term of order \(m\) contains a directed transfer path of length \(m\). An acyclic graph has no path longer than \(q\).
2274 \end{proof}
2275 
2276 \FloatBarrier
2277 \section{Generator defects and functional calculus}\label{extended:section:03}\label{extended:sec:defects}
2278 
2279 Let \(X_f\) and \(X_c\) be complex Banach spaces. Let \(A_f\in\mathcal B(X_f)\), \(A_c\in\mathcal B(X_c)\), and \(C\in\mathcal B(X_f,X_c)\). Define
2280 \[
2281 \Delta=CA_f-A_cC
2282 \]
2283 and
2284 \[
2285 \mathfrak D(t)=C\e^{tA_f}-\e^{tA_c}C.
2286 \]
2287 
2288 \begin{theorem}[Duhamel defect identity]\label{extended:record:010}\label{extended:thm:duhamel_defect}
2289 For every \(t\geq0\),
2290 \[
2291 \mathfrak D(t)
2292 =
2293 \int_0^t
2294 \e^{(t-s)A_c}\Delta\e^{sA_f}
2295 \dd s.
2296 \]
2297 If
2298 \[
2299 \norm{\e^{tA_f}}\leq M_f\e^{\omega_f t},
2300 \qquad
2301 \norm{\e^{tA_c}}\leq M_c\e^{\omega_c t},
2302 \]
2303 then
2304 \[
2305 \norm{\mathfrak D(t)}
2306 \leq
2307 M_fM_c\norm\Delta
2308 \int_0^t
2309 \e^{\omega_c(t-s)+\omega_fs}
2310 \dd s.
2311 \]
2312 For contraction semigroups,
2313 \[
2314 \norm{\mathfrak D(t)}\leq t\norm\Delta.
2315 \]
2316 \end{theorem}
2317 
2318 \begin{proof}
2319 For fixed \(t\), set
2320 \[
2321 F(s)=\e^{(t-s)A_c}C\e^{sA_f}.
2322 \]
2323 Then
2324 \[
2325 F'(s)=\e^{(t-s)A_c}\Delta\e^{sA_f}.
2326 \]
2327 Integrating from zero to \(t\) gives the identity. The estimates follow by taking norms under the integral.
2328 \end{proof}
2329 
2330 \begin{proposition}[Infinitesimal recovery of the defect]\label{extended:record:011}\label{extended:prop:infinitesimal_defect}
2331 The generator defect is the first order semigroup defect:
2332 \[
2333 \lim_{t\to0}
2334 \frac{\mathfrak D(t)}{t}
2335 =
2336 \Delta
2337 \]
2338 in operator norm. Consequently,
2339 \[
2340 C\e^{tA_f}=\e^{tA_c}C
2341 \quad\hbox{for every }t
2342 \]
2343 if and only if \(CA_f=A_cC\).
2344 \end{proposition}
2345 
2346 \begin{proof}
2347 The operator norm expansions
2348 \[
2349 \e^{tA_f}=I+tA_f+o(t),
2350 \qquad
2351 \e^{tA_c}=I+tA_c+o(t)
2352 \]
2353 give the limit. The equivalence follows from this limit and \cref{extended:thm:duhamel_defect}.
2354 \end{proof}
2355 
2356 \begin{theorem}[Resolvent defect identity]\label{extended:record:012}\label{extended:thm:resolvent_defect}
2357 Let \(z\) belong to the resolvent sets of \(A_f\) and \(A_c\). With
2358 \[
2359 R_f(z)=(zI-A_f)^{-1},
2360 \qquad
2361 R_c(z)=(zI-A_c)^{-1},
2362 \]
2363 one has
2364 \[
2365 CR_f(z)-R_c(z)C
2366 =
2367 R_c(z)\Delta R_f(z).
2368 \]
2369 \end{theorem}
2370 
2371 \begin{proof}
2372 Multiply the proposed identity on the left by \(zI-A_c\) and on the right by \(zI-A_f\). Both sides reduce to
2373 \[
2374 (zI-A_c)C-C(zI-A_f)=\Delta.
2375 \]
2376 \end{proof}
2377 
2378 \begin{theorem}[Holomorphic defect formula]\label{extended:record:013}\label{extended:thm:holomorphic_defect}
2379 Let \(\Gamma\) be the positively oriented rectifiable boundary of a finite union of pairwise disjoint bounded Jordan domains. Assume that both spectra lie inside this union and that \(\Gamma\) lies in the common resolvent set. Thus its winding number is one about every spectral point. If \(g\) is holomorphic on a neighborhood of the closure of the union, then
2380 \[
2381 Cg(A_f)-g(A_c)C
2382 =
2383 \frac{1}{2\pi\ii}
2384 \int_\Gamma
2385  g(z)R_c(z)\Delta R_f(z)
2386 \dd z.
2387 \]
2388 In particular,
2389 \[
2390 \begin{aligned}
2391 \norm{Cg(A_f)-g(A_c)C}
2392 \leq{}&
2393 \frac{\operatorname{length}(\Gamma)}{2\pi}
2394 \sup_{z\in\Gamma}\abs{g(z)}
2395 \sup_{z\in\Gamma}\norm{R_c(z)}
2396 \sup_{z\in\Gamma}\norm{R_f(z)}
2397 \norm\Delta.
2398 \end{aligned}
2399 \]
2400 \end{theorem}
2401 
2402 \begin{proof}
2403 Apply the Dunford integral formula to both operators and use \cref{extended:thm:resolvent_defect} under the contour integral. The norm estimate is immediate.
2404 \end{proof}
2405 
2406 \begin{corollary}[Exact naturality]\label{extended:record:014}\label{extended:cor:exact_naturality}
2407 If \(CA_f=A_cC\), then
2408 \[
2409 Cg(A_f)=g(A_c)C
2410 \]
2411 for every holomorphic function covered by \cref{extended:thm:holomorphic_defect}. The conclusion includes exponentials, resolvents, and polynomials.
2412 \end{corollary}
2413 
2414 \begin{proof}
2415 Set \(\Delta=0\) in \cref{extended:thm:holomorphic_defect}.
2416 \end{proof}
2417 
2418 \begin{figure}[t]
2419 \centering
2420 \includegraphics[width=.68\textwidth]{duhamel_defect.pdf}
2421 \caption{A finite semigroup defect and a direct Duhamel bound. The bound is designed to be uniform rather than sharp in this example.}
2422 \label{extended:fig:duhamel_defect}
2423 \end{figure}
2424 
2425 \FloatBarrier
2426 \section{Refinement towers and limiting evolution}\label{extended:section:04}\label{extended:sec:refinement_towers}
2427 \begin{interpretation}
2428 The limiting argument names its topology, reconstruction maps, and summability weights. Convergence of reconstructed trajectories is distinct from closure under the semigroup law. The range-leakage condition identifies the additional obstruction to that closure.
2429 \end{interpretation}
2430 
2431 
2432 The exact inverse limit and the approximate completion problem require different hypotheses. Exact intertwining preserves the inverse limit coordinate by coordinate. Approximate intertwining gives a Cauchy family only after reconstruction maps are chosen.
2433 
2434 \subsection{Exact inverse limits}
2435 
2436 Let
2437 \[
2438 X_0\xleftarrow{C_0}X_1\xleftarrow{C_1}X_2\xleftarrow{}\cdots
2439 \]
2440 be Banach spaces with bounded connecting maps. Define the bounded inverse limit
2441 \[
2442 X_\infty
2443 =
2444 \left\{
2445 (x_\ell)_{\ell\geq0}:
2446 C_\ell x_{\ell+1}=x_\ell,
2447 \quad
2448 \sup_\ell\norm{x_\ell}<\infty
2449 \right\}
2450 \]
2451 with the supremum norm.
2452 
2453 \begin{lemma}[Completeness of the bounded inverse limit]\label{extended:record:015}\label{extended:lem:inverse_limit_banach}
2454 The space \(X_\infty\) is Banach.
2455 \end{lemma}
2456 
2457 \begin{proof}
2458 The product of the spaces \(X_\ell\), equipped with the bounded supremum norm, is Banach. The equations \(C_\ell x_{\ell+1}=x_\ell\) define a closed linear subspace.
2459 \end{proof}
2460 
2461 \begin{theorem}[Exact projective limit semigroup]\label{extended:record:016}\label{extended:thm:exact_projective_limit}
2462 For each \(\ell\), let \(T_\ell(t)\) be a strongly continuous semigroup on \(X_\ell\). Assume
2463 \[
2464 C_\ell T_{\ell+1}(t)=T_\ell(t)C_\ell
2465 \]
2466 for every \(\ell\) and \(t\geq0\), and assume the uniform bound
2467 \[
2468 \sup_\ell\norm{T_\ell(t)}\leq M\e^{\omega t}.
2469 \]
2470 Define
2471 \[
2472 T_\infty(t)(x_\ell)_\ell
2473 =
2474 (T_\ell(t)x_\ell)_\ell.
2475 \]
2476 Then \(T_\infty(t)\) is a bounded semigroup on \(X_\infty\). It is strongly continuous if
2477 \[
2478 \lim_{t\downarrow0}
2479 \sup_\ell
2480 \norm{T_\ell(t)x_\ell-x_\ell}=0
2481 \]
2482 for every \((x_\ell)_\ell\in X_\infty\).
2483 
2484 If \(T_\ell(t)=\e^{tA_\ell}\), the generators are bounded, and
2485 \[
2486 \sup_\ell\norm{A_\ell}<\infty,
2487 \qquad
2488 C_\ell A_{\ell+1}=A_\ell C_\ell,
2489 \]
2490 then the continuity condition holds and
2491 \[
2492 A_\infty(x_\ell)_\ell=(A_\ell x_\ell)_\ell
2493 \]
2494 is the bounded generator of \(T_\infty\).
2495 \end{theorem}
2496 
2497 \begin{proof}
2498 Exact intertwining shows that \((T_\ell(t)x_\ell)_\ell\) remains compatible. The uniform growth bound makes the coordinatewise operator bounded on the supremum norm. The semigroup law holds in every coordinate. The stated continuity condition is exactly strong continuity in the supremum norm.
2499 
2500 For bounded generators with common bound \(M_A\),
2501 \[
2502 \sup_\ell
2503 \norm{(\e^{tA_\ell}-I)x_\ell}
2504 \leq
2505 (\e^{tM_A}-1)\sup_\ell\norm{x_\ell}.
2506 \]
2507 This tends to zero with \(t\). Exact generator intertwining makes \((A_\ell x_\ell)_\ell\) compatible, and coordinatewise differentiation identifies the generator.
2508 \end{proof}
2509 
2510 \begin{remark}\label{extended:record:017}
2511 Strong continuity on each level does not by itself imply strong continuity in the supremum norm of the inverse limit. The additional uniform condition in \cref{extended:thm:exact_projective_limit} cannot be omitted without replacing the topology.
2512 \end{remark}
2513 
2514 \subsection{Summable adjacent defects}
2515 
2516 Assume now that each \(C_\ell:X_{\ell+1}\to X_\ell\) has a bounded right inverse
2517 \[
2518 R_\ell:X_\ell\to X_{\ell+1},
2519 \qquad
2520 C_\ell R_\ell=I_{X_\ell}.
2521 \]
2522 For \(m<n\), set
2523 \[
2524 C_{m,n}=C_mC_{m+1}\cdots C_{n-1},
2525 \qquad
2526 R_{n,m}=R_{n-1}\cdots R_m.
2527 \]
2528 The empty products are
2529 \[
2530 C_{m,m}=I_{X_m},
2531 \qquad
2532 R_{m,m}=I_{X_m}.
2533 \]
2534 
2535 \begin{definition}[Reconstructed fine evolution]\label{extended:record:018}
2536 For a semigroup \(T_n(t)\) on \(X_n\), define the evolution seen from level \(m\) by
2537 \[
2538 U_{m,n}(t)=C_{m,n}T_n(t)R_{n,m}.
2539 \]
2540 \end{definition}
2541 
2542 \begin{theorem}[Weighted summable defect completion]\label{extended:record:019}\label{extended:thm:summable_defect_completion}
2543 Define the adjacent semigroup defect
2544 \[
2545 E_n(t)=C_nT_{n+1}(t)-T_n(t)C_n.
2546 \]
2547 Fix a coarse level \(m\) and a time horizon \(T>0\). For \(n\geq m\), set
2548 \[
2549 w_{m,n}
2550 =
2551 \norm{C_{m,n}}\,
2552 \norm{R_nR_{n,m}}.
2553 \]
2554 If
2555 \[
2556 \sum_{n=m}^{\infty}
2557 w_{m,n}
2558 \sup_{0\leq t\leq T}\norm{E_n(t)}<\infty,
2559 \]
2560 then \(U_{m,n}(t)\) converges in operator norm uniformly for \(0\leq t\leq T\). Its limit is denoted by \(U_m(t)\), and
2561 \[
2562 \sup_{0\leq t\leq T}
2563 \norm{U_m(t)-U_{m,n}(t)}
2564 \leq
2565 \sum_{j=n}^{\infty}w_{m,j}
2566 \sup_{0\leq t\leq T}\norm{E_j(t)}.
2567 \]
2568 \end{theorem}
2569 
2570 \begin{proof}
2571 Since \(C_nR_n=I\),
2572 \[
2573 \begin{aligned}
2574 U_{m,n+1}(t)-U_{m,n}(t)
2575 ={}&
2576 C_{m,n}
2577 \bigl(C_nT_{n+1}(t)-T_n(t)C_n\bigr)
2578 R_nR_{n,m}.
2579 \end{aligned}
2580 \]
2581 Therefore
2582 \[
2583 \norm{U_{m,n+1}(t)-U_{m,n}(t)}
2584 \leq
2585 w_{m,n}\norm{E_n(t)}.
2586 \]
2587 The weighted summability hypothesis makes the sequence uniformly Cauchy. Summing the tail estimates gives the stated bound.
2588 \end{proof}
2589 
2590 \begin{corollary}[Contractive summable defect completion]\label{extended:record:020}\label{extended:cor:contractive_defect_completion}
2591 Assume
2592 \[
2593 \norm{C_\ell}\leq1,
2594 \qquad
2595 \norm{R_\ell}\leq1,
2596 \qquad
2597 \norm{T_\ell(t)}\leq1
2598 \]
2599 for every \(\ell\) and \(t\geq0\). If
2600 \[
2601 \sum_{\ell=0}^{\infty}
2602 \sup_{0\leq t\leq T}\norm{E_\ell(t)}<\infty,
2603 \]
2604 then the conclusion of \cref{extended:thm:summable_defect_completion} holds, and
2605 \[
2606 \sup_{0\leq t\leq T}
2607 \norm{U_m(t)-U_{m,n}(t)}
2608 \leq
2609 \sum_{\ell=n}^{\infty}
2610 \sup_{0\leq t\leq T}\norm{E_\ell(t)}.
2611 \]
2612 \end{corollary}
2613 
2614 \begin{proof}
2615 Every weight \(w_{m,n}\) is at most one.
2616 \end{proof}
2617 
2618 \begin{corollary}[Generator defect criterion]\label{extended:record:021}\label{extended:cor:generator_defect_completion}
2619 Assume
2620 \[
2621 \norm{C_\ell}\leq1,
2622 \qquad
2623 \norm{R_\ell}\leq1,
2624 \]
2625 and suppose \(T_\ell(t)=\e^{tA_\ell}\) are contraction semigroups with bounded generators. Set
2626 \[
2627 \Delta_\ell=C_\ell A_{\ell+1}-A_\ell C_\ell.
2628 \]
2629 If
2630 \[
2631 \sum_{\ell=0}^{\infty}\norm{\Delta_\ell}<\infty,
2632 \]
2633 then the conclusion of \cref{extended:cor:contractive_defect_completion} holds on every bounded time interval, and
2634 \[
2635 \sup_{0\leq t\leq T}
2636 \norm{U_m(t)-U_{m,n}(t)}
2637 \leq
2638 T\sum_{\ell=n}^{\infty}\norm{\Delta_\ell}.
2639 \]
2640 \end{corollary}
2641 
2642 \begin{proof}
2643 Apply the contraction estimate in \cref{extended:thm:duhamel_defect} to every adjacent pair.
2644 \end{proof}
2645 
2646 The limit in \cref{extended:thm:summable_defect_completion} need not satisfy the semigroup law. The obstruction comes from leaving the reconstructed fine subspace between two time steps.
2647 
2648 \begin{theorem}[Asymptotic closure criterion]\label{extended:record:022}\label{extended:thm:asymptotic_closure}
2649 Assume the weighted convergence in \cref{extended:thm:summable_defect_completion} holds for every bounded time interval, and define
2650 \[
2651 P_{m,n}=R_{n,m}C_{m,n}
2652 \]
2653 on \(X_n\). Suppose that for every \(T>0\),
2654 \[
2655 \rho_{m,n}(T)
2656 =
2657 \sup_{0\leq s,t\leq T}
2658 \norm{
2659 C_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m}
2660 }
2661 \longrightarrow0.
2662 \]
2663 Then
2664 \[
2665 U_m(s+t)=U_m(s)U_m(t)
2666 \]
2667 for all \(s,t\geq0\). If the convergence in \cref{extended:thm:summable_defect_completion} is locally uniform and every \(T_n\) is norm continuous, then \(U_m\) is a norm continuous semigroup.
2668 \end{theorem}
2669 
2670 \begin{proof}
2671 At a finite level,
2672 \[
2673 \begin{aligned}
2674 U_{m,n}(s)U_{m,n}(t)-U_{m,n}(s+t)
2675 ={}&
2676 C_{m,n}T_n(s)
2677 (P_{m,n}-I)
2678 T_n(t)R_{n,m}.
2679 \end{aligned}
2680 \]
2681 Its norm is at most \(\rho_{m,n}(T)\) for \(s,t\leq T\). Pass to the limit using uniform convergence. Norm continuity follows from a uniform limit of norm continuous functions on compact intervals.
2682 \end{proof}
2683 
2684 \begin{remark}\label{extended:record:023}
2685 Summable generator defects control consistency between adjacent levels. The quantity \(\rho_{m,n}(T)\) controls closure of the reconstructed fine range under the fine evolution. These are different requirements.
2686 \end{remark}
2687 
2688 \begin{example}[Summable defects do not force a semigroup limit]\label{extended:record:024}\label{extended:ex:summable_not_semigroup}
2689 Let \(X_0=\RR\) and let \(X_n=\RR^2\) for every \(n\geq1\). Set
2690 \[
2691 C_0(x,y)=x,
2692 \qquad
2693 R_0x=(x,0),
2694 \]
2695 and use identity connecting maps and reconstructions above level one. Let \(T_0(t)=I\) and let every higher level carry the rotation group
2696 \[
2697 T_n(t)
2698 =
2699 \begin{pmatrix}
2700 \cos t&-\sin t\\
2701 \sin t&\cos t
2702 \end{pmatrix}.
2703 \]
2704 Only the first adjacent defect is nonzero, so every defect sum is finite. Yet for \(n\geq1\),
2705 \[
2706 U_{0,n}(t)=C_0T_n(t)R_0=\cos t.
2707 \]
2708 The limiting family is not a semigroup because in general
2709 \[
2710 \cos(s+t)\neq\cos s\cos t.
2711 \]
2712 The failure is exactly the excursion from the reconstructed line into its orthogonal complement between two time steps.
2713 \end{example}
2714 
2715 \begin{proposition}[Range leakage criterion]\label{extended:record:025}\label{extended:prop:range_leakage}
2716 Under the contractive assumptions of \cref{extended:cor:contractive_defect_completion}, define
2717 \[
2718 \eta_{m,n}(T)
2719 =
2720 \sup_{0\leq t\leq T}
2721 \norm{(I-P_{m,n})T_n(t)R_{n,m}}.
2722 \]
2723 Then
2724 \[
2725 \rho_{m,n}(T)\leq\eta_{m,n}(T).
2726 \]
2727 Consequently, \(\eta_{m,n}(T)\to0\) for every \(T>0\) is sufficient for the limiting family \(U_m\) to satisfy the semigroup law.
2728 \end{proposition}
2729 
2730 \begin{proof}
2731 For \(s,t\leq T\), the contraction assumptions give
2732 \[
2733 \begin{aligned}
2734 &\norm{
2735 C_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m}
2736 }\\
2737 &\qquad\leq
2738 \norm{(I-P_{m,n})T_n(t)R_{n,m}}.
2739 \end{aligned}
2740 \]
2741 Take the supremum and apply \cref{extended:thm:asymptotic_closure}.
2742 \end{proof}
2743 
2744 \begin{theorem}[Generator control of range leakage]\label{extended:record:026}\label{extended:thm:generator_range_leakage}
2745 Let \(T(t)=\e^{tA}\) on a Banach space \(X\), let \(P\) be a bounded projection, and set \(Q=I-P\). Assume
2746 \[
2747 \norm{T(t)}\leq M\e^{\omega t}
2748 \]
2749 and
2750 \[
2751 \norm{\e^{t(QAQ)|_{QX}}}
2752 \leq M_Q\e^{\omega_Qt}.
2753 \]
2754 Then
2755 \[
2756 \norm{QT(t)P}
2757 \leq
2758 M M_Q\norm P^2\norm{QAP}
2759 \int_0^t
2760 \e^{\omega_Q(t-s)+\omega s}
2761 \dd s.
2762 \]
2763 In particular, for orthogonal projections and contraction bounds,
2764 \[
2765 \sup_{0\leq t\leq T}
2766 \norm{QT(t)P}
2767 \leq
2768 T\norm{QAP}.
2769 \]
2770 \end{theorem}
2771 
2772 \begin{proof}
2773 For \(x\in PX\), write
2774 \[
2775 p(t)=PT(t)x,
2776 \qquad
2777 q(t)=QT(t)x.
2778 \]
2779 The hidden component satisfies
2780 \[
2781 q'(t)=QAQq(t)+QAPp(t),
2782 \qquad
2783 q(0)=0.
2784 \]
2785 Variation of constants gives
2786 \[
2787 q(t)=
2788 \int_0^t
2789 \e^{(t-s)(QAQ)|_{QX}}
2790 QAPp(s)
2791 \dd s.
2792 \]
2793 Since
2794 \(
2795 \norm{p(s)}\leq\norm P\norm{T(s)}\norm x
2796 \)
2797 and \(x=Px\), the first estimate follows. The contraction case is immediate.
2798 \end{proof}
2799 
2800 \begin{corollary}[A usable closure condition for refinement towers]\label{extended:record:027}\label{extended:cor:generator_closure_tower}
2801 Assume the contractive summability hypothesis of \cref{extended:cor:contractive_defect_completion} holds on every bounded time interval. Suppose every \(P_{m,n}\) is an orthogonal projection on a Hilbert space and \(T_n(t)=\e^{tA_n}\) is contractive. If, for every fixed \(m\),
2802 \[
2803 \norm{(I-P_{m,n})A_nP_{m,n}}
2804 \longrightarrow0,
2805 \]
2806 then the limiting family \(U_m\) is a semigroup.
2807 \end{corollary}
2808 
2809 \begin{proof}
2810 A bounded generator of a contraction semigroup on a Hilbert space is dissipative. If \(Q_{m,n}=I-P_{m,n}\) and \(y\in Q_{m,n}X_n\), then
2811 \[
2812 \operatorname{Re}\ip{Q_{m,n}A_nQ_{m,n}y}{y}
2813 =
2814 \operatorname{Re}\ip{A_ny}{y}
2815 \leq0.
2816 \]
2817 Hence the compressed hidden semigroup is contractive. Apply \cref{extended:thm:generator_range_leakage} to \(P_{m,n}\). Since \(P_{m,n}R_{n,m}=R_{n,m}\),
2818 \[
2819 \eta_{m,n}(T)
2820 \leq
2821 T\norm{(I-P_{m,n})A_nP_{m,n}}
2822 \norm{R_{n,m}}.
2823 \]
2824 The right side tends to zero. Use \cref{extended:prop:range_leakage}.
2825 \end{proof}
2826 
2827 \FloatBarrier
2828 \section{Finite Markov transfer and lumping}\label{extended:section:05}\label{extended:sec:markov}
2829 
2830 Probability vectors are columns. A column stochastic matrix \(K\) has nonnegative entries and column sums one. A continuous time generator \(Q\) has nonnegative off-diagonal entries and column sums zero.
2831 
2832 \begin{theorem}[Finite Markov semigroup]\label{extended:record:028}\label{extended:thm:finite_markov}
2833 If \(Q\) is a finite Markov generator, then \(\e^{tQ}\) is column stochastic for every \(t\geq0\).
2834 \end{theorem}
2835 
2836 \begin{proof}
2837 Choose \(a\geq\max_i(-Q_{ii})\). The matrix \(Q+aI\) is nonnegative, so its exponential is nonnegative. Hence
2838 \[
2839 \e^{tQ}=\e^{-at}\e^{t(Q+aI)}
2840 \]
2841 is nonnegative. Since \(\one^TQ=0\), differentiation gives
2842 \[
2843 \one^T\e^{tQ}=\one^T.
2844 \]
2845 \end{proof}
2846 
2847 Let \(\{B_1,\ldots,B_m\}\) be a partition of the fine state set. Its aggregation matrix is
2848 \[
2849 C_{\alpha i}=\one_{\{i\in B_\alpha\}}.
2850 \]
2851 
2852 \begin{definition}[Canonical lumping obstruction]\label{extended:record:029}
2853 For a linear fine transition operator \(K\), define
2854 \[
2855 \Ob_C(K)
2856 =
2857 (CK)|_{\Ker C}
2858 \in
2859 \mathcal B(\Ker C,\RR^m).
2860 \]
2861 \end{definition}
2862 
2863 \begin{theorem}[Descent through an aggregation map]\label{extended:record:030}\label{extended:thm:canonical_lumping_obstruction}
2864 Let \(C:X_f\to X_c\) be a surjective linear map between finite dimensional spaces. A unique operator \(\overline K:X_c\to X_c\) satisfying
2865 \[
2866 CK=\overline KC
2867 \]
2868 exists if and only if
2869 \[
2870 \Ob_C(K)=0.
2871 \]
2872 For any right inverse \(R\) of \(C\), the only possible coarse operator is
2873 \[
2874 \overline K=CKR,
2875 \]
2876 and
2877 \[
2878 CK-\overline KC
2879 =
2880 CK(I-RC).
2881 \]
2882 \end{theorem}
2883 
2884 \begin{proof}
2885 If \(CK=\overline KC\), then \(CK\) vanishes on \(\Ker C\). Conversely, suppose the obstruction vanishes and define
2886 \[
2887 \overline K(Cx)=CKx.
2888 \]
2889 If \(Cx=Cy\), then \(x-y\in\Ker C\), so the value does not depend on the representative. Surjectivity defines \(\overline K\) on all of \(X_c\), and the intertwining relation follows from the definition. It also gives uniqueness. If \(R\) is a right inverse, then
2890 \[
2891 \overline K=\overline KCR=CKR.
2892 \]
2893 Substitution gives the defect identity.
2894 \end{proof}
2895 
2896 \begin{corollary}[Coordinate form of the lumping obstruction]\label{extended:record:031}\label{extended:cor:lumping_obstruction_coordinates}
2897 For the partition aggregation above, \(\Ker C\) is spanned by differences \(e_i-e_j\) with \(i\) and \(j\) in one source block. Also,
2898 \[
2899 \Ob_C(K)(e_i-e_j)
2900 =
2901 \left(
2902 \sum_{r\in B_\alpha}(K_{ri}-K_{rj})
2903 \right)_{\alpha=1}^m.
2904 \]
2905 \end{corollary}
2906 
2907 \begin{proof}
2908 Inside each block, the vectors with coordinate sum zero are spanned by differences of coordinate vectors. Applying \(CK\) gives the displayed aggregated column differences.
2909 \end{proof}
2910 
2911 \begin{theorem}[Characterization of exact discrete lumping]\label{extended:record:032}\label{extended:thm:lumping_characterization}
2912 Let \(K\) be column stochastic. A coarse column stochastic matrix \(\overline K\) satisfying
2913 \[
2914 CK=\overline KC
2915 \]
2916 exists if and only if, for every source block \(B_\beta\), every pair \(i,j\in B_\beta\), and every target block \(B_\alpha\),
2917 \[
2918 \sum_{r\in B_\alpha}K_{ri}
2919 =
2920 \sum_{r\in B_\alpha}K_{rj}.
2921 \]
2922 When it exists, \(\overline K\) is unique and
2923 \[
2924 \overline K_{\alpha\beta}
2925 =
2926 \sum_{r\in B_\alpha}K_{ri}
2927 \]
2928 for any \(i\in B_\beta\).
2929 \end{theorem}
2930 
2931 \begin{proof}
2932 The \(j\)th column of \(CK\) records the transition mass from state \(j\) into every target block. The \(j\)th column of \(\overline KC\) is the column of \(\overline K\) indexed by the source block containing \(j\). Equality therefore holds exactly under the stated condition. The formula gives uniqueness. Summing over \(\alpha\) gives one, so the coarse matrix is column stochastic.
2933 \end{proof}
2934 
2935 \begin{corollary}[Continuous time lumping]\label{extended:record:033}\label{extended:cor:continuous_lumping}
2936 A coarse Markov generator \(\overline Q\) satisfying
2937 \[
2938 CQ=\overline QC
2939 \]
2940 exists if and only if the aggregated rates
2941 \[
2942 \sum_{r\in B_\alpha}Q_{ri}
2943 \]
2944 depend only on the source block of \(i\). In that case
2945 \[
2946 C\e^{tQ}=\e^{t\overline Q}C
2947 \]
2948 for every \(t\geq0\).
2949 \end{corollary}
2950 
2951 \begin{proof}
2952 The characterization is the same column argument as in \cref{extended:thm:lumping_characterization}. The off-diagonal entries of \(\overline Q\) are nonnegative and its column sums are zero. Exact semigroup intertwining follows from \cref{extended:prop:infinitesimal_defect}.
2953 \end{proof}
2954 
2955 \begin{theorem}[Finite horizon lumping defect]\label{extended:record:034}\label{extended:thm:finite_horizon_lumping}
2956 For arbitrary fine and coarse operators \(K\) and \(\overline K\), with the fixed aggregation map \(C\), set
2957 \[
2958 \Delta_K=CK-\overline KC.
2959 \]
2960 Then
2961 \[
2962 CK^n-\overline K^nC
2963 =
2964 \sum_{j=0}^{n-1}
2965 \overline K^{n-1-j}\Delta_KK^j.
2966 \]
2967 If \(K\) and \(\overline K\) are column stochastic, then
2968 \[
2969 \norm{CK^n-\overline K^nC}_{1\to1}
2970 \leq
2971 n\norm{\Delta_K}_{1\to1}.
2972 \]
2973 For every probability vector \(p\), the total variation discrepancy is at most
2974 \[
2975 \frac n2\norm{\Delta_K}_{1\to1}.
2976 \]
2977 \end{theorem}
2978 
2979 \begin{proof}
2980 Sum the telescoping identities
2981 \[
2982 \overline K^{n-1-j}CK^{j+1}
2983 -
2984 \overline K^{n-j}CK^j
2985 =
2986 \overline K^{n-1-j}\Delta_KK^j.
2987 \]
2988 Stochastic matrices are contractions in the \(\ell^1\) norm. The last statement uses total variation equal to one half of the \(\ell^1\) distance.
2989 \end{proof}
2990 
2991 \begin{corollary}[Stationary mass descends]\label{extended:record:035}\label{extended:cor:stationary_descends}
2992 If \(CK=\overline KC\) and \(Kp=p\), then
2993 \[
2994 \overline K(Cp)=Cp.
2995 \]
2996 \end{corollary}
2997 
2998 \begin{proof}
2999 Apply the intertwining identity to \(p\).
3000 \end{proof}
3001 
3002 \begin{figure}[t]
3003 \centering
3004 \includegraphics[width=.72\textwidth]{markov_lumping.pdf}
3005 \caption{Exact lumping agrees at every horizon. A small violation of the block transition condition produces a visible but controlled discrepancy.}
3006 \label{extended:fig:markov_lumping}
3007 \end{figure}
3008 
3009 \FloatBarrier
3010 \section{Conditional expectation and martingale packets}\label{extended:section:06}\label{extended:sec:conditional_expectation}
3011 
3012 Let \(\Omega=\{1,\ldots,n\}\) have strictly positive probabilities \(\pi_i\). For a partition \(\mathcal Q\), let \(B(i)\) denote the block containing \(i\). Define
3013 \[
3014 (P_\mathcal Qf)_i
3015 =
3016 \frac{\sum_{j\in B(i)}\pi_jf_j}
3017 {\sum_{j\in B(i)}\pi_j}.
3018 \]
3019 Equip \(\CC^n\) with
3020 \[
3021 \ip{f}{g}_\pi
3022 =
3023 \sum_i\pi_if_i\overline{g_i}.
3024 \]
3025 
3026 \begin{theorem}[Finite conditional expectation]\label{extended:record:036}\label{extended:thm:finite_conditional_expectation}
3027 The operator \(P_\mathcal Q\) is the orthogonal projector onto the functions that are constant on every block of \(\mathcal Q\). In particular,
3028 \[
3029 P_\mathcal Q^2=P_\mathcal Q,
3030 \qquad
3031 P_\mathcal Q^*=P_\mathcal Q.
3032 \]
3033 \end{theorem}
3034 
3035 \begin{proof}
3036 One application produces a block constant function, so a second application changes nothing. Expanding both weighted inner products block by block gives
3037 \[
3038 \ip{P_\mathcal Qf}{g}_\pi
3039 =
3040 \sum_{B\in\mathcal Q}
3041 \frac{
3042 \left(\sum_{i\in B}\pi_if_i\right)
3043 \overline{\left(\sum_{j\in B}\pi_jg_j\right)}
3044 }{\pi(B)}
3045 =
3046 \ip{f}{P_\mathcal Qg}_\pi.
3047 \]
3048 The range is exactly the block constant subspace.
3049 \end{proof}
3050 
3051 \begin{theorem}[Tower law and martingale packets]\label{extended:record:037}\label{extended:thm:martingale_packets}
3052 Let $P_0,P_1,\ldots,P_r$ come from increasingly fine partitions, so that
3053 \[
3054 \Ran P_0\subseteq\Ran P_1\subseteq\cdots\subseteq\Ran P_r.
3055 \]
3056 Define
3057 \[
3058 D_0=P_0,
3059 \qquad
3060 D_j=P_j-P_{j-1}
3061 \quad(1\leq j\leq r).
3062 \]
3063 Then
3064 \[
3065 P_iP_j=P_{\min(i,j)},
3066 \qquad
3067 D_iD_j=\delta_{ij}D_j,
3068 \qquad
3069 P_r=\sum_{j=0}^rD_j.
3070 \]
3071 Every \(D_j\) is an orthogonal projector, and
3072 \[
3073 \norm{P_rf}_\pi^2
3074 =
3075 \sum_{j=0}^r\norm{D_jf}_\pi^2
3076 \]
3077 and
3078 \[
3079 \norm f_\pi^2
3080 =
3081 \sum_{j=0}^r\norm{D_jf}_\pi^2
3082 +
3083 \norm{f-P_rf}_\pi^2.
3084 \]
3085 \end{theorem}
3086 
3087 \begin{proof}
3088 A coarse average of a fine average is the coarse average. A fine average leaves a coarse constant function unchanged. Hence \(P_iP_j=P_{\min(i,j)}\). Expanding products of the differences gives the orthogonality relations. The sum telescopes to \(P_r\). Orthogonal Pythagorean identities give the norm formulas.
3089 \end{proof}
3090 
3091 \begin{corollary}[Law of total variance]\label{extended:record:038}\label{extended:cor:total_variance}
3092 For a real valued \(f\), let \(P_0f=\Exp_\pi[f]\one\). Then
3093 \[
3094 \Var_\pi(f)
3095 =
3096 \Var_\pi(P_\mathcal Qf)
3097 +
3098 \norm{f-P_\mathcal Qf}_\pi^2.
3099 \]
3100 \end{corollary}
3101 
3102 \begin{proof}
3103 Subtract the mean and apply the orthogonal decomposition into the range of \(P_\mathcal Q\) and its orthogonal complement.
3104 \end{proof}
3105 
3106 \begin{figure}[t]
3107 \centering
3108 \includegraphics[width=.68\textwidth]{martingale_packet_energy.pdf}
3109 \caption{Energy carried by the mean packet and three successive martingale difference packets in a finite weighted example.}
3110 \label{extended:fig:martingale_energy}
3111 \end{figure}
3112 
3113 \begin{standardfact}[General conditional expectations]\label{extended:record:039}
3114 On a probability space, conditional expectation onto a sub sigma algebra is the orthogonal projector in \(L^2\). Nested sigma algebras satisfy the same tower law. Under the usual hypotheses, martingale convergence gives a limit in \(L^2\) and almost surely. Standard treatments include \cite{Williams1991}.
3115 \end{standardfact}
3116 
3117 \FloatBarrier
3118 \section{Gaussian transfer and Hermite packets}\label{extended:section:07}\label{extended:sec:ou_hermite}
3119 
3120 Let \(\gamma\) be the standard Gaussian probability measure on \(\RR\). For \(t\geq0\), define the Ornstein--Uhlenbeck transfer
3121 \[
3122 (\Ocal_t f)(x)
3123 =
3124 \int_\RR
3125 f\left(
3126 \e^{-t}x+\sqrt{1-\e^{-2t}}\,y
3127 \right)
3128 \gamma(\dd y).
3129 \]
3130 Let \(H_n\) be the probabilists' Hermite polynomial defined by
3131 \[
3132 \exp\left(ux-\frac{u^2}{2}\right)
3133 =
3134 \sum_{n=0}^\infty H_n(x)\frac{u^n}{n!},
3135 \]
3136 and set \(h_n=H_n/\sqrt{n!}\).
3137 
3138 \begin{theorem}[Ornstein--Uhlenbeck semigroup]\label{extended:record:040}\label{extended:thm:ou_semigroup}
3139 The operators \((\Ocal_t)_{t\geq0}\) form a positive Markov contraction semigroup on \(L^p(\gamma)\) for every \(1\leq p\leq\infty\). On \(L^2(\gamma)\), the semigroup is self-adjoint and
3140 \[
3141 \Ocal_t h_n=\e^{-nt}h_n.
3142 \]
3143 The family \((h_n)_{n\geq0}\) is an orthonormal basis of \(L^2(\gamma)\).
3144 \end{theorem}
3145 
3146 \begin{proof}
3147 Positivity and preservation of constants follow from the integral formula. If \(X\) and \(Y\) are independent standard Gaussian variables, then
3148 \[
3149 \e^{-s}
3150 \left(
3151 \e^{-t}x+\sqrt{1-\e^{-2t}}X
3152 \right)
3153 +
3154 \sqrt{1-\e^{-2s}}Y
3155 \]
3156 has the same distribution as
3157 \[
3158 \e^{-(s+t)}x
3159 +
3160 \sqrt{1-\e^{-2(s+t)}}Z
3161 \]
3162 with \(Z\) standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of \(\gamma\) give the \(L^p\) contraction. The joint Gaussian pair
3163 \[
3164 \left(
3165 X,
3166 \e^{-t}X+\sqrt{1-\e^{-2t}}Y
3167 \right)
3168 \]
3169 is symmetric under interchange, which proves self-adjointness.
3170 
3171 Apply \(\Ocal_t\) to the Hermite generating function. Gaussian integration gives
3172 \[
3173 \begin{aligned}
3174 \Ocal_t
3175 \exp\left(ux-\frac{u^2}{2}\right)
3176 &=
3177 \exp\left(
3178 \e^{-t}ux-
3179 \frac{\e^{-2t}u^2}{2}
3180 \right)\\
3181 &=
3182 \sum_{n=0}^\infty
3183 \e^{-nt}H_n(x)\frac{u^n}{n!}.
3184 \end{aligned}
3185 \]
3186 Comparison of coefficients proves the eigenvalue formula. The same generating function shows
3187 \[
3188 \int H_mH_n\dd\gamma=n!\delta_{mn}.
3189 \]
3190 The Hermite polynomials span all polynomials, and polynomials are dense in \(L^2(\gamma)\). Hence the normalized family is an orthonormal basis. See \cite{Janson1997} for the general Gaussian Hilbert space setting.
3191 \end{proof}
3192 
3193 \begin{corollary}[Hermite packet reconstruction]\label{extended:record:041}\label{extended:cor:hermite_packets}
3194 Let \(P_n\) be the orthogonal projection onto \(\CC h_n\). Then
3195 \[
3196 I=\sum_{n=0}^\infty P_n
3197 \]
3198 in the strong operator topology and
3199 \[
3200 \Ocal_t
3201 =
3202 \sum_{n=0}^\infty\e^{-nt}P_n.
3203 \]
3204 For \(f\in L^2(\gamma)\),
3205 \[
3206 \norm{\Ocal_t f}_2^2
3207 =
3208 \sum_{n=0}^\infty
3209 \e^{-2nt}\norm{P_nf}_2^2.
3210 \]
3211 \end{corollary}
3212 
3213 \begin{proof}
3214 Apply Parseval's identity and the eigenvalue formula in \cref{extended:thm:ou_semigroup}.
3215 \end{proof}
3216 
3217 \begin{proposition}[Generator and Dirichlet form]\label{extended:record:042}\label{extended:prop:ou_generator}
3218 On the polynomial subspace, the infinitesimal generator of \((\Ocal_t)\) is the differential operator
3219 \[
3220 \Lcal f=f''-xf'.
3221 \]
3222 It satisfies
3223 \[
3224 \Lcal h_n=-nh_n.
3225 \]
3226 For every polynomial \(f\),
3227 \[
3228 \ip{-\Lcal f}{f}_{L^2(\gamma)}
3229 =
3230 \int_\RR\abs{f'(x)}^2\gamma(\dd x).
3231 \]
3232 \end{proposition}
3233 
3234 \begin{proof}
3235 Write
3236 \[
3237 \rho_t=\e^{-t}=1-t+O(t^2),
3238 \qquad
3239 \sigma_t^2=1-\e^{-2t}=2t+O(t^2).
3240 \]
3241 For a polynomial \(f\), Taylor expansion at \(x\) and Gaussian integration give
3242 \[
3243 \begin{aligned}
3244 \Ocal_tf(x)
3245 ={}&f(x)
3246 +f'(x)(\rho_t-1)x
3247 +\frac12f''(x)\sigma_t^2
3248 +o(t)\\
3249 ={}&f(x)+t\bigl(f''(x)-xf'(x)\bigr)+o(t).
3250 \end{aligned}
3251 \]
3252 Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is \(o(t)\), so the expansion may be integrated term by term. Therefore
3253 \[
3254 \left.\frac{\dd}{\dd t}\Ocal_tf\right|_{t=0}
3255 =f''-xf'.
3256 \]
3257 The Hermite eigenvalue identity also follows by differentiating
3258 \(
3259 \Ocal_th_n=\e^{-nt}h_n
3260 \)
3261 at zero. Gaussian integration by parts gives
3262 \[
3263 \int_\RR f'(x)\overline{g(x)}\gamma(\dd x)
3264 =
3265 \int_\RR f(x)\overline{xg(x)-g'(x)}\gamma(\dd x).
3266 \]
3267 Set \(g=f'\) and rearrange.
3268 \end{proof}
3269 
3270 \begin{corollary}[Gaussian spectral gap]\label{extended:record:043}\label{extended:cor:ou_spectral_gap}
3271 If \(f\in L^2(\gamma)\) has mean zero, then
3272 \[
3273 \norm{\Ocal_tf}_2
3274 \leq
3275 \e^{-t}\norm f_2.
3276 \]
3277 For every polynomial \(f\),
3278 \[
3279 \Var_\gamma(f)
3280 \leq
3281 \int_\RR\abs{f'(x)}^2\gamma(\dd x).
3282 \]
3283 \end{corollary}
3284 
3285 \begin{proof}
3286 A mean zero function has no \(h_0\) component, so every surviving Hermite multiplier is at most \(\e^{-t}\). For a polynomial
3287 \[
3288 f-\Exp_\gamma f=\sum_{n\geq1}c_nh_n.
3289 \]
3290 Then
3291 \[
3292 \Var_\gamma(f)=\sum_{n\geq1}\abs{c_n}^2
3293 \leq
3294 \sum_{n\geq1}n\abs{c_n}^2
3295 =
3296 \ip{-\Lcal f}{f}.
3297 \]
3298 Apply \cref{extended:prop:ou_generator}.
3299 \end{proof}
3300 
3301 \begin{corollary}[Fractional Hermite transfer]\label{extended:record:044}\label{extended:cor:fractional_hermite}
3302 Let \(-\Lcal\) be the nonnegative Ornstein--Uhlenbeck number operator with
3303 \[
3304 -\Lcal h_n=nh_n.
3305 \]
3306 For every \(\alpha>0\),
3307 \[
3308 \e^{-t(-\Lcal)^\alpha}f
3309 =
3310 \sum_{n=0}^\infty
3311 \e^{-tn^\alpha}P_nf.
3312 \]
3313 This is a contraction semigroup on \(L^2(\gamma)\).
3314 \end{corollary}
3315 
3316 \begin{proof}
3317 The spectral theorem applies to the orthonormal Hermite basis. Every multiplier has modulus at most one and obeys the scalar semigroup law.
3318 \end{proof}
3319 
3320 \begin{figure}[t]
3321 \centering
3322 \includegraphics[width=.72\textwidth]{ou_hermite.pdf}
3323 \caption{Hermite packet multipliers for the Ornstein--Uhlenbeck semigroup and two fractional powers of its number operator.}
3324 \label{extended:fig:ou_hermite}
3325 \end{figure}
3326 
3327 \FloatBarrier
3328 \section{Bayesian transfer and evidence packets}\label{extended:section:08}\label{extended:sec:bayesian_transfer}
3329 
3330 Let
3331 \[
3332 \Delta_{n-1}
3333 =
3334 \left\{p\in\RR^n:
3335 p_i\geq0,
3336 \quad
3337 \sum_ip_i=1
3338 \right\}
3339 \]
3340 be the probability simplex. Let \(\lambda\in[0,\infty)^n\) have positive evidence
3341 \[
3342 Z_\lambda(p)=\sum_i\lambda_ip_i.
3343 \]
3344 The Bayesian transfer is
3345 \[
3346 \Bcal_\lambda(p)
3347 =
3348 \frac{\lambda\odot p}{Z_\lambda(p)}.
3349 \]
3350 
3351 \begin{proposition}[Scale invariance and composition]\label{extended:record:045}\label{extended:prop:bayes_composition}
3352 For every \(c>0\),
3353 \[
3354 \Bcal_{c\lambda}=\Bcal_\lambda.
3355 \]
3356 If all relevant evidences are positive, then
3357 \[
3358 \Bcal_\mu(\Bcal_\lambda(p))
3359 =
3360 \Bcal_{\lambda\odot\mu}(p).
3361 \]
3362 \end{proposition}
3363 
3364 \begin{proof}
3365 A common likelihood scale cancels between the numerator and denominator. In the second identity, the first evidence appears in both the numerator and the denominator of the second update and cancels.
3366 \end{proof}
3367 
3368 Let \(I=I_1\sqcup\cdots\sqcup I_m\) be a partition. Define packet evidences
3369 \[
3370 Z_\alpha
3371 =
3372 \sum_{i\in I_\alpha}\lambda_ip_i.
3373 \]
3374 For \(Z_\alpha>0\), define
3375 \[
3376 m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta},
3377 \qquad
3378 q_{\alpha,i}=\frac{\lambda_ip_i}{Z_\alpha}
3379 \quad(i\in I_\alpha).
3380 \]
3381 Packets with zero evidence receive posterior mass zero.
3382 
3383 \begin{theorem}[Bayesian packet reconstruction]\label{extended:record:046}\label{extended:thm:bayesian_packet_reconstruction}
3384 The posterior is the disjoint packet synthesis
3385 \[
3386 \Bcal_\lambda(p)
3387 =
3388 \sum_{\alpha:Z_\alpha>0}
3389 m_\alpha q_\alpha.
3390 \]
3391 The local shape \(q_\alpha\) depends only on the prior and likelihood inside packet \(I_\alpha\). Coupling between different packets occurs through the normalization of the packet evidences.
3392 \end{theorem}
3393 
3394 \begin{proof}
3395 For \(i\in I_\alpha\),
3396 \[
3397 m_\alpha q_{\alpha,i}
3398 =
3399 \frac{Z_\alpha}{\sum_\beta Z_\beta}
3400 \frac{\lambda_ip_i}{Z_\alpha}
3401 =
3402 \frac{\lambda_ip_i}{Z_\lambda(p)}.
3403 \]
3404 The packet supports are disjoint.
3405 \end{proof}
3406 
3407 \begin{theorem}[Likelihood perturbation]\label{extended:record:047}\label{extended:thm:likelihood_perturbation}
3408 Let \(\lambda,\mu\geq0\) have positive evidences under a fixed prior \(p\). Set
3409 \[
3410 d_p(\lambda,\mu)
3411 =
3412 \sum_i\abs{\lambda_i-\mu_i}p_i.
3413 \]
3414 Then
3415 \[
3416 \norm{\Bcal_\lambda(p)-\Bcal_\mu(p)}_1
3417 \leq
3418 \frac{2d_p(\lambda,\mu)}
3419 {\max\{Z_\lambda(p),Z_\mu(p)\}}.
3420 \]
3421 \end{theorem}
3422 
3423 \begin{proof}
3424 Add and subtract \(\mu\odot p/Z_\lambda(p)\). This gives
3425 \[
3426 \norm{\Bcal_\lambda(p)-\Bcal_\mu(p)}_1
3427 \leq
3428 \frac{2d_p(\lambda,\mu)}{Z_\lambda(p)}.
3429 \]
3430 Interchanging \(\lambda\) and \(\mu\) gives the same estimate with \(Z_\mu(p)\). Take the smaller of the two upper bounds.
3431 \end{proof}
3432 
3433 \begin{theorem}[Stability with respect to the prior]\label{extended:record:048}\label{extended:thm:prior_stability}
3434 Assume
3435 \[
3436 0<a\leq\lambda_i\leq b
3437 \]
3438 for every \(i\). Then
3439 \[
3440 \norm{\Bcal_\lambda(p)-\Bcal_\lambda(q)}_1
3441 \leq
3442 \frac{2b}{a}\norm{p-q}_1
3443 \]
3444 for all priors \(p,q\in\Delta_{n-1}\).
3445 \end{theorem}
3446 
3447 \begin{proof}
3448 The evidences are at least \(a\). Add and subtract \(\lambda\odot q/Z_\lambda(p)\). The numerator difference is at most \(b\norm{p-q}_1\), and the evidence difference has the same bound. The two terms give the stated constant.
3449 \end{proof}
3450 
3451 \begin{figure}[t]
3452 \centering
3453 \includegraphics[width=.72\textwidth]{bayesian_packets.pdf}
3454 \caption{A prior and posterior on two coarse packets. The likelihood is constant in the first packet and varies inside the second, so only the second local conditional distribution changes.}
3455 \label{extended:fig:bayesian_packets}
3456 \end{figure}
3457 
3458 \FloatBarrier
3459 \section{Bayesian coarsening and finite sufficiency}\label{extended:section:09}\label{extended:sec:bayesian_sufficiency}
3460 
3461 A fixed coarse description will be called sufficient when it reproduces the coarse posterior masses for every positive prior and leaves the conditional distribution inside every packet unchanged. This excludes a coarse rule that hides an unrecorded local update.
3462 
3463 Let \(C\) send a fine probability vector to its block masses
3464 \[
3465 (Cp)_\alpha
3466 =
3467 \sum_{i\in I_\alpha}p_i.
3468 \]
3469 Assume first that every block mass is positive. Write
3470 \[
3471 r_{\alpha,i}
3472 =
3473 \frac{p_i}{(Cp)_\alpha}
3474 \]
3475 for the prior conditional distribution inside packet \(I_\alpha\). Define the prior dependent packet average likelihood
3476 \[
3477 \overline\lambda_\alpha(p)
3478 =
3479 \sum_{i\in I_\alpha}\lambda_ir_{\alpha,i}.
3480 \]
3481 
3482 \begin{theorem}[Exact two stage Bayesian update]\label{extended:record:049}\label{extended:thm:two_stage_bayes}
3483 For every strictly positive prior,
3484 \[
3485 C\Bcal_\lambda(p)
3486 =
3487 \Bcal_{\overline\lambda(p)}(Cp).
3488 \]
3489 For every packet with \(\overline\lambda_\alpha(p)>0\), the posterior conditional distribution is
3490 \[
3491 q_{\alpha,i}
3492 =
3493 \frac{\lambda_ir_{\alpha,i}}
3494 {\overline\lambda_\alpha(p)}.
3495 \]
3496 A packet with zero average likelihood has zero posterior mass. Thus a fine Bayesian update consists of a coarse update by packet average likelihoods followed by independent local updates inside the packets that survive.
3497 \end{theorem}
3498 
3499 \begin{proof}
3500 The posterior mass of packet \(I_\alpha\) is
3501 \[
3502 \frac{\sum_{i\in I_\alpha}\lambda_ip_i}
3503 {\sum_j\lambda_jp_j}
3504 =
3505 \frac{(Cp)_\alpha\overline\lambda_\alpha(p)}
3506 {\sum_\beta(Cp)_\beta\overline\lambda_\beta(p)}.
3507 \]
3508 This is the coarse Bayesian formula. Dividing the fine posterior coordinates by the packet mass gives the local formula.
3509 \end{proof}
3510 
3511 \begin{theorem}[Characterization of fixed coarse sufficiency]\label{extended:record:050}\label{extended:thm:bayesian_sufficiency}
3512 Assume \(\lambda_i>0\) for every state. The following conditions are equivalent.
3513 \begin{enumerate}
3514 \item There is a positive coarse likelihood \(\ell\in(0,\infty)^m\) such that
3515 \[
3516 C\Bcal_\lambda(p)
3517 =
3518 \Bcal_\ell(Cp)
3519 \]
3520 for every strictly positive prior \(p\), and the posterior conditional distribution inside every packet equals its prior conditional distribution.
3521 \item The likelihood is constant on every packet.
3522 \end{enumerate}
3523 When these conditions hold, \(\ell_\alpha\) can be chosen as the common value of \(\lambda\) on \(I_\alpha\), and it is unique up to a common positive scale.
3524 \end{theorem}
3525 
3526 \begin{proof}
3527 If \(\lambda\) is constant on each packet, \cref{extended:thm:two_stage_bayes} has packet averages independent of the prior and its local posterior formula gives \(q_\alpha=r_\alpha\). This proves the first condition.
3528 
3529 Conversely, fix any strictly positive prior. Equality of the local posterior and prior conditionals gives
3530 \[
3531 \frac{\lambda_i r_{\alpha,i}}
3532 {\overline\lambda_\alpha(p)}
3533 =
3534 r_{\alpha,i}
3535 \]
3536 for every \(i\in I_\alpha\). Positivity of the conditional prior implies
3537 \[
3538 \lambda_i=\overline\lambda_\alpha(p)
3539 \]
3540 for every state in the packet. Thus the likelihood is constant on every packet. If two positive coarse likelihoods define the same update for every positive coarse prior, comparison of any two posterior odds shows that their coordinate ratios agree. Hence they differ only by a common positive factor.
3541 \end{proof}
3542 
3543 \begin{theorem}[Coarsest fixed sufficient partition]\label{extended:record:051}\label{extended:thm:coarsest_bayesian_partition}
3544 Let \(\mathcal L\) be a nonempty family of strictly positive likelihood vectors. Define
3545 \[
3546 i\sim_{\mathcal L}j
3547 \quad\Longleftrightarrow\quad
3548 \lambda_i=\lambda_j
3549 \quad\hbox{for every }\lambda\in\mathcal L.
3550 \]
3551 Let \(\mathcal Q_{\mathcal L}\) be the partition into equivalence classes. A partition \(\mathcal Q\) gives a fixed coarse sufficient description in the sense of \cref{extended:thm:bayesian_sufficiency} for every \(\lambda\in\mathcal L\) if and only if every block of \(\mathcal Q\) is contained in a block of \(\mathcal Q_{\mathcal L}\). Hence \(\mathcal Q_{\mathcal L}\) is the unique coarsest partition with this property.
3552 \end{theorem}
3553 
3554 \begin{proof}
3555 Apply \cref{extended:thm:bayesian_sufficiency} to every likelihood in the family. A fixed sufficient description exists exactly when each likelihood is constant on every block of \(\mathcal Q\). This is equivalent to every block of \(\mathcal Q\) being contained in one common level set of the family. The equivalence class partition is itself admissible, and every other admissible partition refines it.
3556 \end{proof}
3557 
3558 \begin{definition}[Within packet information loss]\label{extended:record:052}
3559 For a prior \(p\), define
3560 \[
3561 \eta_i(p)
3562 =
3563 \lambda_i-\overline\lambda_\alpha(p)
3564 \quad\hbox{when }i\in I_\alpha.
3565 \]
3566 Then
3567 \[
3568 \sum_{i\in I_\alpha}r_{\alpha,i}\eta_i(p)=0.
3569 \]
3570 \end{definition}
3571 
3572 \begin{proposition}[Exact local distortion]\label{extended:record:053}\label{extended:prop:local_bayes_distortion}
3573 For every packet with positive mass and evidence,
3574 \[
3575 q_{\alpha,i}-r_{\alpha,i}
3576 =
3577 \frac{r_{\alpha,i}\eta_i(p)}
3578 {\overline\lambda_\alpha(p)}.
3579 \]
3580 Consequently,
3581 \[
3582 \norm{q_\alpha-r_\alpha}_1
3583 =
3584 \frac{
3585 \sum_{i\in I_\alpha}r_{\alpha,i}\abs{\eta_i(p)}
3586 }{\overline\lambda_\alpha(p)}.
3587 \]
3588 If \(\lambda\) is positive on the packet, then
3589 \[
3590 \norm{q_\alpha-r_\alpha}_1
3591 \leq
3592 \frac{\osc_{I_\alpha}(\lambda)}
3593 {\min_{i\in I_\alpha}\lambda_i}.
3594 \]
3595 \end{proposition}
3596 
3597 \begin{proof}
3598 Subtract \(r_{\alpha,i}\) from the local posterior formula in \cref{extended:thm:two_stage_bayes}. The norm identity follows because the factor multiplying \(\eta_i\) is nonnegative. The last estimate uses
3599 \[
3600 \sum_i r_{\alpha,i}\abs{\eta_i}
3601 \leq
3602 \osc_{I_\alpha}(\lambda)
3603 \]
3604 and \(\overline\lambda_\alpha(p)\geq\min_{I_\alpha}\lambda\).
3605 \end{proof}
3606 
3607 \begin{theorem}[Approximate coarse Bayesian rule]\label{extended:record:054}\label{extended:thm:approximate_coarse_bayes}
3608 Choose one positive number \(\ell_\alpha\) for every packet and define the block constant fine likelihood
3609 \[
3610 \widetilde\lambda_i=\ell_\alpha
3611 \quad(i\in I_\alpha).
3612 \]
3613 Then
3614 \[
3615 \norm{
3616 C\Bcal_\lambda(p)-\Bcal_\ell(Cp)
3617 }_1
3618 \leq
3619 \frac{
3620 2\sum_i p_i\abs{\lambda_i-\widetilde\lambda_i}
3621 }{
3622 \max\{Z_\lambda(p),Z_{\widetilde\lambda}(p)\}
3623 }.
3624 \]
3625 \end{theorem}
3626 
3627 \begin{proof}
3628 Block constancy gives
3629 \[
3630 C\Bcal_{\widetilde\lambda}(p)
3631 =
3632 \Bcal_\ell(Cp).
3633 \]
3634 Aggregation is an \(\ell^1\) contraction. Apply \cref{extended:thm:likelihood_perturbation} to the two fine posteriors and then aggregate.
3635 \end{proof}
3636 
3637 \begin{definition}[Relative information]\label{extended:record:055}
3638 For probability vectors \(q\) and \(p\), define
3639 \[
3640 D(q\Vert p)
3641 =
3642 \sum_iq_i\log\frac{q_i}{p_i}.
3643 \]
3644 The convention is \(0\log0=0\). A term with \(q_i>0\) and \(p_i=0\) has value \(+\infty\).
3645 \end{definition}
3646 
3647 \begin{theorem}[Exact information decomposition]\label{extended:record:056}\label{extended:thm:bayesian_information_chain}
3648 Let \(p\) and \(q\) be probability vectors on a finite partition \((I_\alpha)_\alpha\), and assume \(q_i=0\) whenever \(p_i=0\). Let
3649 \[
3650 m=Cp,
3651 \qquad
3652 n=Cq,
3653 \]
3654 and let \(p_\alpha\) and \(q_\alpha\) be the conditional distributions inside packet \(I_\alpha\) whenever the corresponding mass is positive. Then
3655 \[
3656 D(q\Vert p)
3657 =
3658 D(n\Vert m)
3659 +
3660 \sum_{\alpha:n_\alpha>0}
3661  n_\alpha D(q_\alpha\Vert p_\alpha).
3662 \]
3663 The first term is the information retained by the coarse masses. The second term is the exact information carried by changes inside the packets.
3664 \end{theorem}
3665 
3666 \begin{proof}
3667 For \(i\in I_\alpha\), write
3668 \[
3669 p_i=m_\alpha p_{\alpha,i},
3670 \qquad
3671 q_i=n_\alpha q_{\alpha,i}.
3672 \]
3673 Then
3674 \[
3675 \log\frac{q_i}{p_i}
3676 =
3677 \log\frac{n_\alpha}{m_\alpha}
3678 +
3679 \log\frac{q_{\alpha,i}}{p_{\alpha,i}}.
3680 \]
3681 Multiply by \(q_i\), sum first inside each packet, and then sum over the packets. This is the finite relative entropy chain rule. Standard information theoretic treatments include \cite{CoverThomas2006}.
3682 \end{proof}
3683 
3684 \begin{corollary}[Coarse data processing and equality]\label{extended:record:057}\label{extended:cor:coarse_data_processing}
3685 Under the assumptions of \cref{extended:thm:bayesian_information_chain},
3686 \[
3687 D(Cq\Vert Cp)
3688 \leq
3689 D(q\Vert p).
3690 \]
3691 Equality holds if and only if
3692 \[
3693 q_\alpha=p_\alpha
3694 \]
3695 for every packet with positive posterior mass.
3696 \end{corollary}
3697 
3698 \begin{proof}
3699 Every conditional relative information term in \cref{extended:thm:bayesian_information_chain} is nonnegative. It vanishes exactly when the two conditional distributions agree.
3700 \end{proof}
3701 
3702 \begin{corollary}[Information created by a Bayesian update]\label{extended:record:058}\label{extended:cor:bayesian_information_gain}
3703 Let \(q=\Bcal_\lambda(p)\) with positive likelihood and positive prior. Then
3704 \[
3705 D(q\Vert p)
3706 =
3707 \sum_iq_i\log\lambda_i
3708 -
3709 \log Z_\lambda(p).
3710 \]
3711 Inside a packet,
3712 \[
3713 D(q_\alpha\Vert r_\alpha)
3714 =
3715 \sum_{i\in I_\alpha}
3716 q_{\alpha,i}
3717 \log\frac{\lambda_i}{\overline\lambda_\alpha(p)}.
3718 \]
3719 Hence
3720 \[
3721 D(q_\alpha\Vert r_\alpha)
3722 \leq
3723 \log
3724 \frac{\max_{i\in I_\alpha}\lambda_i}
3725 {\min_{i\in I_\alpha}\lambda_i}.
3726 \]
3727 \end{corollary}
3728 
3729 \begin{proof}
3730 The posterior ratio is
3731 \[
3732 \frac{q_i}{p_i}
3733 =
3734 \frac{\lambda_i}{Z_\lambda(p)}.
3735 \]
3736 The local ratio is
3737 \[
3738 \frac{q_{\alpha,i}}{r_{\alpha,i}}
3739 =
3740 \frac{\lambda_i}{\overline\lambda_\alpha(p)}.
3741 \]
3742 Insert these identities into the definition of relative entropy. The final estimate follows because both \(\lambda_i\) and the packet average lie between the packet minimum and maximum.
3743 \end{proof}
3744 
3745 \begin{corollary}[Information criterion for local sufficiency]\label{extended:record:059}\label{extended:cor:information_sufficiency}
3746 For a fixed positive likelihood, the within packet term in \cref{extended:thm:bayesian_information_chain} vanishes for every positive prior if and only if the likelihood is constant on every packet.
3747 \end{corollary}
3748 
3749 \begin{proof}
3750 If the likelihood is block constant, every local posterior equals its local prior. Conversely, choose any positive prior. Vanishing relative entropy gives \(q_\alpha=r_\alpha\). The local posterior formula then forces all likelihood values in that packet to equal their packet average. Repeat for every packet.
3751 \end{proof}
3752 
3753 \begin{figure}[t]
3754 \centering
3755 \includegraphics[width=.72\textwidth]{bayesian_information_split.pdf}
3756 \caption{Relative information separated into a coarse packet contribution and conditional contributions inside the packets.}
3757 \label{extended:fig:bayesian_information_split}
3758 \end{figure}
3759 
3760 \begin{remark}\label{extended:record:060}
3761 The exact coarse likelihood in \cref{extended:thm:two_stage_bayes} depends on the conditional prior inside each packet. \Cref{extended:thm:bayesian_sufficiency} identifies when that dependence disappears. \Cref{extended:thm:bayesian_information_chain} measures what is lost when only the coarse posterior masses are retained.
3762 \end{remark}
3763 
3764 \FloatBarrier
3765 \section{Bayesian flows and hidden Markov recursion}\label{extended:section:10}\label{extended:sec:bayesian_flows}
3766 
3767 Fix a score vector \(g\in\RR^n\). For \(t\geq0\), define
3768 \[
3769 \Phi_t(p)_i
3770 =
3771 \frac{\e^{tg_i}p_i}
3772 {\sum_j\e^{tg_j}p_j}.
3773 \]
3774 
3775 \begin{theorem}[Bayesian flow and replicator equation]\label{extended:record:061}\label{extended:thm:bayesian_flow}
3776 The maps \(\Phi_t\) form a nonlinear semigroup:
3777 \[
3778 \Phi_{s+t}=\Phi_s\circ\Phi_t.
3779 \]
3780 Every coordinate satisfies
3781 \[
3782 \frac{\dd}{\dd t}p_i(t)
3783 =
3784 \bigl(g_i-\ip{g}{p(t)}\bigr)p_i(t).
3785 \]
3786 The support of \(p(t)\) is independent of \(t\), and for any two coordinates in the support,
3787 \[
3788 \frac{p_i(t)}{p_j(t)}
3789 =
3790 \e^{t(g_i-g_j)}
3791 \frac{p_i(0)}{p_j(0)}.
3792 \]
3793 \end{theorem}
3794 
3795 \begin{proof}
3796 The semigroup law follows from multiplication of the likelihoods \(\e^{sg}\) and \(\e^{tg}\). Differentiating the normalized formula gives the replicator equation. A coordinate that starts at zero remains zero, while a positive coordinate remains positive. Dividing two explicit coordinate formulas gives the odds identity.
3797 \end{proof}
3798 
3799 \begin{theorem}[Evidence cumulants and score monotonicity]\label{extended:record:062}\label{extended:thm:evidence_cumulants}
3800 Set
3801 \[
3802 \psi(t)
3803 =
3804 \log\left(
3805 \sum_i\e^{tg_i}p_i(0)
3806 \right).
3807 \]
3808 Then
3809 \[
3810 \psi'(t)=\ip{g}{p(t)},
3811 \qquad
3812 \psi''(t)=\Var_{p(t)}(g)\geq0.
3813 \]
3814 Consequently,
3815 \[
3816 \frac{\dd}{\dd t}\ip{g}{p(t)}
3817 =
3818 \Var_{p(t)}(g).
3819 \]
3820 \end{theorem}
3821 
3822 \begin{proof}
3823 Differentiate the log normalizing constant once and twice. The second derivative is the centered second moment under the tilted distribution.
3824 \end{proof}
3825 
3826 \begin{corollary}[Concentration on the maximal score packet]\label{extended:record:063}\label{extended:cor:bayes_flow_concentration}
3827 Let
3828 \[
3829 G=\max\{g_i:p_i(0)>0\}
3830 \]
3831 and let \(M\) be the set of supported indices with score \(G\). If \(M\) is the full support, then the conclusion below is immediate. Otherwise, suppose
3832 \[
3833 \delta
3834 =
3835 \min\{G-g_i:p_i(0)>0,
3836 \quad i\notin M\}>0.
3837 \]
3838 Then
3839 \[
3840 1-\sum_{i\in M}p_i(t)
3841 \leq
3842 \frac{1-p_M(0)}{p_M(0)}\e^{-\delta t},
3843 \]
3844 where \(p_M(0)=\sum_{i\in M}p_i(0)\).
3845 \end{corollary}
3846 
3847 \begin{proof}
3848 Factor \(\e^{tG}\) from the normalizing constant. The total weight outside \(M\) is at most \((1-p_M(0))\e^{-\delta t}\), while the weight on \(M\) is \(p_M(0)\).
3849 \end{proof}
3850 
3851 \begin{figure}[t]
3852 \centering
3853 \includegraphics[width=.72\textwidth]{bayesian_flow.pdf}
3854 \caption{A finite Bayesian flow. Mass concentrates on the states with the largest score, while every relative odds ratio follows an exact exponential law.}
3855 \label{extended:fig:bayesian_flow}
3856 \end{figure}
3857 
3858 \subsection{Prediction and update}
3859 
3860 Let \(K\) be column stochastic and let \(\lambda^{(r)}\) be the likelihood of the \(r\)th observation. The hidden Markov filter is
3861 \[
3862 \widehat p_r=Kp_{r-1},
3863 \qquad
3864 p_r=\Bcal_{\lambda^{(r)}}(\widehat p_r).
3865 \]
3866 A standard account of finite hidden Markov models appears in \cite{Rabiner1989}.
3867 
3868 \begin{proposition}[Simplex preservation]\label{extended:record:064}\label{extended:prop:hmm_preservation}
3869 Suppose \(p_{r-1}\) is a probability vector and the new evidence is positive. Prediction produces the probability vector \(\widehat p_r\), and normalization produces the probability vector \(p_r\).
3870 \end{proposition}
3871 
3872 \begin{proof}
3873 Column stochasticity preserves nonnegativity and total mass. Bayesian normalization then preserves the same properties.
3874 \end{proof}
3875 
3876 \begin{theorem}[One step filter stability]\label{extended:record:065}\label{extended:thm:hmm_stability}
3877 Let \(K\) and \(\widetilde K\) be column stochastic. Assume
3878 \[
3879 0<a\leq\lambda_i\leq b.
3880 \]
3881 Then
3882 \[
3883 \norm{
3884 \Bcal_\lambda(Kp)-\Bcal_\lambda(\widetilde Kq)
3885 }_1
3886 \leq
3887 \frac{2b}{a}
3888 \left(
3889 \norm{p-q}_1
3890 +
3891 \norm{K-\widetilde K}_{1\to1}
3892 \right).
3893 \]
3894 \end{theorem}
3895 
3896 \begin{proof}
3897 Apply \cref{extended:thm:prior_stability} to \(Kp\) and \(\widetilde Kq\). Then use
3898 \[
3899 \norm{Kp-\widetilde Kq}_1
3900 \leq
3901 \norm{K(p-q)}_1
3902 +
3903 \norm{(K-\widetilde K)q}_1
3904 \]
3905 and the stochastic contraction property.
3906 \end{proof}
3907 
3908 \begin{example}[Beta binomial transfer]\label{extended:record:066}
3909 A beta prior with parameters \((a,b)\) and binomial data containing \(s\) successes in \(n\) trials gives the posterior parameters
3910 \[
3911 (a+s,b+n-s).
3912 \]
3913 The update is additive in the two exponent coordinates. This is a finite dimensional conjugate realization of likelihood composition. Standard Bayesian references include \cite{BernardoSmith1994}.
3914 \end{example}
3915 
3916 \FloatBarrier
3917 \section{Fractional functional calculus}\label{extended:section:11}\label{extended:sec:fractional}
3918 
3919 Let \(H\) be a finite dimensional Hilbert space and let \(L=L^*\geq0\). Write its spectral resolution as
3920 \[
3921 L=\sum_j\lambda_jP_j.
3922 \]
3923 For \(\alpha>0\), define
3924 \[
3925 L^\alpha
3926 =
3927 \sum_j\lambda_j^\alpha P_j
3928 \]
3929 and
3930 \[
3931 T_\alpha(t)
3932 =
3933 \e^{-tL^\alpha}
3934 =
3935 \sum_j\e^{-t\lambda_j^\alpha}P_j.
3936 \]
3937 
3938 \begin{theorem}[Fractional packet calculus]\label{extended:record:067}\label{extended:thm:fractional_packet_calculus}
3939 For \(\alpha,\beta>0\),
3940 \[
3941 L^\alpha L^\beta=L^{\alpha+\beta}.
3942 \]
3943 For \(s,t\geq0\),
3944 \[
3945 T_\alpha(s)T_\alpha(t)=T_\alpha(s+t).
3946 \]
3947 Every spectral packet \(P_jH\) reduces \(L^\alpha\) and \(T_\alpha(t)\).
3948 \end{theorem}
3949 
3950 \begin{proof}
3951 The claims hold on each spectral packet and the projectors are mutually orthogonal.
3952 \end{proof}
3953 
3954 \begin{theorem}[Balakrishnan formula in finite dimension]\label{extended:record:068}\label{extended:thm:balakrishnan}
3955 For \(0<\alpha<1\),
3956 \[
3957 L^\alpha
3958 =
3959 \frac{\sin(\pi\alpha)}{\pi}
3960 \int_0^\infty
3961 s^{\alpha-1}L(sI+L)^{-1}
3962 \dd s.
3963 \]
3964 The integral converges in operator norm.
3965 \end{theorem}
3966 
3967 \begin{proof}
3968 It is enough to prove the formula on an eigenvector with eigenvalue \(x\geq0\). For \(x>0\), the substitution \(s=xu\) gives
3969 \[
3970 \frac{\sin(\pi\alpha)}{\pi}
3971 \int_0^\infty
3972 s^{\alpha-1}\frac{x}{s+x}\dd s
3973 =x^\alpha
3974 \frac{\sin(\pi\alpha)}{\pi}
3975 \int_0^\infty\frac{u^{\alpha-1}}{1+u}\dd u
3976 =x^\alpha.
3977 \]
3978 The value at \(x=0\) is zero. There are finitely many eigenvalues, so the operator integral converges. This is the finite spectral form of the fractional power construction in \cite{Balakrishnan1960}.
3979 \end{proof}
3980 
3981 \begin{theorem}[Exact fractional naturality]\label{extended:record:069}\label{extended:thm:exact_fractional_naturality}
3982 Let \(L_f\) and \(L_c\) be finite dimensional positive self-adjoint operators and let \(C\) satisfy
3983 \[
3984 CL_f=L_cC.
3985 \]
3986 Then, for every \(\alpha>0\) and \(t\geq0\),
3987 \[
3988 CL_f^\alpha=L_c^\alpha C
3989 \]
3990 and
3991 \[
3992 C\e^{-tL_f^\alpha}
3993 =
3994 \e^{-tL_c^\alpha}C.
3995 \]
3996 \end{theorem}
3997 
3998 \begin{proof}
3999 The union of the two finite spectra is a finite set. There is a polynomial that agrees with \(x^\alpha\) on that set, and another polynomial that agrees with \(\e^{-tx^\alpha}\). Exact intertwining passes to every polynomial by induction.
4000 \end{proof}
4001 
4002 \begin{theorem}[Fractional defect bound]\label{extended:record:070}\label{extended:thm:fractional_defect}
4003 Let \(0<\alpha<1\). Suppose
4004 \[
4005 L_f\geq mI,
4006 \qquad
4007 L_c\geq mI
4008 \]
4009 for some \(m>0\). Define
4010 \[
4011 \Delta_L=CL_f-L_cC.
4012 \]
4013 Then
4014 \[
4015 \norm{CL_f^\alpha-L_c^\alpha C}
4016 \leq
4017 \alpha m^{\alpha-1}\norm{\Delta_L}.
4018 \]
4019 Consequently,
4020 \[
4021 \norm{
4022 C\e^{-tL_f^\alpha}
4023 -
4024 \e^{-tL_c^\alpha}C
4025 }
4026 \leq
4027 \alpha t m^{\alpha-1}\norm{\Delta_L}.
4028 \]
4029 \end{theorem}
4030 
4031 \begin{proof}
4032 Set
4033 \[
4034 R_f(s)=(sI+L_f)^{-1},
4035 \qquad
4036 R_c(s)=(sI+L_c)^{-1}.
4037 \]
4038 The resolvent identity gives
4039 \[
4040 CR_f(s)-R_c(s)C
4041 =
4042 -R_c(s)\Delta_LR_f(s).
4043 \]
4044 Since \(LR=I-sR\),
4045 \[
4046 CL_fR_f(s)-L_cR_c(s)C
4047 =
4048 sR_c(s)\Delta_LR_f(s).
4049 \]
4050 Insert this into \cref{extended:thm:balakrishnan}. The spectral lower bound gives
4051 \[
4052 \norm{R_f(s)},\norm{R_c(s)}
4053 \leq
4054 \frac{1}{s+m}.
4055 \]
4056 Therefore
4057 \[
4058 \begin{aligned}
4059 \norm{CL_f^\alpha-L_c^\alpha C}
4060 &\leq
4061 \frac{\sin(\pi\alpha)}{\pi}
4062 \norm{\Delta_L}
4063 \int_0^\infty
4064 \frac{s^\alpha}{(s+m)^2}\dd s\\
4065 &=
4066 \alpha m^{\alpha-1}\norm{\Delta_L}.
4067 \end{aligned}
4068 \]
4069 The semigroup estimate follows from \cref{extended:thm:duhamel_defect}, applied to the contraction generators \(-L_f^\alpha\) and \(-L_c^\alpha\).
4070 \end{proof}
4071 
4072 \begin{remark}\label{extended:record:071}
4073 The factor \(m^{\alpha-1}\) records the singular behavior of \(x^\alpha\) near zero. Without a common positive lower bound, a uniform Lipschitz estimate of this form cannot be expected.
4074 \end{remark}
4075 
4076 \begin{standardfact}[Stable subordination]\label{extended:record:072}\label{extended:fact:stable_subordination}
4077 Let \(0<\alpha<1\) and \(t>0\). There is a probability measure \(\eta_{\alpha,t}\) on \([0,\infty)\) such that
4078 \[
4079 \int_0^\infty \e^{-s\lambda}\eta_{\alpha,t}(\dd s)
4080 =
4081 \e^{-t\lambda^\alpha}
4082 \]
4083 for every \(\lambda\geq0\). Consequently, for a finite dimensional positive self-adjoint operator,
4084 \[
4085 \e^{-tL^\alpha}
4086 =
4087 \int_0^\infty
4088 \e^{-sL}\eta_{\alpha,t}(\dd s)
4089 \]
4090 in operator norm. The measures satisfy the convolution law
4091 \[
4092 \eta_{\alpha,s}*\eta_{\alpha,t}
4093 =
4094 \eta_{\alpha,s+t}.
4095 \]
4096 \end{standardfact}
4097 
4098 The existence of \(\eta_{\alpha,t}\) follows from complete monotonicity of \(\e^{-t\lambda^\alpha}\). The operator identity is obtained by applying the scalar Laplace transform on every spectral packet. This is the finite spectral form of Bochner subordination. See \cite{SchillingSongVondracek2012}.
4099 
4100 \begin{theorem}[Poisson subordination]\label{extended:record:073}\label{extended:thm:poisson_subordination}
4101 For \(t>0\),
4102 \[
4103 \e^{-t\sqrt L}
4104 =
4105 \frac{t}{2\sqrt\pi}
4106 \int_0^\infty
4107 s^{-3/2}
4108 \e^{-t^2/(4s)}
4109 \e^{-sL}
4110 \dd s.
4111 \]
4112 The integral converges in operator norm.
4113 \end{theorem}
4114 
4115 \begin{proof}
4116 On a spectral packet with eigenvalue \(\lambda\), the formula reduces to
4117 \[
4118 \e^{-t\sqrt\lambda}
4119 =
4120 \frac{t}{2\sqrt\pi}
4121 \int_0^\infty
4122 s^{-3/2}\e^{-t^2/(4s)-\lambda s}\dd s.
4123 \]
4124 This scalar identity follows from the Gaussian integral. Recombine the finitely many spectral packets.
4125 \end{proof}
4126 
4127 \begin{figure}[t]
4128 \centering
4129 \includegraphics[width=.72\textwidth]{fractional_multipliers.pdf}
4130 \caption{Fractional semigroup multipliers at time one. Smaller exponents preserve more of the high spectral range.}
4131 \label{extended:fig:fractional_multipliers}
4132 \end{figure}
4133 
4134 \begin{remark}\label{extended:record:074}
4135 For sectorial and nonnormal operators, fractional powers require a resolvent or contour calculus together with domain control. The finite self-adjoint theory proved here does not remove those analytic requirements. A broader account appears in \cite{SchillingSongVondracek2012}.
4136 \end{remark}
4137 
4138 \FloatBarrier
4139 \section{Green operators and finite difference transfer}\label{extended:section:12}\label{extended:sec:green_pde}
4140 
4141 Consider
4142 \[
4143 -u''=f
4144 \quad\hbox{on }(0,1),
4145 \qquad
4146 u(0)=u(1)=0.
4147 \]
4148 The continuum Green kernel is
4149 \[
4150 G(x,y)=\min(x,y)-xy.
4151 \]
4152 Thus
4153 \[
4154 u(x)=\int_0^1G(x,y)f(y)\dd y.
4155 \]
4156 
4157 For \(n\) interior nodes, let
4158 \[
4159 h=\frac{1}{n+1},
4160 \qquad
4161 x_i=ih,
4162 \]
4163 and define
4164 \[
4165 L_h
4166 =
4167 \frac{1}{h^2}
4168 \begin{pmatrix}
4169 2&-1&&\\
4170 -1&2&\ddots&\\
4171 &\ddots&\ddots&-1\\
4172 &&-1&2
4173 \end{pmatrix}.
4174 \]
4175 
4176 \begin{theorem}[Exact nodal Green reconstruction]\label{extended:record:075}\label{extended:thm:nodal_green}
4177 For \(1\leq i,j\leq n\),
4178 \[
4179 \frac{1}{h}(L_h^{-1})_{ij}
4180 =
4181 G(x_i,x_j).
4182 \]
4183 \end{theorem}
4184 
4185 \begin{proof}
4186 The inverse of the unscaled tridiagonal matrix has entry
4187 \[
4188 \frac{\min(i,j)(n+1-\max(i,j))}{n+1}.
4189 \]
4190 Multiplying by \(h^2\) and dividing by \(h\) gives
4191 \[
4192 h\min(i,j)
4193 \left(1-\frac{\max(i,j)}{n+1}\right)
4194 =
4195 \min(x_i,x_j)-x_ix_j.
4196 \]
4197 \end{proof}
4198 
4199 \begin{corollary}[Discrete Green quadrature]\label{extended:record:076}\label{extended:cor:discrete_green_quadrature}
4200 If \(u^h\) solves
4201 \[
4202 L_hu^h=f^h,
4203 \]
4204 then every nodal value has the exact representation
4205 \[
4206 u_i^h
4207 =
4208 h\sum_{j=1}^nG(x_i,x_j)f_j^h.
4209 \]
4210 Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.
4211 \end{corollary}
4212 
4213 \begin{proof}
4214 By \cref{extended:thm:nodal_green},
4215 \[
4216 (L_h^{-1})_{ij}=hG(x_i,x_j).
4217 \]
4218 Multiply the inverse matrix by the source vector.
4219 \end{proof}
4220 
4221 \begin{proposition}[Discrete maximum bound]\label{extended:record:077}\label{extended:prop:discrete_inverse_bound}
4222 The inverse \(L_h^{-1}\) has nonnegative entries and
4223 \[
4224 \norm{L_h^{-1}}_{\infty\to\infty}
4225 =
4226 \max_{1\leq i\leq n}
4227 \frac{x_i(1-x_i)}{2}
4228 \leq
4229 \frac18.
4230 \]
4231 \end{proposition}
4232 
4233 \begin{proof}
4234 Nonnegativity follows from the entry formula in \cref{extended:thm:nodal_green}. Let \(v_i=x_i(1-x_i)/2\). Since the second difference is exact on quadratics,
4235 \[
4236 L_hv=\one.
4237 \]
4238 Hence the \(i\)th row sum of \(L_h^{-1}\) is \(v_i\). For a nonnegative matrix, the infinity operator norm is the largest row sum.
4239 \end{proof}
4240 
4241 \begin{theorem}[Second order Poisson solution error]\label{extended:record:078}\label{extended:thm:poisson_error}
4242 Suppose \(u\in C^4[0,1]\) solves the Dirichlet problem above. Let
4243 \[
4244 f^h_i=f(x_i)
4245 \]
4246 and let \(u^h\) solve
4247 \[
4248 L_hu^h=f^h.
4249 \]
4250 Then
4251 \[
4252 \max_i\abs{u^h_i-u(x_i)}
4253 \leq
4254 \frac{h^2}{96}
4255 \norm{u^{(4)}}_\infty.
4256 \]
4257 \end{theorem}
4258 
4259 \begin{proof}
4260 Taylor expansion with remainder gives
4261 \[
4262 \norm{L_h(u(x_i))_i-f^h}_\infty
4263 \leq
4264 \frac{h^2}{12}\norm{u^{(4)}}_\infty.
4265 \]
4266 Apply \(L_h^{-1}\) and use \cref{extended:prop:discrete_inverse_bound}.
4267 \end{proof}
4268 
4269 \begin{figure}[t]
4270 \centering
4271 \includegraphics[width=.72\textwidth]{green_kernel.pdf}
4272 \caption{The continuum Dirichlet Green section and the scaled discrete inverse. The nodal values agree exactly.}
4273 \label{extended:fig:green_kernel}
4274 \end{figure}
4275 
4276 The discrete eigenpairs are
4277 \[
4278 \lambda_{k,h}
4279 =
4280 \frac{4}{h^2}
4281 \sin^2\left(\frac{k\pi h}{2}\right),
4282 \qquad
4283 v_{k,h}(i)=\sin(k\pi x_i).
4284 \]
4285 
4286 \begin{theorem}[Low mode eigenvalue control]\label{extended:record:079}\label{extended:thm:low_mode}
4287 For \(kh\leq1\),
4288 \[
4289 0\leq(k\pi)^2-\lambda_{k,h}
4290 \leq
4291 \frac{(k\pi)^4}{12}h^2.
4292 \]
4293 Consequently, for every fixed \(k\),
4294 \[
4295 \lambda_{k,h}=(k\pi)^2+O(h^2).
4296 \]
4297 \end{theorem}
4298 
4299 \begin{proof}
4300 Set \(x=k\pi h/2\). The condition \(kh\leq1\) gives \(0\leq x\leq\pi/2\). On this interval,
4301 \[
4302 \sin x\geq x-\frac{x^3}{6},
4303 \]
4304 so
4305 \[
4306 0\leq x^2-\sin^2x\leq\frac{x^4}{3}.
4307 \]
4308 Multiply by \(4/h^2\).
4309 \end{proof}
4310 
4311 \begin{corollary}[Heat and fractional mode transfer]\label{extended:record:080}\label{extended:cor:mode_transfer}
4312 Let \(\lambda_k=(k\pi)^2\). For \(t\geq0\),
4313 \[
4314 0\leq
4315 \e^{-t\lambda_{k,h}}-\e^{-t\lambda_k}
4316 \leq
4317 \frac{t(k\pi)^4}{12}h^2\e^{-t\lambda_{k,h}}.
4318 \]
4319 For \(0<\alpha\leq1\) and fixed \(k\),
4320 \[
4321 \lambda_{k,h}^\alpha
4322 =
4323 \lambda_k^\alpha+O(h^2)
4324 \]
4325 and
4326 \[
4327 \e^{-t\lambda_{k,h}^\alpha}
4328 =
4329 \e^{-t\lambda_k^\alpha}+O(th^2)
4330 \]
4331 uniformly for \(t\) in bounded intervals.
4332 \end{corollary}
4333 
4334 \begin{proof}
4335 The ordinary heat estimate follows from the mean value theorem and \cref{extended:thm:low_mode}. For fixed \(k\), the eigenvalues stay uniformly away from zero once \(h\) is small. The function \(x^\alpha\) is then Lipschitz on a common compact interval containing both eigenvalues. Apply the mean value theorem twice.
4336 \end{proof}
4337 
4338 Finite difference schemes and their stability theory are treated more broadly in \cite{Strikwerda2004}. The present result uses the special one-dimensional Green structure to obtain an explicit constant.
4339 
4340 \FloatBarrier
4341 \section{Finite state Feynman--Kac transfer}\label{extended:section:13}\label{extended:sec:feynman_kac}
4342 \begin{interpretation}
4343 The potential below is real. Its exponential is a potential weight in general, and a survival weight when the potential is nonnegative. This distinction does not change the finite transfer formula.
4344 \end{interpretation}
4345 
4346 Let \(K\) be column stochastic. Let \(V\) be a real potential on the states and let
4347 \[
4348 D_V=\diag(V).
4349 \]
4350 For functions written as columns, the backward Markov operator is \(K^T\).
4351 
4352 \begin{theorem}[Discrete Feynman--Kac formula]\label{extended:record:081}\label{extended:thm:discrete_feynman_kac}
4353 Define
4354 \[
4355 M_V=\diag(\e^{-V})K^T.
4356 \]
4357 For terminal data \(f\),
4358 \[
4359 (M_V^nf)_i
4360 =
4361 \Exp_i\left[
4362 \exp\left(
4363 -\sum_{r=0}^{n-1}V(X_r)
4364 \right)
4365 f(X_n)
4366 \right].
4367 \]
4368 \end{theorem}
4369 
4370 \begin{proof}
4371 For one step,
4372 \[
4373 (M_Vf)_i
4374 =
4375 \e^{-V(i)}\sum_jK_{ji}f_j.
4376 \]
4377 Iterating expands the matrix product over all paths. The product of transition entries is the path probability, and the diagonal factors give the exponential potential weight.
4378 \end{proof}
4379 
4380 \begin{theorem}[Continuous time finite state formula]\label{extended:record:082}\label{extended:thm:continuous_feynman_kac}
4381 Let \(Q\) be a finite Markov generator and let \((X_t)_{t\geq0}\) be the corresponding chain. Then
4382 \[
4383 \left(
4384 \e^{t(Q^T-D_V)}f
4385 \right)_i
4386 =
4387 \Exp_i\left[
4388 \exp\left(
4389 -\int_0^tV(X_s)\dd s
4390 \right)
4391 f(X_t)
4392 \right].
4393 \]
4394 If \(V\geq v_0\), then
4395 \[
4396 \norm{\e^{t(Q^T-D_V)}}_{\infty\to\infty}
4397 \leq
4398 \e^{-v_0t}.
4399 \]
4400 \end{theorem}
4401 
4402 \begin{proof}
4403 Conditioning over the first small time interval gives the backward equation
4404 \[
4405 \dot u=(Q^T-D_V)u,
4406 \qquad
4407 u(0)=f.
4408 \]
4409 The finite system has a unique solution, namely the displayed matrix exponential. The expectation bound follows from
4410 \[
4411 \exp\left(-\int_0^tV(X_s)\dd s\right)
4412 \leq
4413 \e^{-v_0t}.
4414 \]
4415 This finite argument is the matrix form of the Feynman--Kac principle originating in \cite{Kac1949}.
4416 \end{proof}
4417 
4418 \begin{theorem}[Alternating propagation and killing]\label{extended:record:083}\label{extended:thm:feynman_kac_product}
4419 For every \(t\geq0\),
4420 \[
4421 \e^{t(Q^T-D_V)}
4422 =
4423 \lim_{n\to\infty}
4424 \left(
4425 \e^{-tD_V/n}\e^{tQ^T/n}
4426 \right)^n
4427 \]
4428 in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.
4429 \end{theorem}
4430 
4431 \begin{proof}
4432 Set \(B=-D_V\) and \(C=Q^T\). For bounded matrices,
4433 \[
4434 \e^{tB/n}\e^{tC/n}
4435 =
4436 I+\frac{t}{n}(B+C)+O(n^{-2})
4437 \]
4438 uniformly in operator norm. The standard telescoping comparison with
4439 \(
4440 (I+t(B+C)/n)^n
4441 \)
4442 shows that the product converges to \(\e^{t(B+C)}\). This is the displayed factor order, so the limit follows.
4443 \end{proof}
4444 
4445 \begin{corollary}[Positivity and composition]\label{extended:record:084}\label{extended:cor:feynman_kac_semigroup}
4446 The operators
4447 \[
4448 F_V(t)=\e^{t(Q^T-D_V)}
4449 \]
4450 form a positive semigroup. If \(f\geq0\), then \(F_V(t)f\geq0\).
4451 \end{corollary}
4452 
4453 \begin{proof}
4454 Both claims follow from the expectation formula.
4455 \end{proof}
4456 
4457 \FloatBarrier
4458 \section{Schur--Feshbach elimination}\label{extended:section:14}\label{extended:sec:feshbach}
4459 
4460 Let \(X=X_P\oplus X_Q\) and write
4461 \[
4462 A=
4463 \begin{pmatrix}
4464 A_{PP}&A_{PQ}\\
4465 A_{QP}&A_{QQ}
4466 \end{pmatrix}.
4467 \]
4468 For \(zI-A_{QQ}\) invertible, define
4469 \[
4470 \Sigma_P(z)
4471 =
4472 A_{PQ}(zI-A_{QQ})^{-1}A_{QP}
4473 \]
4474 and
4475 \[
4476 F_P(z)
4477 =
4478 zI-A_{PP}-\Sigma_P(z).
4479 \]
4480 
4481 \begin{theorem}[Schur--Feshbach resolvent]\label{extended:record:085}\label{extended:thm:feshbach_resolvent}
4482 Assume \(zI-A_{QQ}\) is invertible. Then \(zI-A\) is invertible if and only if \(F_P(z)\) is invertible. In that case,
4483 \[
4484 P(zI-A)^{-1}P
4485 =
4486 F_P(z)^{-1}.
4487 \]
4488 In finite dimension,
4489 \[
4490 \det(zI-A)
4491 =
4492 \det(zI-A_{QQ})\det F_P(z).
4493 \]
4494 \end{theorem}
4495 
4496 \begin{proof}
4497 Set
4498 \[
4499 L_z=
4500 \begin{pmatrix}
4501 I&-A_{PQ}(zI-A_{QQ})^{-1}\\
4502 0&I
4503 \end{pmatrix}
4504 \]
4505 and
4506 \[
4507 S_z=
4508 \begin{pmatrix}
4509 F_P(z)&0\\
4510 -A_{QP}&zI-A_{QQ}
4511 \end{pmatrix}.
4512 \]
4513 Direct multiplication gives \(zI-A=L_zS_z\). The first factor is invertible. The claims follow from triangular inversion and the determinant of the second factor. This is the finite block form of the Feshbach reduction \cite{Feshbach1958}.
4514 \end{proof}
4515 
4516 \subsection{Time domain memory}
4517 
4518 For the homogeneous equation \(\dot x=Ax\), variation of constants gives
4519 \[
4520 x_Q(t)
4521 =
4522 \e^{tA_{QQ}}x_Q(0)
4523 +
4524 \int_0^t
4525 \e^{(t-s)A_{QQ}}A_{QP}x_P(s)
4526 \dd s.
4527 \]
4528 Substitution into the retained equation gives
4529 \[
4530 \begin{aligned}
4531 \dot x_P(t)
4532 ={}&
4533 A_{PP}x_P(t)
4534 +
4535 A_{PQ}\e^{tA_{QQ}}x_Q(0)\\
4536 &+
4537 \int_0^t
4538 \Kcal(t-s)x_P(s)
4539 \dd s,
4540 \end{aligned}
4541 \]
4542 where
4543 \[
4544 \Kcal(t)
4545 =
4546 A_{PQ}\e^{tA_{QQ}}A_{QP}.
4547 \]
4548 
4549 \begin{theorem}[Memory and self-energy]\label{extended:record:086}\label{extended:thm:memory_self_energy}
4550 If \(z\) lies to the right of the spectrum of \(A_{QQ}\), then
4551 \[
4552 \int_0^\infty
4553 \e^{-zt}\Kcal(t)\dd t
4554 =
4555 \Sigma_P(z).
4556 \]
4557 If, for constants \(M\geq0\) and \(\gamma>0\),
4558 \[
4559 \norm{\e^{tA_{QQ}}}
4560 \leq
4561 M\e^{-\gamma t},
4562 \]
4563 then
4564 \[
4565 \norm{\Kcal(t)}
4566 \leq
4567 M\norm{A_{PQ}}\norm{A_{QP}}\e^{-\gamma t}
4568 \]
4569 and
4570 \[
4571 \norm{\Kcal}_{L^1(0,\infty)}
4572 \leq
4573 \frac{M}{\gamma}
4574 \norm{A_{PQ}}\norm{A_{QP}}.
4575 \]
4576 \end{theorem}
4577 
4578 \begin{proof}
4579 If \(X_Q=\{0\}\), the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write \(B=A_{QQ}\), which is a bounded operator on the hidden Banach space, and put \(s(B)=\max\operatorname{Re}\Spec(B)\). Choose a real \(\beta\) with \(s(B)<\beta<\operatorname{Re}z\). Spectral mapping and the spectral-radius formula give \(r(\e^B)=\e^{s(B)}<\e^\beta\), hence \(\norm{\e^{nB}}\leq M_\beta\e^{\beta n}\) for every integer \(n\geq0\), after increasing \(M_\beta\) to cover the finitely many initial values. Write \(t=n+r\), where \(0\leq r<1\). Then
4580 \[
4581 \norm{\e^{tB}}\leq M_\beta\exp(\norm B+\max(0,-\beta))\e^{\beta t}.
4582 \]
4583 Thus \(\e^{-zt}\e^{tB}\) is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for \(R=\int_0^\infty\e^{-zt}\e^{tB}\dd t\), so \(R=(zI-B)^{-1}\). This is the bounded-generator growth principle in \cite[Chapter IV, Corollary 3.12]{EngelNagel2000}. Multiplication by the coupling blocks gives the first identity. The remaining estimates follow directly from the assumed positive decay rate.
4584 \end{proof}
4585 
4586 \begin{theorem}[Criterion for exact local closure]\label{extended:record:087}\label{extended:thm:local_closure}
4587 Assume \(x_Q(0)=0\) and \(d=\dim X_Q<\infty\). The following conditions are equivalent.
4588 \begin{enumerate}
4589 \item For every retained initial state,
4590 \[
4591 P\e^{tA}P
4592 =
4593 \e^{tA_{PP}}P
4594 \]
4595 for all \(t\geq0\).
4596 \item \(\Sigma_P(z)=0\) for every sufficiently large \(z\).
4597 \item \(\Kcal(t)=0\) for every \(t\geq0\).
4598 \item If \(d=\dim X_Q\), then
4599 \[
4600 A_{PQ}A_{QQ}^kA_{QP}=0
4601 \]
4602 for \(0\leq k<d\).
4603 \end{enumerate}
4604 \end{theorem}
4605 
4606 \begin{proof}
4607 By \cref{extended:thm:laplace_packet_resolvent}, the Laplace transform of the projected propagator is the projected resolvent. By \cref{extended:thm:feshbach_resolvent}, the first condition is equivalent to
4608 \[
4609 (zI-A_{PP}-\Sigma_P(z))^{-1}
4610 =
4611 (zI-A_{PP})^{-1}
4612 \]
4613 for all sufficiently large \(z\), which is equivalent to the second condition. The second and third conditions are equivalent by \cref{extended:thm:memory_self_energy} and uniqueness of the Laplace transform.
4614 
4615 For large \(z\),
4616 \[
4617 \Sigma_P(z)
4618 =
4619 \sum_{k=0}^{\infty}
4620 \frac{A_{PQ}A_{QQ}^kA_{QP}}{z^{k+1}}.
4621 \]
4622 Thus the second condition implies the moment identities. Conversely, the first \(d\) identities imply all later ones by the Cayley--Hamilton theorem, so the series vanishes.
4623 \end{proof}
4624 
4625 \begin{remark}\label{extended:record:088}
4626 A finite hidden block produces a strictly proper rational self-energy. It tends to zero as \(z\) tends to infinity. Hence a nonzero constant self-energy cannot arise from hidden dynamics alone. A constant correction must already be included in the retained generator.
4627 \end{remark}
4628 
4629 \FloatBarrier
4630 \section{Memory modes and controlled compression}\label{extended:section:15}\label{extended:sec:memory_compression}
4631 \begin{interpretation}
4632 Stable hidden modes may carry polynomial factors when a Jordan block is present. A spectral gap controls eventual decay, while nonnormal coupling can produce substantial intermediate memory. The compression certificate below is optimal for its declared primary-block selection problem.
4633 \end{interpretation}
4634 \begin{figure}[htbp]
4635 \centering
4636 \includegraphics[width=.94\textwidth]{monograph_figures/exact_jordan_memory.pdf}
4637 \caption{A two-coordinate Jordan hidden block has an exponential decay factor and a polynomial factor. The nilpotent polynomial is checked exactly at the displayed rational times. The exponential curve is a plotted illustration of the formula.}\label{monograph:fig:exact_jordan_memory}
4638 \end{figure}
4639 
4640 
4641 
4642 Finite hidden systems produce rational self-energies and finite modal memory. This is the state space viewpoint familiar from linear systems theory \cite{Kailath1980}. The same formulas also underlie the Mori--Zwanzig representation of reduced dynamics \cite{Zwanzig1961,ChorinHaldKupferman2000}.
4643 
4644 \begin{theorem}[Polynomial exponential memory of a finite hidden system]\label{extended:record:089}\label{extended:thm:modal_memory}
4645 Let \(X_Q\) be a finite dimensional complex space. Let \(\mu_1,\ldots,\mu_r\) be the distinct eigenvalues of \(A_{QQ}\), let \(Q_j\) be the corresponding primary spectral projections, and set
4646 \[
4647 N_j=(A_{QQ}-\mu_jI)Q_j.
4648 \]
4649 Let \(\nu_j\) be the least positive integer for which \(N_j^{\nu_j}=0\), and define
4650 \[
4651 B_{j,k}
4652 =
4653 A_{PQ}N_j^kQ_jA_{QP}
4654 \qquad
4655 (0\leq k<\nu_j).
4656 \]
4657 Then
4658 \[
4659 \Kcal(t)
4660 =
4661 \sum_{j=1}^r
4662 \e^{\mu_jt}
4663 \sum_{k=0}^{\nu_j-1}
4664 \frac{t^k}{k!}B_{j,k}
4665 \]
4666 and
4667 \[
4668 \Sigma_P(z)
4669 =
4670 \sum_{j=1}^r
4671 \sum_{k=0}^{\nu_j-1}
4672 \frac{B_{j,k}}{(z-\mu_j)^{k+1}}.
4673 \]
4674 Suppose \(\operatorname{Re}\mu_j<0\) for every \(j\). If \(J\subseteq\{1,\ldots,r\}\) and \(\Kcal_J\) retains the complete primary blocks indexed by \(J\), then
4675 \[
4676 \norm{\Kcal-\Kcal_J}_{L^1(0,\infty)}
4677 \leq
4678 \sum_{j\notin J}
4679 \sum_{k=0}^{\nu_j-1}
4680 \frac{\norm{B_{j,k}}}
4681 {(-\operatorname{Re}\mu_j)^{k+1}}.
4682 \]
4683 \end{theorem}
4684 
4685 \begin{proof}
4686 The primary decomposition gives
4687 \[
4688 A_{QQ}Q_j=\mu_jQ_j+N_j
4689 \]
4690 with commuting scalar and nilpotent parts on \(Q_jX_Q\). Therefore
4691 \[
4692 \e^{tA_{QQ}}Q_j
4693 =
4694 \e^{\mu_jt}
4695 \sum_{k=0}^{\nu_j-1}
4696 \frac{t^k}{k!}N_j^kQ_j.
4697 \]
4698 Multiplication by \(A_{PQ}\) and \(A_{QP}\) gives the memory formula. On the \(j\)th primary space, the nilpotent resolvent expansion is
4699 \[
4700 (zI-A_{QQ})^{-1}Q_j
4701 =
4702 \sum_{k=0}^{\nu_j-1}
4703 \frac{N_j^kQ_j}{(z-\mu_j)^{k+1}}.
4704 \]
4705 Multiplication by the coupling blocks gives the self-energy formula for every \(z\) outside the hidden spectrum. Equivalently, on a right half plane it is the Laplace transform identity
4706 \[
4707 \int_0^\infty
4708 \e^{-zt}\e^{\mu_jt}
4709 \frac{t^k}{k!}
4710 \dd t
4711 =
4712 \frac{1}{(z-\mu_j)^{k+1}}.
4713 \]
4714 Under stability,
4715 \[
4716 \int_0^\infty
4717 \e^{\operatorname{Re}\mu_jt}
4718 \frac{t^k}{k!}
4719 \dd t
4720 =
4721 \frac{1}{(-\operatorname{Re}\mu_j)^{k+1}}.
4722 \]
4723 Sum the omitted terms.
4724 \end{proof}
4725 
4726 \begin{corollary}[Optimal selection for the certified primary block bound]\label{extended:record:090}\label{extended:cor:optimal_primary_selection}
4727 Under the stable assumptions of \cref{extended:thm:modal_memory}, define the primary block weights
4728 \[
4729 w_j
4730 =
4731 \sum_{k=0}^{\nu_j-1}
4732 \frac{\norm{B_{j,k}}}
4733 {(-\operatorname{Re}\mu_j)^{k+1}}.
4734 \]
4735 Among all sets \(J\) containing \(k\) primary blocks, the certified upper bound
4736 \[
4737 \sum_{j\notin J}w_j
4738 \]
4739 is minimized by retaining the \(k\) largest weights. If
4740 \[
4741 w_{(1)}\geq w_{(2)}\geq\cdots\geq w_{(r)},
4742 \]
4743 then the least number of retained primary blocks certified to give \(L^1\) error at most \(\varepsilon\) is the least \(k\) for which
4744 \[
4745 \sum_{j>k}w_{(j)}\leq\varepsilon.
4746 \]
4747 \end{corollary}
4748 
4749 \begin{proof}
4750 If a retained set contains a block with smaller weight than an omitted block, exchanging them reduces the omitted weight sum. Repeating the exchange gives the set of the \(k\) largest weights. The tolerance statement is the corresponding sorted tail condition.
4751 \end{proof}
4752 
4753 \begin{corollary}[Normal hidden dynamics]\label{extended:record:091}\label{extended:cor:normal_modal_memory}
4754 Suppose \(X_Q\) is a finite dimensional Hilbert space. If \(A_{QQ}\) is normal with spectral resolution
4755 \[
4756 A_{QQ}=\sum_{j=1}^r\mu_jQ_j,
4757 \]
4758 then
4759 \[
4760 \Kcal(t)=\sum_{j=1}^r\e^{\mu_jt}B_j,
4761 \qquad
4762 \Sigma_P(z)=\sum_{j=1}^r\frac{B_j}{z-\mu_j},
4763 \]
4764 where
4765 \[
4766 B_j=A_{PQ}Q_jA_{QP}.
4767 \]
4768 If \(\operatorname{Re}\mu_j<0\), then
4769 \[
4770 \norm{\Kcal-\Kcal_J}_{L^1(0,\infty)}
4771 \leq
4772 \sum_{j\notin J}
4773 \frac{\norm{B_j}}{-\operatorname{Re}\mu_j}.
4774 \]
4775 \end{corollary}
4776 
4777 \begin{proof}
4778 Normality removes every nilpotent part, so \(\nu_j=1\). Apply \cref{extended:thm:modal_memory}.
4779 \end{proof}
4780 
4781 \begin{corollary}[Pole order and hidden Jordan depth]\label{extended:record:092}\label{extended:cor:pole_jordan_depth}
4782 Every finite hidden block produces a strictly proper rational self-energy. At an eigenvalue \(\mu_j\), its pole order is at most the size of the largest Jordan block of \(A_{QQ}\) at \(\mu_j\). Equality holds when the corresponding highest coefficient \(B_{j,\nu_j-1}\) is nonzero.
4783 \end{corollary}
4784 
4785 \begin{proof}
4786 Read the pole order from the partial fraction expansion in \cref{extended:thm:modal_memory}.
4787 \end{proof}
4788 
4789 \begin{figure}[t]
4790 \centering
4791 \includegraphics[width=.72\textwidth]{jordan_memory.pdf}
4792 \caption{A stable Jordan block produces a polynomial exponential memory mode. A diagonal hidden block produces only pure exponentials.}
4793 \label{extended:fig:jordan_memory}
4794 \end{figure}
4795 
4796 \begin{corollary}[Finite auxiliary state realization]\label{extended:record:093}\label{extended:cor:finite_auxiliary_realization}
4797 Every kernel of the form
4798 \[
4799 \Kcal(t)=B\e^{tD}C
4800 \]
4801 can be represented exactly by adjoining the finite auxiliary state
4802 \[
4803 \dot y=Dy+Cx_P,
4804 \qquad
4805 \dot x_P=A_{PP}x_P+By,
4806 \qquad
4807 y(0)=0.
4808 \]
4809 A nonzero auxiliary initial state adds the explicit forcing \(B\e^{tD}y(0)\). The self-energy of the augmented system is
4810 \[
4811 B(zI-D)^{-1}C.
4812 \]
4813 \end{corollary}
4814 
4815 \begin{proof}
4816 Eliminate \(y\) by variation of constants or apply \cref{extended:thm:feshbach_resolvent} to the augmented block matrix.
4817 \end{proof}
4818 
4819 \begin{theorem}[Volterra stability under kernel compression]\label{extended:record:094}\label{extended:thm:volterra_compression}
4820 Let \(S(t)\) be a strongly continuous semigroup on a Banach space \(X\), fix \(T>0\), and assume
4821 \[
4822 \norm{S(t)}\leq M_T
4823 \quad(0\leq t\leq T).
4824 \]
4825 Let \(K,\widetilde K\in L^1((0,T),\mathcal B(X))\), let \(g\in L^1((0,T),X)\), and let \(x,\widetilde x\in C([0,T],X)\). Suppose these are mild solutions with the same initial state and forcing, except that \(K\) is replaced by \(\widetilde K\):
4826 \[
4827 \begin{aligned}
4828 x(t)
4829 ={}&S(t)x_0+
4830 \int_0^tS(t-s)g(s)\dd s\\
4831 &+
4832 \int_0^tS(t-s)
4833 \int_0^sK(s-r)x(r)\dd r\dd s.
4834 \end{aligned}
4835 \]
4836 Assume
4837 \[
4838 \sup_{0\leq t\leq T}\norm{\widetilde x(t)}\leq B.
4839 \]
4840 Set
4841 \[
4842 L=\norm K_{L^1(0,T)},
4843 \qquad
4844 \varepsilon=\norm{K-\widetilde K}_{L^1(0,T)}.
4845 \]
4846 Then
4847 \[
4848 \sup_{0\leq t\leq T}
4849 \norm{x(t)-\widetilde x(t)}
4850 \leq
4851 M_TBT\varepsilon
4852 \exp(M_TLT).
4853 \]
4854 \end{theorem}
4855 
4856 \begin{proof}
4857 Let
4858 \[
4859 E(t)=\sup_{0\leq r\leq t}\norm{x(r)-\widetilde x(r)}.
4860 \]
4861 Subtract the two mild equations. Fubini's theorem and the triangle inequality give
4862 \[
4863 E(t)
4864 \leq
4865 M_TL\int_0^tE(r)\dd r
4866 +
4867 M_TBt\varepsilon.
4868 \]
4869 For \(t\leq T\), replace the last factor \(t\) by \(T\) and apply Gronwall's inequality.
4870 \end{proof}
4871 
4872 \begin{corollary}[Trajectory bound for modal truncation]\label{extended:record:095}\label{extended:cor:modal_trajectory_bound}
4873 Under the assumptions of \cref{extended:thm:modal_memory,extended:thm:volterra_compression}, replacing \(\Kcal\) by \(\Kcal_J\) changes the retained trajectory by at most
4874 \[
4875 M_TBT
4876 \exp(M_TLT)
4877 \sum_{j\notin J}
4878 \sum_{k=0}^{\nu_j-1}
4879 \frac{\norm{B_{j,k}}}
4880 {(-\operatorname{Re}\mu_j)^{k+1}}.
4881 \]
4882 \end{corollary}
4883 
4884 \begin{proof}
4885 Insert the \(L^1\) error from \cref{extended:thm:modal_memory} into \cref{extended:thm:volterra_compression}.
4886 \end{proof}
4887 
4888 \begin{figure}[t]
4889 \centering
4890 \includegraphics[width=.72\textwidth]{memory_compression.pdf}
4891 \caption{Full and truncated memory kernels with their pointwise difference. The bound uses integrated error.}
4892 \label{extended:fig:memory_compression}
4893 \end{figure}
4894 
4895 \begin{remark}\label{extended:record:096}
4896 A finite hidden realization is exact for rational self-energy. A general memory kernel may require infinitely many modes or a continuum of modes. The compression theorem gives a stability estimate once an \(L^1\) approximation has been supplied. The largest weight rule is optimal for the certified blockwise bound. It need not minimize the true kernel error when different modal terms cancel.
4897 \end{remark}
4898 
4899 \FloatBarrier
4900 \section{Survival, moments, and competing risks}\label{extended:section:16}\label{extended:sec:survival}
4901 
4902 Let \(A\) be a transient state subgenerator. Its off-diagonal entries are nonnegative and its column sums are nonpositive. Let \(p_0\) be a transient probability vector. Define
4903 \[
4904 p(t)=\e^{tA}p_0,
4905 \qquad
4906 S(t)=\one^Tp(t),
4907 \qquad
4908 a^T=-\one^TA.
4909 \]
4910 The row vector \(a^T\) contains the exit rates.
4911 
4912 \begin{theorem}[State dependent killing as survival]\label{extended:record:097}\label{extended:thm:killing_survival}
4913 Let \(Q\) be a finite Markov generator, let \(V\geq0\), and set
4914 \[
4915 A=Q-D_V.
4916 \]
4917 For an initial distribution \(p_0\), define
4918 \[
4919 S_V(t)=\one^T\e^{tA}p_0.
4920 \]
4921 Then
4922 \[
4923 S_V(t)
4924 =
4925 \Exp_{p_0}
4926 \left[
4927 \exp\left(
4928 -\int_0^tV(X_s)\dd s
4929 \right)
4930 \right].
4931 \]
4932 Also,
4933 \[
4934 -S_V'(t)
4935 =
4936 \one^TD_V\e^{tA}p_0.
4937 \]
4938 The matrix \(A=Q-D_V\) has nonnegative off-diagonal entries and nonpositive column sums, so it is the subgenerator of the killed, possibly nontransient chain. Thus the potential in the finite Feynman--Kac formula is a state dependent killing rate in the forward sub-Markov semigroup. Almost-sure absorption follows under the additional spectral stability hypothesis used below.
4939 \end{theorem}
4940 
4941 \begin{proof}
4942 Average \cref{extended:thm:continuous_feynman_kac} with terminal function \(f=\one\) against the initial distribution. Transposition gives
4943 \[
4944 p_0^T\e^{t(Q^T-D_V)}\one
4945 =
4946 \one^T\e^{t(Q-D_V)}p_0.
4947 \]
4948 Since \(\one^TQ=0\), differentiation gives
4949 \[
4950 S_V'(t)
4951 =
4952 -\one^TD_V\e^{tA}p_0.
4953 \]
4954 \end{proof}
4955 
4956 \begin{theorem}[Survival and hazard identities]\label{extended:record:098}\label{extended:thm:survival_hazard}
4957 The event density and hazard are
4958 \[
4959 f(t)=a^T\e^{tA}p_0=-S'(t),
4960 \qquad
4961 h(t)=\frac{f(t)}{S(t)}
4962 \]
4963 whenever \(S(t)>0\). The survival function is nonincreasing and
4964 \[
4965 S(t)
4966 =
4967 \exp\left(
4968 -\int_0^th(s)\dd s
4969 \right).
4970 \]
4971 \end{theorem}
4972 
4973 \begin{proof}
4974 Because \(A\) is a Metzler matrix, \(\e^{tA}\) is nonnegative. The column sum assumption makes \(a^T\) nonnegative. Differentiate \(S(t)\):
4975 \[
4976 S'(t)=\one^TA\e^{tA}p_0=-a^T\e^{tA}p_0.
4977 \]
4978 The remaining identity follows by integrating \((\log S)'=-h\).
4979 \end{proof}
4980 
4981 \begin{theorem}[Laplace transform and moments]\label{extended:record:099}\label{extended:thm:survival_moments}
4982 Assume the spectrum of \(A\) lies in the open left half plane. Then absorption occurs with probability one. For \(s\geq0\),
4983 \[
4984 \Exp[\e^{-s\tau}]
4985 =
4986 a^T(sI-A)^{-1}p_0.
4987 \] For every integer \(m\geq1\),
4988 \[
4989 \Exp[\tau^m]
4990 =
4991 m!\one^T(-A)^{-m}p_0.
4992 \]
4993 \end{theorem}
4994 
4995 \begin{proof}
4996 The Laplace transform follows by integrating the density in \cref{extended:thm:survival_hazard}. Stability of \(A\) gives
4997 \[
4998 \int_0^\infty
4999 \e^{-st}\e^{tA}\dd t
5000 =
5001 (sI-A)^{-1}.
5002 \]
5003 At \(s=0\),
5004 \[
5005 a^T(-A)^{-1}p_0
5006 =
5007 \one^Tp_0=1.
5008 \]
5009 For the moments,
5010 \[
5011 \int_0^\infty t^m\e^{tA}\dd t
5012 =
5013 m!(-A)^{-m-1}.
5014 \]
5015 Multiplying by \(a^T=\one^T(-A)\) gives the result. These are the standard matrix formulas for phase type laws \cite{Neuts1981,AsmussenNermanOlsson1996}.
5016 \end{proof}
5017 
5018 \begin{proposition}[Residual life as a restarted semigroup]\label{extended:record:100}\label{extended:prop:residual_life}
5019 For \(S(t)>0\), define the conditional transient distribution
5020 \[
5021 p_t
5022 =
5023 \frac{\e^{tA}p_0}{S(t)}.
5024 \]
5025 Then
5026 \[
5027 \frac{S(t+s)}{S(t)}
5028 =
5029 \one^T\e^{sA}p_t.
5030 \]
5031 \end{proposition}
5032 
5033 \begin{proof}
5034 Use the semigroup law in the numerator \(S(t+s)=\one^T\e^{sA}\e^{tA}p_0\).
5035 \end{proof}
5036 
5037 \begin{theorem}[Competing risk packets]\label{extended:record:101}\label{extended:thm:competing_risks}
5038 Suppose the exit row splits into nonnegative cause rows
5039 \[
5040 a^T
5041 =
5042 \sum_{r=1}^q a_r^T.
5043 \]
5044 Define
5045 \[
5046 f_r(t)=a_r^T\e^{tA}p_0.
5047 \]
5048 Then
5049 \[
5050 f(t)=\sum_{r=1}^qf_r(t).
5051 \]
5052 The cause specific hazard is
5053 \[
5054 h_r(t)=\frac{f_r(t)}{S(t)},
5055 \]
5056 and
5057 \[
5058 h(t)=\sum_{r=1}^qh_r(t).
5059 \]
5060 If the spectrum of \(A\) lies in the open left half-plane, the eventual probability of cause \(r\) is
5061 \[
5062 \int_0^\infty f_r(t)\dd t
5063 =
5064 a_r^T(-A)^{-1}p_0.
5065 \]
5066 \end{theorem}
5067 
5068 \begin{proof}
5069 Every identity follows by linearity and the stable resolvent integral.
5070 \end{proof}
5071 
5072 \begin{figure}[t]
5073 \centering
5074 \includegraphics[width=.72\textwidth]{survival_causes.pdf}
5075 \caption{Survival and two cause specific event densities from one transient semigroup.}
5076 \label{extended:fig:survival_causes}
5077 \end{figure}
5078 
5079 \begin{remark}\label{extended:record:102}
5080 A finite phase type hazard need not be monotone. The semigroup formulas guarantee nonnegative density and decreasing survival, not monotonicity of the hazard.
5081 \end{remark}
5082 
5083 \FloatBarrier
5084 \section{Finite illustrations and computational checks}\label{extended:section:17}\label{extended:sec:finite_checks}
5085 
5086 The figures in this part use small matrices chosen to make each mechanism visible. They are not evidence for claims outside the proved finite setting. The accompanying verification script recomputes the following identities with independently generated data.
5087 
5088 \begin{enumerate}
5089 \item Packet block convolution, the block differential system, and the Laplace resolvent formula.
5090 \item The Duhamel and resolvent defect identities.
5091 \item Exact and perturbed Markov lumping.
5092 \item Conditional expectation, tower laws, martingale packet energy, and Hermite multipliers.
5093 \item Bayesian reconstruction, sufficiency, local distortion, and information decomposition.
5094 \item The evidence variance identity along a Bayesian flow.
5095 \item The Balakrishnan formula and the fractional defect estimate.
5096 \item Nodal Green reconstruction and the Poisson error bound.
5097 \item Discrete Feynman--Kac path enumeration.
5098 \item The Schur--Feshbach resolvent and memory transform.
5099 \item Jordan memory modes, modal truncation, and the Volterra stability estimate.
5100 \item Feynman--Kac killing, survival transforms, moments, and competing risk masses.
5101 \end{enumerate}
5102 
5103 \begin{table}[t]
5104 \centering
5105 \begin{tabular}{lc}
5106 \toprule
5107 Identity & Typical residual \\
5108 \midrule
5109 Packet block convolution & below \(3\times10^{-14}\) \\
5110 Duhamel defect & below \(3\times10^{-12}\) \\
5111 Markov lumping & below \(3\times10^{-14}\) \\
5112 Bayesian reconstruction & below \(3\times10^{-15}\) \\
5113 Fractional spectral law & below \(3\times10^{-12}\) \\
5114 Nodal Green identity & below \(3\times10^{-13}\) \\
5115 Feynman--Kac enumeration & below \(3\times10^{-14}\) \\
5116 Schur--Feshbach resolvent & below \(3\times10^{-12}\) \\
5117 Memory Laplace identity & below \(3\times10^{-11}\) \\
5118 Survival moment formulas & below \(3\times10^{-11}\) \\
5119 \bottomrule
5120 \end{tabular}
5121 \caption{Numerical tolerances in the finite verification suite. The proofs do not depend on these computations.}
5122 \label{extended:tab:verification_tolerances}
5123 \end{table}
5124 
5125 \FloatBarrier
5126 \section{Limits and further questions}\label{extended:section:18}\label{extended:sec:limits}
5127 
5128 The bounded inverse limit theorem isolates one clean route to continuum evolution. Unbounded generators require graph norms, domain compatibility, and a generation theorem. The range leakage theorem converts the closure quantity into an off-diagonal generator estimate for bounded orthogonal refinement models. Concrete discretizations still require a uniform estimate for that off-diagonal block.
5129 
5130 The finite Bayesian sufficiency theorem is exact, but it concerns one likelihood and one fixed partition. The relative entropy chain rule measures both retained and discarded information for that update. Statistical sufficiency for a parametric family requires a common packet structure that preserves the full parameter dependence. In continuous models the corresponding object is a sufficient sigma algebra or statistic.
5131 
5132 The fractional defect theorem uses a common spectral gap. Removing that gap requires weaker norms or a Hölder estimate near the null space. For sectorial operators, the same calculation should be carried out through resolvent bounds on a sector, with domains tracked explicitly. The finite spectral statement does not settle those issues.
5133 
5134 The Green and Poisson results exploit a one-dimensional second difference matrix. Higher dimensions, irregular meshes, and variable coefficients require a separate stability theory. The packet language remains useful only after coercivity, consistency, and boundary approximation have been established.
5135 
5136 Memory compression is exact once a finite hidden realization is known and quantitative once an integrable kernel approximation is supplied. The primary decomposition shows that a finite hidden block always produces polynomial exponential modes, including the effects of Jordan chains. Choosing a short approximation remains a different problem. Strongly nonnormal systems can exhibit transient growth and may require balanced or rational approximation methods. General Volterra kernels need not admit a short finite realization.
5137 
5138 Finally, Markov lumping, Bayesian sufficiency, and memory closure answer different questions. Lumpability asks whether coarse probabilities evolve autonomously. Bayesian sufficiency asks whether a fixed coarse likelihood reproduces the posterior packet masses without hiding a change in the conditional distribution inside a packet. Memory closure asks whether discarded dynamics can be represented without history. Similar notation should not obscure these distinctions.
5139 
5140 \FloatBarrier
5141 \section{Conclusion}\label{extended:section:19}\label{extended:sec:conclusion}
5142 
5143 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.
5144 
5145 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.
5146 
5147 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.
5148 
5149 Exact elimination returns hidden packets 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.
5150 
5151 This part therefore separates four operations that are often compressed into one informal idea of coarse dynamics:
5152 \[
5153 \begin{aligned}
5154 \text{intertwining}
5155 &\longrightarrow
5156 \text{refinement control}\\
5157 &\longrightarrow
5158 \text{statistical sufficiency}\\
5159 &\longrightarrow
5160 \text{memory closure}.
5161 \end{aligned}
5162 \]
5163 Each arrow has its own hypotheses and its own obstruction.
5164 
5165 
5166 
5167 
5168 
5169 \clearpage
5170 \part*{Part III\quad Retained results and operator calculus}\addcontentsline{toc}{part}{Part III: Retained results and operator calculus}
5171 \setcounter{section}{0}\setcounter{figure}{0}\setcounter{table}{0}
5172 \renewcommand{\thesection}{R\arabic{section}}
5173 \renewcommand{\theHsection}{R.\arabic{section}}
5174 \renewcommand{\thefigure}{R\arabic{figure}}
5175 \renewcommand{\theHfigure}{R.\arabic{figure}}
5176 \renewcommand{\thetable}{R\arabic{table}}
5177 \renewcommand{\theHtable}{R.\arabic{table}}
5178 \FloatBarrier
5179 \section{Transfer paths and product-topology limits}\label{earlier:section:01}
5180 Let $T(t)$ be an arbitrary strongly continuous semigroup on a complex Banach space $X$, with generator $A$, and suppose $\norm{T(t)}\leq M e^{\omega t}$ for $M\geq1$ and $\omega\in\mathbb R$. Let $(P_\alpha)_{\alpha\in I_0}$ be a finite family of bounded pairwise orthogonal idempotents summing to $I$, and put $T_{\beta\alpha}(t)=P_\beta T(t)P_\alpha$. The generator need not be bounded. In the following resolvent statement the integral is strong: apply the integrand to each $x\in X$ and take its Bochner integral in $X$. It is not asserted to be a Bochner integral in the operator-norm space $\mathcal B(X)$. The general Laplace formula is \cite[Chapter II, Theorem 1.10]{EngelNagel2000}.
5181 \begin{theorem}[Packet convolution]\label{earlier:record:001}\label{earlier:thm:packet_convolution}
5182 For all \(s,t\geq0\) and all packets \(\alpha,\beta\),
5183 \[
5184 \boxed{
5185 T_{\beta\alpha}(s+t)
5186 =
5187 \sum_{\gamma\in I_0}
5188 T_{\beta\gamma}(s)T_{\gamma\alpha}(t).
5189 }
5190 \]
5191 The original semigroup is reconstructed from its blocks by
5192 \[
5193 T(t)=\sum_{\alpha,\beta\in I_0}T_{\beta\alpha}(t).
5194 \]
5195 \end{theorem}
5196 
5197 \begin{proof}
5198 Insert the identity between the two semigroup factors:
5199 \[
5200 \begin{aligned}
5201 P_\beta T(s+t)P_\alpha
5202 &=P_\beta T(s)\left(\sum_\gamma P_\gamma\right)T(t)P_\alpha\\
5203 &=\sum_\gamma
5204 P_\beta T(s)P_\gamma T(t)P_\alpha.
5205 \end{aligned}
5206 \]
5207 The reconstruction formula follows by inserting the packet identity on both sides of \(T(t)\).
5208 \end{proof}
5209 
5210 \begin{proposition}[Packet resolvent convolution]\label{earlier:record:002}\label{earlier:prop:packet_resolvent}
5211 Let \(z\) belong to the resolvent set of \(A\). If \(\operatorname{Re}z\) is larger than a growth bound for the semigroup, then
5212 \[
5213 P_\beta(zI-A)^{-1}P_\alpha
5214 =
5215 \int_0^{\infty}\e^{-zt}T_{\beta\alpha}(t)\dd t.
5216 \]
5217 \end{proposition}
5218 
5219 \begin{proof}
5220 The standard Laplace representation of the resolvent is
5221 \[
5222 (zI-A)^{-1}
5223 =
5224 \int_0^{\infty}\e^{-zt}T(t)\dd t.
5225 \]
5226 Multiplication by the bounded packet projections may be passed through the Bochner integral.
5227 \end{proof}
5228 
5229 
5230 For the path-order statement, now assume $A$ is bounded as in Part II. A directed edge from $\alpha$ to $\beta$ means that $P_\beta A P_\alpha$ is nonzero, with $\alpha\ne\beta$. A packet always has the length-zero path to itself.
5231 \begin{corollary}[Minimum transfer order]\label{earlier:record:003}\label{earlier:cor:minimum_transfer_order}
5232 Suppose every directed packet path from \(\alpha\) to \(\beta\) uses at least \(m_0\) off-diagonal transfers. Then
5233 \[
5234 P_\beta\e^{tA}P_\alpha
5235 =O(t^{m_0})
5236 \quad\text{as }t\downarrow0.
5237 \]
5238 If there is no directed path from \(\alpha\) to \(\beta\), the block vanishes for all \(t\).
5239 \end{corollary}
5240 
5241 \begin{proof}
5242 Every nonzero term in \cref{extended:thm:dyson_paths} contains at least \(m_0\) time integrations. The simplex \(0<r_1<\cdots<r_m<t\) has volume \(t^m/m!\), so the first possible order is \(t^{m_0}\). If there is no path, every packet term vanishes.
5243 \end{proof}
5244 
5245 
5246 Let $C_j:X_{j+1}\to X_j$ be bounded linear maps, and put $C_{\ell,m}=C_\ell\cdots C_{m-1}$ for $\ell<m$, with $C_{j,j}=I$. No common ambient space is assumed.
5247 \begin{theorem}[Multilevel telescoping identity]\label{earlier:record:004}\label{earlier:thm:multilevel_telescoping}
5248 Let \(T_j\) be bounded operators on \(X_j\), and set
5249 \[
5250 E_j=C_jT_{j+1}-T_jC_j.
5251 \]
5252 Then
5253 \[
5254 \boxed{
5255 C_{\ell,m}T_m-T_\ell C_{\ell,m}
5256 =
5257 \sum_{j=\ell}^{m-1}
5258 C_{\ell,j}E_jC_{j+1,m}.
5259 }
5260 \]
5261 \end{theorem}
5262 
5263 \begin{proof}
5264 Expand each summand as
5265 \[
5266 C_{\ell,j+1}T_{j+1}C_{j+1,m}
5267 -
5268 C_{\ell,j}T_jC_{j,m}.
5269 \]
5270 The intermediate terms cancel. The first surviving term is \(C_{\ell,m}T_m\), and the last is \(-T_\ell C_{\ell,m}\).
5271 \end{proof}
5272 
5273 \begin{corollary}[Summable generator defects]\label{earlier:record:005}\label{earlier:cor:summable_defects}
5274 Suppose \(C_j\) are contractions and \((T_j(t))\) are contraction semigroups with bounded generators \(A_j\). Set
5275 \[
5276 \delta_j=\norm{C_jA_{j+1}-A_jC_j}.
5277 \]
5278 Then
5279 \[
5280 \norm{C_{\ell,m}T_m(t)-T_\ell(t)C_{\ell,m}}
5281 \leq
5282  t\sum_{j=\ell}^{m-1}\delta_j.
5283 \]
5284 If \(\sum_{j\geq0}\delta_j<\infty\), the right side tends to zero as \(\ell\to\infty\), uniformly in \(m>\ell\) and uniformly for \(t\) in a fixed bounded interval.
5285 \end{corollary}
5286 
5287 \begin{proof}
5288 By \cref{extended:thm:duhamel_defect}, the interlevel semigroup defect has norm at most \(t\delta_j\). Apply \cref{earlier:thm:multilevel_telescoping} and use the contraction bounds on all coarsening maps.
5289 \end{proof}
5290 
5291 \begin{definition}[Projective limit space]\label{earlier:record:006}\label{earlier:def:projective_limit}
5292 Let \((X_j,C_j)\) be a sequence of Banach spaces and bounded linear maps \(C_j:X_{j+1}\to X_j\). Its projective limit is
5293 \[
5294 X_\infty
5295 =
5296 \left\{
5297 (x_j)_{j\geq0}\in\prod_{j\geq0}X_j:
5298 C_jx_{j+1}=x_j\text{ for every }j
5299 \right\}.
5300 \]
5301 We equip \(X_\infty\) with the subspace topology inherited from the product topology.
5302 \end{definition}
5303 
5304 \begin{theorem}[Exact projective limit semigroup]\label{earlier:record:007}\label{earlier:thm:projective_limit_semigroup}
5305 Suppose each \(X_j\) carries a strongly continuous semigroup \((T_j(t))_{t\geq0}\), and
5306 \[
5307 C_jT_{j+1}(t)=T_j(t)C_j
5308 \]
5309 for every \(j\) and every \(t\geq0\). Then
5310 \[
5311 T_\infty(t)(x_j)_{j\geq0}
5312 =
5313 (T_j(t)x_j)_{j\geq0}
5314 \]
5315 defines a strongly continuous semigroup on \(X_\infty\).
5316 \end{theorem}
5317 
5318 \begin{proof}
5319 Exact intertwining gives
5320 \[
5321 C_jT_{j+1}(t)x_{j+1}
5322 =
5323 T_j(t)C_jx_{j+1}
5324 =
5325 T_j(t)x_j.
5326 \]
5327 Thus the coordinatewise evolution remains in the projective limit. The semigroup law holds in every coordinate. Strong continuity in the product topology follows because every coordinate map
5328 \[
5329 t\longmapsto T_j(t)x_j
5330 \]
5331 is continuous.
5332 \end{proof}
5333 
5334 \begin{corollary}[Coordinate generators]\label{earlier:record:008}\label{earlier:cor:projective_generators}
5335 Assume the setting of \cref{earlier:thm:projective_limit_semigroup}. If \(x=(x_j)\) satisfies \(x_j\in\operatorname{Dom}(A_j)\) and the sequence \((A_jx_j)\) belongs to \(X_\infty\), then \(x\) lies in the generator domain of \(T_\infty\) and
5336 \[
5337 A_\infty x=(A_jx_j)_{j\geq0}.
5338 \]
5339 \end{corollary}
5340 
5341 \begin{proof}
5342 The difference quotients converge in every coordinate to \(A_jx_j\). The assumed compatibility places the coordinatewise limit in \(X_\infty\), which is exactly convergence in the product topology.
5343 \end{proof}
5344 
5345 
5346 The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement.
5347 \FloatBarrier
5348 \section{Reversible coarse dynamics}\label{earlier:section:02}
5349 Let $Q$ be a finite Markov generator acting on column probability vectors, let $(B_a)$ be a partition into nonempty blocks, and let $C_{ai}=\one_{i\in B_a}$. Strong lumpability means $CQ=\overline Q C$ for a coarse Markov generator. For observables, the backward generator is $Q^T$. Weighted $L^2$ inner products use the stated stationary probability.
5350 \begin{theorem}[Reversibility descends]\label{earlier:record:009}\label{earlier:thm:reversibility_descends}
5351 Assume \(Q\) is strongly lumpable and reversible with respect to a positive stationary distribution \(\pi\), so
5352 \[
5353 Q_{ij}\pi_j=Q_{ji}\pi_i.
5354 \]
5355 Define
5356 \[
5357 \overline\pi_a=\sum_{i\in B_a}\pi_i.
5358 \]
5359 Then \(\overline\pi\) is stationary for \(\overline Q\), and
5360 \[
5361 \overline Q_{ab}\overline\pi_b
5362 =
5363 \overline Q_{ba}\overline\pi_a.
5364 \]
5365 Thus the coarse chain is reversible.
5366 \end{theorem}
5367 
5368 \begin{proof}
5369 Stationarity follows from
5370 \[
5371 \overline Q\overline\pi
5372 =\overline QC\pi
5373 =CQ\pi=0.
5374 \]
5375 For reversibility, use the aggregate rate formula:
5376 \[
5377 \begin{aligned}
5378 \overline Q_{ab}\overline\pi_b
5379 &=\sum_{j\in B_b}\pi_j\sum_{i\in B_a}Q_{ij}\\
5380 &=\sum_{i\in B_a}\sum_{j\in B_b}\pi_iQ_{ji}\\
5381 &=\overline Q_{ba}\overline\pi_a.
5382 \end{aligned}
5383 \]
5384 \end{proof}
5385 
5386 \begin{corollary}[Coarse Dirichlet form]\label{earlier:record:010}\label{earlier:cor:coarse_dirichlet_form}
5387 Under the hypotheses of \cref{earlier:thm:reversibility_descends}, let \(Jg\) be the fine function that is constant with value \(g_a\) on \(B_a\). Then
5388 \[
5389 -\ip{Jg}{Q^TJg}_{L^2(\pi)}
5390 =
5391 -\ip{g}{\overline Q^Tg}_{L^2(\overline\pi)}.
5392 \]
5393 \end{corollary}
5394 
5395 \begin{proof}
5396 Expand the left side and group the fine indices by their coarse blocks. The aggregate rate criterion converts the resulting sums into the coarse generator entries.
5397 \end{proof}
5398 
5399 
5400 \FloatBarrier
5401 \section{Prior-restricted sufficiency and filtering}\label{earlier:section:03}
5402 Let $L$ be a strictly positive finite likelihood and let $\Bcal_L(p)=L\odot p/\ip{L}{p}$ for a probability vector $p$. Fix probability vectors $r^{(a)}$ supported on $B_a$, and restrict the priors to $p_i=q_a r_i^{(a)}$ for $i\in B_a$, with $q$ a coarse probability vector.
5403 \begin{proposition}[Sufficiency relative to a prior disintegration]\label{earlier:record:011}\label{earlier:prop:relative_sufficiency}
5404 Define
5405 \[
5406 \overline L_a
5407 =
5408 \sum_{i\in B_a}L_ir_i^{(a)}.
5409 \]
5410 For every prior in the declared family,
5411 \[
5412 C\Bcal_L(p)
5413 =
5414 \Bcal_{\overline L}(q).
5415 \]
5416 \end{proposition}
5417 
5418 \begin{proof}
5419 The evidence in block \(B_a\) is
5420 \[
5421 Z_a
5422 =q_a\sum_{i\in B_a}L_ir_i^{(a)}
5423 =q_a\overline L_a.
5424 \]
5425 Apply \cref{extended:thm:bayesian_packet_reconstruction}.
5426 \end{proof}
5427 
5428 \begin{lemma}[Normalization bound]\label{earlier:record:012}\label{earlier:lem:normalization_bound}
5429 Let \(a,b\in[0,\infty)^m\) be nonzero, and set \(A=\norm a_1\) and \(B=\norm b_1\). Then
5430 \[
5431 \boxed{
5432 \norm{a/A-b/B}_1
5433 \leq
5434 \frac{2\norm{a-b}_1}{\max\{A,B\}}.
5435 }
5436 \]
5437 \end{lemma}
5438 
5439 \begin{proof}
5440 Using \(A\) in the denominator gives
5441 \[
5442 \norm{a/A-b/B}_1
5443 \leq
5444 \frac{\norm{a-b}_1}{A}
5445 +
5446 \norm b_1\abs{1/A-1/B}.
5447 \]
5448 The second term equals \(\abs{B-A}/A\), which is at most \(\norm{a-b}_1/A\). Interchanging \(a\) and \(b\) gives the analogous bound with \(B\). Taking the smaller of the two bounds gives the result.
5449 \end{proof}
5450 
5451 
5452 For an arbitrary prior $p$, put $q=Cp$ and $Z(p,L)=\ip{L}{p}$. Define $\ell_a(p)=\sum_{i\in B_a}L_ip_i/q_a$ when $q_a>0$. Choose any positive value when $q_a=0$. Only the product $q_a\ell_a(p)$ enters the formulas.
5453 \begin{theorem}[Exact information loss formula]\label{earlier:record:013}\label{earlier:thm:information_loss}
5454 For every prior,
5455 \[
5456 \boxed{
5457 C\Bcal_L(p)
5458 =
5459 \Bcal_{\ell(p)}(q).
5460 }
5461 \]
5462 Let \(\overline L\) be any fixed positive coarse likelihood and set
5463 \[
5464 d(p\mid L,\overline L)
5465 =
5466 \sum_{a=1}^mq_a\abs{\ell_a(p)-\overline L_a}.
5467 \]
5468 Then
5469 \[
5470 \norm{
5471 C\Bcal_L(p)-\Bcal_{\overline L}(q)
5472 }_1
5473 \leq
5474 \frac{2d(p\mid L,\overline L)}
5475 {\max\{Z(p,L),\ip{\overline L}{q}\}}.
5476 \]
5477 \end{theorem}
5478 
5479 \begin{proof}
5480 The packet evidence in block \(a\) is \(q_a\ell_a(p)\), which proves the exact identity. Apply \cref{earlier:lem:normalization_bound} to
5481 \[
5482 a=q\odot\ell(p),
5483 \qquad
5484 b=q\odot\overline L.
5485 \]
5486 Their \(\ell^1\) norms are the two evidences, and \(\norm{a-b}_1=d(p\mid L,\overline L)\).
5487 \end{proof}
5488 
5489 \begin{corollary}[Oscillation bound]\label{earlier:record:014}\label{earlier:cor:bayes_oscillation}
5490 Suppose \(\overline L_a\) is any representative likelihood for block \(B_a\), and set
5491 \[
5492 \omega_a
5493 =
5494 \max_{i\in B_a}\abs{L_i-\overline L_a}.
5495 \]
5496 Then
5497 \[
5498 d(p\mid L,\overline L)
5499 \leq
5500 \sum_aq_a\omega_a.
5501 \]
5502 Thus small within block likelihood oscillation gives a small coarse posterior defect whenever the evidence is bounded away from zero.
5503 \end{corollary}
5504 
5505 \begin{proof}
5506 The conditional average \(\ell_a(p)\) lies in the convex hull of the likelihood values inside \(B_a\). Hence
5507 \[
5508 \abs{\ell_a(p)-\overline L_a}\leq\omega_a.
5509 \]
5510 Multiply by \(q_a\) and sum.
5511 \end{proof}
5512 
5513 
5514 Along the Bayesian flow $p_i(t)=e^{tg_i}p_i(0)/\sum_j e^{tg_j}p_j(0)$, the following statement holds for every real observable $h$.
5515 \begin{corollary}[Covariance identity]\label{earlier:record:015}\label{earlier:cor:bayesian_covariance}
5516 For any observable \(h\in\RR^n\),
5517 \[
5518 \frac{\dd}{\dd t}\EE_{p(t)}[h]
5519 =
5520 \Cov_{p(t)}(h,g).
5521 \]
5522 In particular,
5523 \[
5524 \frac{\dd}{\dd t}\EE_{p(t)}[g]
5525 =
5526 \Var_{p(t)}(g)\geq0.
5527 \]
5528 \end{corollary}
5529 
5530 \begin{proof}
5531 Differentiate \(\sum_ih_ip_i(t)\) and insert the replicator equation.
5532 \end{proof}
5533 
5534 
5535 For stochastic matrices $K,\overline K$ and positive likelihoods $L_y,\overline L_y$, put $\Phi_y=\Bcal_{L_y}\circ K$ and $\overline\Phi_y=\Bcal_{\overline L_y}\circ\overline K$.
5536 \begin{theorem}[Exact coarse hidden Markov filter]\label{earlier:record:016}\label{earlier:thm:exact_coarse_filter}
5537 Assume
5538 \[
5539 CK=\overline KC
5540 \]
5541 and every likelihood \(L_y\) is constant on each coarse block with block values \(\overline L_y\). Then
5542 \[
5543 C\Phi_y(p)=\overline\Phi_y(Cp)
5544 \]
5545 for every prior. For any observation sequence \(y_1,\ldots,y_r\), the complete filtering recursion commutes with aggregation:
5546 \[
5547 C\Phi_{y_r}\circ\cdots\circ\Phi_{y_1}
5548 =
5549 \overline\Phi_{y_r}\circ\cdots\circ\overline\Phi_{y_1}C.
5550 \]
5551 \end{theorem}
5552 
5553 \begin{proof}
5554 Exact Markov lumping gives \(CKp=\overline KCp\). The Bayesian sufficiency criterion gives exact commutation of the update. Their composition proves the one step identity. Iterate it over the observation sequence.
5555 \end{proof}
5556 
5557 \begin{theorem}[One step filter defect]\label{earlier:record:017}\label{earlier:thm:filter_defect}
5558 Let \(r=Kp\), \(q=Cr\), and let \(\ell_y(r)\) be the exact coarse likelihood obtained by averaging \(L_y\) inside each block with respect to \(r\). Assume
5559 \[
5560 0<\overline m\leq(\overline L_y)_a\leq\overline M.
5561 \]
5562 Then
5563 \[
5564 \begin{aligned}
5565 \norm{C\Phi_y(p)-\overline\Phi_y(Cp)}_1
5566 \leq{}&
5567 \frac{2d(r\mid L_y,\overline L_y)}
5568 {\max\{\ip{L_y}{r},\ip{\overline L_y}{q}\}}\\
5569 &+
5570 \frac{2\overline M}{\overline m}
5571 \norm{CKp-\overline KCp}_1.
5572 \end{aligned}
5573 \]
5574 \end{theorem}
5575 
5576 \begin{proof}
5577 Insert the intermediate posterior \(\Bcal_{\overline L_y}(CKp)\). The first difference is bounded by \cref{earlier:thm:information_loss}. The second is bounded by \cref{extended:thm:prior_stability} applied to the coarse likelihood.
5578 \end{proof}
5579 
5580 
5581 \FloatBarrier
5582 \section{Fractional modes and killed transfer}\label{earlier:section:04}
5583 For the one-dimensional Dirichlet difference operator, let $h=1/M$ with an integer $M\geq2$, let $1\leq k\leq M-1$, and set $\lambda_{k,h}=4h^{-2}\sin^2(k\pi h/2)$. The retained explicit coefficient below is valid for every discrete mode.
5584 \begin{corollary}[Fractional low mode dispersion]\label{earlier:record:018}\label{earlier:cor:fractional_low_mode_dispersion}
5585 Let \(0<\alpha\leq1\). Then
5586 \[
5587 0
5588 \leq
5589 (k\pi)^{2\alpha}-\lambda_{k,h}^\alpha
5590 \leq
5591 \frac{\alpha(4k^2)^{\alpha-1}(k\pi)^4}{12}h^2.
5592 \]
5593 For every \(t\geq0\),
5594 \[
5595 \abs{
5596 \e^{-t\lambda_{k,h}^\alpha}
5597 -
5598 \e^{-t(k\pi)^{2\alpha}}
5599 }
5600 \leq
5601 \frac{t\alpha(4k^2)^{\alpha-1}(k\pi)^4}{12}h^2.
5602 \]
5603 \end{corollary}
5604 
5605 \begin{proof}
5606 The chord bound \(\sin x\geq2x/\pi\) on \([0,\pi/2]\) gives \(\lambda_{k,h}\geq4k^2\). Since \(x\mapsto x^\alpha\) is increasing and has decreasing derivative,
5607 \[
5608 (k\pi)^{2\alpha}-\lambda_{k,h}^\alpha
5609 \leq
5610 \alpha\lambda_{k,h}^{\alpha-1}
5611 \bigl((k\pi)^2-\lambda_{k,h}\bigr).
5612 \]
5613 Use \cref{extended:thm:low_mode} and the lower bound for \(\lambda_{k,h}\). The exponential estimate follows from
5614 \[
5615 \abs{\e^{-ta}-\e^{-tb}}
5616 \leq t\abs{a-b}
5617 \]
5618 for nonnegative \(a\) and \(b\).
5619 \end{proof}
5620 
5621 
5622 
5623 Let $Q$ and $\overline Q$ be finite Markov generators in the column convention. All times in this section satisfy $t\geq0$.
5624 \begin{theorem}[Potential perturbation bound]\label{earlier:record:019}\label{earlier:thm:feynman_kac_potential_bound}
5625 Let \(V\) and \(W\) be nonnegative diagonal matrices. In the supremum norm,
5626 \[
5627 \boxed{
5628 \norm{
5629 \e^{t(Q^T-V)}
5630 -
5631 \e^{t(Q^T-W)}
5632 }_{\infty\to\infty}
5633 \leq
5634  t\norm{V-W}_{\infty\to\infty}.
5635 }
5636 \]
5637 \end{theorem}
5638 
5639 \begin{proof}
5640 Both killed Markov semigroups are contractions in the supremum norm. Apply the Duhamel identity with generator defect \(W-V\).
5641 \end{proof}
5642 
5643 \begin{theorem}[Coarse Feynman--Kac transfer]\label{earlier:record:020}\label{earlier:thm:coarse_feynman_kac}
5644 Assume \(CQ=\overline QC\), and let \(J=C^T\) lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so
5645 \[
5646 VJ=J\overline V.
5647 \]
5648 Then
5649 \[
5650 \boxed{
5651 \e^{t(Q^T-V)}J
5652 =
5653 J\e^{t(\overline Q^T-\overline V)}
5654 }
5655 \]
5656 for every \(t\geq0\).
5657 \end{theorem}
5658 
5659 \begin{proof}
5660 Transpose the Markov intertwining relation to obtain
5661 \[
5662 Q^TJ=J\overline Q^T.
5663 \]
5664 Together with \(VJ=J\overline V\), this gives
5665 \[
5666 (Q^T-V)J=J(\overline Q^T-\overline V).
5667 \]
5668 The exponential intertwining follows from the power series.
5669 \end{proof}
5670 
5671 
5672 \FloatBarrier
5673 \section{Memory compression, rational realization, and phase-type consequences}\label{earlier:section:05}
5674 Fix $T>0$. For the next estimate let $A$ be a bounded operator on a Banach space $X$, let $\Kcal,\widetilde\Kcal\in L^1((0,T),\mathcal B(X))$, and let $f$ be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II.
5675 \begin{theorem}[Stability of memory compression]\label{earlier:record:021}\label{earlier:thm:memory_compression_stability}
5676 Let \(x\) and \(\widetilde x\) solve
5677 \[
5678 \dot x(t)
5679 =Ax(t)+\int_0^t\Kcal(t-s)x(s)\dd s+f(t)
5680 \]
5681 and
5682 \[
5683 \dot{\widetilde x}(t)
5684 =A\widetilde x(t)
5685 +\int_0^t\widetilde\Kcal(t-s)\widetilde x(s)\dd s
5686 +f(t)
5687 \]
5688 with the same initial value. Fix \(T>0\), and assume
5689 \[
5690 \sup_{0\leq t\leq T}\norm{\widetilde x(t)}\leq M.
5691 \]
5692 Set
5693 \[
5694 a=\norm A,
5695 \qquad
5696 k=\norm{\Kcal}_{L^1(0,T)},
5697 \qquad
5698 \delta=\norm{\Kcal-\widetilde\Kcal}_{L^1(0,T)}.
5699 \]
5700 Then
5701 \[
5702 \boxed{
5703 \sup_{0\leq t\leq T}
5704 \norm{x(t)-\widetilde x(t)}
5705 \leq
5706 MT\delta\e^{(a+k)T}.
5707 }
5708 \]
5709 \end{theorem}
5710 
5711 \begin{proof}
5712 Let \(e=x-\widetilde x\). Integrating the difference equation and changing the order of integration gives
5713 \[
5714 \norm{e(t)}
5715 \leq
5716 (a+k)\int_0^t\norm{e(r)}\dd r
5717 +Mt\delta.
5718 \]
5719 Since \(t\leq T\), Gronwall's inequality gives
5720 \[
5721 \norm{e(t)}
5722 \leq
5723 MT\delta\e^{(a+k)t}.
5724 \]
5725 Take the supremum over \([0,T]\).
5726 \end{proof}
5727 
5728 
5729 
5730 For the exact closure criterion of Part II, allowing an arbitrary bounded local generator $B$ does not enlarge the class: differentiating $P e^{tA}P=e^{tB}P$ at $t=0$ forces $B=A_{PP}$ on the retained space. Thus its zero-memory criterion excludes every alternative bounded local generator.
5731 
5732 A finite auxiliary realization of a matrix kernel means $\Kcal(t)=B e^{tD}C$, with finite matrices of compatible sizes. Laplace transforms are taken in a right half-plane of absolute convergence. The rational characterization concerns continuous kernels of exponential order, so uniqueness of the ordinary Laplace transform applies.
5733 \begin{theorem}[Rational memory realization]\label{earlier:record:022}\label{earlier:thm:rational_memory_realization}
5734 A matrix valued memory transform has a finite auxiliary realization if and only if it is a strictly proper rational matrix function. More precisely,
5735 \[
5736 \boxed{
5737 \widehat\Kcal(z)=B(zI-D)^{-1}C
5738 }
5739 \]
5740 if and only if
5741 \[
5742 \boxed{
5743 \Kcal(t)=B\e^{tD}C.
5744 }
5745 \]
5746 \end{theorem}
5747 
5748 \begin{proof}
5749 The forward implication follows by taking the Laplace transform. The resulting matrix is rational and tends to zero as \(z\) tends to infinity.
5750 
5751 Conversely, decompose every entry of a strictly proper rational matrix into partial fractions. A term
5752 \[
5753 \frac{c}{(z-\lambda)^r}
5754 \]
5755 is an entry of the resolvent of an \(r\) by \(r\) Jordan block with eigenvalue \(\lambda\). Taking a direct sum of these blocks and choosing the input and output maps entry by entry gives matrices \(B,D,C\) with the required resolvent. Inverse Laplace transformation gives the time domain formula.
5756 \end{proof}
5757 
5758 \begin{corollary}[Exponential sum memory]\label{earlier:record:023}\label{earlier:cor:exponential_sum_memory}
5759 If
5760 \[
5761 \Kcal(t)=\sum_{j=1}^{m}B_j\e^{\lambda_jt}C_j,
5762 \]
5763 then \(\Kcal\) has a finite auxiliary realization with one auxiliary block for each exponential. Repeated poles produce polynomial factors in \(t\) and are realized by Jordan blocks.
5764 \end{corollary}
5765 
5766 \begin{proof}
5767 If \(B_j\) and \(C_j\) have an intermediate space of dimension \(d_j\), use a block diagonal matrix \(D\) with blocks \(\lambda_j I_{d_j}\), or the required Jordan blocks when poles repeat. Assemble the matrices \(B_j\) and \(C_j\) into the output and input maps.
5768 \end{proof}
5769 
5770 
5771 For the survival consequences, $A$ is a finite Metzler matrix with nonpositive column sums and spectrum in the open left half-plane, $p_0$ is a probability vector, $S(t)=\one^T e^{tA}p_0$, and $f(t)=-S'(t)$. The one-phase model has $\lambda>0$. The survival Laplace identity of Part II also holds for every complex $s$ with $\operatorname{Re}s\geq0$: stability gives absolute convergence of $\int_0^\infty e^{-st}e^{tA}\,dt=(sI-A)^{-1}$, and multiplying by the exit row and $p_0$ gives the same transform.
5772 \begin{corollary}[Phase type rationality]\label{earlier:record:024}\label{earlier:cor:phase_type_rationality}
5773 The Laplace transform of every finite phase type distribution is rational. If \(A\) is diagonalizable, then \(S(t)\) and \(f(t)\) are finite linear combinations of exponential functions. Jordan blocks add polynomial factors in \(t\).
5774 \end{corollary}
5775 
5776 \begin{proof}
5777 The resolvent \((sI-A)^{-1}\) is rational. The time domain statement follows from the Jordan form of \(A\).
5778 \end{proof}
5779 
5780 \begin{corollary}[One phase model]\label{earlier:record:025}\label{earlier:cor:one_phase_model}
5781 If \(A=(-\lambda)\) and \(p_0=1\), then
5782 \[
5783 S(t)=\e^{-\lambda t},
5784 \qquad
5785 f(t)=\lambda\e^{-\lambda t},
5786 \qquad
5787 h(t)=\lambda.
5788 \]
5789 \end{corollary}
5790 
5791 \begin{proof}
5792 Insert the scalar generator into \cref{extended:thm:survival_hazard}.
5793 \end{proof}
5794 
5795 
5796 \FloatBarrier
5797 \section{Earlier exact examples}\label{earlier:section:06}
5798 These five short examples are retained from the earlier human edition. Their numerical parameters are independent of the later visual guide's worked models.
5799 \begin{example}[One transfer between two packets]\label{earlier:record:026}\label{earlier:ex:two_packet_transfer}
5800 Let
5801 \[
5802 A=
5803 \begin{pmatrix}
5804 a&0\\
5805 c&b
5806 \end{pmatrix}
5807 \]
5808 and let the two coordinate axes be the packet spaces. If \(a\neq b\), then
5809 \[
5810 \e^{tA}
5811 =
5812 \begin{pmatrix}
5813 \e^{at}&0\\
5814 \displaystyle c\frac{\e^{at}-\e^{bt}}{a-b}&\e^{bt}
5815 \end{pmatrix}.
5816 \]
5817 The lower left block is produced by one transfer from the first packet to the second. Its expansion begins with \(ct\), in agreement with \cref{earlier:cor:minimum_transfer_order}. When \(a=b\), the same block is \(ct\e^{at}\).
5818 \end{example}
5819 
5820 \begin{example}[A four state lumpable chain]\label{earlier:record:027}\label{earlier:ex:four_state_lumping}
5821 Partition four states into \(B_1=\{1,2\}\) and \(B_2=\{3,4\}\). For positive rates \(a,b,u,v\), set
5822 \[
5823 Q=
5824 \begin{pmatrix}
5825 -u-a&u&b/2&b/2\\
5826 u&-u-a&b/2&b/2\\
5827 a/2&a/2&-v-b&v\\
5828 a/2&a/2&v&-v-b
5829 \end{pmatrix}.
5830 \]
5831 The rates \(u\) and \(v\) move inside the coarse blocks. Every state in \(B_1\) leaves for \(B_2\) at total rate \(a\), and every state in \(B_2\) leaves for \(B_1\) at total rate \(b\). Hence
5832 \[
5833 \overline Q=
5834 \begin{pmatrix}
5835 -a&b\\
5836 a&-b
5837 \end{pmatrix}
5838 \]
5839 and \(CQ=\overline QC\). The internal rates disappear from the coarse generator even though they remain present in the fine evolution.
5840 \end{example}
5841 
5842 \begin{example}[A visible likelihood loss]\label{earlier:record:028}\label{earlier:ex:bayes_information_loss}
5843 Let \(B_1=\{1,2\}\), let \(B_2=\{3\}\), and take
5844 \[
5845 p=(3/10,2/10,5/10),
5846 \qquad
5847 L=(1,3,2).
5848 \]
5849 The coarse prior is \((1/2,1/2)\). The exact block likelihoods are
5850 \[
5851 \ell_1(p)=\frac{(3/10)+3(2/10)}{1/2}=\frac95,
5852 \qquad
5853 \ell_2(p)=2.
5854 \]
5855 Thus the exact coarse posterior is \((9/19,10/19)\). If the first block is replaced by the fixed representative value \(2\), the approximate coarse posterior is \((1/2,1/2)\). The \(\ell^1\) defect is \(1/19\), while \cref{earlier:thm:information_loss} gives the valid upper bound \(1/10\).
5856 \end{example}
5857 
5858 \begin{example}[One hidden mode]\label{earlier:record:029}\label{earlier:ex:one_hidden_mode}
5859 Consider
5860 \[
5861 A=
5862 \begin{pmatrix}
5863 -a&c\\
5864 d&-b
5865 \end{pmatrix},
5866 \qquad
5867 a,b>0,
5868 \]
5869 and retain the first coordinate. The memory kernel and self-energy are
5870 \[
5871 \Kcal(t)=cd\e^{-bt},
5872 \qquad
5873 \Sigma(z)=\frac{cd}{z+b}.
5874 \]
5875 The retained coordinate has an exact time local closure for every retained initial value precisely when \(cd=0\). When \(cd\neq0\), one auxiliary scalar state realizes the memory exactly.
5876 \end{example}
5877 
5878 \begin{example}[Two sequential phases]\label{earlier:record:030}\label{earlier:ex:two_phase_survival}
5879 Let
5880 \[
5881 A=
5882 \begin{pmatrix}
5883 -\lambda&0\\
5884 \lambda&-\mu
5885 \end{pmatrix},
5886 \qquad
5887 p_0=
5888 \begin{pmatrix}1\\0\end{pmatrix},
5889 \]
5890 where \(\lambda\neq\mu\) are positive. The first phase moves to the second at rate \(\lambda\), and the second is absorbed at rate \(\mu\). Direct calculation gives
5891 \[
5892 S(t)=
5893 \frac{\mu\e^{-\lambda t}-\lambda\e^{-\mu t}}{\mu-\lambda},
5894 \]
5895 \[
5896 f(t)=
5897 \frac{\lambda\mu}{\mu-\lambda}
5898 \left(\e^{-\lambda t}-\e^{-\mu t}\right).
5899 \]
5900 The ratio \(f(t)/S(t)\) is generally not constant. A time local Markov model on two transient phases can therefore produce a nonconstant hazard after the phase coordinate is hidden.
5901 \end{example}
5902 
5903 \FloatBarrier
5904 \section{Holomorphic operator substitution}\label{migration:section:01}\label{migration:sec:operator_substitution}
5905 \begin{readerroadmap}
5906 The scalar function, zero-free domain, and logarithm branch are fixed before substituting an operator. The bounded Banach-space functional calculus here is separate from the finite canonical-jet contract of Part I. Keep those spaces and hypotheses visible when using the reciprocal family.
5907 \end{readerroadmap}
5908 
5909 
5910 This section presents the operator-substitution result transferred from Chapter IV. Fix an integer $N\geq1$ and the entire scalar function
5911 \[
5912 \Dfun_N(z)=\sum_{m=0}^{\infty}\frac{(-1)^m z^{Nm}}{(Nm)!}.
5913 \]
5914 More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain $U$, choose $g=\log\Dfun_N$ once and define $\Sfun_N^{(r)}(z)=\exp(-r g(z))$ for complex $r$. The bounded Banach-space result below uses classical holomorphic functional calculus. It is separate from the finite canonical-jet calculus formalized in Part I.
5915 
5916 \begin{standardfact}[Holomorphic functional calculus]\label{migration:record:001}\label{migration:sf:holomorphic_calculus}
5917 Let $A$ be a bounded operator on a complex Banach space.  If $f$ is holomorphic on a neighborhood of $\Spec(A)$, then
5918 \[
5919 f(A)
5920 =
5921 \frac1{2\pi i}\int_\Gamma f(z)(zI-A)^{-1}\,dz
5922 \]
5923 for an admissible positively oriented contour $\Gamma$ lying in the holomorphy domain and the resolvent set, with winding number one about the spectrum and zero outside the domain. The calculus preserves sums, products, and composition.  We use the standard form from \cite{DunfordSchwartz1958}. An accessible proof is given in \cite[Appendix A]{TaylorBanachAlgebras}.
5924 \end{standardfact}
5925 
5926 \begin{proposition}[Operator reciprocal family]\label{migration:record:002}\label{migration:prop:operator_reciprocal}
5927 Let $A$ be bounded and suppose $\Spec(A)$ lies in a simply connected region on which $\Dfun_N$ has no zeros.  Fix a holomorphic branch $g=\log\Dfun_N$ there.  Then
5928 \[
5929 \Sfun_N^{(r)}(A)
5930 =
5931 \exp\bigl(-r g(A)\bigr)
5932 =
5933 \bigl(\Dfun_N^{-r}\bigr)(A)
5934 \]
5935 is well defined and
5936 \[
5937 \Sfun_N^{(r)}(A)\Sfun_N^{(s)}(A)
5938 =
5939 \Sfun_N^{(r+s)}(A).
5940 \]
5941 If $(P_i)$ is a finite complete family of bounded idempotents, then
5942 \[
5943 P_i f(A)P_j
5944 =
5945 \frac1{2\pi i}
5946 \int_\Gamma
5947 f(z)P_i(zI-A)^{-1}P_j\,dz
5948 \]
5949 for every holomorphic $f$ on a neighborhood of the spectrum.
5950 \end{proposition}
5951 
5952 \begin{proof}
5953 The spectral mapping theorem makes $\Dfun_N(A)$ invertible.  The product identity follows from multiplicativity of the functional calculus and the fixed logarithm.  Applying $P_i$ and $P_j$ to the Cauchy integral formula gives the packet formula.
5954 \end{proof}
5955 
5956 
5957 \clearpage
5958 \part*{Part IV\quad Worked visual atlas}\addcontentsline{toc}{part}{Part IV: Worked visual atlas}
5959 \input{atlas-pages.tex}
5960 \clearpage\bibliographystyle{alpha}\bibliography{references}
5961 \end{document}