Paper VI: Transfer Semigroups: Stochastic, Bayesian, Fractional, and PDE Realizations

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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 
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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 \begin{document}
247 \KomorebiCover
248 \KomorebiOpeningNote{\textbf{Part I.} The complete released 33-result formal core, including nonnormal memory and canonical finite holomorphic naturality.
249 
250 \medskip
251 \textbf{Part II.} The complete strengthened mathematical development: bounded refinement towers, Bayesian information, Hermite transfer, quantitative memory compression, and competing risks.
252 
253 \medskip
254 \textbf{Part III.} Distinct retained results from the earlier human edition, exact examples, and the bounded-operator reciprocal calculus received from Paper IV.
255 
256 \medskip
257 \textbf{Part IV.} Forty-two visual pages work through exact finite models, analytical examples, and the boundaries of the formal contract.}
258 
259 \section*{Reading this edition}\addcontentsline{toc}{section}{Reading this edition}
260 Paper VI has several substantial source branches. This edition preserves the latest formal-core manuscript in Part I and the complete strengthened standalone manuscript in Part II. Part III retains twenty-four earlier results with distinct statements or hypotheses, five exact examples, and the holomorphic operator-substitution section migrated from Paper IV. Overlap is intentional and is not a claim of independent novelty for each proof environment.
261 
262 The formal denominator is exactly the thirty-three scoped results of Part I. The inherited September 6, 2026 release records 341 explicit Lean theorem declarations across 47 proof modules and an audit of 782 namespace declarations. These are different units. No fresh Lean compilation was available for this October edition, and the formal count does not automatically include Parts II or III. The Lean guide gives the exact targets and specialization boundaries.
263 
264 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.
265 
266 The current checks, historical compiler evidence, and numerical regressions are recorded separately in the accompanying package. Finite checks and illustrations have bounded scopes. They do not replace proofs. The September 12 worked visual appendix, its exact models, and additional guides to the strengthened and retained material are supplied in the visual companion.
267 
268 Read Part I for the common operator language and the exact released 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.
269 
270 The final mathematical audit makes explicit the hypotheses needed for contour winding, positive memory decay, finite moment closure, and eventual absorption. It also clarifies summability across varying spaces. Every substantive change is recorded against preserved sources. Historical source test counts retain their historical dates.
271 \clearpage
272 \tableofcontents
273 \clearpage
274 \listoffigures
275 \listoftables
276 \clearpage
277 \pagenumbering{arabic}
278 \part*{Part I\quad The released formal core}\addcontentsline{toc}{part}{Part I: The released formal core}
279 \begin{center}\large Projector-Resolved Seed Calculus VI\par\normalsize Transfer Semigroups: Stochastic, Bayesian, Fractional, and PDE Realizations\end{center}
280 
281 
282 
283 \begin{abstract}
284 This paper 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.
285 \end{abstract}
286 
287 \begin{center}
288 \fbox{\begin{minipage}{0.93\textwidth}
289 \textbf{Scope and claim boundary.} The paper 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 paper.
290 \end{minipage}}
291 \end{center}
292 
293 
294 
295 \FloatBarrier
296 \section{The transfer-semigroup realization}\label{core:section:01}
297 
298 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
299 \[
300 T(t)=e^{tA},\qquad t\ge 0,
301 \]
302 while a positive operator $L$ is more naturally written
303 \[
304 H(t)=e^{-tL}.
305 \]
306 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 $e^{-tL}$.
307 
308 The common operator chain is
309 \[
310 \boxed{
311 \text{seed}\longrightarrow\text{transfer}\longrightarrow\text{project}
312 \longrightarrow\text{condition}\longrightarrow\text{refine}
313 \longrightarrow\text{complete}\longrightarrow\text{reconstruct}.
314 }
315 \]
316 
317 \begin{figure}[htbp]
318 \centering
319 \includegraphics[width=0.98\textwidth]{01_transfer_semigroup_architecture.pdf}
320 \caption{The common operator language used throughout the paper. Semigroups, resolvents, normalized Bayesian maps, and fractional powers are distinct realizations of the same transfer stage.}
321 \label{core:fig:architecture}
322 \end{figure}
323 
Statement: Transfer-semigroup seed system324 \begin{definition}[Transfer-semigroup seed system]\label{core:record:001}
325 A finite transfer-semigroup seed system is a tuple
326 \[
327 \mathfrak S=(X,s,A,\{P_\alpha\}_{\alpha\in I},\tau),
328 \]
329 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.
330 \end{definition}
331 
Statement: Finite semigroup law · Go to proof · Read statement and proof side by side332 \begin{theorem}[Finite semigroup law]\label{core:record:002}
333 For every matrix $A$ and all $s,t\in\bbR$,
334 \[
335 \boxed{e^{(s+t)A}=e^{sA}e^{tA}.}
336 \]
337 Moreover, $T(0)=I$ and $T'(t)=AT(t)=T(t)A$.
338 \end{theorem}
339 
Proof block: Finite semigroup law · Go to statement · Read statement and proof side by side340 \begin{proof}
341 Both exponentials are absolutely convergent matrix power series. Since every power of $A$ commutes with every other power, the Cauchy product gives
342 \[
343 e^{sA}e^{tA}
344 =\sum_{m,n\ge0}\frac{s^mt^n}{m!n!}A^{m+n}
345 =\sum_{r\ge0}\frac{(s+t)^r}{r!}A^r.
346 \]
347 Termwise differentiation is valid in finite dimension and gives the derivative identities.
348 \end{proof}
349 
Statement: Infinite-dimensional generators350 \begin{boundary}[Infinite-dimensional generators]\label{core:record:003}
351 For an unbounded operator, writing $e^{tA}$ is not a definition unless a semigroup-generation theorem applies. The finite matrix identities in this paper remain algebraically correct models, but a continuum realization must state the operator domain, closure, growth bound, and topology of convergence.
352 \end{boundary}
353 
354 \FloatBarrier
355 \section{Packet blocks of a semigroup}\label{core:section:02}
356 \begin{keyidea}
357 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.
358 \end{keyidea}
359 \begin{figure}[htbp]
360 \centering
361 \includegraphics[width=.94\textwidth]{monograph_figures/exact_packet_composition.pdf}
362 \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}
363 \end{figure}
364 
365 
366 
367 Let
368 \[
369 P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,
370 \qquad
371 \sum_{\alpha\in I}P_\alpha=I.
372 \]
373 For any operator $B$, define
374 \[
375 B_{\beta\alpha}=P_\beta BP_\alpha.
376 \]
377 Then
378 \[
379 B=\sum_{\alpha,\beta}B_{\beta\alpha}.
380 \]
381 For the semigroup, write
382 \[
383 T_{\beta\alpha}(t)=P_\beta e^{tA}P_\alpha.
384 \]
385 
Statement: Packet semigroup convolution · Go to proof · Read statement and proof side by side386 \begin{theorem}[Packet semigroup convolution]\label{core:record:004}
387 For every $s,t$ and every pair of packets $\alpha,\beta$,
388 \[
389 \boxed{
390 T_{\beta\alpha}(s+t)
391 =\sum_{\gamma\in I}T_{\beta\gamma}(s)T_{\gamma\alpha}(t).
392 }
393 \]
394 Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.
395 \end{theorem}
396 
Proof block: Packet semigroup convolution · Go to statement · Read statement and proof side by side397 \begin{proof}
398 Insert the resolution of the identity between the two semigroup factors:
399 \[
400 \begin{aligned}
401 P_\beta e^{(s+t)A}P_\alpha
402 &=P_\beta e^{sA}\left(\sum_\gamma P_\gamma\right)e^{tA}P_\alpha\\
403 &=\sum_\gamma
404 (P_\beta e^{sA}P_\gamma)(P_\gamma e^{tA}P_\alpha).
405 \end{aligned}
406 \]
407 \end{proof}
408 
Statement: Reducing packets · Go to proof · Read statement and proof side by side409 \begin{corollary}[Reducing packets]\label{core:record:005}
410 If $P_\beta AP_\alpha=0$ for every $\beta\ne\alpha$, then
411 \[
412 P_\beta e^{tA}P_\alpha=0
413 \quad(\beta\ne\alpha)
414 \]
415 for every $t$. Each packet evolves independently under $P_\alpha AP_\alpha$.
416 \end{corollary}
417 
Proof block: Reducing packets · Go to statement · Read statement and proof side by side418 \begin{proof}
419 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.
420 \end{proof}
421 
422 \begin{figure}[htbp]
423 \centering
424 \includegraphics[width=0.66\textwidth]{02_packet_semigroup_blocks.pdf}
425 \caption{Norms of the packet blocks $P_\beta e^{tA}P_\alpha$. Off-diagonal entries quantify transfer between packets.}
426 \label{core:fig:packet-blocks}
427 \end{figure}
428 
429 \begin{remark}[Path expansion]\label{core:record:006}
430 Expanding the exponential and inserting $I=\sum_\alpha P_\alpha$ between every pair of generators gives
431 \[
432 P_{\alpha_m}A^mP_{\alpha_0}
433 =
434 \sum_{\alpha_1,\ldots,\alpha_{m-1}}
435 P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
436 \]
437 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.
438 \end{remark}
439 
440 \FloatBarrier
441 \section{Refinement defects and the Duhamel identity}\label{core:section:03}
442 \begin{interpretation}
443 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.
444 \end{interpretation}
445 
446 
447 Let $X_f$ and $X_c$ be fine and coarse spaces, with generators $A_f$ and $A_c$, and let
448 \[
449 C:X_f\to X_c
450 \]
451 be a coarsening map. Define the generator-level refinement defect
452 \[
453 \boxed{\Delta=CA_f-A_cC.}
454 \]
455 The corresponding semigroup defect is
456 \[
457 \mathfrak D(t)=Ce^{tA_f}-e^{tA_c}C.
458 \]
459 
Statement: Exact Duhamel refinement identity · Go to proof · Read statement and proof side by side460 \begin{theorem}[Exact Duhamel refinement identity]\label{core:record:007}
461 For every $t\ge0$,
462 \[
463 \boxed{
464 Ce^{tA_f}-e^{tA_c}C
465 =
466 \int_0^t e^{(t-r)A_c}\Delta e^{rA_f}\,dr.
467 }
468 \]
469 In particular, $\Delta=0$ implies exact semigroup intertwining at every time.
470 \end{theorem}
471 
Proof block: Exact Duhamel refinement identity · Go to statement · Read statement and proof side by side472 \begin{proof}
473 Set
474 \[
475 F(r)=e^{(t-r)A_c}Ce^{rA_f}.
476 \]
477 Then
478 \[
479 F'(r)=e^{(t-r)A_c}(CA_f-A_cC)e^{rA_f}.
480 \]
481 Integrating from $0$ to $t$ gives $F(t)-F(0)=Ce^{tA_f}-e^{tA_c}C$ and proves the formula.
482 \end{proof}
483 
Statement: Norm bound484 \begin{corollary}[Norm bound]\label{core:record:008}
485 Suppose
486 \[
487 \norm{e^{tA_f}}\le M_f e^{\omega_f t},
488 \qquad
489 \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
490 \]
491 Then
492 \[
493 \norm{\mathfrak D(t)}
494 \le
495 M_fM_c\norm{\Delta}
496 \int_0^t e^{\omega_c(t-r)}e^{\omega_fr}\,dr.
497 \]
498 For contraction semigroups,
499 \[
500 \boxed{\norm{\mathfrak D(t)}\le t\norm{\Delta}.}
501 \]
502 \end{corollary}
503 
504 \begin{figure}[htbp]
505 \centering
506 \includegraphics[width=0.80\textwidth]{14_semigroup_refinement_defect.pdf}
507 \caption{Exact generator intertwining gives a vanishing semigroup defect. A controlled perturbation produces a nonzero defect propagated by the Duhamel identity.}
508 \label{core:fig:semigroup-defect}
509 \end{figure}
510 
511 \subsection{Resolvent form}
512 
513 For $z$ in the common resolvent set, let
514 \[
515 R_f(z)=(zI-A_f)^{-1},
516 \qquad
517 R_c(z)=(zI-A_c)^{-1}.
518 \]
519 
Statement: Resolvent refinement identity · Go to proof · Read statement and proof side by side520 \begin{theorem}[Resolvent refinement identity]\label{core:record:009}
521 \[
522 \boxed{
523 CR_f(z)-R_c(z)C
524 =R_c(z)\Delta R_f(z).
525 }
526 \]
527 \end{theorem}
528 
Proof block: Resolvent refinement identity · Go to statement · Read statement and proof side by side529 \begin{proof}
530 Multiply the left side by $zI-A_c$ on the left and $zI-A_f$ on the right. The result is
531 \[
532 (zI-A_c)C-C(zI-A_f)=CA_f-A_cC=\Delta.
533 \]
534 Multiplication by the two resolvents recovers the stated identity.
535 \end{proof}
536 
537 \FloatBarrier
538 \section{Markov seeds and stochastic transfer}\label{core:section:04}
539 
540 Use column probability vectors. A finite continuous-time Markov generator $Q$ has nonnegative off-diagonal entries and column sums zero. The stochastic semigroup is
541 \[
542 K(t)=e^{tQ}.
543 \]
544 
Statement: Finite Markov semigroup · Go to proof · Read statement and proof side by side545 \begin{theorem}[Finite Markov semigroup]\label{core:record:010}
546 If $Q$ has nonnegative off-diagonal entries and column sums zero, then $e^{tQ}$ has nonnegative entries and column sums one for every $t\ge0$.
547 \end{theorem}
548 
Proof block: Finite Markov semigroup · Go to statement · Read statement and proof side by side549 \begin{proof}
550 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
551 \[
552 e^{tQ}=e^{-\lambda t}e^{tB}
553 \]
554 is nonnegative. Since $\bm 1^TQ=0$, differentiation gives
555 \[
556 \frac{d}{dt}\bm 1^Te^{tQ}=\bm 1^TQe^{tQ}=0,
557 \]
558 so $\bm 1^Te^{tQ}=\bm 1^T$.
559 \end{proof}
560 
561 \begin{figure}[htbp]
562 \centering
563 \includegraphics[width=0.78\textwidth]{03_random_walk_transfer.pdf}
564 \caption{The point mass $\delta_0$ propagated by repeated symmetric random-walk transfer.}
565 \label{core:fig:random-walk}
566 \end{figure}
567 
568 \begin{example}[Symmetric random walk]\label{core:record:011}
569 Starting from $\delta_0$ and applying the transfer
570 \[
571 T=\frac12(S+S^{-1})
572 \]
573 for $n$ steps gives
574 \[
575 \bbP(X_n=k)=
576 2^{-n}\binom{n}{(n+k)/2}
577 \]
578 when $k\equiv n\pmod2$, and zero otherwise. Hence
579 \[
580 \Exp[X_n]=0,
581 \qquad
582 \Var(X_n)=n.
583 \]
584 The parity packet is exact: all mass remains in the one residue class permitted by $n$.
585 \end{example}
586 
587 \FloatBarrier
588 \section{Markov lumping as exact refinement}\label{core:section:05}
589 
590 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
591 \[
592 \Delta=CK-\bar KC.
593 \]
594 
Statement: Exact lumpability · Go to proof · Read statement and proof side by side595 \begin{theorem}[Exact lumpability]\label{core:record:012}
596 If $\Delta=0$, then
597 \[
598 \boxed{CK^n=\bar K^nC}
599 \]
600 for every $n\ge0$. Thus fine evolution followed by coarse reconstruction equals coarse evolution from the beginning.
601 \end{theorem}
602 
Proof block: Exact lumpability · Go to statement · Read statement and proof side by side603 \begin{proof}
604 The result is immediate for $n=0$. If $CK^n=\bar K^nC$, then
605 \[
606 CK^{n+1}=\bar K^nCK=\bar K^{n+1}C.
607 \]
608 \end{proof}
609 
Statement: Finite-horizon defect formula · Go to proof · Read statement and proof side by side610 \begin{theorem}[Finite-horizon defect formula]\label{core:record:013}
611 For arbitrary $\Delta=CK-\bar KC$,
612 \[
613 \boxed{
614 CK^n-\bar K^nC
615 =
616 \sum_{j=0}^{n-1}\bar K^{n-1-j}\Delta K^j.
617 }
618 \]
619 If $K$ and $\bar K$ are contractions in the chosen norm, then
620 \[
621 \norm{CK^n-\bar K^nC}\le n\norm{\Delta}.
622 \]
623 \end{theorem}
624 
Proof block: Finite-horizon defect formula · Go to statement · Read statement and proof side by side625 \begin{proof}
626 The identity follows by telescoping:
627 \[
628 \bar K^{n-1-j}CK^{j+1}-\bar K^{n-j}CK^j
629 =\bar K^{n-1-j}(CK-\bar KC)K^j.
630 \]
631 Summing from $j=0$ to $n-1$ leaves $CK^n-\bar K^nC$. The norm bound follows termwise.
632 \end{proof}
633 
634 \begin{figure}[htbp]
635 \centering
636 \includegraphics[width=0.82\textwidth]{04_markov_lumping.pdf}
637 \caption{Exact lumping agrees at every horizon. A small generator-level or one-step defect creates a controlled coarse discrepancy.}
638 \label{core:fig:lumping}
639 \end{figure}
640 
641 \FloatBarrier
642 \section{Conditional expectation as a stochastic projector}\label{core:section:06}
643 \begin{readerroadmap}
644 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.
645 \end{readerroadmap}
646 
647 
648 Let $\Omega=\{1,\ldots,n\}$ carry strictly positive probabilities $\pi_i$, and let $\mathcal Q=\{B_1,\ldots,B_m\}$ be a partition. Define
649 \[
650 (P_{\mathcal Q}f)_i
651 =
652 \frac{\sum_{j\in B(i)}\pi_jf_j}{\sum_{j\in B(i)}\pi_j},
653 \]
654 where $B(i)$ is the block containing $i$.
655 
Statement: Finite conditional-expectation projector · Go to proof · Read statement and proof side by side656 \begin{theorem}[Finite conditional-expectation projector]\label{core:record:014}
657 With the weighted inner product
658 \[
659 \langle f,g\rangle_\pi=\sum_i\pi_if_i\overline{g_i},
660 \]
661 the operator $P_{\mathcal Q}$ satisfies
662 \[
663 P_{\mathcal Q}^2=P_{\mathcal Q},
664 \qquad
665 \langle P_{\mathcal Q}f,g\rangle_\pi
666 =
667 \langle f,P_{\mathcal Q}g\rangle_\pi.
668 \]
669 Its range is the space of functions constant on each partition block.
670 \end{theorem}
671 
Proof block: Finite conditional-expectation projector · Go to statement · Read statement and proof side by side672 \begin{proof}
673 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
674 \[
675 \sum_{B\in\mathcal Q}
676 \frac{1}{\pi(B)}
677 \left(\sum_{i\in B}\pi_i f_i\right)
678 \overline{\left(\sum_{j\in B}\pi_j g_j\right)}.
679 \]
680 \end{proof}
681 
Statement: Tower law and martingale packets · Go to proof · Read statement and proof side by side682 \begin{theorem}[Tower law and martingale packets]\label{core:record:015}
683 If $\mathcal Q_c$ is coarser than $\mathcal Q_f$, then
684 \[
685 \boxed{
686 P_cP_f=P_fP_c=P_c.
687 }
688 \]
689 For a nested tower $P_0\le P_1\le\cdots\le P_r$, define
690 \[
691 D_0=P_0,
692 \qquad
693 D_j=P_j-P_{j-1}.
694 \]
695 Then
696 \[
697 D_iD_j=\delta_{ij}D_j,
698 \qquad
699 P_r=\sum_{j=0}^rD_j.
700 \]
701 \end{theorem}
702 
Proof block: Tower law and martingale packets · Go to statement · Read statement and proof side by side703 \begin{proof}
704 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 $P_iP_j=P_{\min(i,j)}$.
705 \end{proof}
706 
707 \begin{figure}[htbp]
708 \centering
709 \includegraphics[width=0.86\textwidth]{08_hermite_packets.pdf}
710 \caption{Hermite--Gaussian packets, which diagonalize the Ornstein--Uhlenbeck semigroup. The finite numerical release verifies 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.}
711 \label{core:fig:hermite}
712 \end{figure}
713 
Statement: Gaussian chaos realization714 \begin{classical}[Gaussian chaos realization]\label{core:record:016}
715 For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally:
716 \[
717 T_tH_n=e^{-nt}H_n.
718 \]
719 Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.
720 \end{classical}
721 
722 \FloatBarrier
723 \section{Bayesian conditioning as a normalized nonlinear operator}\label{core:section:07}
724 
725 Let $p=(p_i)$ be a probability vector and let $\lambda=(\lambda_i)$ be a nonnegative likelihood with positive evidence
726 \[
727 Z_\lambda(p)=\sum_i\lambda_ip_i.
728 \]
729 Define
730 \[
731 \boxed{
732 \cB_\lambda(p)
733 =
734 \frac{\lambda\odot p}{Z_\lambda(p)}.
735 }
736 \]
737 This map is nonlinear solely because of its scalar reconstruction factor.
738 
Statement: Independent-likelihood composition · Go to proof · Read statement and proof side by side739 \begin{theorem}[Independent-likelihood composition]\label{core:record:017}
740 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
741 \[
742 \boxed{
743 \cB_\mu(\cB_\lambda(p))
744 =
745 \cB_{\lambda\odot\mu}(p).
746 }
747 \]
748 \end{theorem}
749 
Proof block: Independent-likelihood composition · Go to statement · Read statement and proof side by side750 \begin{proof}
751 The first normalization appears in both numerator and denominator of the second update and cancels:
752 \[
753 \frac{\mu_i(\lambda_ip_i/Z_\lambda)}{\sum_j\mu_j\lambda_jp_j/Z_\lambda}
754 =
755 \frac{\mu_i\lambda_ip_i}{\sum_j\mu_j\lambda_jp_j}.
756 \]
757 \end{proof}
758 
759 \subsection{Packet evidence and reconstruction}
760 
761 Let the state indices be partitioned into packets $I_\alpha$. Define
762 \[
763 Z_\alpha=\sum_{i\in I_\alpha}\lambda_ip_i,
764 \qquad
765 m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta},
766 \qquad
767 q_{\alpha,i}=\begin{cases}
768 \lambda_ip_i/Z_\alpha,&Z_\alpha>0,\ i\in I_\alpha,\\
769 0,&\text{otherwise}.
770 \end{cases}
771 \]
772 
Statement: Bayesian packet reconstruction · Go to proof · Read statement and proof side by side773 \begin{theorem}[Bayesian packet reconstruction]\label{core:record:018}
774 \[
775 \boxed{
776 \cB_\lambda(p)
777 =
778 \sum_\alpha m_\alpha q_\alpha.
779 }
780 \]
781 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.
782 \end{theorem}
783 
Proof block: Bayesian packet reconstruction · Go to statement · Read statement and proof side by side784 \begin{proof}
785 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 $i\in I_\alpha$,
786 \[
787 m_\alpha q_{\alpha,i}
788 =
789 \frac{Z_\alpha}{Z}\frac{\lambda_ip_i}{Z_\alpha}
790 =
791 \frac{\lambda_ip_i}{Z}.
792 \]
793 The packet supports are disjoint, so summing reconstructs the full posterior.
794 \end{proof}
795 
796 \begin{figure}[htbp]
797 \centering
798 \includegraphics[width=0.82\textwidth]{05_bayesian_packet_update.pdf}
799 \caption{A finite prior and posterior split into two state packets. Evidence changes both the local packet shape and the global packet mass.}
800 \label{core:fig:bayes-packet}
801 \end{figure}
802 
Statement: Likelihood perturbation bound · Go to proof · Read statement and proof side by side803 \begin{theorem}[Likelihood perturbation bound]\label{core:record:019}
804 Let $p$ be fixed and let $\lambda,\mu\ge0$ have positive evidences $Z_\lambda,Z_\mu$. Then
805 \[
806 \boxed{
807 \norm{\cB_\lambda(p)-\cB_\mu(p)}_1
808 \le
809 \frac{2\sum_i|\lambda_i-\mu_i|p_i}{\min(Z_\lambda,Z_\mu)}.
810 }
811 \]
812 \end{theorem}
813 
Proof block: Likelihood perturbation bound · Go to statement · Read statement and proof side by side814 \begin{proof}
815 Add and subtract $\mu\odot p/Z_\lambda$ and use the triangle inequality:
816 \[
817 \norm{\frac{\lambda\odot p}{Z_\lambda}-\frac{\mu\odot p}{Z_\mu}}_1
818 \le
819 \frac{\norm{(\lambda-\mu)\odot p}_1}{Z_\lambda}
820 +
821 \norm{\mu\odot p}_1\left|\frac1{Z_\lambda}-\frac1{Z_\mu}\right|.
822 \]
823 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.
824 \end{proof}
825 
826 \FloatBarrier
827 \section{Exact Bayesian refinement and information-loss packets}\label{core:section:08}
828 \begin{interpretation}
829 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.
830 \end{interpretation}
831 \begin{figure}[htbp]
832 \centering
833 \includegraphics[width=.94\textwidth]{monograph_figures/exact_bayesian_coarsening.pdf}
834 \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}
835 \end{figure}
836 
837 
838 
839 Let the fine state set $I$ be partitioned into nonempty coarse packets $I_a$, $a\in A$.  The coarsening map on probability vectors is
840 \[
841 (C_\pi p)_a=P_a=\sum_{i\in I_a}p_i.
842 \]
843 For a fine likelihood $\lambda=(\lambda_i)_{i\in I}$ and a prior $p$, define the effective coarse likelihood
844 \[
845 r_a(p)
846 =
847 \frac{\sum_{i\in I_a}\lambda_i p_i}{P_a}
848 \qquad(P_a>0).
849 \]
850 Its value on a zero-mass packet is irrelevant and may be set to zero.
851 
Statement: Exact effective-likelihood factorization · Go to proof · Read statement and proof side by side852 \begin{theorem}[Exact effective-likelihood factorization]\label{core:record:020}
853 Whenever the fine evidence is positive,
854 \[
855 \boxed{
856 C_\pi\cB_\lambda(p)
857 =
858 \cB_{r(p)}(C_\pi p).}
859 \]
860 Thus the only fine information needed for the coarse posterior, beyond the coarse prior masses, is the vector of conditional packet evidences $r(p)$.
861 \end{theorem}
862 
Proof block: Exact effective-likelihood factorization · Go to statement · Read statement and proof side by side863 \begin{proof}
864 For $P=C_\pi p$ and $Z=\sum_i\lambda_ip_i=\sum_aP_ar_a(p)$,
865 \[
866 \bigl(C_\pi\cB_\lambda(p)\bigr)_a
867 =
868 \frac{\sum_{i\in I_a}\lambda_ip_i}{Z}
869 =
870 \frac{P_ar_a(p)}{\sum_bP_br_b(p)}
871 =
872 \bigl(\cB_{r(p)}(P)\bigr)_a.
873 \]
874 \end{proof}
875 
876 The next result gives the exact necessary and sufficient condition for a coarse likelihood to work for every prior.
877 
Statement: Bayesian refinement criterion · Go to proof · Read statement and proof side by side878 \begin{theorem}[Bayesian refinement criterion]\label{core:record:021}
879 Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive.  The identity
880 \[
881 \boxed{
882 C_\pi\cB_\lambda(p)
883 =
884 \cB_{\bar\lambda}(C_\pi p)}
885 \]
886 holds for every fine prior $p$ if and only if there exists one scalar $c>0$ such that
887 \[
888 \boxed{
889 \lambda_i=c\,\bar\lambda_a
890 \qquad(i\in I_a).}
891 \]
892 Equivalently, the likelihood is constant on each coarse packet, with the coarse values determined up to the globally invisible Bayesian scaling $c$.
893 \end{theorem}
894 
Proof block: Bayesian refinement criterion · Go to statement · Read statement and proof side by side895 \begin{proof}
896 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 $c$.
897 
898 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 $\lambda_a$.
899 
900 Now use priors supported on one state of packet $a$ and one state of packet $b$.  Equality of the two coarse posterior odds gives
901 \[
902 \frac{\lambda_a}{\lambda_b}
903 =
904 \frac{\bar\lambda_a}{\bar\lambda_b}.
905 \]
906 This holds for every pair of coarse packets, so $\lambda_a=c\bar\lambda_a$ for one common $c>0$.
907 \end{proof}
908 
909 \begin{remark}[The one-packet boundary]\label{core:record:022}
910 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.
911 \end{remark}
912 
913 When sufficiency fails, the defect still has an exact finite factorization.  Fix a declared coarse likelihood $\bar\lambda$, put
914 \[
915 P=C_\pi p,
916 \qquad
917 \delta(p)=r(p)-\bar\lambda,
918 \qquad
919 \bar b=\cB_{\bar\lambda}(P),
920 \qquad
921 Z_r=\langle r(p),P\rangle.
922 \]
923 
Statement: Bayesian information-loss packet · Go to proof · Read statement and proof side by side924 \begin{theorem}[Bayesian information-loss packet]\label{core:record:023}
925 Assume nonnegative fine and declared coarse likelihoods, with both $Z_r>0$ and $Z_{\bar\lambda}>0$. The coarsening defect is
926 \[
927 \boxed{
928 C_\pi\cB_\lambda(p)
929 -
930 \cB_{\bar\lambda}(P)
931 =
932 \frac{1}{Z_r}
933 \left(\diag(P)-\bar bP^{\mathsf T}\right)
934 \delta(p).}
935 \]
936 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,
937 \[
938 \boxed{
939 \left\|C_\pi\cB_\lambda(p)-\cB_{\bar\lambda}(P)\right\|_1
940 \le
941 \frac{2\sum_aP_a|\delta_a(p)|}
942 {\min\{Z_r,Z_{\bar\lambda}\}}}
943 \]
944 and
945 \[
946 \sum_aP_a|\delta_a(p)|
947 \le
948 \sum_i p_i\left|\lambda_i-\bar\lambda_{\pi(i)}\right|.
949 \]
950 \end{theorem}
951 
Proof block: Bayesian information-loss packet · Go to statement · Read statement and proof side by side952 \begin{proof}
953 The effective-likelihood theorem reduces the defect to
954 \[
955 \cB_{\bar\lambda+\delta}(P)-\cB_{\bar\lambda}(P).
956 \]
957 Since $Z_r=Z_{\bar\lambda}+P^{\mathsf T}\delta$, direct subtraction gives
958 \[
959 \frac{1}{Z_r}
960 \left(\diag(P)-\bar bP^{\mathsf T}\right)\delta.
961 \]
962 Also,
963 \[
964 \left(\diag(P)-\bar bP^{\mathsf T}\right)\bar\lambda
965 =P\odot\bar\lambda-\bar bZ_{\bar\lambda}=0.
966 \]
967 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 $P_a>0$,
968 \[
969 P_a|\delta_a|
970 =
971 P_a\left|\sum_{i\in I_a}\frac{p_i}{P_a}
972 (\lambda_i-\bar\lambda_a)\right|
973 \le
974 \sum_{i\in I_a}p_i|\lambda_i-\bar\lambda_a|,
975 \]
976 and summing over $a$ proves the second inequality.
977 \end{proof}
978 
Statement: Nested Bayesian packet tower · Go to proof · Read statement and proof side by side979 \begin{corollary}[Nested Bayesian packet tower]\label{core:record:024}
980 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.
981 \end{corollary}
982 
Proof block: Nested Bayesian packet tower · Go to statement · Read statement and proof side by side983 \begin{proof}
984 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.
985 \end{proof}
986 
Statement: Canonical coarsest likelihood-resolving partition · Go to proof · Read statement and proof side by side987 \begin{corollary}[Canonical coarsest likelihood-resolving partition]\label{core:record:025}
988 For a finite family of likelihoods $\{\lambda^{(y)}\}_{y\in Y}$, define
989 \[
990 i\sim j
991 \quad\Longleftrightarrow\quad
992 \lambda_i^{(y)}=\lambda_j^{(y)}
993 \text{ for every }y\in Y.
994 \]
995 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.
996 \end{corollary}
997 
Proof block: Canonical coarsest likelihood-resolving partition · Go to statement · Read statement and proof side by side998 \begin{proof}
999 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.
1000 \end{proof}
1001 
1002 \begin{figure}[htbp]
1003 \centering
1004 \includegraphics[width=0.92\textwidth]{17_bayesian_refinement_square.pdf}
1005 \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.}
1006 \label{core:fig:bayes-refinement}
1007 \end{figure}
1008 
1009 \begin{figure}[htbp]
1010 \centering
1011 \includegraphics[width=0.94\textwidth]{18_bayesian_information_loss.pdf}
1012 \caption{When packet sufficiency fails, the exact defect factors through the difference between the conditional effective likelihood and the declared coarse likelihood.}
1013 \label{core:fig:bayes-loss}
1014 \end{figure}
1015 
Statement: Bayesian refinement1016 \begin{certificate}[Bayesian refinement]\label{core:record:026}
1017 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.
1018 \end{certificate}
1019 
Statement: Finite exact theorem1020 \begin{boundary}[Finite exact theorem]\label{core:record:027}
1021 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.
1022 \end{boundary}
1023 
1024 
1025 \FloatBarrier
1026 \section{Bayesian flows, conjugacy, and hidden states}\label{core:section:09}
1027 
1028 Fix a score vector $g$. The likelihood flow is
1029 \[
1030 \lambda_i(t)=e^{tg_i},
1031 \]
1032 and the Bayesian flow is
1033 \[
1034 \Phi_t(p)_i
1035 =
1036 \frac{e^{tg_i}p_i}{\sum_je^{tg_j}p_j}.
1037 \]
1038 
Statement: Bayesian nonlinear semigroup · Go to proof · Read statement and proof side by side1039 \begin{theorem}[Bayesian nonlinear semigroup]\label{core:record:028}
1040 \[
1041 \boxed{
1042 \Phi_{s+t}=\Phi_s\circ\Phi_t.
1043 }
1044 \]
1045 Its generator is the centered multiplicative field
1046 \[
1047 \boxed{
1048 \frac{d}{dt}p_i(t)
1049 =
1050 \bigl(g_i-\langle g,p(t)\rangle\bigr)p_i(t).
1051 }
1052 \]
1053 \end{theorem}
1054 
Proof block: Bayesian nonlinear semigroup · Go to statement · Read statement and proof side by side1055 \begin{proof}
1056 The semigroup law is the likelihood-composition theorem with $e^{sg}e^{tg}=e^{(s+t)g}$. Differentiating the normalized exponential formula gives
1057 \[
1058 \dot p_i=g_ip_i-p_i\frac{\sum_jg_je^{tg_j}p_j(0)}{\sum_je^{tg_j}p_j(0)}.
1059 \]
1060 The quotient is $\langle g,p(t)\rangle$.
1061 \end{proof}
1062 
1063 \begin{figure}[htbp]
1064 \centering
1065 \includegraphics[width=0.80\textwidth]{07_bayesian_flow.pdf}
1066 \caption{Continuous likelihood accumulation produces a nonlinear semigroup on the probability simplex.}
1067 \label{core:fig:bayesian-flow}
1068 \end{figure}
1069 
1070 \begin{example}[Beta--Binomial conjugate packet]\label{core:record:029}
1071 A $\operatorname{Beta}(a,b)$ prior and $s$ successes in $n$ trials produce
1072 \[
1073 \operatorname{Beta}(a+s,b+n-s).
1074 \]
1075 The prior exponents are additive packets: the likelihood increments them by $(s,n-s)$. In the numerical example,
1076 \[
1077 (a,b)=(2,3),\quad (s,n)=(7,10),
1078 \]
1079 so the posterior is $\operatorname{Beta}(9,6)$ with mean $0.6$.
1080 \end{example}
1081 
1082 \begin{figure}[htbp]
1083 \centering
1084 \includegraphics[width=0.80\textwidth]{06_beta_binomial_conjugacy.pdf}
1085 \caption{A conjugate Bayesian update viewed as exact transfer of exponent packets.}
1086 \label{core:fig:beta-conjugacy}
1087 \end{figure}
1088 
1089 \subsection{Hidden Markov filtering}
1090 
1091 Let $K$ be column stochastic and let $\lambda^{(r)}$ be the likelihood associated with the $r$th observation. The filter recursion is
1092 \[
1093 \widehat p_r=Kp_{r-1},
1094 \qquad
1095 p_r=\cB_{\lambda^{(r)}}(\widehat p_r).
1096 \]
1097 
Statement: Prediction-update reconstruction · Go to proof · Read statement and proof side by side1098 \begin{proposition}[Prediction-update reconstruction]\label{core:record:030}
1099 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.
1100 \end{proposition}
1101 
Proof block: Prediction-update reconstruction · Go to statement · Read statement and proof side by side1102 \begin{proof}
1103 Column stochasticity gives $\widehat p_r\ge0$ and $\bm1^T\widehat p_r=1$. Bayesian normalization then gives $p_r\ge0$ and $\bm1^Tp_r=1$.
1104 \end{proof}
1105 
1106 \begin{figure}[htbp]
1107 \centering
1108 \includegraphics[width=0.80\textwidth]{15_hidden_markov_filter.pdf}
1109 \caption{A hidden Markov filter alternates a linear prediction semigroup with nonlinear Bayesian normalization.}
1110 \label{core:fig:hmm}
1111 \end{figure}
1112 
1113 \FloatBarrier
1114 \section{Fractional powers as spectral packet transfer}\label{core:section:10}
1115 \begin{readerroadmap}
1116 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.
1117 \end{readerroadmap}
1118 
1119 
1120 Let $L=L^*\ge0$ on a finite-dimensional Hilbert space. Its spectral resolution is
1121 \[
1122 L=\sum_j\lambda_jP_j.
1123 \]
1124 For $\alpha>0$, define
1125 \[
1126 \boxed{L^\alpha=\sum_j\lambda_j^\alpha P_j.}
1127 \]
1128 The associated fractional semigroup is
1129 \[
1130 T_\alpha(t)=e^{-tL^\alpha}
1131 =\sum_je^{-t\lambda_j^\alpha}P_j.
1132 \]
1133 
Statement: Fractional packet calculus · Go to proof · Read statement and proof side by side1134 \begin{theorem}[Fractional packet calculus]\label{core:record:031}
1135 For $\alpha,\beta>0$,
1136 \[
1137 L^\alpha L^\beta=L^{\alpha+\beta},
1138 \]
1139 and for $s,t\ge0$,
1140 \[
1141 \boxed{T_\alpha(s)T_\alpha(t)=T_\alpha(s+t).}
1142 \]
1143 Every spectral packet $P_j$ reduces $L^\alpha$ and $T_\alpha(t)$.
1144 \end{theorem}
1145 
Proof block: Fractional packet calculus · Go to statement · Read statement and proof side by side1146 \begin{proof}
1147 All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example,
1148 \[
1149 L^\alpha L^\beta
1150 =\sum_{j,k}\lambda_j^\alpha\lambda_k^\beta P_jP_k
1151 =\sum_j\lambda_j^{\alpha+\beta}P_j.
1152 \]
1153 \end{proof}
1154 
1155 \begin{figure}[htbp]
1156 \centering
1157 \includegraphics[width=0.80\textwidth]{09_fractional_semigroup.pdf}
1158 \caption{Different spectral exponents produce different transfer profiles from the same point seed.}
1159 \label{core:fig:fractional}
1160 \end{figure}
1161 
1162 \subsection{Square-root subordination}
1163 
Statement: Poisson subordination · Go to proof · Read statement and proof side by side1164 \begin{theorem}[Poisson subordination]\label{core:record:032}
1165 For $t>0$ and $L\ge0$,
1166 \[
1167 \boxed{
1168 e^{-t\sqrt L}
1169 =
1170 \frac{t}{2\sqrt\pi}
1171 \int_0^\infty
1172 s^{-3/2}e^{-t^2/(4s)}e^{-sL}\,ds.
1173 }
1174 \]
1175 The integral converges in operator norm in finite dimension.
1176 \end{theorem}
1177 
Proof block: Poisson subordination · Go to statement · Read statement and proof side by side1178 \begin{proof}
1179 By the spectral theorem it suffices to prove the scalar identity
1180 \[
1181 e^{-t\sqrt\lambda}
1182 =
1183 \frac{t}{2\sqrt\pi}
1184 \int_0^\infty
1185 s^{-3/2}e^{-t^2/(4s)-\lambda s}\,ds
1186 \]
1187 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.
1188 \end{proof}
1189 
1190 \subsection{Fractional naturality under a bounded refinement defect}
1191 
1192 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
1193 \[
1194 D=CA-BC,
1195 \qquad
1196 D_\alpha=CA^\alpha-B^\alpha C,
1197 \qquad 0<\alpha<1.
1198 \]
1199 
Statement: Fractional intertwining theorem · Go to proof · Read statement and proof side by side1200 \begin{theorem}[Fractional intertwining theorem]\label{core:record:033}\label{core:thm:fractional-naturality}
1201 For bounded nonnegative self-adjoint $A$ and $B$,
1202 \[
1203 \boxed{
1204 D_\alpha
1205 =
1206 \frac{\sin(\pi\alpha)}{\pi}
1207 \int_0^\infty
1208  t^\alpha(tI+B)^{-1}D(tI+A)^{-1}\,dt.
1209 }
1210 \]
1211 The improper integral converges in operator norm.  Moreover,
1212 \[
1213 \boxed{
1214 \norm{D_\alpha}
1215 \le
1216 K_\alpha\norm{C}^{1-\alpha}\norm{D}^{\alpha},
1217 \qquad
1218 K_\alpha=
1219 \frac{2^{1-\alpha}\sin(\pi\alpha)}
1220 {\pi\alpha(1-\alpha)}.
1221 }
1222 \]
1223 If $A,B\ge mI$ for some $m>0$, then the stronger linear estimate holds:
1224 \[
1225 \boxed{
1226 \norm{D_\alpha}
1227 \le
1228 \alpha m^{\alpha-1}\norm{D}.
1229 }
1230 \]
1231 In particular, exact generator intertwining $CA=BC$ implies exact fractional intertwining $CA^\alpha=B^\alpha C$.
1232 \end{theorem}
1233 
Proof block: Fractional intertwining theorem · Go to statement · Read statement and proof side by side1234 \begin{proof}
1235 For $0<\alpha<1$, the Balakrishnan formula gives
1236 \[
1237 A^\alpha
1238 =
1239 \frac{\sin(\pi\alpha)}{\pi}
1240 \int_0^\infty t^{\alpha-1}A(tI+A)^{-1}\,dt,
1241 \]
1242 and likewise for $B$.  Using
1243 \[
1244 A(tI+A)^{-1}=I-t(tI+A)^{-1}
1245 \]
1246 and the rectangular resolvent identity
1247 \[
1248 (tI+B)^{-1}C-C(tI+A)^{-1}
1249 =
1250 (tI+B)^{-1}(CA-BC)(tI+A)^{-1},
1251 \]
1252 the difference of the two Balakrishnan integrals becomes the displayed resolvent-defect formula.
1253 
1254 For small $t$,
1255 \[
1256 \norm{CA(tI+A)^{-1}-B(tI+B)^{-1}C}
1257 \le 2\norm C,
1258 \]
1259 because the two positive resolvent factors are contractions.  For large $t$, the resolvent identity gives the bound $\norm D/t$.  Splitting the integral at
1260 \[
1261 r=\frac{\norm D}{2\norm C}
1262 \]
1263 when $C$ and $D$ are nonzero yields
1264 \[
1265 \begin{aligned}
1266 \norm{D_\alpha}
1267 &\le
1268 \frac{\sin(\pi\alpha)}{\pi}
1269 \left(
1270 2\norm C\int_0^r t^{\alpha-1}\,dt
1271 +
1272 \norm D\int_r^\infty t^{\alpha-2}\,dt
1273 \right)\\
1274 &=
1275 K_\alpha\norm C^{1-\alpha}\norm D^\alpha.
1276 \end{aligned}
1277 \]
1278 The zero cases are immediate.
1279 
1280 If $A,B\ge mI$, then both resolvents have norm at most $(t+m)^{-1}$, so
1281 \[
1282 \norm{D_\alpha}
1283 \le
1284 \frac{\sin(\pi\alpha)}{\pi}\norm D
1285 \int_0^\infty\frac{t^\alpha}{(t+m)^2}\,dt.
1286 \]
1287 After $t=mu$, the beta integral equals
1288 \[
1289 m^{\alpha-1}\frac{\alpha\pi}{\sin(\pi\alpha)},
1290 \]
1291 which gives the linear estimate.
1292 \end{proof}
1293 
Statement: Fractional semigroup refinement · Go to proof · Read statement and proof side by side1294 \begin{corollary}[Fractional semigroup refinement]\label{core:record:034}\label{core:cor:fractional-semigroup-refinement}
1295 For $s\ge0$,
1296 \[
1297 \boxed{
1298 Ce^{-sA^\alpha}-e^{-sB^\alpha}C
1299 =
1300 -\int_0^s
1301  e^{-(s-r)B^\alpha}D_\alpha e^{-rA^\alpha}\,dr.
1302 }
1303 \]
1304 Hence
1305 \[
1306 \norm{Ce^{-sA^\alpha}-e^{-sB^\alpha}C}
1307 \le s\norm{D_\alpha}.
1308 \]
1309 Under the common gap $A,B\ge mI$,
1310 \[
1311 \boxed{
1312 \norm{Ce^{-sA^\alpha}-e^{-sB^\alpha}C}
1313 \le
1314 s e^{-sm^\alpha}\alpha m^{\alpha-1}\norm D.
1315 }
1316 \]
1317 \end{corollary}
1318 
Proof block: Fractional semigroup refinement · Go to statement · Read statement and proof side by side1319 \begin{proof}
1320 Differentiate
1321 \[
1322 r\longmapsto e^{-(s-r)B^\alpha}Ce^{-rA^\alpha}
1323 \]
1324 and integrate from $0$ to $s$.  Positive fractional semigroups are contractions; under the gap, the product of their two norm bounds is $e^{-sm^\alpha}$.
1325 \end{proof}
1326 
Statement: Fractional refinement towers1327 \begin{corollary}[Fractional refinement towers]\label{core:record:035}
1328 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
1329 \[
1330 \sum_\ell\norm{D_\ell}^{\alpha}<\infty,
1331 \]
1332 then the fractional defects
1333 \[
1334 C_\ell A_{\ell+1}^{\alpha}-A_\ell^{\alpha}C_\ell
1335 \]
1336 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.
1337 \end{corollary}
1338 
1339 \begin{figure}[htbp]
1340 \centering
1341 \includegraphics[width=0.94\textwidth]{19_fractional_naturality.pdf}
1342 \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.}
1343 \label{core:fig:fractional-naturality}
1344 \end{figure}
1345 
Statement: Unbounded and nonnormal generators1346 \begin{boundary}[Unbounded and nonnormal generators]\label{core:record:036}
1347 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.
1348 \end{boundary}
1349 
Statement: Non-self-adjoint fractional powers1350 \begin{boundary}[Non-self-adjoint fractional powers]\label{core:record:037}
1351 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.
1352 \end{boundary}
1353 
1354 \FloatBarrier
1355 \section{Green operators and the Poisson seed}\label{core:section:11}
1356 
1357 On the interval $(0,1)$ with Dirichlet boundary conditions, the Green kernel for $-u''=f$ is
1358 \[
1359 \boxed{G(x,y)=\min(x,y)-xy.}
1360 \]
1361 Thus the point source $\delta_y$ generates the solution kernel, and a general source is reconstructed by
1362 \[
1363 u(x)=\int_0^1G(x,y)f(y)\,dy.
1364 \]
1365 
1366 For $n$ interior points with spacing $h=1/(n+1)$, define
1367 \[
1368 L_h=\frac1{h^2}
1369 \begin{pmatrix}
1370 2&-1&&\\
1371 -1&2&\ddots&\\
1372 &\ddots&\ddots&-1\\
1373 &&-1&2
1374 \end{pmatrix}.
1375 \]
1376 
Statement: Exact discrete Green reconstruction at grid nodes · Go to proof · Read statement and proof side by side1377 \begin{theorem}[Exact discrete Green reconstruction at grid nodes]\label{core:record:038}
1378 Let $x_i=ih$. Then
1379 \[
1380 \boxed{
1381 \frac1h(L_h^{-1})_{ij}
1382 =G(x_i,x_j).
1383 }
1384 \]
1385 Thus the scaled discrete inverse reproduces the continuum Dirichlet Green kernel exactly at the grid nodes.
1386 \end{theorem}
1387 
Proof block: Exact discrete Green reconstruction at grid nodes · Go to statement · Read statement and proof side by side1388 \begin{proof}
1389 The inverse of the unscaled tridiagonal matrix has entries
1390 \[
1391 \frac{\min(i,j)(n+1-\max(i,j))}{n+1}.
1392 \]
1393 Multiplying by $h^2$ and then dividing by $h$ gives
1394 \[
1395 h\min(i,j)\left(1-\frac{\max(i,j)}{n+1}\right)
1396 =\min(x_i,x_j)-x_ix_j.
1397 \]
1398 \end{proof}
1399 
1400 \begin{figure}[htbp]
1401 \centering
1402 \includegraphics[width=0.80\textwidth]{10_poisson_green_kernel.pdf}
1403 \caption{A section of the continuum Green kernel and the scaled discrete inverse. They agree at the grid nodes.}
1404 \label{core:fig:green}
1405 \end{figure}
1406 
1407 \subsection{Low-mode convergence}
1408 
1409 The $k$th discrete eigenvalue is
1410 \[
1411 \lambda_{k,h}=\frac4{h^2}\sin^2\left(\frac{k\pi h}{2}\right),
1412 \]
1413 whereas the continuum eigenvalue is $(k\pi)^2$.
1414 
Statement: Second-order low-mode convergence · Go to proof · Read statement and proof side by side1415 \begin{theorem}[Second-order low-mode convergence]\label{core:record:039}
1416 For fixed $k$,
1417 \[
1418 \lambda_{k,h}=(k\pi)^2+O(h^2).
1419 \]
1420 More precisely, for sufficiently small $h$,
1421 \[
1422 0\le (k\pi)^2-\lambda_{k,h}
1423 \le \frac{(k\pi)^4}{12}h^2.
1424 \]
1425 \end{theorem}
1426 
Proof block: Second-order low-mode convergence · Go to statement · Read statement and proof side by side1427 \begin{proof}
1428 For $x\ge0$, $0\le x^2-\sin^2x\le x^4/3$. Set $x=k\pi h/2$ and multiply by $4/h^2$.
1429 \end{proof}
1430 
1431 \begin{figure}[htbp]
1432 \centering
1433 \includegraphics[width=0.72\textwidth]{11_pde_refinement_convergence.pdf}
1434 \caption{The first finite-difference eigenvalue converges with observed order approximately two.}
1435 \label{core:fig:pde-convergence}
1436 \end{figure}
1437 
1438 \FloatBarrier
1439 \section{Finite-state Feynman--Kac transfer}\label{core:section:12}
1440 
1441 Let $K$ be a column-stochastic matrix, let $V$ be a potential on the states, and let $f$ be terminal data. Define
1442 \[
1443 M_V=\diag(e^{-V})K^T.
1444 \]
1445 The backward Feynman--Kac value after $n$ steps is
1446 \[
1447 u_n=M_V^nf.
1448 \]
1449 
Statement: Discrete Feynman Kac path formula · Go to proof · Read statement and proof side by side1450 \begin{theorem}[Discrete Feynman--Kac path formula]\label{core:record:040}
1451 For an initial state $i$,
1452 \[
1453 \boxed{
1454 (u_n)_i
1455 =
1456 \Exp_i\left[
1457 \exp\left(-\sum_{r=0}^{n-1}V(X_r)\right)f(X_n)
1458 \right].
1459 }
1460 \]
1461 \end{theorem}
1462 
Proof block: Discrete Feynman Kac path formula · Go to statement · Read statement and proof side by side1463 \begin{proof}
1464 For $n=1$,
1465 \[
1466 (M_Vf)_i=e^{-V(i)}\sum_jK_{j i}f_j,
1467 \]
1468 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 $\prod_re^{-V(i_r)}$.
1469 \end{proof}
1470 
1471 \begin{figure}[htbp]
1472 \centering
1473 \includegraphics[width=0.78\textwidth]{12_feynman_kac_transfer.pdf}
1474 \caption{Potential-weighted transfer values computed by the finite Feynman--Kac operator. Exhaustive path enumeration agrees with the matrix formula in the supplied tests.}
1475 \label{core:fig:feynman-kac}
1476 \end{figure}
1477 
1478 \FloatBarrier
1479 \section{Exact elimination of latent packets}\label{core:section:13}
1480 \begin{keyidea}
1481 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.
1482 \end{keyidea}
1483 
1484 
1485 Let $X=P X\oplus Q X$ and write
1486 \[
1487 A=
1488 \begin{pmatrix}
1489 A_{PP}&A_{PQ}\\
1490 A_{QP}&A_{QQ}
1491 \end{pmatrix}.
1492 \]
1493 
Statement: Schur Feshbach projected resolvent · Go to proof · Read statement and proof side by side1494 \begin{theorem}[Schur--Feshbach projected resolvent]\label{core:record:041}
1495 Assume both $zI-A$ and $zI-A_{QQ}$ are invertible. Define
1496 \[
1497 \boxed{
1498 F_P(z)
1499 =zI-A_{PP}-A_{PQ}(zI-A_{QQ})^{-1}A_{QP}.
1500 }
1501 \]
1502 Then
1503 \[
1504 \boxed{
1505 \left.P(zI-A)^{-1}\right|_{\Ran P}=F_P(z)^{-1}.
1506 }
1507 \]
1508 \end{theorem}
1509 
Proof block: Schur Feshbach projected resolvent · Go to statement · Read statement and proof side by side1510 \begin{proof}
1511 Solve the hidden block equation
1512 \[
1513 (zI-A_{QQ})x_Q=A_{QP}x_P+y_Q
1514 \]
1515 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.
1516 \end{proof}
1517 
1518 The self-energy
1519 \[
1520 \Sigma_P(z)=A_{PQ}(zI-A_{QQ})^{-1}A_{QP}
1521 \]
1522 contains the exact effect of the eliminated packets. A bare truncation $A_{PP}$ discards this term and is generally not exact.
1523 
1524 \subsection{Time-domain memory}
1525 
1526 For the homogeneous system $\dot x=Ax$, variation of constants gives
1527 \[
1528 x_Q(t)=e^{tA_{QQ}}x_Q(0)
1529 +\int_0^te^{(t-r)A_{QQ}}A_{QP}x_P(r)\,dr.
1530 \]
1531 Substitution yields
1532 \[
1533 \dot x_P(t)
1534 =A_{PP}x_P(t)+A_{PQ}e^{tA_{QQ}}x_Q(0)
1535 +\int_0^t\cK(t-r)x_P(r)\,dr,
1536 \]
1537 where
1538 \[
1539 \boxed{
1540 \cK(t)=A_{PQ}e^{tA_{QQ}}A_{QP}.
1541 }
1542 \]
1543 
Statement: Memory/self-energy correspondence · Go to proof · Read statement and proof side by side1544 \begin{theorem}[Memory/self-energy correspondence]\label{core:record:042}
1545 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
1546 \[
1547 \boxed{
1548 \int_0^\infty e^{-zt}\cK(t)\,dt
1549 =A_{PQ}(zI-A_{QQ})^{-1}A_{QP}
1550 =\Sigma_P(z).
1551 }
1552 \]
1553 \end{theorem}
1554 
Proof block: Memory/self-energy correspondence · Go to statement · Read statement and proof side by side1555 \begin{proof}
1556 Put $B=A_{QQ}$ and $T_z(t)=e^{-zt}e^{tB}=e^{t(B-zI)}$.
1557 The generalized-eigenspace decomposition in finite dimensions writes $T_z$ as a
1558 finite sum of constant matrix coefficients times $t^k e^{(\lambda-z)t}$.
1559 This follows by expanding the nilpotent part on each generalized eigenspace;
1560 equivalently, apply the decomposition to left multiplication by $B$ on the matrix
1561 algebra and evaluate the exponential at $I$.
1562 The strict spectral inequality makes every such term absolutely integrable on
1563 $(0,\infty)$ and forces it to tend to zero. Hence $T_z$ is Bochner integrable
1564 and $T_z(t)\to0$, including when $B$ is nonnormal.
1565 
1566 Let $J=\int_0^\infty T_z(t)\,dt$. Integrating both derivative identities
1567 $T_z'=(B-zI)T_z=T_z(B-zI)$ and using $T_z(0)=I$ gives
1568 \[
1569 (zI-B)J=J(zI-B)=I.
1570 \]
1571 Thus $J=(zI-B)^{-1}$. Continuous left and right multiplication carry the
1572 Bochner integral through the two rectangular transfer blocks, yielding the claim.
1573 \end{proof}
1574 
1575 \begin{figure}[htbp]
1576 \centering
1577 \includegraphics[width=0.78\textwidth]{13_latent_memory_kernel.pdf}
1578 \caption{The norm of the exact memory kernel produced by eliminated packets.}
1579 \label{core:fig:memory}
1580 \end{figure}
1581 
1582 \FloatBarrier
1583 \section{Survival and hazard semigroups}\label{core:section:14}
1584 
1585 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
1586 \[
1587 p(t)=e^{tA}p_0,
1588 \qquad
1589 S(t)=\bm1^Tp(t).
1590 \]
1591 The exit-rate row vector is
1592 \[
1593 a^T=-\bm1^TA\ge0.
1594 \]
1595 
Statement: Semigroup survival identities · Go to proof · Read statement and proof side by side1596 \begin{theorem}[Semigroup survival identities]\label{core:record:043}
1597 For $t\ge0$, the finite-time exit density and, wherever $S(t)>0$, the hazard are
1598 \[
1599 \boxed{
1600 f(t)=a^Te^{tA}p_0=-S'(t),
1601 \qquad
1602 h(t)=\frac{f(t)}{S(t)}.
1603 }
1604 \]
1605 Consequently, $S$ is nonincreasing and
1606 \[
1607 S(t)=\exp\left(-\int_0^th(r)\,dr\right)
1608 \]
1609 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.
1610 \end{theorem}
1611 
Proof block: Semigroup survival identities · Go to statement · Read statement and proof side by side1612 \begin{proof}
1613 Differentiate:
1614 \[
1615 S'(t)=\bm1^TAe^{tA}p_0=-a^Te^{tA}p_0.
1616 \]
1617 The hazard identity is the definition of conditional instantaneous exit rate. Integrating $(\log S)'=-h$ gives the final formula.
1618 \end{proof}
1619 
1620 \begin{figure}[htbp]
1621 \centering
1622 \includegraphics[width=0.78\textwidth]{16_survival_hazard.pdf}
1623 \caption{Survival and instantaneous hazard produced by one transient transfer semigroup.}
1624 \label{core:fig:survival}
1625 \end{figure}
1626 
1627 \FloatBarrier
1628 \section{One common operator language}\label{core:section:15}
1629 
1630 The preceding constructions are not identical, but they use the same packet operations. The domain dictionary is summarized in \cref{core:tab:dictionary}.
1631 
1632 \begin{table}[htbp]
1633 \centering
1634 \small
1635 \begin{tabularx}{\textwidth}{L{0.17\textwidth} L{0.19\textwidth} Y Y}
1636 \toprule
1637 Realization & Seed & Transfer & Packet/reconstruction law\\
1638 \midrule
1639 Markov & point mass $\delta_i$ & $K^n$ or $e^{tQ}$ & state lumping and $CK^n=\bar K^nC$\\
1640 Conditional & random variable & $P_{\mathcal Q}$ & nested partitions and tower law\\
1641 Bayesian & prior $p$ & $\cB_\lambda$ & packet evidences and normalized synthesis\\
1642 Fractional & spectral source & $e^{-tL^\alpha}$ & spectral projectors $P_j$\\
1643 PDE & source $f$ or $\delta_y$ & resolvent/Green operator & eigenmode or spatial packet synthesis\\
1644 Feynman--Kac & terminal data $f$ & $(\diag(e^{-V})K^T)^n$ & weighted path reconstruction\\
1645 Latent system & retained state & Feshbach/self-energy & hidden packets return as memory\\
1646 Survival & initial transient law & $e^{tA}$ & sum over transient packets\\
1647 \bottomrule
1648 \end{tabularx}
1649 \caption{The transfer-semigroup dictionary in the common operator language.}
1650 \label{core:tab:dictionary}
1651 \end{table}
1652 
1653 For clarity, the finite-dimensional holomorphic calculus used below has the
1654 following canonical definition. For $A$ acting on $\mathbb C^d$, its generalized
1655 eigenspaces $E_\mu=\ker(A-\mu I)^d$ form a direct-sum decomposition over the
1656 eigenvalues. Define the unique linear operator $F(A)$ by
1657 \[
1658 \left.F(A)\right|_{E_\mu}
1659 =\left.\sum_{k=0}^{d-1}\frac{F^{(k)}(\mu)}{k!}(A-\mu I)^k\right|_{E_\mu}.
1660 \]
1661 The formula is independent of a basis and of increasing the truncation index:
1662 the extra nilpotent powers vanish. It depends only on the finite Taylor jets of
1663 $F$ at the spectrum. If $F$ is holomorphic on an open neighborhood $U$ of the
1664 spectrum, each eigenvalue admits a radius $r_\mu>0$ with
1665 $\overline{D(\mu,r_\mu)}\subset U$, and Cauchy's formula gives
1666 \[
1667 \frac{F^{(k)}(\mu)}{k!}
1668 =\frac{1}{2\pi i}\oint_{|z-\mu|=r_\mu}
1669   \frac{F(z)}{(z-\mu)^{k+1}}\,dz.
1670 \]
1671 Consequently the same canonical operator is obtained from any such local
1672 Cauchy circles, even when $U$ is disconnected. This is the standard finite
1673 Taylor-jet definition of the holomorphic matrix function.
1674 
Statement: Naturality under exact intertwining · Go to proof · Read statement and proof side by side1675 \begin{theorem}[Naturality under exact intertwining]\label{core:record:044}
1676 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
1677 \[
1678 CA_f=A_cC,
1679 \]
1680 then
1681 \[
1682 \boxed{CF(A_f)=F(A_c)C.}
1683 \]
1684 In particular, exact generator refinement propagates simultaneously to semigroups, resolvents, integer powers, and analytic transfer functions.
1685 \end{theorem}
1686 
Proof block: Naturality under exact intertwining · Go to statement · Read statement and proof side by side1687 \begin{proof}
1688 The intertwining identity implies
1689 $C(A_f-\mu I)^k=(A_c-\mu I)^kC$ for every $k\ge0$.
1690 It therefore sends the generalized $\mu$-eigenspace of $A_f$ into that of $A_c$.
1691 Use a common truncation index at least as large as both dimensions in the
1692 canonical formula above. Multiplying its finite sum by $C$ now gives the desired
1693 identity on every generalized eigenspace. The direct-sum decomposition gives
1694 it on the full space. The argument covers nilpotent Jordan parts and uses only
1695 local spectral Taylor jets; it does not require one Taylor series to converge
1696 on both spectra. The zero-dimensional case is immediate.
1697 \end{proof}
1698 
1699 \begin{remark}[Where nonlinearity enters]\label{core:record:045}
1700 Bayesian conditioning does not fit the preceding linear functional calculus because normalization depends on the current state. It nevertheless obeys an exact composition law and has a closed packet reconstruction. Paper III supplies the general nonlinear coordinate theorem; the Bayesian map is a particularly transparent probabilistic realization.
1701 \end{remark}
1702 
1703 \FloatBarrier
1704 \section{Numerical certificates}\label{core:section:16}
1705 
1706 All calculations in this release 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.
1707 
1708 \begin{table}[htbp]
1709 \centering
1710 \small
1711 \begin{tabular}{L{0.56\textwidth} r}
1712 \toprule
1713 Certificate & Value\\
1714 \midrule
1715 Packet-block semigroup composition residual & $2.554\times10^{-15}$\\
1716 Operator reconstruction residual & $0$\\
1717 Random-walk mass error & $0$\\
1718 Exact Markov generator-lumping defect & $5.551\times10^{-17}$\\
1719 Exact finite-horizon lumping error & $2.220\times10^{-16}$\\
1720 Perturbed ten-step lumping error & $6.523\times10^{-3}$\\
1721 Linear defect bound for that example & $2.000\times10^{-1}$\\
1722 Bayesian composition residual & $5.551\times10^{-17}$\\
1723 Bayesian packet reconstruction residual & $0$\\
1724 Bayesian refinement commutation residual & $1.110\times10^{-16}$\\
1725 Bayesian defect-factorization residual & $7.286\times10^{-17}$\\
1726 Bayesian flow generator residual & $1.959\times10^{-11}$\\
1727 Conditional-expectation tower residual & $0$\\
1728 Hermite Gram residual & $9.992\times10^{-15}$\\
1729 Fractional semigroup residual & $2.082\times10^{-17}$\\
1730 Maximum square-root subordination error & $1.665\times10^{-16}$\\
1731 Exact fractional-naturality residual & $0$\\
1732 Fractional-power defect / gap certificate & $0.51059 / 0.95079$\\
1733 Fractional-semigroup defect / certificate & $0.12969 / 0.44480$\\
1734 Observed low-mode PDE convergence order & $1.999854$\\
1735 Feynman--Kac enumeration residual & $1.665\times10^{-16}$\\
1736 Feshbach projected-resolvent residual & $5.551\times10^{-17}$\\
1737 Memory Laplace-transform residual & $6.939\times10^{-18}$\\
1738 Duhamel refinement identity residual & $2.108\times10^{-16}$\\
1739 Survival monotonicity violation & $0$\\
1740 Hidden-Markov normalization error & $1.110\times10^{-16}$\\
1741 \bottomrule
1742 \end{tabular}
1743 \caption{Selected results from the machine-readable numerical record.}
1744 \label{core:tab:numerics}
1745 \end{table}
1746 
Statement: Test suite1747 \begin{certificate}[Test suite]\label{core:record:046}
1748 The September 6 release records the following automated-test result:
1749 \[
1750 \boxed{54/54\text{ automated tests}.}
1751 \]
1752 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.
1753 \end{certificate}
1754 
1755 \FloatBarrier
1756 \section{Remaining conjectural extensions}\label{core:section:17}
1757 
1758 The finite theory closes several operator identities but does not settle the corresponding universal continuum questions.
1759 
Statement: Transfer-semigroup realization1760 \begin{conjecture}[Transfer-semigroup realization]\label{core:record:047}
1761 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.
1762 \end{conjecture}
1763 
Statement: Uniform refinement control1764 \begin{conjecture}[Uniform refinement control]\label{core:record:048}
1765 For a nested family of physically or analytically meaningful discretizations, there exist norms and constants independent of resolution such that
1766 \[
1767 \norm{C_\ell e^{tA_{\ell+1}}-e^{tA_\ell}C_\ell}
1768 \le
1769 M\,t\,\norm{C_\ell A_{\ell+1}-A_\ell C_\ell}
1770 \]
1771 on every bounded time interval. Summable generator defects would then produce a coherent inverse-limit evolution.
1772 \end{conjecture}
1773 
Statement: Latent-packet closure1774 \begin{conjecture}[Latent-packet closure]\label{core:record:049}
1775 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.
1776 \end{conjecture}
1777 
Statement: What remains outside Paper VI1778 \begin{boundary}[What remains outside Paper VI]\label{core:record:050}
1779 The following issues are not proved here:
1780 \begin{enumerate}[leftmargin=2em]
1781 \item existence and uniqueness for arbitrary stochastic differential equations;
1782 \item posterior consistency or asymptotic normality in infinite statistical models;
1783 \item domain invariance for unbounded fractional operators;
1784 \item tightness and compactness required for stochastic continuum limits;
1785 \item nonlinear PDE well-posedness beyond the stated finite and linear models;
1786 \item an optimal packet choice for a general Markov, Bayesian, or PDE problem.
1787 \end{enumerate}
1788 These belong to later realization-specific work or to external analytic theory.
1789 \end{boundary}
1790 
1791 \FloatBarrier
1792 \section{Computational implementation}\label{core:section:18}
1793 
1794 The accompanying Python package implements the formulas and finite models used in the paper.
1795 \begin{center}
1796 \texttt{projector\_transfer\_semigroups}
1797 \end{center}
1798 The computational organization mirrors the paper:
1799 \[
1800 \begin{array}{c|l}
1801 \texttt{packets.py} & \text{projectors, blocks, reconstruction, block composition}\\
1802 \texttt{semigroup.py} & \text{matrix semigroups, defects, Feshbach operators, memory}\\
1803 \texttt{markov.py} & \text{random walks, exact lumping, survival}\\
1804 \texttt{bayes.py} & \text{normalized updates, packet evidence, conjugacy, flows, HMM}\\
1805 \texttt{conditional.py} & \text{conditional expectation and Gaussian chaos}\\
1806 \texttt{fractional.py} & \text{spectral powers, subordination, and fractional refinement certificates}\\
1807 \texttt{pde.py} & \text{Dirichlet Laplacians, Green kernels, eigenvalue error}\\
1808 \texttt{feynman\_kac.py} & \text{weighted transfer and exhaustive path enumeration}.
1809 \end{array}
1810 \]
1811 
1812 \subsection{Representative algorithms}
1813 
1814 \begin{lstlisting}[caption={Normalized finite Bayesian update.}]
1815 def bayes_update(prior, likelihood):
1816     p = normalize(prior)
1817     evidence = likelihood @ p
1818     if evidence <= 0:
1819         raise ValueError("positive evidence required")
1820     return likelihood * p / evidence
1821 \end{lstlisting}
1822 
1823 \begin{lstlisting}[caption={Packet block composition.}]
1824 def block_composition(left, right):
1825     result = []
1826     for beta in range(len(left)):
1827         row = []
1828         for alpha in range(len(left)):
1829             value = sum(
1830                 left[beta][gamma] @ right[gamma][alpha]
1831                 for gamma in range(len(left))
1832             )
1833             row.append(value)
1834         result.append(row)
1835     return result
1836 \end{lstlisting}
1837 
1838 \begin{lstlisting}[caption={Finite Schur-Feshbach effective operator.}]
1839 def feshbach_effective_operator(A, retained, z):
1840     App, Apq, Aqp, Aqq = split_blocks(A, retained)
1841     return z*I - App - Apq @ inv(z*I - Aqq) @ Aqp
1842 \end{lstlisting}
1843 
1844 The full implementation, tests, tables, figure-generation code, and exact numerical record are included with the release rather than embedded in the typeset paper.
1845 
1846 \FloatBarrier
1847 \section{Conclusion}\label{core:section:19}
1848 
1849 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 now characterized by packetwise likelihood sufficiency. Fractional calculus is spectral packet reweighting, and bounded positive refinement defects now 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.
1850 
1851 The central structural identity is
1852 \[
1853 \boxed{
1854 \text{generator defect}
1855 \longrightarrow
1856 \text{Duhamel semigroup defect}
1857 \longrightarrow
1858 \text{controlled refinement error}.
1859 }
1860 \]
1861 Together with exact packet reconstruction, this makes the semigroup paper the natural bridge from the foundational calculus to the quantum, computational, and arithmetic realizations that follow.
1862 
1863 \clearpage
1864 \appendix
1865 \FloatBarrier
1866 \section{Lean verification and scope}\label{core:section:20}\label{core:sec:lean-status}
1867 \noindent\textbf{Fifth checked block, 6 September 2026.}
1868 The release contains 341 explicit Lean theorem declarations across 47 proof modules,
1869 including 31 additional declarations since the fourth checked block. The September 6 release records a successful full
1870 build and namespace audit with Lean 4.34.0-rc2 and pinned mathlib. The audit
1871 inspects 782 declarations, including definitions and generated auxiliaries; only
1872 \texttt{propext}, \texttt{Classical.choice} and \texttt{Quot.sound} are allowed.
1873 These are different counting units from the paper's 33 theorem-like results.
1874 
1875 The ledger has \textbf{33 complete, zero partial and zero pending results}.
1876 The last two results, 13.2 and 15.1, are now complete. The three separately stated
1877 conjectures remain open and are excluded from this count.
1878 
1879 ``Complete'' means the stated scope follows from the named checked targets
1880 and the ordinary specializations recorded in the ledger. Prose-to-Lean correspondence
1881 is manually reviewed. The fractional targets use complex Hilbert spaces;
1882 complexification for real operators is an ordinary mathematical specialization,
1883 not a separately checked Lean construction. Completion of the theorem ledger does
1884 not certify every prose sentence, numerical figure or open conjecture.
1885 
1886 \small
1887 \begin{longtable}{@{}p{0.08\textwidth}p{0.68\textwidth}p{0.17\textwidth}@{}}
1888 \toprule Result & Manuscript statement & Lean scope\\
1889 \midrule\endfirsthead
1890 \toprule Result & Manuscript statement & Lean scope\\\midrule\endhead
1891 \bottomrule\endfoot
1892 1.2 & Finite semigroup law & Complete\\
1893 2.1 & Packet semigroup convolution & Complete\\
1894 2.2 & Reducing packets & Complete\\
1895 3.1 & Exact Duhamel refinement identity & Complete\\
1896 3.2 & Norm bound & Complete\\
1897 3.3 & Resolvent refinement identity & Complete\\
1898 4.1 & Finite Markov semigroup & Complete\\
1899 5.1 & Exact lumpability & Complete\\
1900 5.2 & Finite-horizon defect formula & Complete\\
1901 6.1 & Finite conditional-expectation projector & Complete\\
1902 6.2 & Tower law and martingale packets & Complete\\
1903 7.1 & Independent-likelihood composition & Complete\\
1904 7.2 & Bayesian packet reconstruction & Complete\\
1905 7.3 & Likelihood perturbation bound & Complete\\
1906 8.1 & Exact effective-likelihood factorization & Complete\\
1907 8.2 & Bayesian refinement criterion & Complete\\
1908 8.4 & Bayesian information-loss packet & Complete\\
1909 8.5 & Nested Bayesian packet tower & Complete\\
1910 8.6 & Canonical coarsest likelihood-resolving partition & Complete\\
1911 9.1 & Bayesian nonlinear semigroup & Complete\\
1912 9.3 & Prediction-update reconstruction & Complete\\
1913 10.1 & Fractional packet calculus & Complete\\
1914 10.2 & Poisson subordination & Complete\\
1915 10.3 & Fractional intertwining theorem & Complete\\
1916 10.4 & Fractional semigroup refinement & Complete\\
1917 10.5 & Fractional refinement towers & Complete\\
1918 11.1 & Exact discrete Green reconstruction at grid nodes & Complete\\
1919 11.2 & Second-order low-mode convergence & Complete\\
1920 12.1 & Discrete Feynman--Kac path formula & Complete\\
1921 13.1 & Schur--Feshbach projected resolvent & Complete\\
1922 13.2 & Memory/self-energy correspondence & Complete\\
1923 14.1 & Semigroup survival identities & Complete\\
1924 15.1 & Naturality under exact intertwining & Complete\\
1925 \end{longtable}
1926 \normalsize
1927 
1928 \paragraph{General nonnormal memory (13.2).}
1929 The actual damped exponential is decomposed into finitely many terms of the form
1930 $t^k e^{(\mu-z)t}$. Integrability and convergence to zero follow from
1931 $\operatorname{Re}\mu<\operatorname{Re}z$. The formal proof uses generalized
1932 eigenspaces of left multiplication on the matrix algebra, so nilpotent Jordan
1933 parts are included. Both inverse identities follow by integrating the two
1934 exponential derivative formulas. A continuous linear sandwich map gives the
1935 literal rectangular matrix memory integral and the actual inverse. No normality
1936 or externally supplied growth bound is required.
1937 
1938 \paragraph{Canonical holomorphic naturality (15.1).}
1939 The standard finite Taylor-jet calculus defined before the theorem is constructed
1940 using the direct sum of generalized eigenspaces. Uniqueness, cutoff stability and
1941 exact rectangular naturality are proved. Cauchy's formula identifies the local
1942 coefficients with actual circle integrals; admissible radii exist in every stated
1943 open holomorphic neighborhood, and changing them leaves the canonical operator
1944 unchanged. The argument permits disconnected neighborhoods and different matrix
1945 dimensions. It makes no assertion about general Banach-algebra contour deformation.
1946 
1947 \paragraph{Fractional results (10.3--10.5).}
1948 The inherited fourth block proves the scalar beta normalization, the Lebesgue
1949 integral representation of actual positive fractional powers, the literal
1950 resolvent-defect formula, both printed norm constants and exact intertwining.
1951 The rectangular complex-Hilbert-space semigroup proof substitutes those constants
1952 into Duhamel's identity. The varying-space tower proof gives summability of
1953 fractional-defect norms and uniform convergence of time-defect norm series on
1954 bounded time intervals. Here ``summable'' means summable in norm. Operators with
1955 unrelated codomains are not added together; a separate fixed-space theorem proves
1956 operator-series convergence when such a sum is defined.
1957 
1958 \paragraph{Verification and numerical evidence.}
1959 The September 6 release records that the full pinned build, project-wide axiom audit, all 54 companion tests and six
1960 verifier controls passed. No fresh Lean build is asserted for this October edition. The 54 tests comprise the original 37 and 17 added boundary
1961 and validation cases. The 19 numerical figure pairs are unchanged. The earlier
1962 79-field numerical comparison remains historical evidence, not a rerun in that September 6
1963 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
1964 previous checked ledgers accompany the source archive. Numerical tests are finite
1965 regressions and are not substitutes for the Lean proofs.
1966 
1967 
1968 \clearpage
1969 \part*{Part II\quad Extended mathematical development}\addcontentsline{toc}{part}{Part II: Extended mathematical development}
1970 \setcounter{section}{0}\setcounter{figure}{0}\setcounter{table}{0}
1971 \renewcommand{\thesection}{E\arabic{section}}
1972 \renewcommand{\theHsection}{E.\arabic{section}}
1973 \renewcommand{\thefigure}{E\arabic{figure}}
1974 \renewcommand{\theHfigure}{E.\arabic{figure}}
1975 \renewcommand{\thetable}{E\arabic{table}}
1976 \renewcommand{\theHtable}{E.\arabic{table}}
1977 \graphicspath{{extension_figures/}}
1978 \begin{abstract}
1979 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.
1980 \end{abstract}
1981 
1982 
1983 \FloatBarrier
1984 \section{Introduction}\label{extended:section:01}\label{extended:sec:introduction}
1985 
1986 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}.
1987 
1988 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.
1989 
1990 The first part of the article treats 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
1991 \[
1992 \Delta=CA_f-A_cC
1993 \]
1994 produces the entire semigroup defect through a Duhamel integral. The same defect controls resolvents and holomorphic functions of the generators.
1995 
1996 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.
1997 
1998 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.
1999 
2000 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.
2001 
2002 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
2003 \[
2004 \norm{CL_f^\alpha-L_c^\alpha C}
2005 \leq
2006 \alpha m^{\alpha-1}
2007 \norm{CL_f-L_cC},
2008 \qquad 0<\alpha<1.
2009 \]
2010 A Duhamel argument then controls the fractional semigroup defect. This closes a finite version of fractional naturality under explicit spectral hypotheses.
2011 
2012 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.
2013 
2014 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 the article.
2015 
2016 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.
2017 
2018 \begin{table}[t]
2019 \centering
2020 \small
2021 \begin{tabular}{>{\raggedright\arraybackslash}p{.18\textwidth}>{\raggedright\arraybackslash}p{.29\textwidth}>{\raggedright\arraybackslash}p{.43\textwidth}}
2022 \toprule
2023 Realization & Transfer & Exact coarse condition \\
2024 \midrule
2025 Operator evolution & $\e^{tA}$ & generator intertwining \\
2026 Markov chain & $K^n$ or $\e^{tQ}$ & equality of aggregated transition masses \\
2027 Conditional expectation & $P_{\mathcal Q}$ & nesting of the underlying partitions \\
2028 Bayesian update & $\Bcal_\lambda$ & likelihood constant inside each coarse packet \\
2029 Fractional diffusion & $\e^{-tL^\alpha}$ & spectral or resolvent intertwining \\
2030 Hidden dynamics & projected resolvent & vanishing self energy, or an explicit memory state \\
2031 Survival law & transient semigroup & summation over all transient states \\
2032 \bottomrule
2033 \end{tabular}
2034 \caption{The transfer operations studied in the article and the condition that makes each coarse description exact.}
2035 \label{extended:tab:transfer_dictionary}
2036 \end{table}
2037 
2038 \FloatBarrier
2039 \section{Packet resolved semigroups}\label{extended:section:02}\label{extended:sec:packet_semigroups}
2040 
2041 Let \(X\) be a complex Banach space. A finite packet system is a family of bounded idempotents
2042 \[
2043 \Pcal=(P_\alpha)_{\alpha\in I}
2044 \]
2045 such that
2046 \[
2047 P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,
2048 \qquad
2049 \sum_{\alpha\in I}P_\alpha=I_X.
2050 \]
2051 No orthogonality is needed in this section.
2052 
Statement: Packet resolved transfer system2053 \begin{definition}[Packet resolved transfer system]\label{extended:record:001}
2054 A packet resolved transfer system is a triple
2055 \[
2056 (X,A,\Pcal)
2057 \]
2058 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.
2059 \end{definition}
2060 
Statement: Packet blocks2061 \begin{definition}[Packet blocks]\label{extended:record:002}
2062 For \(B\in\mathcal B(X)\), define
2063 \[
2064 B_{\beta\alpha}=P_\beta BP_\alpha.
2065 \]
2066 If \(A\in\mathcal B(X)\) and \(T(t)=\e^{tA}\), write
2067 \[
2068 T_{\beta\alpha}(t)=P_\beta T(t)P_\alpha.
2069 \]
2070 \end{definition}
2071 
Statement: Semigroup and packet convolution · Go to proof · Read statement and proof side by side2072 \begin{theorem}[Semigroup and packet convolution]\label{extended:record:003}\label{extended:thm:packet_convolution}
2073 For every bounded operator \(A\), the family \(T(t)=\e^{tA}\) is a uniformly continuous group and
2074 \[
2075 T(s+t)=T(s)T(t).
2076 \]
2077 For every \(\alpha,\beta\in I\),
2078 \[
2079 T_{\beta\alpha}(s+t)
2080 =
2081 \sum_{\gamma\in I}
2082 T_{\beta\gamma}(s)T_{\gamma\alpha}(t).
2083 \]
2084 \end{theorem}
2085 
Proof block: Semigroup and packet convolution · Go to statement · Read statement and proof side by side2086 \begin{proof}
2087 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.
2088 \end{proof}
2089 
Statement: Block differential system · Go to proof · Read statement and proof side by side2090 \begin{proposition}[Block differential system]\label{extended:record:004}\label{extended:prop:block_differential}
2091 The packet blocks are differentiable in operator norm and satisfy
2092 \[
2093 \frac{\dd}{\dd t}T_{\beta\alpha}(t)
2094 =
2095 \sum_{\gamma\in I}
2096 A_{\beta\gamma}T_{\gamma\alpha}(t)
2097 =
2098 \sum_{\gamma\in I}
2099 T_{\beta\gamma}(t)A_{\gamma\alpha}.
2100 \]
2101 Their initial values are
2102 \[
2103 T_{\beta\alpha}(0)=\delta_{\beta\alpha}P_\alpha.
2104 \]
2105 \end{proposition}
2106 
Proof block: Block differential system · Go to statement · Read statement and proof side by side2107 \begin{proof}
2108 Differentiate \(T(t)\) in operator norm and insert the packet resolution between \(A\) and \(T(t)\), then between \(T(t)\) and \(A\).
2109 \end{proof}
2110 
Statement: Laplace resolvent and packet transform · Go to proof · Read statement and proof side by side2111 \begin{theorem}[Laplace resolvent and packet transform]\label{extended:record:005}\label{extended:thm:laplace_packet_resolvent}
2112 Let \(A\in\mathcal B(X)\), let \(T(t)=\e^{tA}\), and let
2113 \[
2114 \operatorname{Re}z>\norm A.
2115 \]
2116 Then the Bochner integral converges in operator norm and
2117 \[
2118 (zI-A)^{-1}
2119 =
2120 \int_0^\infty \e^{-zt}T(t)\dd t.
2121 \]
2122 For every pair of packets,
2123 \[
2124 P_\beta(zI-A)^{-1}P_\alpha
2125 =
2126 \int_0^\infty
2127 \e^{-zt}T_{\beta\alpha}(t)\dd t.
2128 \]
2129 \end{theorem}
2130 
Proof block: Laplace resolvent and packet transform · Go to statement · Read statement and proof side by side2131 \begin{proof}
2132 The estimate \(\norm{T(t)}\leq\e^{t\norm A}\) gives absolute convergence of the operator valued integral. Let
2133 \[
2134 R(z)=\int_0^\infty\e^{-zt}T(t)\dd t.
2135 \]
2136 Since \(T'(t)=AT(t)=T(t)A\), integration by parts gives
2137 \[
2138 (zI-A)R(z)
2139 =
2140 -\int_0^\infty
2141 \frac{\dd}{\dd t}
2142 \bigl(\e^{-zt}T(t)\bigr)
2143 \dd t
2144 =I.
2145 \]
2146 The same calculation on the other side gives
2147 \(
2148 R(z)(zI-A)=I
2149 \).
2150 Multiplication by \(P_\beta\) and \(P_\alpha\) may be passed under the norm convergent integral, which gives the packet formula.
2151 \end{proof}
2152 
Statement: Resolvent expansion over packet paths · Go to proof · Read statement and proof side by side2153 \begin{corollary}[Resolvent expansion over packet paths]\label{extended:record:006}\label{extended:cor:resolvent_packet_paths}
2154 With \(A=D+N\) as in the directed path decomposition, let
2155 \[
2156 R_D(z)=(zI-D)^{-1}.
2157 \]
2158 If
2159 \[
2160 \norm{R_D(z)N}<1,
2161 \]
2162 then
2163 \[
2164 (zI-A)^{-1}
2165 =
2166 \sum_{m=0}^\infty
2167 \bigl(R_D(z)N\bigr)^mR_D(z).
2168 \]
2169 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 \(\beta\).
2170 \end{corollary}
2171 
Proof block: Resolvent expansion over packet paths · Go to statement · Read statement and proof side by side2172 \begin{proof}
2173 Factor
2174 \[
2175 zI-A=(zI-D)(I-R_D(z)N)
2176 \]
2177 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.
2178 \end{proof}
2179 
Statement: Reducing packet criterion · Go to proof · Read statement and proof side by side2180 \begin{theorem}[Reducing packet criterion]\label{extended:record:007}\label{extended:thm:reducing_packets}
2181 The following conditions are equivalent.
2182 \begin{enumerate}
2183 \item \(AP_\alpha=P_\alpha A\) for every \(\alpha\).
2184 \item \(A_{\beta\alpha}=0\) whenever \(\beta\neq\alpha\).
2185 \item \(T_{\beta\alpha}(t)=0\) whenever \(\beta\neq\alpha\) and \(t\in\RR\).
2186 \item Every packet space \(P_\alpha X\) is invariant under \(T(t)\) for every \(t\).
2187 \end{enumerate}
2188 When these conditions hold,
2189 \[
2190 T(t)|_{P_\alpha X}
2191 =
2192 \e^{tA_{\alpha\alpha}}.
2193 \]
2194 \end{theorem}
2195 
Proof block: Reducing packet criterion · Go to statement · Read statement and proof side by side2196 \begin{proof}
2197 The first two conditions are equivalent because the packet resolution gives
2198 \[
2199 AP_\alpha=\sum_\beta A_{\beta\alpha},
2200 \qquad
2201 P_\alpha A=\sum_\gamma A_{\alpha\gamma}.
2202 \]
2203 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.
2204 \end{proof}
2205 
2206 \begin{figure}[H]
2207 \centering
2208 \includegraphics[width=.66\textwidth]{packet_blocks_t0_0.pdf}
2209 \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}
2210 \end{figure}
2211 \begin{figure}[H]
2212 \centering
2213 \includegraphics[width=.66\textwidth]{packet_blocks_t0_8.pdf}
2214 \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}
2215 \end{figure}
2216 \begin{figure}[H]
2217 \centering
2218 \includegraphics[width=.66\textwidth]{packet_blocks_t2_0.pdf}
2219 \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}
2220 \end{figure}
2221 
2222 
2223 \subsection{Directed path expansion}
2224 
2225 Set
2226 \[
2227 D=\sum_{\alpha\in I}P_\alpha AP_\alpha,
2228 \qquad
2229 N=A-D.
2230 \]
2231 The operator \(D\) preserves every packet and \(N\) contains only transfers between distinct packets.
2232 
Statement: Dyson expansion over packet paths · Go to proof · Read statement and proof side by side2233 \begin{theorem}[Dyson expansion over packet paths]\label{extended:record:008}\label{extended:thm:dyson_paths}
2234 For \(t\geq0\),
2235 \[
2236 \begin{aligned}
2237 \e^{tA}
2238 ={}&\e^{tD}
2239 +
2240 \sum_{m=1}^{\infty}
2241 \int_{0<s_1<\cdots<s_m<t}
2242 \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D}
2243 \cdots
2244 N\e^{s_1D}
2245 \dd s_1\cdots\dd s_m.
2246 \end{aligned}
2247 \]
2248 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 \(\beta\).
2249 \end{theorem}
2250 
Proof block: Dyson expansion over packet paths · Go to statement · Read statement and proof side by side2251 \begin{proof}
2252 Write the variation of constants identity
2253 \[
2254 \e^{tA}
2255 =
2256 \e^{tD}
2257 +
2258 \int_0^t\e^{(t-s)D}N\e^{sA}\dd s.
2259 \]
2260 Iterating the identity gives the displayed series. The norm of the term of order \(m\) is at most
2261 \[
2262 \e^{t\norm D}\frac{(t\norm N)^m}{m!}.
2263 \]
2264 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.
2265 \end{proof}
2266 
Statement: Termination on an acyclic transfer graph · Go to proof · Read statement and proof side by side2267 \begin{corollary}[Termination on an acyclic transfer graph]\label{extended:record:009}\label{extended:cor:acyclic_termination}
2268 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.
2269 \end{corollary}
2270 
Proof block: Termination on an acyclic transfer graph · Go to statement · Read statement and proof side by side2271 \begin{proof}
2272 Every nonzero term of order \(m\) contains a directed transfer path of length \(m\). An acyclic graph has no path longer than \(q\).
2273 \end{proof}
2274 
2275 \FloatBarrier
2276 \section{Generator defects and functional calculus}\label{extended:section:03}\label{extended:sec:defects}
2277 
2278 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
2279 \[
2280 \Delta=CA_f-A_cC
2281 \]
2282 and
2283 \[
2284 \mathfrak D(t)=C\e^{tA_f}-\e^{tA_c}C.
2285 \]
2286 
Statement: Duhamel defect identity · Go to proof · Read statement and proof side by side2287 \begin{theorem}[Duhamel defect identity]\label{extended:record:010}\label{extended:thm:duhamel_defect}
2288 For every \(t\geq0\),
2289 \[
2290 \mathfrak D(t)
2291 =
2292 \int_0^t
2293 \e^{(t-s)A_c}\Delta\e^{sA_f}
2294 \dd s.
2295 \]
2296 If
2297 \[
2298 \norm{\e^{tA_f}}\leq M_f\e^{\omega_f t},
2299 \qquad
2300 \norm{\e^{tA_c}}\leq M_c\e^{\omega_c t},
2301 \]
2302 then
2303 \[
2304 \norm{\mathfrak D(t)}
2305 \leq
2306 M_fM_c\norm\Delta
2307 \int_0^t
2308 \e^{\omega_c(t-s)+\omega_fs}
2309 \dd s.
2310 \]
2311 For contraction semigroups,
2312 \[
2313 \norm{\mathfrak D(t)}\leq t\norm\Delta.
2314 \]
2315 \end{theorem}
2316 
Proof block: Duhamel defect identity · Go to statement · Read statement and proof side by side2317 \begin{proof}
2318 For fixed \(t\), set
2319 \[
2320 F(s)=\e^{(t-s)A_c}C\e^{sA_f}.
2321 \]
2322 Then
2323 \[
2324 F'(s)=\e^{(t-s)A_c}\Delta\e^{sA_f}.
2325 \]
2326 Integrating from zero to \(t\) gives the identity. The estimates follow by taking norms under the integral.
2327 \end{proof}
2328 
Statement: Infinitesimal recovery of the defect · Go to proof · Read statement and proof side by side2329 \begin{proposition}[Infinitesimal recovery of the defect]\label{extended:record:011}\label{extended:prop:infinitesimal_defect}
2330 The generator defect is the first order semigroup defect:
2331 \[
2332 \lim_{t\to0}
2333 \frac{\mathfrak D(t)}{t}
2334 =
2335 \Delta
2336 \]
2337 in operator norm. Consequently,
2338 \[
2339 C\e^{tA_f}=\e^{tA_c}C
2340 \quad\hbox{for every }t
2341 \]
2342 if and only if \(CA_f=A_cC\).
2343 \end{proposition}
2344 
Proof block: Infinitesimal recovery of the defect · Go to statement · Read statement and proof side by side2345 \begin{proof}
2346 The operator norm expansions
2347 \[
2348 \e^{tA_f}=I+tA_f+o(t),
2349 \qquad
2350 \e^{tA_c}=I+tA_c+o(t)
2351 \]
2352 give the limit. The equivalence follows from this limit and \cref{extended:thm:duhamel_defect}.
2353 \end{proof}
2354 
Statement: Resolvent defect identity · Go to proof · Read statement and proof side by side2355 \begin{theorem}[Resolvent defect identity]\label{extended:record:012}\label{extended:thm:resolvent_defect}
2356 Let \(z\) belong to the resolvent sets of \(A_f\) and \(A_c\). With
2357 \[
2358 R_f(z)=(zI-A_f)^{-1},
2359 \qquad
2360 R_c(z)=(zI-A_c)^{-1},
2361 \]
2362 one has
2363 \[
2364 CR_f(z)-R_c(z)C
2365 =
2366 R_c(z)\Delta R_f(z).
2367 \]
2368 \end{theorem}
2369 
Proof block: Resolvent defect identity · Go to statement · Read statement and proof side by side2370 \begin{proof}
2371 Multiply the proposed identity on the left by \(zI-A_c\) and on the right by \(zI-A_f\). Both sides reduce to
2372 \[
2373 (zI-A_c)C-C(zI-A_f)=\Delta.
2374 \]
2375 \end{proof}
2376 
Statement: Holomorphic defect formula · Go to proof · Read statement and proof side by side2377 \begin{theorem}[Holomorphic defect formula]\label{extended:record:013}\label{extended:thm:holomorphic_defect}
2378 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
2379 \[
2380 Cg(A_f)-g(A_c)C
2381 =
2382 \frac{1}{2\pi\ii}
2383 \int_\Gamma
2384  g(z)R_c(z)\Delta R_f(z)
2385 \dd z.
2386 \]
2387 In particular,
2388 \[
2389 \begin{aligned}
2390 \norm{Cg(A_f)-g(A_c)C}
2391 \leq{}&
2392 \frac{\operatorname{length}(\Gamma)}{2\pi}
2393 \sup_{z\in\Gamma}\abs{g(z)}
2394 \sup_{z\in\Gamma}\norm{R_c(z)}
2395 \sup_{z\in\Gamma}\norm{R_f(z)}
2396 \norm\Delta.
2397 \end{aligned}
2398 \]
2399 \end{theorem}
2400 
Proof block: Holomorphic defect formula · Go to statement · Read statement and proof side by side2401 \begin{proof}
2402 Apply the Dunford integral formula to both operators and use \cref{extended:thm:resolvent_defect} under the contour integral. The norm estimate is immediate.
2403 \end{proof}
2404 
Statement: Exact naturality · Go to proof · Read statement and proof side by side2405 \begin{corollary}[Exact naturality]\label{extended:record:014}\label{extended:cor:exact_naturality}
2406 If \(CA_f=A_cC\), then
2407 \[
2408 Cg(A_f)=g(A_c)C
2409 \]
2410 for every holomorphic function covered by \cref{extended:thm:holomorphic_defect}. The conclusion includes exponentials, resolvents, and polynomials.
2411 \end{corollary}
2412 
Proof block: Exact naturality · Go to statement · Read statement and proof side by side2413 \begin{proof}
2414 Set \(\Delta=0\) in \cref{extended:thm:holomorphic_defect}.
2415 \end{proof}
2416 
2417 \begin{figure}[t]
2418 \centering
2419 \includegraphics[width=.68\textwidth]{duhamel_defect.pdf}
2420 \caption{A finite semigroup defect and a direct Duhamel bound. The bound is designed to be uniform rather than sharp in this example.}
2421 \label{extended:fig:duhamel_defect}
2422 \end{figure}
2423 
2424 \FloatBarrier
2425 \section{Refinement towers and limiting evolution}\label{extended:section:04}\label{extended:sec:refinement_towers}
2426 \begin{interpretation}
2427 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.
2428 \end{interpretation}
2429 
2430 
2431 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.
2432 
2433 \subsection{Exact inverse limits}
2434 
2435 Let
2436 \[
2437 X_0\xleftarrow{C_0}X_1\xleftarrow{C_1}X_2\xleftarrow{}\cdots
2438 \]
2439 be Banach spaces with bounded connecting maps. Define the bounded inverse limit
2440 \[
2441 X_\infty
2442 =
2443 \left\{
2444 (x_\ell)_{\ell\geq0}:
2445 C_\ell x_{\ell+1}=x_\ell,
2446 \quad
2447 \sup_\ell\norm{x_\ell}<\infty
2448 \right\}
2449 \]
2450 with the supremum norm.
2451 
Statement: Completeness of the bounded inverse limit · Go to proof · Read statement and proof side by side2452 \begin{lemma}[Completeness of the bounded inverse limit]\label{extended:record:015}\label{extended:lem:inverse_limit_banach}
2453 The space \(X_\infty\) is Banach.
2454 \end{lemma}
2455 
Proof block: Completeness of the bounded inverse limit · Go to statement · Read statement and proof side by side2456 \begin{proof}
2457 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.
2458 \end{proof}
2459 
Statement: Exact projective limit semigroup · Go to proof · Read statement and proof side by side2460 \begin{theorem}[Exact projective limit semigroup]\label{extended:record:016}\label{extended:thm:exact_projective_limit}
2461 For each \(\ell\), let \(T_\ell(t)\) be a strongly continuous semigroup on \(X_\ell\). Assume
2462 \[
2463 C_\ell T_{\ell+1}(t)=T_\ell(t)C_\ell
2464 \]
2465 for every \(\ell\) and \(t\geq0\), and assume the uniform bound
2466 \[
2467 \sup_\ell\norm{T_\ell(t)}\leq M\e^{\omega t}.
2468 \]
2469 Define
2470 \[
2471 T_\infty(t)(x_\ell)_\ell
2472 =
2473 (T_\ell(t)x_\ell)_\ell.
2474 \]
2475 Then \(T_\infty(t)\) is a bounded semigroup on \(X_\infty\). It is strongly continuous if
2476 \[
2477 \lim_{t\downarrow0}
2478 \sup_\ell
2479 \norm{T_\ell(t)x_\ell-x_\ell}=0
2480 \]
2481 for every \((x_\ell)_\ell\in X_\infty\).
2482 
2483 If \(T_\ell(t)=\e^{tA_\ell}\), the generators are bounded, and
2484 \[
2485 \sup_\ell\norm{A_\ell}<\infty,
2486 \qquad
2487 C_\ell A_{\ell+1}=A_\ell C_\ell,
2488 \]
2489 then the continuity condition holds and
2490 \[
2491 A_\infty(x_\ell)_\ell=(A_\ell x_\ell)_\ell
2492 \]
2493 is the bounded generator of \(T_\infty\).
2494 \end{theorem}
2495 
Proof block: Exact projective limit semigroup · Go to statement · Read statement and proof side by side2496 \begin{proof}
2497 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.
2498 
2499 For bounded generators with common bound \(M_A\),
2500 \[
2501 \sup_\ell
2502 \norm{(\e^{tA_\ell}-I)x_\ell}
2503 \leq
2504 (\e^{tM_A}-1)\sup_\ell\norm{x_\ell}.
2505 \]
2506 This tends to zero with \(t\). Exact generator intertwining makes \((A_\ell x_\ell)_\ell\) compatible, and coordinatewise differentiation identifies the generator.
2507 \end{proof}
2508 
2509 \begin{remark}\label{extended:record:017}
2510 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.
2511 \end{remark}
2512 
2513 \subsection{Summable adjacent defects}
2514 
2515 Assume now that each \(C_\ell:X_{\ell+1}\to X_\ell\) has a bounded right inverse
2516 \[
2517 R_\ell:X_\ell\to X_{\ell+1},
2518 \qquad
2519 C_\ell R_\ell=I_{X_\ell}.
2520 \]
2521 For \(m<n\), set
2522 \[
2523 C_{m,n}=C_mC_{m+1}\cdots C_{n-1},
2524 \qquad
2525 R_{n,m}=R_{n-1}\cdots R_m.
2526 \]
2527 The empty products are
2528 \[
2529 C_{m,m}=I_{X_m},
2530 \qquad
2531 R_{m,m}=I_{X_m}.
2532 \]
2533 
Statement: Reconstructed fine evolution2534 \begin{definition}[Reconstructed fine evolution]\label{extended:record:018}
2535 For a semigroup \(T_n(t)\) on \(X_n\), define the evolution seen from level \(m\) by
2536 \[
2537 U_{m,n}(t)=C_{m,n}T_n(t)R_{n,m}.
2538 \]
2539 \end{definition}
2540 
Statement: Weighted summable defect completion · Go to proof · Read statement and proof side by side2541 \begin{theorem}[Weighted summable defect completion]\label{extended:record:019}\label{extended:thm:summable_defect_completion}
2542 Define the adjacent semigroup defect
2543 \[
2544 E_n(t)=C_nT_{n+1}(t)-T_n(t)C_n.
2545 \]
2546 Fix a coarse level \(m\) and a time horizon \(T>0\). For \(n\geq m\), set
2547 \[
2548 w_{m,n}
2549 =
2550 \norm{C_{m,n}}\,
2551 \norm{R_nR_{n,m}}.
2552 \]
2553 If
2554 \[
2555 \sum_{n=m}^{\infty}
2556 w_{m,n}
2557 \sup_{0\leq t\leq T}\norm{E_n(t)}<\infty,
2558 \]
2559 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
2560 \[
2561 \sup_{0\leq t\leq T}
2562 \norm{U_m(t)-U_{m,n}(t)}
2563 \leq
2564 \sum_{j=n}^{\infty}w_{m,j}
2565 \sup_{0\leq t\leq T}\norm{E_j(t)}.
2566 \]
2567 \end{theorem}
2568 
Proof block: Weighted summable defect completion · Go to statement · Read statement and proof side by side2569 \begin{proof}
2570 Since \(C_nR_n=I\),
2571 \[
2572 \begin{aligned}
2573 U_{m,n+1}(t)-U_{m,n}(t)
2574 ={}&
2575 C_{m,n}
2576 \bigl(C_nT_{n+1}(t)-T_n(t)C_n\bigr)
2577 R_nR_{n,m}.
2578 \end{aligned}
2579 \]
2580 Therefore
2581 \[
2582 \norm{U_{m,n+1}(t)-U_{m,n}(t)}
2583 \leq
2584 w_{m,n}\norm{E_n(t)}.
2585 \]
2586 The weighted summability hypothesis makes the sequence uniformly Cauchy. Summing the tail estimates gives the stated bound.
2587 \end{proof}
2588 
Statement: Contractive summable defect completion · Go to proof · Read statement and proof side by side2589 \begin{corollary}[Contractive summable defect completion]\label{extended:record:020}\label{extended:cor:contractive_defect_completion}
2590 Assume
2591 \[
2592 \norm{C_\ell}\leq1,
2593 \qquad
2594 \norm{R_\ell}\leq1,
2595 \qquad
2596 \norm{T_\ell(t)}\leq1
2597 \]
2598 for every \(\ell\) and \(t\geq0\). If
2599 \[
2600 \sum_{\ell=0}^{\infty}
2601 \sup_{0\leq t\leq T}\norm{E_\ell(t)}<\infty,
2602 \]
2603 then the conclusion of \cref{extended:thm:summable_defect_completion} holds, and
2604 \[
2605 \sup_{0\leq t\leq T}
2606 \norm{U_m(t)-U_{m,n}(t)}
2607 \leq
2608 \sum_{\ell=n}^{\infty}
2609 \sup_{0\leq t\leq T}\norm{E_\ell(t)}.
2610 \]
2611 \end{corollary}
2612 
Proof block: Contractive summable defect completion · Go to statement · Read statement and proof side by side2613 \begin{proof}
2614 Every weight \(w_{m,n}\) is at most one.
2615 \end{proof}
2616 
Statement: Generator defect criterion · Go to proof · Read statement and proof side by side2617 \begin{corollary}[Generator defect criterion]\label{extended:record:021}\label{extended:cor:generator_defect_completion}
2618 Assume
2619 \[
2620 \norm{C_\ell}\leq1,
2621 \qquad
2622 \norm{R_\ell}\leq1,
2623 \]
2624 and suppose \(T_\ell(t)=\e^{tA_\ell}\) are contraction semigroups with bounded generators. Set
2625 \[
2626 \Delta_\ell=C_\ell A_{\ell+1}-A_\ell C_\ell.
2627 \]
2628 If
2629 \[
2630 \sum_{\ell=0}^{\infty}\norm{\Delta_\ell}<\infty,
2631 \]
2632 then the conclusion of \cref{extended:cor:contractive_defect_completion} holds on every bounded time interval, and
2633 \[
2634 \sup_{0\leq t\leq T}
2635 \norm{U_m(t)-U_{m,n}(t)}
2636 \leq
2637 T\sum_{\ell=n}^{\infty}\norm{\Delta_\ell}.
2638 \]
2639 \end{corollary}
2640 
Proof block: Generator defect criterion · Go to statement · Read statement and proof side by side2641 \begin{proof}
2642 Apply the contraction estimate in \cref{extended:thm:duhamel_defect} to every adjacent pair.
2643 \end{proof}
2644 
2645 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.
2646 
Statement: Asymptotic closure criterion · Go to proof · Read statement and proof side by side2647 \begin{theorem}[Asymptotic closure criterion]\label{extended:record:022}\label{extended:thm:asymptotic_closure}
2648 Assume the weighted convergence in \cref{extended:thm:summable_defect_completion} holds for every bounded time interval, and define
2649 \[
2650 P_{m,n}=R_{n,m}C_{m,n}
2651 \]
2652 on \(X_n\). Suppose that for every \(T>0\),
2653 \[
2654 \rho_{m,n}(T)
2655 =
2656 \sup_{0\leq s,t\leq T}
2657 \norm{
2658 C_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m}
2659 }
2660 \longrightarrow0.
2661 \]
2662 Then
2663 \[
2664 U_m(s+t)=U_m(s)U_m(t)
2665 \]
2666 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.
2667 \end{theorem}
2668 
Proof block: Asymptotic closure criterion · Go to statement · Read statement and proof side by side2669 \begin{proof}
2670 At a finite level,
2671 \[
2672 \begin{aligned}
2673 U_{m,n}(s)U_{m,n}(t)-U_{m,n}(s+t)
2674 ={}&
2675 C_{m,n}T_n(s)
2676 (P_{m,n}-I)
2677 T_n(t)R_{n,m}.
2678 \end{aligned}
2679 \]
2680 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.
2681 \end{proof}
2682 
2683 \begin{remark}\label{extended:record:023}
2684 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.
2685 \end{remark}
2686 
2687 \begin{example}[Summable defects do not force a semigroup limit]\label{extended:record:024}\label{extended:ex:summable_not_semigroup}
2688 Let \(X_0=\RR\) and let \(X_n=\RR^2\) for every \(n\geq1\). Set
2689 \[
2690 C_0(x,y)=x,
2691 \qquad
2692 R_0x=(x,0),
2693 \]
2694 and use identity connecting maps and reconstructions above level one. Let \(T_0(t)=I\) and let every higher level carry the rotation group
2695 \[
2696 T_n(t)
2697 =
2698 \begin{pmatrix}
2699 \cos t&-\sin t\\
2700 \sin t&\cos t
2701 \end{pmatrix}.
2702 \]
2703 Only the first adjacent defect is nonzero, so every defect sum is finite. Yet for \(n\geq1\),
2704 \[
2705 U_{0,n}(t)=C_0T_n(t)R_0=\cos t.
2706 \]
2707 The limiting family is not a semigroup because in general
2708 \[
2709 \cos(s+t)\neq\cos s\cos t.
2710 \]
2711 The failure is exactly the excursion from the reconstructed line into its orthogonal complement between two time steps.
2712 \end{example}
2713 
Statement: Range leakage criterion · Go to proof · Read statement and proof side by side2714 \begin{proposition}[Range leakage criterion]\label{extended:record:025}\label{extended:prop:range_leakage}
2715 Under the contractive assumptions of \cref{extended:cor:contractive_defect_completion}, define
2716 \[
2717 \eta_{m,n}(T)
2718 =
2719 \sup_{0\leq t\leq T}
2720 \norm{(I-P_{m,n})T_n(t)R_{n,m}}.
2721 \]
2722 Then
2723 \[
2724 \rho_{m,n}(T)\leq\eta_{m,n}(T).
2725 \]
2726 Consequently, \(\eta_{m,n}(T)\to0\) for every \(T>0\) is sufficient for the limiting family \(U_m\) to satisfy the semigroup law.
2727 \end{proposition}
2728 
Proof block: Range leakage criterion · Go to statement · Read statement and proof side by side2729 \begin{proof}
2730 For \(s,t\leq T\), the contraction assumptions give
2731 \[
2732 \begin{aligned}
2733 &\norm{
2734 C_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m}
2735 }\\
2736 &\qquad\leq
2737 \norm{(I-P_{m,n})T_n(t)R_{n,m}}.
2738 \end{aligned}
2739 \]
2740 Take the supremum and apply \cref{extended:thm:asymptotic_closure}.
2741 \end{proof}
2742 
Statement: Generator control of range leakage · Go to proof · Read statement and proof side by side2743 \begin{theorem}[Generator control of range leakage]\label{extended:record:026}\label{extended:thm:generator_range_leakage}
2744 Let \(T(t)=\e^{tA}\) on a Banach space \(X\), let \(P\) be a bounded projection, and set \(Q=I-P\). Assume
2745 \[
2746 \norm{T(t)}\leq M\e^{\omega t}
2747 \]
2748 and
2749 \[
2750 \norm{\e^{t(QAQ)|_{QX}}}
2751 \leq M_Q\e^{\omega_Qt}.
2752 \]
2753 Then
2754 \[
2755 \norm{QT(t)P}
2756 \leq
2757 M M_Q\norm P^2\norm{QAP}
2758 \int_0^t
2759 \e^{\omega_Q(t-s)+\omega s}
2760 \dd s.
2761 \]
2762 In particular, for orthogonal projections and contraction bounds,
2763 \[
2764 \sup_{0\leq t\leq T}
2765 \norm{QT(t)P}
2766 \leq
2767 T\norm{QAP}.
2768 \]
2769 \end{theorem}
2770 
Proof block: Generator control of range leakage · Go to statement · Read statement and proof side by side2771 \begin{proof}
2772 For \(x\in PX\), write
2773 \[
2774 p(t)=PT(t)x,
2775 \qquad
2776 q(t)=QT(t)x.
2777 \]
2778 The hidden component satisfies
2779 \[
2780 q'(t)=QAQq(t)+QAPp(t),
2781 \qquad
2782 q(0)=0.
2783 \]
2784 Variation of constants gives
2785 \[
2786 q(t)=
2787 \int_0^t
2788 \e^{(t-s)(QAQ)|_{QX}}
2789 QAPp(s)
2790 \dd s.
2791 \]
2792 Since
2793 \(
2794 \norm{p(s)}\leq\norm P\norm{T(s)}\norm x
2795 \)
2796 and \(x=Px\), the first estimate follows. The contraction case is immediate.
2797 \end{proof}
2798 
Statement: A usable closure condition for refinement towers · Go to proof · Read statement and proof side by side2799 \begin{corollary}[A usable closure condition for refinement towers]\label{extended:record:027}\label{extended:cor:generator_closure_tower}
2800 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 \(m\),
2801 \[
2802 \norm{(I-P_{m,n})A_nP_{m,n}}
2803 \longrightarrow0,
2804 \]
2805 then the limiting family \(U_m\) is a semigroup.
2806 \end{corollary}
2807 
Proof block: A usable closure condition for refinement towers · Go to statement · Read statement and proof side by side2808 \begin{proof}
2809 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
2810 \[
2811 \operatorname{Re}\ip{Q_{m,n}A_nQ_{m,n}y}{y}
2812 =
2813 \operatorname{Re}\ip{A_ny}{y}
2814 \leq0.
2815 \]
2816 Hence the compressed hidden semigroup is contractive. Apply \cref{extended:thm:generator_range_leakage} to \(P_{m,n}\). Since \(P_{m,n}R_{n,m}=R_{n,m}\),
2817 \[
2818 \eta_{m,n}(T)
2819 \leq
2820 T\norm{(I-P_{m,n})A_nP_{m,n}}
2821 \norm{R_{n,m}}.
2822 \]
2823 The right side tends to zero. Use \cref{extended:prop:range_leakage}.
2824 \end{proof}
2825 
2826 \FloatBarrier
2827 \section{Finite Markov transfer and lumping}\label{extended:section:05}\label{extended:sec:markov}
2828 
2829 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.
2830 
Statement: Finite Markov semigroup · Go to proof · Read statement and proof side by side2831 \begin{theorem}[Finite Markov semigroup]\label{extended:record:028}\label{extended:thm:finite_markov}
2832 If \(Q\) is a finite Markov generator, then \(\e^{tQ}\) is column stochastic for every \(t\geq0\).
2833 \end{theorem}
2834 
Proof block: Finite Markov semigroup · Go to statement · Read statement and proof side by side2835 \begin{proof}
2836 Choose \(a\geq\max_i(-Q_{ii})\). The matrix \(Q+aI\) is nonnegative, so its exponential is nonnegative. Hence
2837 \[
2838 \e^{tQ}=\e^{-at}\e^{t(Q+aI)}
2839 \]
2840 is nonnegative. Since \(\one^TQ=0\), differentiation gives
2841 \[
2842 \one^T\e^{tQ}=\one^T.
2843 \]
2844 \end{proof}
2845 
2846 Let \(\{B_1,\ldots,B_m\}\) be a partition of the fine state set. Its aggregation matrix is
2847 \[
2848 C_{\alpha i}=\one_{\{i\in B_\alpha\}}.
2849 \]
2850 
Statement: Canonical lumping obstruction2851 \begin{definition}[Canonical lumping obstruction]\label{extended:record:029}
2852 For a linear fine transition operator \(K\), define
2853 \[
2854 \Ob_C(K)
2855 =
2856 (CK)|_{\Ker C}
2857 \in
2858 \mathcal B(\Ker C,\RR^m).
2859 \]
2860 \end{definition}
2861 
Statement: Descent through an aggregation map · Go to proof · Read statement and proof side by side2862 \begin{theorem}[Descent through an aggregation map]\label{extended:record:030}\label{extended:thm:canonical_lumping_obstruction}
2863 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
2864 \[
2865 CK=\overline KC
2866 \]
2867 exists if and only if
2868 \[
2869 \Ob_C(K)=0.
2870 \]
2871 For any right inverse \(R\) of \(C\), the only possible coarse operator is
2872 \[
2873 \overline K=CKR,
2874 \]
2875 and
2876 \[
2877 CK-\overline KC
2878 =
2879 CK(I-RC).
2880 \]
2881 \end{theorem}
2882 
Proof block: Descent through an aggregation map · Go to statement · Read statement and proof side by side2883 \begin{proof}
2884 If \(CK=\overline KC\), then \(CK\) vanishes on \(\Ker C\). Conversely, suppose the obstruction vanishes and define
2885 \[
2886 \overline K(Cx)=CKx.
2887 \]
2888 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
2889 \[
2890 \overline K=\overline KCR=CKR.
2891 \]
2892 Substitution gives the defect identity.
2893 \end{proof}
2894 
Statement: Coordinate form of the lumping obstruction · Go to proof · Read statement and proof side by side2895 \begin{corollary}[Coordinate form of the lumping obstruction]\label{extended:record:031}\label{extended:cor:lumping_obstruction_coordinates}
2896 For the partition aggregation above, \(\Ker C\) is spanned by differences \(e_i-e_j\) with \(i\) and \(j\) in one source block. Also,
2897 \[
2898 \Ob_C(K)(e_i-e_j)
2899 =
2900 \left(
2901 \sum_{r\in B_\alpha}(K_{ri}-K_{rj})
2902 \right)_{\alpha=1}^m.
2903 \]
2904 \end{corollary}
2905 
Proof block: Coordinate form of the lumping obstruction · Go to statement · Read statement and proof side by side2906 \begin{proof}
2907 Inside each block, the vectors with coordinate sum zero are spanned by differences of coordinate vectors. Applying \(CK\) gives the displayed aggregated column differences.
2908 \end{proof}
2909 
Statement: Characterization of exact discrete lumping · Go to proof · Read statement and proof side by side2910 \begin{theorem}[Characterization of exact discrete lumping]\label{extended:record:032}\label{extended:thm:lumping_characterization}
2911 Let \(K\) be column stochastic. A coarse column stochastic matrix \(\overline K\) satisfying
2912 \[
2913 CK=\overline KC
2914 \]
2915 exists if and only if, for every source block \(B_\beta\), every pair \(i,j\in B_\beta\), and every target block \(B_\alpha\),
2916 \[
2917 \sum_{r\in B_\alpha}K_{ri}
2918 =
2919 \sum_{r\in B_\alpha}K_{rj}.
2920 \]
2921 When it exists, \(\overline K\) is unique and
2922 \[
2923 \overline K_{\alpha\beta}
2924 =
2925 \sum_{r\in B_\alpha}K_{ri}
2926 \]
2927 for any \(i\in B_\beta\).
2928 \end{theorem}
2929 
Proof block: Characterization of exact discrete lumping · Go to statement · Read statement and proof side by side2930 \begin{proof}
2931 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.
2932 \end{proof}
2933 
Statement: Continuous time lumping · Go to proof · Read statement and proof side by side2934 \begin{corollary}[Continuous time lumping]\label{extended:record:033}\label{extended:cor:continuous_lumping}
2935 A coarse Markov generator \(\overline Q\) satisfying
2936 \[
2937 CQ=\overline QC
2938 \]
2939 exists if and only if the aggregated rates
2940 \[
2941 \sum_{r\in B_\alpha}Q_{ri}
2942 \]
2943 depend only on the source block of \(i\). In that case
2944 \[
2945 C\e^{tQ}=\e^{t\overline Q}C
2946 \]
2947 for every \(t\geq0\).
2948 \end{corollary}
2949 
Proof block: Continuous time lumping · Go to statement · Read statement and proof side by side2950 \begin{proof}
2951 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}.
2952 \end{proof}
2953 
Statement: Finite horizon lumping defect · Go to proof · Read statement and proof side by side2954 \begin{theorem}[Finite horizon lumping defect]\label{extended:record:034}\label{extended:thm:finite_horizon_lumping}
2955 For arbitrary fine and coarse operators \(K\) and \(\overline K\), with the fixed aggregation map \(C\), set
2956 \[
2957 \Delta_K=CK-\overline KC.
2958 \]
2959 Then
2960 \[
2961 CK^n-\overline K^nC
2962 =
2963 \sum_{j=0}^{n-1}
2964 \overline K^{n-1-j}\Delta_KK^j.
2965 \]
2966 If \(K\) and \(\overline K\) are column stochastic, then
2967 \[
2968 \norm{CK^n-\overline K^nC}_{1\to1}
2969 \leq
2970 n\norm{\Delta_K}_{1\to1}.
2971 \]
2972 For every probability vector \(p\), the total variation discrepancy is at most
2973 \[
2974 \frac n2\norm{\Delta_K}_{1\to1}.
2975 \]
2976 \end{theorem}
2977 
Proof block: Finite horizon lumping defect · Go to statement · Read statement and proof side by side2978 \begin{proof}
2979 Sum the telescoping identities
2980 \[
2981 \overline K^{n-1-j}CK^{j+1}
2982 -
2983 \overline K^{n-j}CK^j
2984 =
2985 \overline K^{n-1-j}\Delta_KK^j.
2986 \]
2987 Stochastic matrices are contractions in the \(\ell^1\) norm. The last statement uses total variation equal to one half of the \(\ell^1\) distance.
2988 \end{proof}
2989 
Statement: Stationary mass descends · Go to proof · Read statement and proof side by side2990 \begin{corollary}[Stationary mass descends]\label{extended:record:035}\label{extended:cor:stationary_descends}
2991 If \(CK=\overline KC\) and \(Kp=p\), then
2992 \[
2993 \overline K(Cp)=Cp.
2994 \]
2995 \end{corollary}
2996 
Proof block: Stationary mass descends · Go to statement · Read statement and proof side by side2997 \begin{proof}
2998 Apply the intertwining identity to \(p\).
2999 \end{proof}
3000 
3001 \begin{figure}[t]
3002 \centering
3003 \includegraphics[width=.72\textwidth]{markov_lumping.pdf}
3004 \caption{Exact lumping agrees at every horizon. A small violation of the block transition condition produces a visible but controlled discrepancy.}
3005 \label{extended:fig:markov_lumping}
3006 \end{figure}
3007 
3008 \FloatBarrier
3009 \section{Conditional expectation and martingale packets}\label{extended:section:06}\label{extended:sec:conditional_expectation}
3010 
3011 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
3012 \[
3013 (P_\mathcal Qf)_i
3014 =
3015 \frac{\sum_{j\in B(i)}\pi_jf_j}
3016 {\sum_{j\in B(i)}\pi_j}.
3017 \]
3018 Equip \(\CC^n\) with
3019 \[
3020 \ip{f}{g}_\pi
3021 =
3022 \sum_i\pi_if_i\overline{g_i}.
3023 \]
3024 
Statement: Finite conditional expectation · Go to proof · Read statement and proof side by side3025 \begin{theorem}[Finite conditional expectation]\label{extended:record:036}\label{extended:thm:finite_conditional_expectation}
3026 The operator \(P_\mathcal Q\) is the orthogonal projector onto the functions that are constant on every block of \(\mathcal Q\). In particular,
3027 \[
3028 P_\mathcal Q^2=P_\mathcal Q,
3029 \qquad
3030 P_\mathcal Q^*=P_\mathcal Q.
3031 \]
3032 \end{theorem}
3033 
Proof block: Finite conditional expectation · Go to statement · Read statement and proof side by side3034 \begin{proof}
3035 One application produces a block constant function, so a second application changes nothing. Expanding both weighted inner products block by block gives
3036 \[
3037 \ip{P_\mathcal Qf}{g}_\pi
3038 =
3039 \sum_{B\in\mathcal Q}
3040 \frac{
3041 \left(\sum_{i\in B}\pi_if_i\right)
3042 \overline{\left(\sum_{j\in B}\pi_jg_j\right)}
3043 }{\pi(B)}
3044 =
3045 \ip{f}{P_\mathcal Qg}_\pi.
3046 \]
3047 The range is exactly the block constant subspace.
3048 \end{proof}
3049 
Statement: Tower law and martingale packets · Go to proof · Read statement and proof side by side3050 \begin{theorem}[Tower law and martingale packets]\label{extended:record:037}\label{extended:thm:martingale_packets}
3051 Let $P_0,P_1,\ldots,P_r$ come from increasingly fine partitions, so that
3052 \[
3053 \Ran P_0\subseteq\Ran P_1\subseteq\cdots\subseteq\Ran P_r.
3054 \]
3055 Define
3056 \[
3057 D_0=P_0,
3058 \qquad
3059 D_j=P_j-P_{j-1}
3060 \quad(1\leq j\leq r).
3061 \]
3062 Then
3063 \[
3064 P_iP_j=P_{\min(i,j)},
3065 \qquad
3066 D_iD_j=\delta_{ij}D_j,
3067 \qquad
3068 P_r=\sum_{j=0}^rD_j.
3069 \]
3070 Every \(D_j\) is an orthogonal projector, and
3071 \[
3072 \norm{P_rf}_\pi^2
3073 =
3074 \sum_{j=0}^r\norm{D_jf}_\pi^2
3075 \]
3076 and
3077 \[
3078 \norm f_\pi^2
3079 =
3080 \sum_{j=0}^r\norm{D_jf}_\pi^2
3081 +
3082 \norm{f-P_rf}_\pi^2.
3083 \]
3084 \end{theorem}
3085 
Proof block: Tower law and martingale packets · Go to statement · Read statement and proof side by side3086 \begin{proof}
3087 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.
3088 \end{proof}
3089 
Statement: Law of total variance · Go to proof · Read statement and proof side by side3090 \begin{corollary}[Law of total variance]\label{extended:record:038}\label{extended:cor:total_variance}
3091 For a real valued \(f\), let \(P_0f=\Exp_\pi[f]\one\). Then
3092 \[
3093 \Var_\pi(f)
3094 =
3095 \Var_\pi(P_\mathcal Qf)
3096 +
3097 \norm{f-P_\mathcal Qf}_\pi^2.
3098 \]
3099 \end{corollary}
3100 
Proof block: Law of total variance · Go to statement · Read statement and proof side by side3101 \begin{proof}
3102 Subtract the mean and apply the orthogonal decomposition into the range of \(P_\mathcal Q\) and its orthogonal complement.
3103 \end{proof}
3104 
3105 \begin{figure}[t]
3106 \centering
3107 \includegraphics[width=.68\textwidth]{martingale_packet_energy.pdf}
3108 \caption{Energy carried by the mean packet and three successive martingale difference packets in a finite weighted example.}
3109 \label{extended:fig:martingale_energy}
3110 \end{figure}
3111 
Statement: General conditional expectations3112 \begin{standardfact}[General conditional expectations]\label{extended:record:039}
3113 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}.
3114 \end{standardfact}
3115 
3116 \FloatBarrier
3117 \section{Gaussian transfer and Hermite packets}\label{extended:section:07}\label{extended:sec:ou_hermite}
3118 
3119 Let \(\gamma\) be the standard Gaussian probability measure on \(\RR\). For \(t\geq0\), define the Ornstein Uhlenbeck transfer
3120 \[
3121 (\Ocal_t f)(x)
3122 =
3123 \int_\RR
3124 f\left(
3125 \e^{-t}x+\sqrt{1-\e^{-2t}}\,y
3126 \right)
3127 \gamma(\dd y).
3128 \]
3129 Let \(H_n\) be the probabilists' Hermite polynomial defined by
3130 \[
3131 \exp\left(ux-\frac{u^2}{2}\right)
3132 =
3133 \sum_{n=0}^\infty H_n(x)\frac{u^n}{n!},
3134 \]
3135 and set \(h_n=H_n/\sqrt{n!}\).
3136 
Statement: Ornstein Uhlenbeck semigroup · Go to proof · Read statement and proof side by side3137 \begin{theorem}[Ornstein Uhlenbeck semigroup]\label{extended:record:040}\label{extended:thm:ou_semigroup}
3138 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
3139 \[
3140 \Ocal_t h_n=\e^{-nt}h_n.
3141 \]
3142 The family \((h_n)_{n\geq0}\) is an orthonormal basis of \(L^2(\gamma)\).
3143 \end{theorem}
3144 
Proof block: Ornstein Uhlenbeck semigroup · Go to statement · Read statement and proof side by side3145 \begin{proof}
3146 Positivity and preservation of constants follow from the integral formula. If \(X\) and \(Y\) are independent standard Gaussian variables, then
3147 \[
3148 \e^{-s}
3149 \left(
3150 \e^{-t}x+\sqrt{1-\e^{-2t}}X
3151 \right)
3152 +
3153 \sqrt{1-\e^{-2s}}Y
3154 \]
3155 has the same distribution as
3156 \[
3157 \e^{-(s+t)}x
3158 +
3159 \sqrt{1-\e^{-2(s+t)}}Z
3160 \]
3161 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
3162 \[
3163 \left(
3164 X,
3165 \e^{-t}X+\sqrt{1-\e^{-2t}}Y
3166 \right)
3167 \]
3168 is symmetric under interchange, which proves self adjointness.
3169 
3170 Apply \(\Ocal_t\) to the Hermite generating function. Gaussian integration gives
3171 \[
3172 \begin{aligned}
3173 \Ocal_t
3174 \exp\left(ux-\frac{u^2}{2}\right)
3175 &=
3176 \exp\left(
3177 \e^{-t}ux-
3178 \frac{\e^{-2t}u^2}{2}
3179 \right)\\
3180 &=
3181 \sum_{n=0}^\infty
3182 \e^{-nt}H_n(x)\frac{u^n}{n!}.
3183 \end{aligned}
3184 \]
3185 Comparison of coefficients proves the eigenvalue formula. The same generating function shows
3186 \[
3187 \int H_mH_n\dd\gamma=n!\delta_{mn}.
3188 \]
3189 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.
3190 \end{proof}
3191 
Statement: Hermite packet reconstruction · Go to proof · Read statement and proof side by side3192 \begin{corollary}[Hermite packet reconstruction]\label{extended:record:041}\label{extended:cor:hermite_packets}
3193 Let \(P_n\) be the orthogonal projection onto \(\CC h_n\). Then
3194 \[
3195 I=\sum_{n=0}^\infty P_n
3196 \]
3197 in the strong operator topology and
3198 \[
3199 \Ocal_t
3200 =
3201 \sum_{n=0}^\infty\e^{-nt}P_n.
3202 \]
3203 For \(f\in L^2(\gamma)\),
3204 \[
3205 \norm{\Ocal_t f}_2^2
3206 =
3207 \sum_{n=0}^\infty
3208 \e^{-2nt}\norm{P_nf}_2^2.
3209 \]
3210 \end{corollary}
3211 
Proof block: Hermite packet reconstruction · Go to statement · Read statement and proof side by side3212 \begin{proof}
3213 Apply Parseval's identity and the eigenvalue formula in \cref{extended:thm:ou_semigroup}.
3214 \end{proof}
3215 
Statement: Generator and Dirichlet form · Go to proof · Read statement and proof side by side3216 \begin{proposition}[Generator and Dirichlet form]\label{extended:record:042}\label{extended:prop:ou_generator}
3217 On the polynomial subspace, the infinitesimal generator of \((\Ocal_t)\) is the differential operator
3218 \[
3219 \Lcal f=f''-xf'.
3220 \]
3221 It satisfies
3222 \[
3223 \Lcal h_n=-nh_n.
3224 \]
3225 For every polynomial \(f\),
3226 \[
3227 \ip{-\Lcal f}{f}_{L^2(\gamma)}
3228 =
3229 \int_\RR\abs{f'(x)}^2\gamma(\dd x).
3230 \]
3231 \end{proposition}
3232 
Proof block: Generator and Dirichlet form · Go to statement · Read statement and proof side by side3233 \begin{proof}
3234 Write
3235 \[
3236 \rho_t=\e^{-t}=1-t+O(t^2),
3237 \qquad
3238 \sigma_t^2=1-\e^{-2t}=2t+O(t^2).
3239 \]
3240 For a polynomial \(f\), Taylor expansion at \(x\) and Gaussian integration give
3241 \[
3242 \begin{aligned}
3243 \Ocal_tf(x)
3244 ={}&f(x)
3245 +f'(x)(\rho_t-1)x
3246 +\frac12f''(x)\sigma_t^2
3247 +o(t)\\
3248 ={}&f(x)+t\bigl(f''(x)-xf'(x)\bigr)+o(t).
3249 \end{aligned}
3250 \]
3251 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
3252 \[
3253 \left.\frac{\dd}{\dd t}\Ocal_tf\right|_{t=0}
3254 =f''-xf'.
3255 \]
3256 The Hermite eigenvalue identity also follows by differentiating
3257 \(
3258 \Ocal_th_n=\e^{-nt}h_n
3259 \)
3260 at zero. Gaussian integration by parts gives
3261 \[
3262 \int_\RR f'(x)\overline{g(x)}\gamma(\dd x)
3263 =
3264 \int_\RR f(x)\overline{xg(x)-g'(x)}\gamma(\dd x).
3265 \]
3266 Set \(g=f'\) and rearrange.
3267 \end{proof}
3268 
Statement: Gaussian spectral gap · Go to proof · Read statement and proof side by side3269 \begin{corollary}[Gaussian spectral gap]\label{extended:record:043}\label{extended:cor:ou_spectral_gap}
3270 If \(f\in L^2(\gamma)\) has mean zero, then
3271 \[
3272 \norm{\Ocal_tf}_2
3273 \leq
3274 \e^{-t}\norm f_2.
3275 \]
3276 For every polynomial \(f\),
3277 \[
3278 \Var_\gamma(f)
3279 \leq
3280 \int_\RR\abs{f'(x)}^2\gamma(\dd x).
3281 \]
3282 \end{corollary}
3283 
Proof block: Gaussian spectral gap · Go to statement · Read statement and proof side by side3284 \begin{proof}
3285 A mean zero function has no \(h_0\) component, so every surviving Hermite multiplier is at most \(\e^{-t}\). For a polynomial
3286 \[
3287 f-\Exp_\gamma f=\sum_{n\geq1}c_nh_n.
3288 \]
3289 Then
3290 \[
3291 \Var_\gamma(f)=\sum_{n\geq1}\abs{c_n}^2
3292 \leq
3293 \sum_{n\geq1}n\abs{c_n}^2
3294 =
3295 \ip{-\Lcal f}{f}.
3296 \]
3297 Apply \cref{extended:prop:ou_generator}.
3298 \end{proof}
3299 
Statement: Fractional Hermite transfer · Go to proof · Read statement and proof side by side3300 \begin{corollary}[Fractional Hermite transfer]\label{extended:record:044}\label{extended:cor:fractional_hermite}
3301 Let \(-\Lcal\) be the nonnegative Ornstein Uhlenbeck number operator with
3302 \[
3303 -\Lcal h_n=nh_n.
3304 \]
3305 For every \(\alpha>0\),
3306 \[
3307 \e^{-t(-\Lcal)^\alpha}f
3308 =
3309 \sum_{n=0}^\infty
3310 \e^{-tn^\alpha}P_nf.
3311 \]
3312 This is a contraction semigroup on \(L^2(\gamma)\).
3313 \end{corollary}
3314 
Proof block: Fractional Hermite transfer · Go to statement · Read statement and proof side by side3315 \begin{proof}
3316 The spectral theorem applies to the orthonormal Hermite basis. Every multiplier has modulus at most one and obeys the scalar semigroup law.
3317 \end{proof}
3318 
3319 \begin{figure}[t]
3320 \centering
3321 \includegraphics[width=.72\textwidth]{ou_hermite.pdf}
3322 \caption{Hermite packet multipliers for the Ornstein Uhlenbeck semigroup and two fractional powers of its number operator.}
3323 \label{extended:fig:ou_hermite}
3324 \end{figure}
3325 
3326 \FloatBarrier
3327 \section{Bayesian transfer and evidence packets}\label{extended:section:08}\label{extended:sec:bayesian_transfer}
3328 
3329 Let
3330 \[
3331 \Delta_{n-1}
3332 =
3333 \left\{p\in\RR^n:
3334 p_i\geq0,
3335 \quad
3336 \sum_ip_i=1
3337 \right\}
3338 \]
3339 be the probability simplex. Let \(\lambda\in[0,\infty)^n\) have positive evidence
3340 \[
3341 Z_\lambda(p)=\sum_i\lambda_ip_i.
3342 \]
3343 The Bayesian transfer is
3344 \[
3345 \Bcal_\lambda(p)
3346 =
3347 \frac{\lambda\odot p}{Z_\lambda(p)}.
3348 \]
3349 
Statement: Scale invariance and composition · Go to proof · Read statement and proof side by side3350 \begin{proposition}[Scale invariance and composition]\label{extended:record:045}\label{extended:prop:bayes_composition}
3351 For every \(c>0\),
3352 \[
3353 \Bcal_{c\lambda}=\Bcal_\lambda.
3354 \]
3355 If all relevant evidences are positive, then
3356 \[
3357 \Bcal_\mu(\Bcal_\lambda(p))
3358 =
3359 \Bcal_{\lambda\odot\mu}(p).
3360 \]
3361 \end{proposition}
3362 
Proof block: Scale invariance and composition · Go to statement · Read statement and proof side by side3363 \begin{proof}
3364 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.
3365 \end{proof}
3366 
3367 Let \(I=I_1\sqcup\cdots\sqcup I_m\) be a partition. Define packet evidences
3368 \[
3369 Z_\alpha
3370 =
3371 \sum_{i\in I_\alpha}\lambda_ip_i.
3372 \]
3373 For \(Z_\alpha>0\), define
3374 \[
3375 m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta},
3376 \qquad
3377 q_{\alpha,i}=\frac{\lambda_ip_i}{Z_\alpha}
3378 \quad(i\in I_\alpha).
3379 \]
3380 Packets with zero evidence receive posterior mass zero.
3381 
Statement: Bayesian packet reconstruction · Go to proof · Read statement and proof side by side3382 \begin{theorem}[Bayesian packet reconstruction]\label{extended:record:046}\label{extended:thm:bayesian_packet_reconstruction}
3383 The posterior is the disjoint packet synthesis
3384 \[
3385 \Bcal_\lambda(p)
3386 =
3387 \sum_{\alpha:Z_\alpha>0}
3388 m_\alpha q_\alpha.
3389 \]
3390 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.
3391 \end{theorem}
3392 
Proof block: Bayesian packet reconstruction · Go to statement · Read statement and proof side by side3393 \begin{proof}
3394 For \(i\in I_\alpha\),
3395 \[
3396 m_\alpha q_{\alpha,i}
3397 =
3398 \frac{Z_\alpha}{\sum_\beta Z_\beta}
3399 \frac{\lambda_ip_i}{Z_\alpha}
3400 =
3401 \frac{\lambda_ip_i}{Z_\lambda(p)}.
3402 \]
3403 The packet supports are disjoint.
3404 \end{proof}
3405 
Statement: Likelihood perturbation · Go to proof · Read statement and proof side by side3406 \begin{theorem}[Likelihood perturbation]\label{extended:record:047}\label{extended:thm:likelihood_perturbation}
3407 Let \(\lambda,\mu\geq0\) have positive evidences under a fixed prior \(p\). Set
3408 \[
3409 d_p(\lambda,\mu)
3410 =
3411 \sum_i\abs{\lambda_i-\mu_i}p_i.
3412 \]
3413 Then
3414 \[
3415 \norm{\Bcal_\lambda(p)-\Bcal_\mu(p)}_1
3416 \leq
3417 \frac{2d_p(\lambda,\mu)}
3418 {\max\{Z_\lambda(p),Z_\mu(p)\}}.
3419 \]
3420 \end{theorem}
3421 
Proof block: Likelihood perturbation · Go to statement · Read statement and proof side by side3422 \begin{proof}
3423 Add and subtract \(\mu\odot p/Z_\lambda(p)\). This gives
3424 \[
3425 \norm{\Bcal_\lambda(p)-\Bcal_\mu(p)}_1
3426 \leq
3427 \frac{2d_p(\lambda,\mu)}{Z_\lambda(p)}.
3428 \]
3429 Interchanging \(\lambda\) and \(\mu\) gives the same estimate with \(Z_\mu(p)\). Take the smaller of the two upper bounds.
3430 \end{proof}
3431 
Statement: Stability with respect to the prior · Go to proof · Read statement and proof side by side3432 \begin{theorem}[Stability with respect to the prior]\label{extended:record:048}\label{extended:thm:prior_stability}
3433 Assume
3434 \[
3435 0<a\leq\lambda_i\leq b
3436 \]
3437 for every \(i\). Then
3438 \[
3439 \norm{\Bcal_\lambda(p)-\Bcal_\lambda(q)}_1
3440 \leq
3441 \frac{2b}{a}\norm{p-q}_1
3442 \]
3443 for all priors \(p,q\in\Delta_{n-1}\).
3444 \end{theorem}
3445 
Proof block: Stability with respect to the prior · Go to statement · Read statement and proof side by side3446 \begin{proof}
3447 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.
3448 \end{proof}
3449 
3450 \begin{figure}[t]
3451 \centering
3452 \includegraphics[width=.72\textwidth]{bayesian_packets.pdf}
3453 \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.}
3454 \label{extended:fig:bayesian_packets}
3455 \end{figure}
3456 
3457 \FloatBarrier
3458 \section{Bayesian coarsening and finite sufficiency}\label{extended:section:09}\label{extended:sec:bayesian_sufficiency}
3459 
3460 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.
3461 
3462 Let \(C\) send a fine probability vector to its block masses
3463 \[
3464 (Cp)_\alpha
3465 =
3466 \sum_{i\in I_\alpha}p_i.
3467 \]
3468 Assume first that every block mass is positive. Write
3469 \[
3470 r_{\alpha,i}
3471 =
3472 \frac{p_i}{(Cp)_\alpha}
3473 \]
3474 for the prior conditional distribution inside packet \(I_\alpha\). Define the prior dependent packet average likelihood
3475 \[
3476 \overline\lambda_\alpha(p)
3477 =
3478 \sum_{i\in I_\alpha}\lambda_ir_{\alpha,i}.
3479 \]
3480 
Statement: Exact two stage Bayesian update · Go to proof · Read statement and proof side by side3481 \begin{theorem}[Exact two stage Bayesian update]\label{extended:record:049}\label{extended:thm:two_stage_bayes}
3482 For every strictly positive prior,
3483 \[
3484 C\Bcal_\lambda(p)
3485 =
3486 \Bcal_{\overline\lambda(p)}(Cp).
3487 \]
3488 For every packet with \(\overline\lambda_\alpha(p)>0\), the posterior conditional distribution is
3489 \[
3490 q_{\alpha,i}
3491 =
3492 \frac{\lambda_ir_{\alpha,i}}
3493 {\overline\lambda_\alpha(p)}.
3494 \]
3495 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.
3496 \end{theorem}
3497 
Proof block: Exact two stage Bayesian update · Go to statement · Read statement and proof side by side3498 \begin{proof}
3499 The posterior mass of packet \(I_\alpha\) is
3500 \[
3501 \frac{\sum_{i\in I_\alpha}\lambda_ip_i}
3502 {\sum_j\lambda_jp_j}
3503 =
3504 \frac{(Cp)_\alpha\overline\lambda_\alpha(p)}
3505 {\sum_\beta(Cp)_\beta\overline\lambda_\beta(p)}.
3506 \]
3507 This is the coarse Bayesian formula. Dividing the fine posterior coordinates by the packet mass gives the local formula.
3508 \end{proof}
3509 
Statement: Characterization of fixed coarse sufficiency · Go to proof · Read statement and proof side by side3510 \begin{theorem}[Characterization of fixed coarse sufficiency]\label{extended:record:050}\label{extended:thm:bayesian_sufficiency}
3511 Assume \(\lambda_i>0\) for every state. The following conditions are equivalent.
3512 \begin{enumerate}
3513 \item There is a positive coarse likelihood \(\ell\in(0,\infty)^m\) such that
3514 \[
3515 C\Bcal_\lambda(p)
3516 =
3517 \Bcal_\ell(Cp)
3518 \]
3519 for every strictly positive prior \(p\), and the posterior conditional distribution inside every packet equals its prior conditional distribution.
3520 \item The likelihood is constant on every packet.
3521 \end{enumerate}
3522 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.
3523 \end{theorem}
3524 
Proof block: Characterization of fixed coarse sufficiency · Go to statement · Read statement and proof side by side3525 \begin{proof}
3526 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.
3527 
3528 Conversely, fix any strictly positive prior. Equality of the local posterior and prior conditionals gives
3529 \[
3530 \frac{\lambda_i r_{\alpha,i}}
3531 {\overline\lambda_\alpha(p)}
3532 =
3533 r_{\alpha,i}
3534 \]
3535 for every \(i\in I_\alpha\). Positivity of the conditional prior implies
3536 \[
3537 \lambda_i=\overline\lambda_\alpha(p)
3538 \]
3539 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.
3540 \end{proof}
3541 
Statement: Coarsest fixed sufficient partition · Go to proof · Read statement and proof side by side3542 \begin{theorem}[Coarsest fixed sufficient partition]\label{extended:record:051}\label{extended:thm:coarsest_bayesian_partition}
3543 Let \(\mathcal L\) be a nonempty family of strictly positive likelihood vectors. Define
3544 \[
3545 i\sim_{\mathcal L}j
3546 \quad\Longleftrightarrow\quad
3547 \lambda_i=\lambda_j
3548 \quad\hbox{for every }\lambda\in\mathcal L.
3549 \]
3550 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.
3551 \end{theorem}
3552 
Proof block: Coarsest fixed sufficient partition · Go to statement · Read statement and proof side by side3553 \begin{proof}
3554 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.
3555 \end{proof}
3556 
Statement: Within packet information loss3557 \begin{definition}[Within packet information loss]\label{extended:record:052}
3558 For a prior \(p\), define
3559 \[
3560 \eta_i(p)
3561 =
3562 \lambda_i-\overline\lambda_\alpha(p)
3563 \quad\hbox{when }i\in I_\alpha.
3564 \]
3565 Then
3566 \[
3567 \sum_{i\in I_\alpha}r_{\alpha,i}\eta_i(p)=0.
3568 \]
3569 \end{definition}
3570 
Statement: Exact local distortion · Go to proof · Read statement and proof side by side3571 \begin{proposition}[Exact local distortion]\label{extended:record:053}\label{extended:prop:local_bayes_distortion}
3572 For every packet with positive mass and evidence,
3573 \[
3574 q_{\alpha,i}-r_{\alpha,i}
3575 =
3576 \frac{r_{\alpha,i}\eta_i(p)}
3577 {\overline\lambda_\alpha(p)}.
3578 \]
3579 Consequently,
3580 \[
3581 \norm{q_\alpha-r_\alpha}_1
3582 =
3583 \frac{
3584 \sum_{i\in I_\alpha}r_{\alpha,i}\abs{\eta_i(p)}
3585 }{\overline\lambda_\alpha(p)}.
3586 \]
3587 If \(\lambda\) is positive on the packet, then
3588 \[
3589 \norm{q_\alpha-r_\alpha}_1
3590 \leq
3591 \frac{\osc_{I_\alpha}(\lambda)}
3592 {\min_{i\in I_\alpha}\lambda_i}.
3593 \]
3594 \end{proposition}
3595 
Proof block: Exact local distortion · Go to statement · Read statement and proof side by side3596 \begin{proof}
3597 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
3598 \[
3599 \sum_i r_{\alpha,i}\abs{\eta_i}
3600 \leq
3601 \osc_{I_\alpha}(\lambda)
3602 \]
3603 and \(\overline\lambda_\alpha(p)\geq\min_{I_\alpha}\lambda\).
3604 \end{proof}
3605 
Statement: Approximate coarse Bayesian rule · Go to proof · Read statement and proof side by side3606 \begin{theorem}[Approximate coarse Bayesian rule]\label{extended:record:054}\label{extended:thm:approximate_coarse_bayes}
3607 Choose one positive number \(\ell_\alpha\) for every packet and define the block constant fine likelihood
3608 \[
3609 \widetilde\lambda_i=\ell_\alpha
3610 \quad(i\in I_\alpha).
3611 \]
3612 Then
3613 \[
3614 \norm{
3615 C\Bcal_\lambda(p)-\Bcal_\ell(Cp)
3616 }_1
3617 \leq
3618 \frac{
3619 2\sum_i p_i\abs{\lambda_i-\widetilde\lambda_i}
3620 }{
3621 \max\{Z_\lambda(p),Z_{\widetilde\lambda}(p)\}
3622 }.
3623 \]
3624 \end{theorem}
3625 
Proof block: Approximate coarse Bayesian rule · Go to statement · Read statement and proof side by side3626 \begin{proof}
3627 Block constancy gives
3628 \[
3629 C\Bcal_{\widetilde\lambda}(p)
3630 =
3631 \Bcal_\ell(Cp).
3632 \]
3633 Aggregation is an \(\ell^1\) contraction. Apply \cref{extended:thm:likelihood_perturbation} to the two fine posteriors and then aggregate.
3634 \end{proof}
3635 
Statement: Relative information3636 \begin{definition}[Relative information]\label{extended:record:055}
3637 For probability vectors \(q\) and \(p\), define
3638 \[
3639 D(q\Vert p)
3640 =
3641 \sum_iq_i\log\frac{q_i}{p_i}.
3642 \]
3643 The convention is \(0\log0=0\). A term with \(q_i>0\) and \(p_i=0\) has value \(+\infty\).
3644 \end{definition}
3645 
Statement: Exact information decomposition · Go to proof · Read statement and proof side by side3646 \begin{theorem}[Exact information decomposition]\label{extended:record:056}\label{extended:thm:bayesian_information_chain}
3647 Let \(p\) and \(q\) be probability vectors on a finite partition \((I_\alpha)_\alpha\), and assume \(q_i=0\) whenever \(p_i=0\). Let
3648 \[
3649 m=Cp,
3650 \qquad
3651 n=Cq,
3652 \]
3653 and let \(p_\alpha\) and \(q_\alpha\) be the conditional distributions inside packet \(I_\alpha\) whenever the corresponding mass is positive. Then
3654 \[
3655 D(q\Vert p)
3656 =
3657 D(n\Vert m)
3658 +
3659 \sum_{\alpha:n_\alpha>0}
3660  n_\alpha D(q_\alpha\Vert p_\alpha).
3661 \]
3662 The first term is the information retained by the coarse masses. The second term is the exact information carried by changes inside the packets.
3663 \end{theorem}
3664 
Proof block: Exact information decomposition · Go to statement · Read statement and proof side by side3665 \begin{proof}
3666 For \(i\in I_\alpha\), write
3667 \[
3668 p_i=m_\alpha p_{\alpha,i},
3669 \qquad
3670 q_i=n_\alpha q_{\alpha,i}.
3671 \]
3672 Then
3673 \[
3674 \log\frac{q_i}{p_i}
3675 =
3676 \log\frac{n_\alpha}{m_\alpha}
3677 +
3678 \log\frac{q_{\alpha,i}}{p_{\alpha,i}}.
3679 \]
3680 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}.
3681 \end{proof}
3682 
Statement: Coarse data processing and equality · Go to proof · Read statement and proof side by side3683 \begin{corollary}[Coarse data processing and equality]\label{extended:record:057}\label{extended:cor:coarse_data_processing}
3684 Under the assumptions of \cref{extended:thm:bayesian_information_chain},
3685 \[
3686 D(Cq\Vert Cp)
3687 \leq
3688 D(q\Vert p).
3689 \]
3690 Equality holds if and only if
3691 \[
3692 q_\alpha=p_\alpha
3693 \]
3694 for every packet with positive posterior mass.
3695 \end{corollary}
3696 
Proof block: Coarse data processing and equality · Go to statement · Read statement and proof side by side3697 \begin{proof}
3698 Every conditional relative information term in \cref{extended:thm:bayesian_information_chain} is nonnegative. It vanishes exactly when the two conditional distributions agree.
3699 \end{proof}
3700 
Statement: Information created by a Bayesian update · Go to proof · Read statement and proof side by side3701 \begin{corollary}[Information created by a Bayesian update]\label{extended:record:058}\label{extended:cor:bayesian_information_gain}
3702 Let \(q=\Bcal_\lambda(p)\) with positive likelihood and positive prior. Then
3703 \[
3704 D(q\Vert p)
3705 =
3706 \sum_iq_i\log\lambda_i
3707 -
3708 \log Z_\lambda(p).
3709 \]
3710 Inside a packet,
3711 \[
3712 D(q_\alpha\Vert r_\alpha)
3713 =
3714 \sum_{i\in I_\alpha}
3715 q_{\alpha,i}
3716 \log\frac{\lambda_i}{\overline\lambda_\alpha(p)}.
3717 \]
3718 Hence
3719 \[
3720 D(q_\alpha\Vert r_\alpha)
3721 \leq
3722 \log
3723 \frac{\max_{i\in I_\alpha}\lambda_i}
3724 {\min_{i\in I_\alpha}\lambda_i}.
3725 \]
3726 \end{corollary}
3727 
Proof block: Information created by a Bayesian update · Go to statement · Read statement and proof side by side3728 \begin{proof}
3729 The posterior ratio is
3730 \[
3731 \frac{q_i}{p_i}
3732 =
3733 \frac{\lambda_i}{Z_\lambda(p)}.
3734 \]
3735 The local ratio is
3736 \[
3737 \frac{q_{\alpha,i}}{r_{\alpha,i}}
3738 =
3739 \frac{\lambda_i}{\overline\lambda_\alpha(p)}.
3740 \]
3741 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.
3742 \end{proof}
3743 
Statement: Information criterion for local sufficiency · Go to proof · Read statement and proof side by side3744 \begin{corollary}[Information criterion for local sufficiency]\label{extended:record:059}\label{extended:cor:information_sufficiency}
3745 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.
3746 \end{corollary}
3747 
Proof block: Information criterion for local sufficiency · Go to statement · Read statement and proof side by side3748 \begin{proof}
3749 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.
3750 \end{proof}
3751 
3752 \begin{figure}[t]
3753 \centering
3754 \includegraphics[width=.72\textwidth]{bayesian_information_split.pdf}
3755 \caption{Relative information separated into a coarse packet contribution and conditional contributions inside the packets.}
3756 \label{extended:fig:bayesian_information_split}
3757 \end{figure}
3758 
3759 \begin{remark}\label{extended:record:060}
3760 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.
3761 \end{remark}
3762 
3763 \FloatBarrier
3764 \section{Bayesian flows and hidden Markov recursion}\label{extended:section:10}\label{extended:sec:bayesian_flows}
3765 
3766 Fix a score vector \(g\in\RR^n\). For \(t\geq0\), define
3767 \[
3768 \Phi_t(p)_i
3769 =
3770 \frac{\e^{tg_i}p_i}
3771 {\sum_j\e^{tg_j}p_j}.
3772 \]
3773 
Statement: Bayesian flow and replicator equation · Go to proof · Read statement and proof side by side3774 \begin{theorem}[Bayesian flow and replicator equation]\label{extended:record:061}\label{extended:thm:bayesian_flow}
3775 The maps \(\Phi_t\) form a nonlinear semigroup:
3776 \[
3777 \Phi_{s+t}=\Phi_s\circ\Phi_t.
3778 \]
3779 Every coordinate satisfies
3780 \[
3781 \frac{\dd}{\dd t}p_i(t)
3782 =
3783 \bigl(g_i-\ip{g}{p(t)}\bigr)p_i(t).
3784 \]
3785 The support of \(p(t)\) is independent of \(t\), and for any two coordinates in the support,
3786 \[
3787 \frac{p_i(t)}{p_j(t)}
3788 =
3789 \e^{t(g_i-g_j)}
3790 \frac{p_i(0)}{p_j(0)}.
3791 \]
3792 \end{theorem}
3793 
Proof block: Bayesian flow and replicator equation · Go to statement · Read statement and proof side by side3794 \begin{proof}
3795 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.
3796 \end{proof}
3797 
Statement: Evidence cumulants and score monotonicity · Go to proof · Read statement and proof side by side3798 \begin{theorem}[Evidence cumulants and score monotonicity]\label{extended:record:062}\label{extended:thm:evidence_cumulants}
3799 Set
3800 \[
3801 \psi(t)
3802 =
3803 \log\left(
3804 \sum_i\e^{tg_i}p_i(0)
3805 \right).
3806 \]
3807 Then
3808 \[
3809 \psi'(t)=\ip{g}{p(t)},
3810 \qquad
3811 \psi''(t)=\Var_{p(t)}(g)\geq0.
3812 \]
3813 Consequently,
3814 \[
3815 \frac{\dd}{\dd t}\ip{g}{p(t)}
3816 =
3817 \Var_{p(t)}(g).
3818 \]
3819 \end{theorem}
3820 
Proof block: Evidence cumulants and score monotonicity · Go to statement · Read statement and proof side by side3821 \begin{proof}
3822 Differentiate the log normalizing constant once and twice. The second derivative is the centered second moment under the tilted distribution.
3823 \end{proof}
3824 
Statement: Concentration on the maximal score packet · Go to proof · Read statement and proof side by side3825 \begin{corollary}[Concentration on the maximal score packet]\label{extended:record:063}\label{extended:cor:bayes_flow_concentration}
3826 Let
3827 \[
3828 G=\max\{g_i:p_i(0)>0\}
3829 \]
3830 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
3831 \[
3832 \delta
3833 =
3834 \min\{G-g_i:p_i(0)>0,
3835 \quad i\notin M\}>0.
3836 \]
3837 Then
3838 \[
3839 1-\sum_{i\in M}p_i(t)
3840 \leq
3841 \frac{1-p_M(0)}{p_M(0)}\e^{-\delta t},
3842 \]
3843 where \(p_M(0)=\sum_{i\in M}p_i(0)\).
3844 \end{corollary}
3845 
Proof block: Concentration on the maximal score packet · Go to statement · Read statement and proof side by side3846 \begin{proof}
3847 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 \(p_M(0)\).
3848 \end{proof}
3849 
3850 \begin{figure}[t]
3851 \centering
3852 \includegraphics[width=.72\textwidth]{bayesian_flow.pdf}
3853 \caption{A finite Bayesian flow. Mass concentrates on the states with the largest score, while every relative odds ratio follows an exact exponential law.}
3854 \label{extended:fig:bayesian_flow}
3855 \end{figure}
3856 
3857 \subsection{Prediction and update}
3858 
3859 Let \(K\) be column stochastic and let \(\lambda^{(r)}\) be the likelihood of the \(r\)th observation. The hidden Markov filter is
3860 \[
3861 \widehat p_r=Kp_{r-1},
3862 \qquad
3863 p_r=\Bcal_{\lambda^{(r)}}(\widehat p_r).
3864 \]
3865 A standard account of finite hidden Markov models appears in \cite{Rabiner1989}.
3866 
Statement: Simplex preservation · Go to proof · Read statement and proof side by side3867 \begin{proposition}[Simplex preservation]\label{extended:record:064}\label{extended:prop:hmm_preservation}
3868 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 \(p_r\).
3869 \end{proposition}
3870 
Proof block: Simplex preservation · Go to statement · Read statement and proof side by side3871 \begin{proof}
3872 Column stochasticity preserves nonnegativity and total mass. Bayesian normalization then preserves the same properties.
3873 \end{proof}
3874 
Statement: One step filter stability · Go to proof · Read statement and proof side by side3875 \begin{theorem}[One step filter stability]\label{extended:record:065}\label{extended:thm:hmm_stability}
3876 Let \(K\) and \(\widetilde K\) be column stochastic. Assume
3877 \[
3878 0<a\leq\lambda_i\leq b.
3879 \]
3880 Then
3881 \[
3882 \norm{
3883 \Bcal_\lambda(Kp)-\Bcal_\lambda(\widetilde Kq)
3884 }_1
3885 \leq
3886 \frac{2b}{a}
3887 \left(
3888 \norm{p-q}_1
3889 +
3890 \norm{K-\widetilde K}_{1\to1}
3891 \right).
3892 \]
3893 \end{theorem}
3894 
Proof block: One step filter stability · Go to statement · Read statement and proof side by side3895 \begin{proof}
3896 Apply \cref{extended:thm:prior_stability} to \(Kp\) and \(\widetilde Kq\). Then use
3897 \[
3898 \norm{Kp-\widetilde Kq}_1
3899 \leq
3900 \norm{K(p-q)}_1
3901 +
3902 \norm{(K-\widetilde K)q}_1
3903 \]
3904 and the stochastic contraction property.
3905 \end{proof}
3906 
3907 \begin{example}[Beta binomial transfer]\label{extended:record:066}
3908 A beta prior with parameters \((a,b)\) and binomial data containing \(s\) successes in \(n\) trials gives the posterior parameters
3909 \[
3910 (a+s,b+n-s).
3911 \]
3912 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}.
3913 \end{example}
3914 
3915 \FloatBarrier
3916 \section{Fractional functional calculus}\label{extended:section:11}\label{extended:sec:fractional}
3917 
3918 Let \(H\) be a finite dimensional Hilbert space and let \(L=L^*\geq0\). Write its spectral resolution as
3919 \[
3920 L=\sum_j\lambda_jP_j.
3921 \]
3922 For \(\alpha>0\), define
3923 \[
3924 L^\alpha
3925 =
3926 \sum_j\lambda_j^\alpha P_j
3927 \]
3928 and
3929 \[
3930 T_\alpha(t)
3931 =
3932 \e^{-tL^\alpha}
3933 =
3934 \sum_j\e^{-t\lambda_j^\alpha}P_j.
3935 \]
3936 
Statement: Fractional packet calculus · Go to proof · Read statement and proof side by side3937 \begin{theorem}[Fractional packet calculus]\label{extended:record:067}\label{extended:thm:fractional_packet_calculus}
3938 For \(\alpha,\beta>0\),
3939 \[
3940 L^\alpha L^\beta=L^{\alpha+\beta}.
3941 \]
3942 For \(s,t\geq0\),
3943 \[
3944 T_\alpha(s)T_\alpha(t)=T_\alpha(s+t).
3945 \]
3946 Every spectral packet \(P_jH\) reduces \(L^\alpha\) and \(T_\alpha(t)\).
3947 \end{theorem}
3948 
Proof block: Fractional packet calculus · Go to statement · Read statement and proof side by side3949 \begin{proof}
3950 The claims hold on each spectral packet and the projectors are mutually orthogonal.
3951 \end{proof}
3952 
Statement: Balakrishnan formula in finite dimension · Go to proof · Read statement and proof side by side3953 \begin{theorem}[Balakrishnan formula in finite dimension]\label{extended:record:068}\label{extended:thm:balakrishnan}
3954 For \(0<\alpha<1\),
3955 \[
3956 L^\alpha
3957 =
3958 \frac{\sin(\pi\alpha)}{\pi}
3959 \int_0^\infty
3960 s^{\alpha-1}L(sI+L)^{-1}
3961 \dd s.
3962 \]
3963 The integral converges in operator norm.
3964 \end{theorem}
3965 
Proof block: Balakrishnan formula in finite dimension · Go to statement · Read statement and proof side by side3966 \begin{proof}
3967 It is enough to prove the formula on an eigenvector with eigenvalue \(x\geq0\). For \(x>0\), the substitution \(s=xu\) gives
3968 \[
3969 \frac{\sin(\pi\alpha)}{\pi}
3970 \int_0^\infty
3971 s^{\alpha-1}\frac{x}{s+x}\dd s
3972 =x^\alpha
3973 \frac{\sin(\pi\alpha)}{\pi}
3974 \int_0^\infty\frac{u^{\alpha-1}}{1+u}\dd u
3975 =x^\alpha.
3976 \]
3977 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}.
3978 \end{proof}
3979 
Statement: Exact fractional naturality · Go to proof · Read statement and proof side by side3980 \begin{theorem}[Exact fractional naturality]\label{extended:record:069}\label{extended:thm:exact_fractional_naturality}
3981 Let \(L_f\) and \(L_c\) be finite dimensional positive self adjoint operators and let \(C\) satisfy
3982 \[
3983 CL_f=L_cC.
3984 \]
3985 Then, for every \(\alpha>0\) and \(t\geq0\),
3986 \[
3987 CL_f^\alpha=L_c^\alpha C
3988 \]
3989 and
3990 \[
3991 C\e^{-tL_f^\alpha}
3992 =
3993 \e^{-tL_c^\alpha}C.
3994 \]
3995 \end{theorem}
3996 
Proof block: Exact fractional naturality · Go to statement · Read statement and proof side by side3997 \begin{proof}
3998 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.
3999 \end{proof}
4000 
Statement: Fractional defect bound · Go to proof · Read statement and proof side by side4001 \begin{theorem}[Fractional defect bound]\label{extended:record:070}\label{extended:thm:fractional_defect}
4002 Let \(0<\alpha<1\). Suppose
4003 \[
4004 L_f\geq mI,
4005 \qquad
4006 L_c\geq mI
4007 \]
4008 for some \(m>0\). Define
4009 \[
4010 \Delta_L=CL_f-L_cC.
4011 \]
4012 Then
4013 \[
4014 \norm{CL_f^\alpha-L_c^\alpha C}
4015 \leq
4016 \alpha m^{\alpha-1}\norm{\Delta_L}.
4017 \]
4018 Consequently,
4019 \[
4020 \norm{
4021 C\e^{-tL_f^\alpha}
4022 -
4023 \e^{-tL_c^\alpha}C
4024 }
4025 \leq
4026 \alpha t m^{\alpha-1}\norm{\Delta_L}.
4027 \]
4028 \end{theorem}
4029 
Proof block: Fractional defect bound · Go to statement · Read statement and proof side by side4030 \begin{proof}
4031 Set
4032 \[
4033 R_f(s)=(sI+L_f)^{-1},
4034 \qquad
4035 R_c(s)=(sI+L_c)^{-1}.
4036 \]
4037 The resolvent identity gives
4038 \[
4039 CR_f(s)-R_c(s)C
4040 =
4041 -R_c(s)\Delta_LR_f(s).
4042 \]
4043 Since \(LR=I-sR\),
4044 \[
4045 CL_fR_f(s)-L_cR_c(s)C
4046 =
4047 sR_c(s)\Delta_LR_f(s).
4048 \]
4049 Insert this into \cref{extended:thm:balakrishnan}. The spectral lower bound gives
4050 \[
4051 \norm{R_f(s)},\norm{R_c(s)}
4052 \leq
4053 \frac{1}{s+m}.
4054 \]
4055 Therefore
4056 \[
4057 \begin{aligned}
4058 \norm{CL_f^\alpha-L_c^\alpha C}
4059 &\leq
4060 \frac{\sin(\pi\alpha)}{\pi}
4061 \norm{\Delta_L}
4062 \int_0^\infty
4063 \frac{s^\alpha}{(s+m)^2}\dd s\\
4064 &=
4065 \alpha m^{\alpha-1}\norm{\Delta_L}.
4066 \end{aligned}
4067 \]
4068 The semigroup estimate follows from \cref{extended:thm:duhamel_defect}, applied to the contraction generators \(-L_f^\alpha\) and \(-L_c^\alpha\).
4069 \end{proof}
4070 
4071 \begin{remark}\label{extended:record:071}
4072 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.
4073 \end{remark}
4074 
Statement: Stable subordination4075 \begin{standardfact}[Stable subordination]\label{extended:record:072}\label{extended:fact:stable_subordination}
4076 Let \(0<\alpha<1\) and \(t>0\). There is a probability measure \(\eta_{\alpha,t}\) on \([0,\infty)\) such that
4077 \[
4078 \int_0^\infty \e^{-s\lambda}\eta_{\alpha,t}(\dd s)
4079 =
4080 \e^{-t\lambda^\alpha}
4081 \]
4082 for every \(\lambda\geq0\). Consequently, for a finite dimensional positive self adjoint operator,
4083 \[
4084 \e^{-tL^\alpha}
4085 =
4086 \int_0^\infty
4087 \e^{-sL}\eta_{\alpha,t}(\dd s)
4088 \]
4089 in operator norm. The measures satisfy the convolution law
4090 \[
4091 \eta_{\alpha,s}*\eta_{\alpha,t}
4092 =
4093 \eta_{\alpha,s+t}.
4094 \]
4095 \end{standardfact}
4096 
4097 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}.
4098 
Statement: Poisson subordination · Go to proof · Read statement and proof side by side4099 \begin{theorem}[Poisson subordination]\label{extended:record:073}\label{extended:thm:poisson_subordination}
4100 For \(t>0\),
4101 \[
4102 \e^{-t\sqrt L}
4103 =
4104 \frac{t}{2\sqrt\pi}
4105 \int_0^\infty
4106 s^{-3/2}
4107 \e^{-t^2/(4s)}
4108 \e^{-sL}
4109 \dd s.
4110 \]
4111 The integral converges in operator norm.
4112 \end{theorem}
4113 
Proof block: Poisson subordination · Go to statement · Read statement and proof side by side4114 \begin{proof}
4115 On a spectral packet with eigenvalue \(\lambda\), the formula reduces to
4116 \[
4117 \e^{-t\sqrt\lambda}
4118 =
4119 \frac{t}{2\sqrt\pi}
4120 \int_0^\infty
4121 s^{-3/2}\e^{-t^2/(4s)-\lambda s}\dd s.
4122 \]
4123 This scalar identity follows from the Gaussian integral. Recombine the finitely many spectral packets.
4124 \end{proof}
4125 
4126 \begin{figure}[t]
4127 \centering
4128 \includegraphics[width=.72\textwidth]{fractional_multipliers.pdf}
4129 \caption{Fractional semigroup multipliers at time one. Smaller exponents preserve more of the high spectral range.}
4130 \label{extended:fig:fractional_multipliers}
4131 \end{figure}
4132 
4133 \begin{remark}\label{extended:record:074}
4134 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}.
4135 \end{remark}
4136 
4137 \FloatBarrier
4138 \section{Green operators and finite difference transfer}\label{extended:section:12}\label{extended:sec:green_pde}
4139 
4140 Consider
4141 \[
4142 -u''=f
4143 \quad\hbox{on }(0,1),
4144 \qquad
4145 u(0)=u(1)=0.
4146 \]
4147 The continuum Green kernel is
4148 \[
4149 G(x,y)=\min(x,y)-xy.
4150 \]
4151 Thus
4152 \[
4153 u(x)=\int_0^1G(x,y)f(y)\dd y.
4154 \]
4155 
4156 For \(n\) interior nodes, let
4157 \[
4158 h=\frac{1}{n+1},
4159 \qquad
4160 x_i=ih,
4161 \]
4162 and define
4163 \[
4164 L_h
4165 =
4166 \frac{1}{h^2}
4167 \begin{pmatrix}
4168 2&-1&&\\
4169 -1&2&\ddots&\\
4170 &\ddots&\ddots&-1\\
4171 &&-1&2
4172 \end{pmatrix}.
4173 \]
4174 
Statement: Exact nodal Green reconstruction · Go to proof · Read statement and proof side by side4175 \begin{theorem}[Exact nodal Green reconstruction]\label{extended:record:075}\label{extended:thm:nodal_green}
4176 For \(1\leq i,j\leq n\),
4177 \[
4178 \frac{1}{h}(L_h^{-1})_{ij}
4179 =
4180 G(x_i,x_j).
4181 \]
4182 \end{theorem}
4183 
Proof block: Exact nodal Green reconstruction · Go to statement · Read statement and proof side by side4184 \begin{proof}
4185 The inverse of the unscaled tridiagonal matrix has entry
4186 \[
4187 \frac{\min(i,j)(n+1-\max(i,j))}{n+1}.
4188 \]
4189 Multiplying by \(h^2\) and dividing by \(h\) gives
4190 \[
4191 h\min(i,j)
4192 \left(1-\frac{\max(i,j)}{n+1}\right)
4193 =
4194 \min(x_i,x_j)-x_ix_j.
4195 \]
4196 \end{proof}
4197 
Statement: Discrete Green quadrature · Go to proof · Read statement and proof side by side4198 \begin{corollary}[Discrete Green quadrature]\label{extended:record:076}\label{extended:cor:discrete_green_quadrature}
4199 If \(u^h\) solves
4200 \[
4201 L_hu^h=f^h,
4202 \]
4203 then every nodal value has the exact representation
4204 \[
4205 u_i^h
4206 =
4207 h\sum_{j=1}^nG(x_i,x_j)f_j^h.
4208 \]
4209 Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.
4210 \end{corollary}
4211 
Proof block: Discrete Green quadrature · Go to statement · Read statement and proof side by side4212 \begin{proof}
4213 By \cref{extended:thm:nodal_green},
4214 \[
4215 (L_h^{-1})_{ij}=hG(x_i,x_j).
4216 \]
4217 Multiply the inverse matrix by the source vector.
4218 \end{proof}
4219 
Statement: Discrete maximum bound · Go to proof · Read statement and proof side by side4220 \begin{proposition}[Discrete maximum bound]\label{extended:record:077}\label{extended:prop:discrete_inverse_bound}
4221 The inverse \(L_h^{-1}\) has nonnegative entries and
4222 \[
4223 \norm{L_h^{-1}}_{\infty\to\infty}
4224 =
4225 \max_{1\leq i\leq n}
4226 \frac{x_i(1-x_i)}{2}
4227 \leq
4228 \frac18.
4229 \]
4230 \end{proposition}
4231 
Proof block: Discrete maximum bound · Go to statement · Read statement and proof side by side4232 \begin{proof}
4233 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,
4234 \[
4235 L_hv=\one.
4236 \]
4237 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.
4238 \end{proof}
4239 
Statement: Second order Poisson solution error · Go to proof · Read statement and proof side by side4240 \begin{theorem}[Second order Poisson solution error]\label{extended:record:078}\label{extended:thm:poisson_error}
4241 Suppose \(u\in C^4[0,1]\) solves the Dirichlet problem above. Let
4242 \[
4243 f^h_i=f(x_i)
4244 \]
4245 and let \(u^h\) solve
4246 \[
4247 L_hu^h=f^h.
4248 \]
4249 Then
4250 \[
4251 \max_i\abs{u^h_i-u(x_i)}
4252 \leq
4253 \frac{h^2}{96}
4254 \norm{u^{(4)}}_\infty.
4255 \]
4256 \end{theorem}
4257 
Proof block: Second order Poisson solution error · Go to statement · Read statement and proof side by side4258 \begin{proof}
4259 Taylor expansion with remainder gives
4260 \[
4261 \norm{L_h(u(x_i))_i-f^h}_\infty
4262 \leq
4263 \frac{h^2}{12}\norm{u^{(4)}}_\infty.
4264 \]
4265 Apply \(L_h^{-1}\) and use \cref{extended:prop:discrete_inverse_bound}.
4266 \end{proof}
4267 
4268 \begin{figure}[t]
4269 \centering
4270 \includegraphics[width=.72\textwidth]{green_kernel.pdf}
4271 \caption{The continuum Dirichlet Green section and the scaled discrete inverse. The nodal values agree exactly.}
4272 \label{extended:fig:green_kernel}
4273 \end{figure}
4274 
4275 The discrete eigenpairs are
4276 \[
4277 \lambda_{k,h}
4278 =
4279 \frac{4}{h^2}
4280 \sin^2\left(\frac{k\pi h}{2}\right),
4281 \qquad
4282 v_{k,h}(i)=\sin(k\pi x_i).
4283 \]
4284 
Statement: Low mode eigenvalue control · Go to proof · Read statement and proof side by side4285 \begin{theorem}[Low mode eigenvalue control]\label{extended:record:079}\label{extended:thm:low_mode}
4286 For \(kh\leq1\),
4287 \[
4288 0\leq(k\pi)^2-\lambda_{k,h}
4289 \leq
4290 \frac{(k\pi)^4}{12}h^2.
4291 \]
4292 Consequently, for every fixed \(k\),
4293 \[
4294 \lambda_{k,h}=(k\pi)^2+O(h^2).
4295 \]
4296 \end{theorem}
4297 
Proof block: Low mode eigenvalue control · Go to statement · Read statement and proof side by side4298 \begin{proof}
4299 Set \(x=k\pi h/2\). The condition \(kh\leq1\) gives \(0\leq x\leq\pi/2\). On this interval,
4300 \[
4301 \sin x\geq x-\frac{x^3}{6},
4302 \]
4303 so
4304 \[
4305 0\leq x^2-\sin^2x\leq\frac{x^4}{3}.
4306 \]
4307 Multiply by \(4/h^2\).
4308 \end{proof}
4309 
Statement: Heat and fractional mode transfer · Go to proof · Read statement and proof side by side4310 \begin{corollary}[Heat and fractional mode transfer]\label{extended:record:080}\label{extended:cor:mode_transfer}
4311 Let \(\lambda_k=(k\pi)^2\). For \(t\geq0\),
4312 \[
4313 0\leq
4314 \e^{-t\lambda_{k,h}}-\e^{-t\lambda_k}
4315 \leq
4316 \frac{t(k\pi)^4}{12}h^2\e^{-t\lambda_{k,h}}.
4317 \]
4318 For \(0<\alpha\leq1\) and fixed \(k\),
4319 \[
4320 \lambda_{k,h}^\alpha
4321 =
4322 \lambda_k^\alpha+O(h^2)
4323 \]
4324 and
4325 \[
4326 \e^{-t\lambda_{k,h}^\alpha}
4327 =
4328 \e^{-t\lambda_k^\alpha}+O(th^2)
4329 \]
4330 uniformly for \(t\) in bounded intervals.
4331 \end{corollary}
4332 
Proof block: Heat and fractional mode transfer · Go to statement · Read statement and proof side by side4333 \begin{proof}
4334 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.
4335 \end{proof}
4336 
4337 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.
4338 
4339 \FloatBarrier
4340 \section{Finite state Feynman Kac transfer}\label{extended:section:13}\label{extended:sec:feynman_kac}
4341 \begin{interpretation}
4342 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.
4343 \end{interpretation}
4344 
4345 Let \(K\) be column stochastic. Let \(V\) be a real potential on the states and let
4346 \[
4347 D_V=\diag(V).
4348 \]
4349 For functions written as columns, the backward Markov operator is \(K^T\).
4350 
Statement: Discrete Feynman Kac formula · Go to proof · Read statement and proof side by side4351 \begin{theorem}[Discrete Feynman Kac formula]\label{extended:record:081}\label{extended:thm:discrete_feynman_kac}
4352 Define
4353 \[
4354 M_V=\diag(\e^{-V})K^T.
4355 \]
4356 For terminal data \(f\),
4357 \[
4358 (M_V^nf)_i
4359 =
4360 \Exp_i\left[
4361 \exp\left(
4362 -\sum_{r=0}^{n-1}V(X_r)
4363 \right)
4364 f(X_n)
4365 \right].
4366 \]
4367 \end{theorem}
4368 
Proof block: Discrete Feynman Kac formula · Go to statement · Read statement and proof side by side4369 \begin{proof}
4370 For one step,
4371 \[
4372 (M_Vf)_i
4373 =
4374 \e^{-V(i)}\sum_jK_{ji}f_j.
4375 \]
4376 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.
4377 \end{proof}
4378 
Statement: Continuous time finite state formula · Go to proof · Read statement and proof side by side4379 \begin{theorem}[Continuous time finite state formula]\label{extended:record:082}\label{extended:thm:continuous_feynman_kac}
4380 Let \(Q\) be a finite Markov generator and let \((X_t)_{t\geq0}\) be the corresponding chain. Then
4381 \[
4382 \left(
4383 \e^{t(Q^T-D_V)}f
4384 \right)_i
4385 =
4386 \Exp_i\left[
4387 \exp\left(
4388 -\int_0^tV(X_s)\dd s
4389 \right)
4390 f(X_t)
4391 \right].
4392 \]
4393 If \(V\geq v_0\), then
4394 \[
4395 \norm{\e^{t(Q^T-D_V)}}_{\infty\to\infty}
4396 \leq
4397 \e^{-v_0t}.
4398 \]
4399 \end{theorem}
4400 
Proof block: Continuous time finite state formula · Go to statement · Read statement and proof side by side4401 \begin{proof}
4402 Conditioning over the first small time interval gives the backward equation
4403 \[
4404 \dot u=(Q^T-D_V)u,
4405 \qquad
4406 u(0)=f.
4407 \]
4408 The finite system has a unique solution, namely the displayed matrix exponential. The expectation bound follows from
4409 \[
4410 \exp\left(-\int_0^tV(X_s)\dd s\right)
4411 \leq
4412 \e^{-v_0t}.
4413 \]
4414 This finite argument is the matrix form of the Feynman Kac principle originating in \cite{Kac1949}.
4415 \end{proof}
4416 
Statement: Alternating propagation and killing · Go to proof · Read statement and proof side by side4417 \begin{theorem}[Alternating propagation and killing]\label{extended:record:083}\label{extended:thm:feynman_kac_product}
4418 For every \(t\geq0\),
4419 \[
4420 \e^{t(Q^T-D_V)}
4421 =
4422 \lim_{n\to\infty}
4423 \left(
4424 \e^{-tD_V/n}\e^{tQ^T/n}
4425 \right)^n
4426 \]
4427 in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.
4428 \end{theorem}
4429 
Proof block: Alternating propagation and killing · Go to statement · Read statement and proof side by side4430 \begin{proof}
4431 Set \(B=-D_V\) and \(C=Q^T\). For bounded matrices,
4432 \[
4433 \e^{tB/n}\e^{tC/n}
4434 =
4435 I+\frac{t}{n}(B+C)+O(n^{-2})
4436 \]
4437 uniformly in operator norm. The standard telescoping comparison with
4438 \(
4439 (I+t(B+C)/n)^n
4440 \)
4441 shows that the product converges to \(\e^{t(B+C)}\). This is the displayed factor order, so the limit follows.
4442 \end{proof}
4443 
Statement: Positivity and composition · Go to proof · Read statement and proof side by side4444 \begin{corollary}[Positivity and composition]\label{extended:record:084}\label{extended:cor:feynman_kac_semigroup}
4445 The operators
4446 \[
4447 F_V(t)=\e^{t(Q^T-D_V)}
4448 \]
4449 form a positive semigroup. If \(f\geq0\), then \(F_V(t)f\geq0\).
4450 \end{corollary}
4451 
Proof block: Positivity and composition · Go to statement · Read statement and proof side by side4452 \begin{proof}
4453 Both claims follow from the expectation formula.
4454 \end{proof}
4455 
4456 \FloatBarrier
4457 \section{Schur Feshbach elimination}\label{extended:section:14}\label{extended:sec:feshbach}
4458 
4459 Let \(X=X_P\oplus X_Q\) and write
4460 \[
4461 A=
4462 \begin{pmatrix}
4463 A_{PP}&A_{PQ}\\
4464 A_{QP}&A_{QQ}
4465 \end{pmatrix}.
4466 \]
4467 For \(zI-A_{QQ}\) invertible, define
4468 \[
4469 \Sigma_P(z)
4470 =
4471 A_{PQ}(zI-A_{QQ})^{-1}A_{QP}
4472 \]
4473 and
4474 \[
4475 F_P(z)
4476 =
4477 zI-A_{PP}-\Sigma_P(z).
4478 \]
4479 
Statement: Schur Feshbach resolvent · Go to proof · Read statement and proof side by side4480 \begin{theorem}[Schur Feshbach resolvent]\label{extended:record:085}\label{extended:thm:feshbach_resolvent}
4481 Assume \(zI-A_{QQ}\) is invertible. Then \(zI-A\) is invertible if and only if \(F_P(z)\) is invertible. In that case,
4482 \[
4483 P(zI-A)^{-1}P
4484 =
4485 F_P(z)^{-1}.
4486 \]
4487 In finite dimension,
4488 \[
4489 \det(zI-A)
4490 =
4491 \det(zI-A_{QQ})\det F_P(z).
4492 \]
4493 \end{theorem}
4494 
Proof block: Schur Feshbach resolvent · Go to statement · Read statement and proof side by side4495 \begin{proof}
4496 Set
4497 \[
4498 L_z=
4499 \begin{pmatrix}
4500 I&-A_{PQ}(zI-A_{QQ})^{-1}\\
4501 0&I
4502 \end{pmatrix}
4503 \]
4504 and
4505 \[
4506 S_z=
4507 \begin{pmatrix}
4508 F_P(z)&0\\
4509 -A_{QP}&zI-A_{QQ}
4510 \end{pmatrix}.
4511 \]
4512 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}.
4513 \end{proof}
4514 
4515 \subsection{Time domain memory}
4516 
4517 For the homogeneous equation \(\dot x=Ax\), variation of constants gives
4518 \[
4519 x_Q(t)
4520 =
4521 \e^{tA_{QQ}}x_Q(0)
4522 +
4523 \int_0^t
4524 \e^{(t-s)A_{QQ}}A_{QP}x_P(s)
4525 \dd s.
4526 \]
4527 Substitution into the retained equation gives
4528 \[
4529 \begin{aligned}
4530 \dot x_P(t)
4531 ={}&
4532 A_{PP}x_P(t)
4533 +
4534 A_{PQ}\e^{tA_{QQ}}x_Q(0)\\
4535 &+
4536 \int_0^t
4537 \Kcal(t-s)x_P(s)
4538 \dd s,
4539 \end{aligned}
4540 \]
4541 where
4542 \[
4543 \Kcal(t)
4544 =
4545 A_{PQ}\e^{tA_{QQ}}A_{QP}.
4546 \]
4547 
Statement: Memory and self energy · Go to proof · Read statement and proof side by side4548 \begin{theorem}[Memory and self energy]\label{extended:record:086}\label{extended:thm:memory_self_energy}
4549 If \(z\) lies to the right of the spectrum of \(A_{QQ}\), then
4550 \[
4551 \int_0^\infty
4552 \e^{-zt}\Kcal(t)\dd t
4553 =
4554 \Sigma_P(z).
4555 \]
4556 If, for constants \(M\geq0\) and \(\gamma>0\),
4557 \[
4558 \norm{\e^{tA_{QQ}}}
4559 \leq
4560 M\e^{-\gamma t},
4561 \]
4562 then
4563 \[
4564 \norm{\Kcal(t)}
4565 \leq
4566 M\norm{A_{PQ}}\norm{A_{QP}}\e^{-\gamma t}
4567 \]
4568 and
4569 \[
4570 \norm{\Kcal}_{L^1(0,\infty)}
4571 \leq
4572 \frac{M}{\gamma}
4573 \norm{A_{PQ}}\norm{A_{QP}}.
4574 \]
4575 \end{theorem}
4576 
Proof block: Memory and self energy · Go to statement · Read statement and proof side by side4577 \begin{proof}
4578 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
4579 \[
4580 \norm{\e^{tB}}\leq M_\beta\exp(\norm B+\max(0,-\beta))\e^{\beta t}.
4581 \]
4582 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.
4583 \end{proof}
4584 
Statement: Criterion for exact local closure · Go to proof · Read statement and proof side by side4585 \begin{theorem}[Criterion for exact local closure]\label{extended:record:087}\label{extended:thm:local_closure}
4586 Assume \(x_Q(0)=0\) and \(d=\dim X_Q<\infty\). The following conditions are equivalent.
4587 \begin{enumerate}
4588 \item For every retained initial state,
4589 \[
4590 P\e^{tA}P
4591 =
4592 \e^{tA_{PP}}P
4593 \]
4594 for all \(t\geq0\).
4595 \item \(\Sigma_P(z)=0\) for every sufficiently large \(z\).
4596 \item \(\Kcal(t)=0\) for every \(t\geq0\).
4597 \item If \(d=\dim X_Q\), then
4598 \[
4599 A_{PQ}A_{QQ}^kA_{QP}=0
4600 \]
4601 for \(0\leq k<d\).
4602 \end{enumerate}
4603 \end{theorem}
4604 
Proof block: Criterion for exact local closure · Go to statement · Read statement and proof side by side4605 \begin{proof}
4606 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
4607 \[
4608 (zI-A_{PP}-\Sigma_P(z))^{-1}
4609 =
4610 (zI-A_{PP})^{-1}
4611 \]
4612 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.
4613 
4614 For large \(z\),
4615 \[
4616 \Sigma_P(z)
4617 =
4618 \sum_{k=0}^{\infty}
4619 \frac{A_{PQ}A_{QQ}^kA_{QP}}{z^{k+1}}.
4620 \]
4621 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.
4622 \end{proof}
4623 
4624 \begin{remark}\label{extended:record:088}
4625 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.
4626 \end{remark}
4627 
4628 \FloatBarrier
4629 \section{Memory modes and controlled compression}\label{extended:section:15}\label{extended:sec:memory_compression}
4630 \begin{interpretation}
4631 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.
4632 \end{interpretation}
4633 \begin{figure}[htbp]
4634 \centering
4635 \includegraphics[width=.94\textwidth]{monograph_figures/exact_jordan_memory.pdf}
4636 \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}
4637 \end{figure}
4638 
4639 
4640 
4641 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}.
4642 
Statement: Polynomial exponential memory of a finite hidden system · Go to proof · Read statement and proof side by side4643 \begin{theorem}[Polynomial exponential memory of a finite hidden system]\label{extended:record:089}\label{extended:thm:modal_memory}
4644 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
4645 \[
4646 N_j=(A_{QQ}-\mu_jI)Q_j.
4647 \]
4648 Let \(\nu_j\) be the least positive integer for which \(N_j^{\nu_j}=0\), and define
4649 \[
4650 B_{j,k}
4651 =
4652 A_{PQ}N_j^kQ_jA_{QP}
4653 \qquad
4654 (0\leq k<\nu_j).
4655 \]
4656 Then
4657 \[
4658 \Kcal(t)
4659 =
4660 \sum_{j=1}^r
4661 \e^{\mu_jt}
4662 \sum_{k=0}^{\nu_j-1}
4663 \frac{t^k}{k!}B_{j,k}
4664 \]
4665 and
4666 \[
4667 \Sigma_P(z)
4668 =
4669 \sum_{j=1}^r
4670 \sum_{k=0}^{\nu_j-1}
4671 \frac{B_{j,k}}{(z-\mu_j)^{k+1}}.
4672 \]
4673 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
4674 \[
4675 \norm{\Kcal-\Kcal_J}_{L^1(0,\infty)}
4676 \leq
4677 \sum_{j\notin J}
4678 \sum_{k=0}^{\nu_j-1}
4679 \frac{\norm{B_{j,k}}}
4680 {(-\operatorname{Re}\mu_j)^{k+1}}.
4681 \]
4682 \end{theorem}
4683 
Proof block: Polynomial exponential memory of a finite hidden system · Go to statement · Read statement and proof side by side4684 \begin{proof}
4685 The primary decomposition gives
4686 \[
4687 A_{QQ}Q_j=\mu_jQ_j+N_j
4688 \]
4689 with commuting scalar and nilpotent parts on \(Q_jX_Q\). Therefore
4690 \[
4691 \e^{tA_{QQ}}Q_j
4692 =
4693 \e^{\mu_jt}
4694 \sum_{k=0}^{\nu_j-1}
4695 \frac{t^k}{k!}N_j^kQ_j.
4696 \]
4697 Multiplication by \(A_{PQ}\) and \(A_{QP}\) gives the memory formula. On the \(j\)th primary space, the nilpotent resolvent expansion is
4698 \[
4699 (zI-A_{QQ})^{-1}Q_j
4700 =
4701 \sum_{k=0}^{\nu_j-1}
4702 \frac{N_j^kQ_j}{(z-\mu_j)^{k+1}}.
4703 \]
4704 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
4705 \[
4706 \int_0^\infty
4707 \e^{-zt}\e^{\mu_jt}
4708 \frac{t^k}{k!}
4709 \dd t
4710 =
4711 \frac{1}{(z-\mu_j)^{k+1}}.
4712 \]
4713 Under stability,
4714 \[
4715 \int_0^\infty
4716 \e^{\operatorname{Re}\mu_jt}
4717 \frac{t^k}{k!}
4718 \dd t
4719 =
4720 \frac{1}{(-\operatorname{Re}\mu_j)^{k+1}}.
4721 \]
4722 Sum the omitted terms.
4723 \end{proof}
4724 
Statement: Optimal selection for the certified primary block bound · Go to proof · Read statement and proof side by side4725 \begin{corollary}[Optimal selection for the certified primary block bound]\label{extended:record:090}\label{extended:cor:optimal_primary_selection}
4726 Under the stable assumptions of \cref{extended:thm:modal_memory}, define the primary block weights
4727 \[
4728 w_j
4729 =
4730 \sum_{k=0}^{\nu_j-1}
4731 \frac{\norm{B_{j,k}}}
4732 {(-\operatorname{Re}\mu_j)^{k+1}}.
4733 \]
4734 Among all sets \(J\) containing \(k\) primary blocks, the certified upper bound
4735 \[
4736 \sum_{j\notin J}w_j
4737 \]
4738 is minimized by retaining the \(k\) largest weights. If
4739 \[
4740 w_{(1)}\geq w_{(2)}\geq\cdots\geq w_{(r)},
4741 \]
4742 then the least number of retained primary blocks certified to give \(L^1\) error at most \(\varepsilon\) is the least \(k\) for which
4743 \[
4744 \sum_{j>k}w_{(j)}\leq\varepsilon.
4745 \]
4746 \end{corollary}
4747 
Proof block: Optimal selection for the certified primary block bound · Go to statement · Read statement and proof side by side4748 \begin{proof}
4749 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.
4750 \end{proof}
4751 
Statement: Normal hidden dynamics · Go to proof · Read statement and proof side by side4752 \begin{corollary}[Normal hidden dynamics]\label{extended:record:091}\label{extended:cor:normal_modal_memory}
4753 Suppose \(X_Q\) is a finite dimensional Hilbert space. If \(A_{QQ}\) is normal with spectral resolution
4754 \[
4755 A_{QQ}=\sum_{j=1}^r\mu_jQ_j,
4756 \]
4757 then
4758 \[
4759 \Kcal(t)=\sum_{j=1}^r\e^{\mu_jt}B_j,
4760 \qquad
4761 \Sigma_P(z)=\sum_{j=1}^r\frac{B_j}{z-\mu_j},
4762 \]
4763 where
4764 \[
4765 B_j=A_{PQ}Q_jA_{QP}.
4766 \]
4767 If \(\operatorname{Re}\mu_j<0\), then
4768 \[
4769 \norm{\Kcal-\Kcal_J}_{L^1(0,\infty)}
4770 \leq
4771 \sum_{j\notin J}
4772 \frac{\norm{B_j}}{-\operatorname{Re}\mu_j}.
4773 \]
4774 \end{corollary}
4775 
Proof block: Normal hidden dynamics · Go to statement · Read statement and proof side by side4776 \begin{proof}
4777 Normality removes every nilpotent part, so \(\nu_j=1\). Apply \cref{extended:thm:modal_memory}.
4778 \end{proof}
4779 
Statement: Pole order and hidden Jordan depth · Go to proof · Read statement and proof side by side4780 \begin{corollary}[Pole order and hidden Jordan depth]\label{extended:record:092}\label{extended:cor:pole_jordan_depth}
4781 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.
4782 \end{corollary}
4783 
Proof block: Pole order and hidden Jordan depth · Go to statement · Read statement and proof side by side4784 \begin{proof}
4785 Read the pole order from the partial fraction expansion in \cref{extended:thm:modal_memory}.
4786 \end{proof}
4787 
4788 \begin{figure}[t]
4789 \centering
4790 \includegraphics[width=.72\textwidth]{jordan_memory.pdf}
4791 \caption{A stable Jordan block produces a polynomial exponential memory mode. A diagonal hidden block produces only pure exponentials.}
4792 \label{extended:fig:jordan_memory}
4793 \end{figure}
4794 
Statement: Finite auxiliary state realization · Go to proof · Read statement and proof side by side4795 \begin{corollary}[Finite auxiliary state realization]\label{extended:record:093}\label{extended:cor:finite_auxiliary_realization}
4796 Every kernel of the form
4797 \[
4798 \Kcal(t)=B\e^{tD}C
4799 \]
4800 can be represented exactly by adjoining the finite auxiliary state
4801 \[
4802 \dot y=Dy+Cx_P,
4803 \qquad
4804 \dot x_P=A_{PP}x_P+By,
4805 \qquad
4806 y(0)=0.
4807 \]
4808 A nonzero auxiliary initial state adds the explicit forcing \(B\e^{tD}y(0)\). The self energy of the augmented system is
4809 \[
4810 B(zI-D)^{-1}C.
4811 \]
4812 \end{corollary}
4813 
Proof block: Finite auxiliary state realization · Go to statement · Read statement and proof side by side4814 \begin{proof}
4815 Eliminate \(y\) by variation of constants or apply \cref{extended:thm:feshbach_resolvent} to the augmented block matrix.
4816 \end{proof}
4817 
Statement: Volterra stability under kernel compression · Go to proof · Read statement and proof side by side4818 \begin{theorem}[Volterra stability under kernel compression]\label{extended:record:094}\label{extended:thm:volterra_compression}
4819 Let \(S(t)\) be a strongly continuous semigroup on a Banach space \(X\), fix \(T>0\), and assume
4820 \[
4821 \norm{S(t)}\leq M_T
4822 \quad(0\leq t\leq T).
4823 \]
4824 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 \(\widetilde K\):
4825 \[
4826 \begin{aligned}
4827 x(t)
4828 ={}&S(t)x_0+
4829 \int_0^tS(t-s)g(s)\dd s\\
4830 &+
4831 \int_0^tS(t-s)
4832 \int_0^sK(s-r)x(r)\dd r\dd s.
4833 \end{aligned}
4834 \]
4835 Assume
4836 \[
4837 \sup_{0\leq t\leq T}\norm{\widetilde x(t)}\leq B.
4838 \]
4839 Set
4840 \[
4841 L=\norm K_{L^1(0,T)},
4842 \qquad
4843 \varepsilon=\norm{K-\widetilde K}_{L^1(0,T)}.
4844 \]
4845 Then
4846 \[
4847 \sup_{0\leq t\leq T}
4848 \norm{x(t)-\widetilde x(t)}
4849 \leq
4850 M_TBT\varepsilon
4851 \exp(M_TLT).
4852 \]
4853 \end{theorem}
4854 
Proof block: Volterra stability under kernel compression · Go to statement · Read statement and proof side by side4855 \begin{proof}
4856 Let
4857 \[
4858 E(t)=\sup_{0\leq r\leq t}\norm{x(r)-\widetilde x(r)}.
4859 \]
4860 Subtract the two mild equations. Fubini's theorem and the triangle inequality give
4861 \[
4862 E(t)
4863 \leq
4864 M_TL\int_0^tE(r)\dd r
4865 +
4866 M_TBt\varepsilon.
4867 \]
4868 For \(t\leq T\), replace the last factor \(t\) by \(T\) and apply Gronwall's inequality.
4869 \end{proof}
4870 
Statement: Trajectory bound for modal truncation · Go to proof · Read statement and proof side by side4871 \begin{corollary}[Trajectory bound for modal truncation]\label{extended:record:095}\label{extended:cor:modal_trajectory_bound}
4872 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
4873 \[
4874 M_TBT
4875 \exp(M_TLT)
4876 \sum_{j\notin J}
4877 \sum_{k=0}^{\nu_j-1}
4878 \frac{\norm{B_{j,k}}}
4879 {(-\operatorname{Re}\mu_j)^{k+1}}.
4880 \]
4881 \end{corollary}
4882 
Proof block: Trajectory bound for modal truncation · Go to statement · Read statement and proof side by side4883 \begin{proof}
4884 Insert the \(L^1\) error from \cref{extended:thm:modal_memory} into \cref{extended:thm:volterra_compression}.
4885 \end{proof}
4886 
4887 \begin{figure}[t]
4888 \centering
4889 \includegraphics[width=.72\textwidth]{memory_compression.pdf}
4890 \caption{Full and truncated memory kernels with their pointwise difference. The bound uses integrated error.}
4891 \label{extended:fig:memory_compression}
4892 \end{figure}
4893 
4894 \begin{remark}\label{extended:record:096}
4895 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.
4896 \end{remark}
4897 
4898 \FloatBarrier
4899 \section{Survival, moments, and competing risks}\label{extended:section:16}\label{extended:sec:survival}
4900 
4901 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
4902 \[
4903 p(t)=\e^{tA}p_0,
4904 \qquad
4905 S(t)=\one^Tp(t),
4906 \qquad
4907 a^T=-\one^TA.
4908 \]
4909 The row vector \(a^T\) contains the exit rates.
4910 
Statement: State dependent killing as survival · Go to proof · Read statement and proof side by side4911 \begin{theorem}[State dependent killing as survival]\label{extended:record:097}\label{extended:thm:killing_survival}
4912 Let \(Q\) be a finite Markov generator, let \(V\geq0\), and set
4913 \[
4914 A=Q-D_V.
4915 \]
4916 For an initial distribution \(p_0\), define
4917 \[
4918 S_V(t)=\one^T\e^{tA}p_0.
4919 \]
4920 Then
4921 \[
4922 S_V(t)
4923 =
4924 \Exp_{p_0}
4925 \left[
4926 \exp\left(
4927 -\int_0^tV(X_s)\dd s
4928 \right)
4929 \right].
4930 \]
4931 Also,
4932 \[
4933 -S_V'(t)
4934 =
4935 \one^TD_V\e^{tA}p_0.
4936 \]
4937 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.
4938 \end{theorem}
4939 
Proof block: State dependent killing as survival · Go to statement · Read statement and proof side by side4940 \begin{proof}
4941 Average \cref{extended:thm:continuous_feynman_kac} with terminal function \(f=\one\) against the initial distribution. Transposition gives
4942 \[
4943 p_0^T\e^{t(Q^T-D_V)}\one
4944 =
4945 \one^T\e^{t(Q-D_V)}p_0.
4946 \]
4947 Since \(\one^TQ=0\), differentiation gives
4948 \[
4949 S_V'(t)
4950 =
4951 -\one^TD_V\e^{tA}p_0.
4952 \]
4953 \end{proof}
4954 
Statement: Survival and hazard identities · Go to proof · Read statement and proof side by side4955 \begin{theorem}[Survival and hazard identities]\label{extended:record:098}\label{extended:thm:survival_hazard}
4956 The event density and hazard are
4957 \[
4958 f(t)=a^T\e^{tA}p_0=-S'(t),
4959 \qquad
4960 h(t)=\frac{f(t)}{S(t)}
4961 \]
4962 whenever \(S(t)>0\). The survival function is nonincreasing and
4963 \[
4964 S(t)
4965 =
4966 \exp\left(
4967 -\int_0^th(s)\dd s
4968 \right).
4969 \]
4970 \end{theorem}
4971 
Proof block: Survival and hazard identities · Go to statement · Read statement and proof side by side4972 \begin{proof}
4973 Because \(A\) is a Metzler matrix, \(\e^{tA}\) is nonnegative. The column sum assumption makes \(a^T\) nonnegative. Differentiate \(S(t)\):
4974 \[
4975 S'(t)=\one^TA\e^{tA}p_0=-a^T\e^{tA}p_0.
4976 \]
4977 The remaining identity follows by integrating \((\log S)'=-h\).
4978 \end{proof}
4979 
Statement: Laplace transform and moments · Go to proof · Read statement and proof side by side4980 \begin{theorem}[Laplace transform and moments]\label{extended:record:099}\label{extended:thm:survival_moments}
4981 Assume the spectrum of \(A\) lies in the open left half plane. Then absorption occurs with probability one. For \(s\geq0\),
4982 \[
4983 \Exp[\e^{-s\tau}]
4984 =
4985 a^T(sI-A)^{-1}p_0.
4986 \] For every integer \(m\geq1\),
4987 \[
4988 \Exp[\tau^m]
4989 =
4990 m!\one^T(-A)^{-m}p_0.
4991 \]
4992 \end{theorem}
4993 
Proof block: Laplace transform and moments · Go to statement · Read statement and proof side by side4994 \begin{proof}
4995 The Laplace transform follows by integrating the density in \cref{extended:thm:survival_hazard}. Stability of \(A\) gives
4996 \[
4997 \int_0^\infty
4998 \e^{-st}\e^{tA}\dd t
4999 =
5000 (sI-A)^{-1}.
5001 \]
5002 At \(s=0\),
5003 \[
5004 a^T(-A)^{-1}p_0
5005 =
5006 \one^Tp_0=1.
5007 \]
5008 For the moments,
5009 \[
5010 \int_0^\infty t^m\e^{tA}\dd t
5011 =
5012 m!(-A)^{-m-1}.
5013 \]
5014 Multiplying by \(a^T=\one^T(-A)\) gives the result. These are the standard matrix formulas for phase type laws \cite{Neuts1981,AsmussenNermanOlsson1996}.
5015 \end{proof}
5016 
Statement: Residual life as a restarted semigroup · Go to proof · Read statement and proof side by side5017 \begin{proposition}[Residual life as a restarted semigroup]\label{extended:record:100}\label{extended:prop:residual_life}
5018 For \(S(t)>0\), define the conditional transient distribution
5019 \[
5020 p_t
5021 =
5022 \frac{\e^{tA}p_0}{S(t)}.
5023 \]
5024 Then
5025 \[
5026 \frac{S(t+s)}{S(t)}
5027 =
5028 \one^T\e^{sA}p_t.
5029 \]
5030 \end{proposition}
5031 
Proof block: Residual life as a restarted semigroup · Go to statement · Read statement and proof side by side5032 \begin{proof}
5033 Use the semigroup law in the numerator \(S(t+s)=\one^T\e^{sA}\e^{tA}p_0\).
5034 \end{proof}
5035 
Statement: Competing risk packets · Go to proof · Read statement and proof side by side5036 \begin{theorem}[Competing risk packets]\label{extended:record:101}\label{extended:thm:competing_risks}
5037 Suppose the exit row splits into nonnegative cause rows
5038 \[
5039 a^T
5040 =
5041 \sum_{r=1}^q a_r^T.
5042 \]
5043 Define
5044 \[
5045 f_r(t)=a_r^T\e^{tA}p_0.
5046 \]
5047 Then
5048 \[
5049 f(t)=\sum_{r=1}^qf_r(t).
5050 \]
5051 The cause specific hazard is
5052 \[
5053 h_r(t)=\frac{f_r(t)}{S(t)},
5054 \]
5055 and
5056 \[
5057 h(t)=\sum_{r=1}^qh_r(t).
5058 \]
5059 If the spectrum of \(A\) lies in the open left half-plane, the eventual probability of cause \(r\) is
5060 \[
5061 \int_0^\infty f_r(t)\dd t
5062 =
5063 a_r^T(-A)^{-1}p_0.
5064 \]
5065 \end{theorem}
5066 
Proof block: Competing risk packets · Go to statement · Read statement and proof side by side5067 \begin{proof}
5068 Every identity follows by linearity and the stable resolvent integral.
5069 \end{proof}
5070 
5071 \begin{figure}[t]
5072 \centering
5073 \includegraphics[width=.72\textwidth]{survival_causes.pdf}
5074 \caption{Survival and two cause specific event densities from one transient semigroup.}
5075 \label{extended:fig:survival_causes}
5076 \end{figure}
5077 
5078 \begin{remark}\label{extended:record:102}
5079 A finite phase type hazard need not be monotone. The semigroup formulas guarantee nonnegative density and decreasing survival, not monotonicity of the hazard.
5080 \end{remark}
5081 
5082 \FloatBarrier
5083 \section{Finite illustrations and computational checks}\label{extended:section:17}\label{extended:sec:finite_checks}
5084 
5085 The figures in the article 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.
5086 
5087 \begin{enumerate}
5088 \item Packet block convolution, the block differential system, and the Laplace resolvent formula.
5089 \item The Duhamel and resolvent defect identities.
5090 \item Exact and perturbed Markov lumping.
5091 \item Conditional expectation, tower laws, martingale packet energy, and Hermite multipliers.
5092 \item Bayesian reconstruction, sufficiency, local distortion, and information decomposition.
5093 \item The evidence variance identity along a Bayesian flow.
5094 \item The Balakrishnan formula and the fractional defect estimate.
5095 \item Nodal Green reconstruction and the Poisson error bound.
5096 \item Discrete Feynman Kac path enumeration.
5097 \item The Schur Feshbach resolvent and memory transform.
5098 \item Jordan memory modes, modal truncation, and the Volterra stability estimate.
5099 \item Feynman Kac killing, survival transforms, moments, and competing risk masses.
5100 \end{enumerate}
5101 
5102 \begin{table}[t]
5103 \centering
5104 \begin{tabular}{lc}
5105 \toprule
5106 Identity & Typical residual \\
5107 \midrule
5108 Packet block convolution & below \(3\times10^{-14}\) \\
5109 Duhamel defect & below \(3\times10^{-12}\) \\
5110 Markov lumping & below \(3\times10^{-14}\) \\
5111 Bayesian reconstruction & below \(3\times10^{-15}\) \\
5112 Fractional spectral law & below \(3\times10^{-12}\) \\
5113 Nodal Green identity & below \(3\times10^{-13}\) \\
5114 Feynman Kac enumeration & below \(3\times10^{-14}\) \\
5115 Schur Feshbach resolvent & below \(3\times10^{-12}\) \\
5116 Memory Laplace identity & below \(3\times10^{-11}\) \\
5117 Survival moment formulas & below \(3\times10^{-11}\) \\
5118 \bottomrule
5119 \end{tabular}
5120 \caption{Numerical tolerances in the finite verification suite. The proofs do not depend on these computations.}
5121 \label{extended:tab:verification_tolerances}
5122 \end{table}
5123 
5124 \FloatBarrier
5125 \section{Limits and further questions}\label{extended:section:18}\label{extended:sec:limits}
5126 
5127 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.
5128 
5129 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.
5130 
5131 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.
5132 
5133 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.
5134 
5135 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.
5136 
5137 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.
5138 
5139 \FloatBarrier
5140 \section{Conclusion}\label{extended:section:19}\label{extended:sec:conclusion}
5141 
5142 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.
5143 
5144 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.
5145 
5146 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.
5147 
5148 Hidden packets provide the final structural lesson. Exact elimination does not erase them. It returns them as a rational self energy in the resolvent and a causal memory kernel in time. The primary decomposition of the hidden generator produces every finite memory kernel as a finite sum of polynomial exponential modes. Stable blocks can be truncated with an integrated error certificate, and the Volterra estimate transfers that certificate to the retained trajectory. State dependent killing identifies the Feynman Kac potential with a survival mechanism. Survival and competing risks are then scalar reconstructions of a transient semigroup.
5149 
5150 The article therefore separates four operations that are often compressed into one informal idea of coarse dynamics:
5151 \[
5152 \begin{aligned}
5153 \text{intertwining}
5154 &\longrightarrow
5155 \text{refinement control}\\
5156 &\longrightarrow
5157 \text{statistical sufficiency}\\
5158 &\longrightarrow
5159 \text{memory closure}.
5160 \end{aligned}
5161 \]
5162 Each arrow has its own hypotheses and its own obstruction.
5163 
5164 
5165 
5166 
5167 
5168 \clearpage
5169 \part*{Part III\quad Retained results and operator calculus}\addcontentsline{toc}{part}{Part III: Retained results and operator calculus}
5170 \setcounter{section}{0}\setcounter{figure}{0}\setcounter{table}{0}
5171 \renewcommand{\thesection}{R\arabic{section}}
5172 \renewcommand{\theHsection}{R.\arabic{section}}
5173 \renewcommand{\thefigure}{R\arabic{figure}}
5174 \renewcommand{\theHfigure}{R.\arabic{figure}}
5175 \renewcommand{\thetable}{R\arabic{table}}
5176 \renewcommand{\theHtable}{R.\arabic{table}}
5177 \FloatBarrier
5178 \section{Transfer paths and product-topology limits}\label{earlier:section:01}
5179 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}.
Statement: Packet convolution · Go to proof · Read statement and proof side by side5180 \begin{theorem}[Packet convolution]\label{earlier:record:001}\label{earlier:thm:packet_convolution}
5181 For all \(s,t\geq0\) and all packets \(\alpha,\beta\),
5182 \[
5183 \boxed{
5184 T_{\beta\alpha}(s+t)
5185 =
5186 \sum_{\gamma\in I_0}
5187 T_{\beta\gamma}(s)T_{\gamma\alpha}(t).
5188 }
5189 \]
5190 The original semigroup is reconstructed from its blocks by
5191 \[
5192 T(t)=\sum_{\alpha,\beta\in I_0}T_{\beta\alpha}(t).
5193 \]
5194 \end{theorem}
5195 
Proof block: Packet convolution · Go to statement · Read statement and proof side by side5196 \begin{proof}
5197 Insert the identity between the two semigroup factors:
5198 \[
5199 \begin{aligned}
5200 P_\beta T(s+t)P_\alpha
5201 &=P_\beta T(s)\left(\sum_\gamma P_\gamma\right)T(t)P_\alpha\\
5202 &=\sum_\gamma
5203 P_\beta T(s)P_\gamma T(t)P_\alpha.
5204 \end{aligned}
5205 \]
5206 The reconstruction formula follows by inserting the packet identity on both sides of \(T(t)\).
5207 \end{proof}
5208 
Statement: Packet resolvent convolution · Go to proof · Read statement and proof side by side5209 \begin{proposition}[Packet resolvent convolution]\label{earlier:record:002}\label{earlier:prop:packet_resolvent}
5210 Let \(z\) belong to the resolvent set of \(A\). If \(\operatorname{Re}z\) is larger than a growth bound for the semigroup, then
5211 \[
5212 P_\beta(zI-A)^{-1}P_\alpha
5213 =
5214 \int_0^{\infty}\e^{-zt}T_{\beta\alpha}(t)\dd t.
5215 \]
5216 \end{proposition}
5217 
Proof block: Packet resolvent convolution · Go to statement · Read statement and proof side by side5218 \begin{proof}
5219 The standard Laplace representation of the resolvent is
5220 \[
5221 (zI-A)^{-1}
5222 =
5223 \int_0^{\infty}\e^{-zt}T(t)\dd t.
5224 \]
5225 Multiplication by the bounded packet projections may be passed through the Bochner integral.
5226 \end{proof}
5227 
5228 
5229 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.
Statement: Minimum transfer order · Go to proof · Read statement and proof side by side5230 \begin{corollary}[Minimum transfer order]\label{earlier:record:003}\label{earlier:cor:minimum_transfer_order}
5231 Suppose every directed packet path from \(\alpha\) to \(\beta\) uses at least \(m_0\) off diagonal transfers. Then
5232 \[
5233 P_\beta\e^{tA}P_\alpha
5234 =O(t^{m_0})
5235 \quad\text{as }t\downarrow0.
5236 \]
5237 If there is no directed path from \(\alpha\) to \(\beta\), the block vanishes for all \(t\).
5238 \end{corollary}
5239 
Proof block: Minimum transfer order · Go to statement · Read statement and proof side by side5240 \begin{proof}
5241 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.
5242 \end{proof}
5243 
5244 
5245 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.
Statement: Several level telescoping identity · Go to proof · Read statement and proof side by side5246 \begin{theorem}[Several level telescoping identity]\label{earlier:record:004}\label{earlier:thm:multilevel_telescoping}
5247 Let \(T_j\) be bounded operators on \(X_j\), and set
5248 \[
5249 E_j=C_jT_{j+1}-T_jC_j.
5250 \]
5251 Then
5252 \[
5253 \boxed{
5254 C_{\ell,m}T_m-T_\ell C_{\ell,m}
5255 =
5256 \sum_{j=\ell}^{m-1}
5257 C_{\ell,j}E_jC_{j+1,m}.
5258 }
5259 \]
5260 \end{theorem}
5261 
Proof block: Several level telescoping identity · Go to statement · Read statement and proof side by side5262 \begin{proof}
5263 Expand each summand as
5264 \[
5265 C_{\ell,j+1}T_{j+1}C_{j+1,m}
5266 -
5267 C_{\ell,j}T_jC_{j,m}.
5268 \]
5269 The intermediate terms cancel. The first surviving term is \(C_{\ell,m}T_m\), and the last is \(-T_\ell C_{\ell,m}\).
5270 \end{proof}
5271 
Statement: Summable generator defects · Go to proof · Read statement and proof side by side5272 \begin{corollary}[Summable generator defects]\label{earlier:record:005}\label{earlier:cor:summable_defects}
5273 Suppose \(C_j\) are contractions and \((T_j(t))\) are contraction semigroups with bounded generators \(A_j\). Set
5274 \[
5275 \delta_j=\norm{C_jA_{j+1}-A_jC_j}.
5276 \]
5277 Then
5278 \[
5279 \norm{C_{\ell,m}T_m(t)-T_\ell(t)C_{\ell,m}}
5280 \leq
5281  t\sum_{j=\ell}^{m-1}\delta_j.
5282 \]
5283 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.
5284 \end{corollary}
5285 
Proof block: Summable generator defects · Go to statement · Read statement and proof side by side5286 \begin{proof}
5287 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.
5288 \end{proof}
5289 
Statement: Projective limit space5290 \begin{definition}[Projective limit space]\label{earlier:record:006}\label{earlier:def:projective_limit}
5291 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
5292 \[
5293 X_\infty
5294 =
5295 \left\{
5296 (x_j)_{j\geq0}\in\prod_{j\geq0}X_j:
5297 C_jx_{j+1}=x_j\text{ for every }j
5298 \right\}.
5299 \]
5300 We equip \(X_\infty\) with the subspace topology inherited from the product topology.
5301 \end{definition}
5302 
Statement: Exact projective limit semigroup · Go to proof · Read statement and proof side by side5303 \begin{theorem}[Exact projective limit semigroup]\label{earlier:record:007}\label{earlier:thm:projective_limit_semigroup}
5304 Suppose each \(X_j\) carries a strongly continuous semigroup \((T_j(t))_{t\geq0}\), and
5305 \[
5306 C_jT_{j+1}(t)=T_j(t)C_j
5307 \]
5308 for every \(j\) and every \(t\geq0\). Then
5309 \[
5310 T_\infty(t)(x_j)_{j\geq0}
5311 =
5312 (T_j(t)x_j)_{j\geq0}
5313 \]
5314 defines a strongly continuous semigroup on \(X_\infty\).
5315 \end{theorem}
5316 
Proof block: Exact projective limit semigroup · Go to statement · Read statement and proof side by side5317 \begin{proof}
5318 Exact intertwining gives
5319 \[
5320 C_jT_{j+1}(t)x_{j+1}
5321 =
5322 T_j(t)C_jx_{j+1}
5323 =
5324 T_j(t)x_j.
5325 \]
5326 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
5327 \[
5328 t\longmapsto T_j(t)x_j
5329 \]
5330 is continuous.
5331 \end{proof}
5332 
Statement: Coordinate generators · Go to proof · Read statement and proof side by side5333 \begin{corollary}[Coordinate generators]\label{earlier:record:008}\label{earlier:cor:projective_generators}
5334 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
5335 \[
5336 A_\infty x=(A_jx_j)_{j\geq0}.
5337 \]
5338 \end{corollary}
5339 
Proof block: Coordinate generators · Go to statement · Read statement and proof side by side5340 \begin{proof}
5341 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.
5342 \end{proof}
5343 
5344 
5345 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.
5346 \FloatBarrier
5347 \section{Reversible coarse dynamics}\label{earlier:section:02}
5348 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.
Statement: Reversibility descends · Go to proof · Read statement and proof side by side5349 \begin{theorem}[Reversibility descends]\label{earlier:record:009}\label{earlier:thm:reversibility_descends}
5350 Assume \(Q\) is strongly lumpable and reversible with respect to a positive stationary distribution \(\pi\), so
5351 \[
5352 Q_{ij}\pi_j=Q_{ji}\pi_i.
5353 \]
5354 Define
5355 \[
5356 \overline\pi_a=\sum_{i\in B_a}\pi_i.
5357 \]
5358 Then \(\overline\pi\) is stationary for \(\overline Q\), and
5359 \[
5360 \overline Q_{ab}\overline\pi_b
5361 =
5362 \overline Q_{ba}\overline\pi_a.
5363 \]
5364 Thus the coarse chain is reversible.
5365 \end{theorem}
5366 
Proof block: Reversibility descends · Go to statement · Read statement and proof side by side5367 \begin{proof}
5368 Stationarity follows from
5369 \[
5370 \overline Q\overline\pi
5371 =\overline QC\pi
5372 =CQ\pi=0.
5373 \]
5374 For reversibility, use the aggregate rate formula:
5375 \[
5376 \begin{aligned}
5377 \overline Q_{ab}\overline\pi_b
5378 &=\sum_{j\in B_b}\pi_j\sum_{i\in B_a}Q_{ij}\\
5379 &=\sum_{i\in B_a}\sum_{j\in B_b}\pi_iQ_{ji}\\
5380 &=\overline Q_{ba}\overline\pi_a.
5381 \end{aligned}
5382 \]
5383 \end{proof}
5384 
Statement: Coarse Dirichlet form · Go to proof · Read statement and proof side by side5385 \begin{corollary}[Coarse Dirichlet form]\label{earlier:record:010}\label{earlier:cor:coarse_dirichlet_form}
5386 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
5387 \[
5388 -\ip{Jg}{Q^TJg}_{L^2(\pi)}
5389 =
5390 -\ip{g}{\overline Q^Tg}_{L^2(\overline\pi)}.
5391 \]
5392 \end{corollary}
5393 
Proof block: Coarse Dirichlet form · Go to statement · Read statement and proof side by side5394 \begin{proof}
5395 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.
5396 \end{proof}
5397 
5398 
5399 \FloatBarrier
5400 \section{Prior-restricted sufficiency and filtering}\label{earlier:section:03}
5401 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.
Statement: Sufficiency relative to a prior disintegration · Go to proof · Read statement and proof side by side5402 \begin{proposition}[Sufficiency relative to a prior disintegration]\label{earlier:record:011}\label{earlier:prop:relative_sufficiency}
5403 Define
5404 \[
5405 \overline L_a
5406 =
5407 \sum_{i\in B_a}L_ir_i^{(a)}.
5408 \]
5409 For every prior in the declared family,
5410 \[
5411 C\Bcal_L(p)
5412 =
5413 \Bcal_{\overline L}(q).
5414 \]
5415 \end{proposition}
5416 
Proof block: Sufficiency relative to a prior disintegration · Go to statement · Read statement and proof side by side5417 \begin{proof}
5418 The evidence in block \(B_a\) is
5419 \[
5420 Z_a
5421 =q_a\sum_{i\in B_a}L_ir_i^{(a)}
5422 =q_a\overline L_a.
5423 \]
5424 Apply \cref{extended:thm:bayesian_packet_reconstruction}.
5425 \end{proof}
5426 
Statement: Normalization bound · Go to proof · Read statement and proof side by side5427 \begin{lemma}[Normalization bound]\label{earlier:record:012}\label{earlier:lem:normalization_bound}
5428 Let \(a,b\in[0,\infty)^m\) be nonzero, and set \(A=\norm a_1\) and \(B=\norm b_1\). Then
5429 \[
5430 \boxed{
5431 \norm{a/A-b/B}_1
5432 \leq
5433 \frac{2\norm{a-b}_1}{\max\{A,B\}}.
5434 }
5435 \]
5436 \end{lemma}
5437 
Proof block: Normalization bound · Go to statement · Read statement and proof side by side5438 \begin{proof}
5439 Using \(A\) in the denominator gives
5440 \[
5441 \norm{a/A-b/B}_1
5442 \leq
5443 \frac{\norm{a-b}_1}{A}
5444 +
5445 \norm b_1\abs{1/A-1/B}.
5446 \]
5447 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.
5448 \end{proof}
5449 
5450 
5451 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.
Statement: Exact information loss formula · Go to proof · Read statement and proof side by side5452 \begin{theorem}[Exact information loss formula]\label{earlier:record:013}\label{earlier:thm:information_loss}
5453 For every prior,
5454 \[
5455 \boxed{
5456 C\Bcal_L(p)
5457 =
5458 \Bcal_{\ell(p)}(q).
5459 }
5460 \]
5461 Let \(\overline L\) be any fixed positive coarse likelihood and set
5462 \[
5463 d(p\mid L,\overline L)
5464 =
5465 \sum_{a=1}^mq_a\abs{\ell_a(p)-\overline L_a}.
5466 \]
5467 Then
5468 \[
5469 \norm{
5470 C\Bcal_L(p)-\Bcal_{\overline L}(q)
5471 }_1
5472 \leq
5473 \frac{2d(p\mid L,\overline L)}
5474 {\max\{Z(p,L),\ip{\overline L}{q}\}}.
5475 \]
5476 \end{theorem}
5477 
Proof block: Exact information loss formula · Go to statement · Read statement and proof side by side5478 \begin{proof}
5479 The packet evidence in block \(a\) is \(q_a\ell_a(p)\), which proves the exact identity. Apply \cref{earlier:lem:normalization_bound} to
5480 \[
5481 a=q\odot\ell(p),
5482 \qquad
5483 b=q\odot\overline L.
5484 \]
5485 Their \(\ell^1\) norms are the two evidences, and \(\norm{a-b}_1=d(p\mid L,\overline L)\).
5486 \end{proof}
5487 
Statement: Oscillation bound · Go to proof · Read statement and proof side by side5488 \begin{corollary}[Oscillation bound]\label{earlier:record:014}\label{earlier:cor:bayes_oscillation}
5489 Suppose \(\overline L_a\) is any representative likelihood for block \(B_a\), and set
5490 \[
5491 \omega_a
5492 =
5493 \max_{i\in B_a}\abs{L_i-\overline L_a}.
5494 \]
5495 Then
5496 \[
5497 d(p\mid L,\overline L)
5498 \leq
5499 \sum_aq_a\omega_a.
5500 \]
5501 Thus small within block likelihood oscillation gives a small coarse posterior defect whenever the evidence is bounded away from zero.
5502 \end{corollary}
5503 
Proof block: Oscillation bound · Go to statement · Read statement and proof side by side5504 \begin{proof}
5505 The conditional average \(\ell_a(p)\) lies in the convex hull of the likelihood values inside \(B_a\). Hence
5506 \[
5507 \abs{\ell_a(p)-\overline L_a}\leq\omega_a.
5508 \]
5509 Multiply by \(q_a\) and sum.
5510 \end{proof}
5511 
5512 
5513 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 $h$.
Statement: Covariance identity · Go to proof · Read statement and proof side by side5514 \begin{corollary}[Covariance identity]\label{earlier:record:015}\label{earlier:cor:bayesian_covariance}
5515 For any observable \(h\in\RR^n\),
5516 \[
5517 \frac{\dd}{\dd t}\EE_{p(t)}[h]
5518 =
5519 \Cov_{p(t)}(h,g).
5520 \]
5521 In particular,
5522 \[
5523 \frac{\dd}{\dd t}\EE_{p(t)}[g]
5524 =
5525 \Var_{p(t)}(g)\geq0.
5526 \]
5527 \end{corollary}
5528 
Proof block: Covariance identity · Go to statement · Read statement and proof side by side5529 \begin{proof}
5530 Differentiate \(\sum_ih_ip_i(t)\) and insert the replicator equation.
5531 \end{proof}
5532 
5533 
5534 For stochastic matrices $K,\overline K$ and positive likelihoods $L_y,\overline L_y$, put $\Phi_y=\Bcal_{L_y}\circ K$ and $\overline\Phi_y=\Bcal_{\overline L_y}\circ\overline K$.
Statement: Exact coarse hidden Markov filter · Go to proof · Read statement and proof side by side5535 \begin{theorem}[Exact coarse hidden Markov filter]\label{earlier:record:016}\label{earlier:thm:exact_coarse_filter}
5536 Assume
5537 \[
5538 CK=\overline KC
5539 \]
5540 and every likelihood \(L_y\) is constant on each coarse block with block values \(\overline L_y\). Then
5541 \[
5542 C\Phi_y(p)=\overline\Phi_y(Cp)
5543 \]
5544 for every prior. For any observation sequence \(y_1,\ldots,y_r\), the complete filtering recursion commutes with aggregation:
5545 \[
5546 C\Phi_{y_r}\circ\cdots\circ\Phi_{y_1}
5547 =
5548 \overline\Phi_{y_r}\circ\cdots\circ\overline\Phi_{y_1}C.
5549 \]
5550 \end{theorem}
5551 
Proof block: Exact coarse hidden Markov filter · Go to statement · Read statement and proof side by side5552 \begin{proof}
5553 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.
5554 \end{proof}
5555 
Statement: One step filter defect · Go to proof · Read statement and proof side by side5556 \begin{theorem}[One step filter defect]\label{earlier:record:017}\label{earlier:thm:filter_defect}
5557 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
5558 \[
5559 0<\overline m\leq(\overline L_y)_a\leq\overline M.
5560 \]
5561 Then
5562 \[
5563 \begin{aligned}
5564 \norm{C\Phi_y(p)-\overline\Phi_y(Cp)}_1
5565 \leq{}&
5566 \frac{2d(r\mid L_y,\overline L_y)}
5567 {\max\{\ip{L_y}{r},\ip{\overline L_y}{q}\}}\\
5568 &+
5569 \frac{2\overline M}{\overline m}
5570 \norm{CKp-\overline KCp}_1.
5571 \end{aligned}
5572 \]
5573 \end{theorem}
5574 
Proof block: One step filter defect · Go to statement · Read statement and proof side by side5575 \begin{proof}
5576 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.
5577 \end{proof}
5578 
5579 
5580 \FloatBarrier
5581 \section{Fractional modes and killed transfer}\label{earlier:section:04}
5582 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.
Statement: Fractional low mode dispersion · Go to proof · Read statement and proof side by side5583 \begin{corollary}[Fractional low mode dispersion]\label{earlier:record:018}\label{earlier:cor:fractional_low_mode_dispersion}
5584 Let \(0<\alpha\leq1\). Then
5585 \[
5586 0
5587 \leq
5588 (k\pi)^{2\alpha}-\lambda_{k,h}^\alpha
5589 \leq
5590 \frac{\alpha(4k^2)^{\alpha-1}(k\pi)^4}{12}h^2.
5591 \]
5592 For every \(t\geq0\),
5593 \[
5594 \abs{
5595 \e^{-t\lambda_{k,h}^\alpha}
5596 -
5597 \e^{-t(k\pi)^{2\alpha}}
5598 }
5599 \leq
5600 \frac{t\alpha(4k^2)^{\alpha-1}(k\pi)^4}{12}h^2.
5601 \]
5602 \end{corollary}
5603 
Proof block: Fractional low mode dispersion · Go to statement · Read statement and proof side by side5604 \begin{proof}
5605 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,
5606 \[
5607 (k\pi)^{2\alpha}-\lambda_{k,h}^\alpha
5608 \leq
5609 \alpha\lambda_{k,h}^{\alpha-1}
5610 \bigl((k\pi)^2-\lambda_{k,h}\bigr).
5611 \]
5612 Use \cref{extended:thm:low_mode} and the lower bound for \(\lambda_{k,h}\). The exponential estimate follows from
5613 \[
5614 \abs{\e^{-ta}-\e^{-tb}}
5615 \leq t\abs{a-b}
5616 \]
5617 for nonnegative \(a\) and \(b\).
5618 \end{proof}
5619 
5620 
5621 
5622 Let $Q$ and $\overline Q$ be finite Markov generators in the column convention. All times in this section satisfy $t\geq0$.
Statement: Potential perturbation bound · Go to proof · Read statement and proof side by side5623 \begin{theorem}[Potential perturbation bound]\label{earlier:record:019}\label{earlier:thm:feynman_kac_potential_bound}
5624 Let \(V\) and \(W\) be nonnegative diagonal matrices. In the supremum norm,
5625 \[
5626 \boxed{
5627 \norm{
5628 \e^{t(Q^T-V)}
5629 -
5630 \e^{t(Q^T-W)}
5631 }_{\infty\to\infty}
5632 \leq
5633  t\norm{V-W}_{\infty\to\infty}.
5634 }
5635 \]
5636 \end{theorem}
5637 
Proof block: Potential perturbation bound · Go to statement · Read statement and proof side by side5638 \begin{proof}
5639 Both killed Markov semigroups are contractions in the supremum norm. Apply the Duhamel identity with generator defect \(W-V\).
5640 \end{proof}
5641 
Statement: Coarse Feynman Kac transfer · Go to proof · Read statement and proof side by side5642 \begin{theorem}[Coarse Feynman Kac transfer]\label{earlier:record:020}\label{earlier:thm:coarse_feynman_kac}
5643 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
5644 \[
5645 VJ=J\overline V.
5646 \]
5647 Then
5648 \[
5649 \boxed{
5650 \e^{t(Q^T-V)}J
5651 =
5652 J\e^{t(\overline Q^T-\overline V)}
5653 }
5654 \]
5655 for every \(t\geq0\).
5656 \end{theorem}
5657 
Proof block: Coarse Feynman Kac transfer · Go to statement · Read statement and proof side by side5658 \begin{proof}
5659 Transpose the Markov intertwining relation to obtain
5660 \[
5661 Q^TJ=J\overline Q^T.
5662 \]
5663 Together with \(VJ=J\overline V\), this gives
5664 \[
5665 (Q^T-V)J=J(\overline Q^T-\overline V).
5666 \]
5667 The exponential intertwining follows from the power series.
5668 \end{proof}
5669 
5670 
5671 \FloatBarrier
5672 \section{Memory compression, rational realization, and phase-type consequences}\label{earlier:section:05}
5673 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.
Statement: Stability of memory compression · Go to proof · Read statement and proof side by side5674 \begin{theorem}[Stability of memory compression]\label{earlier:record:021}\label{earlier:thm:memory_compression_stability}
5675 Let \(x\) and \(\widetilde x\) solve
5676 \[
5677 \dot x(t)
5678 =Ax(t)+\int_0^t\Kcal(t-s)x(s)\dd s+f(t)
5679 \]
5680 and
5681 \[
5682 \dot{\widetilde x}(t)
5683 =A\widetilde x(t)
5684 +\int_0^t\widetilde\Kcal(t-s)\widetilde x(s)\dd s
5685 +f(t)
5686 \]
5687 with the same initial value. Fix \(T>0\), and assume
5688 \[
5689 \sup_{0\leq t\leq T}\norm{\widetilde x(t)}\leq M.
5690 \]
5691 Set
5692 \[
5693 a=\norm A,
5694 \qquad
5695 k=\norm{\Kcal}_{L^1(0,T)},
5696 \qquad
5697 \delta=\norm{\Kcal-\widetilde\Kcal}_{L^1(0,T)}.
5698 \]
5699 Then
5700 \[
5701 \boxed{
5702 \sup_{0\leq t\leq T}
5703 \norm{x(t)-\widetilde x(t)}
5704 \leq
5705 MT\delta\e^{(a+k)T}.
5706 }
5707 \]
5708 \end{theorem}
5709 
Proof block: Stability of memory compression · Go to statement · Read statement and proof side by side5710 \begin{proof}
5711 Let \(e=x-\widetilde x\). Integrating the difference equation and changing the order of integration gives
5712 \[
5713 \norm{e(t)}
5714 \leq
5715 (a+k)\int_0^t\norm{e(r)}\dd r
5716 +Mt\delta.
5717 \]
5718 Since \(t\leq T\), Gronwall's inequality gives
5719 \[
5720 \norm{e(t)}
5721 \leq
5722 MT\delta\e^{(a+k)t}.
5723 \]
5724 Take the supremum over \([0,T]\).
5725 \end{proof}
5726 
5727 
5728 
5729 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.
5730 
5731 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.
Statement: Rational memory realization · Go to proof · Read statement and proof side by side5732 \begin{theorem}[Rational memory realization]\label{earlier:record:022}\label{earlier:thm:rational_memory_realization}
5733 A matrix valued memory transform has a finite auxiliary realization if and only if it is a strictly proper rational matrix function. More precisely,
5734 \[
5735 \boxed{
5736 \widehat\Kcal(z)=B(zI-D)^{-1}C
5737 }
5738 \]
5739 if and only if
5740 \[
5741 \boxed{
5742 \Kcal(t)=B\e^{tD}C.
5743 }
5744 \]
5745 \end{theorem}
5746 
Proof block: Rational memory realization · Go to statement · Read statement and proof side by side5747 \begin{proof}
5748 The forward implication follows by taking the Laplace transform. The resulting matrix is rational and tends to zero as \(z\) tends to infinity.
5749 
5750 Conversely, decompose every entry of a strictly proper rational matrix into partial fractions. A term
5751 \[
5752 \frac{c}{(z-\lambda)^r}
5753 \]
5754 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.
5755 \end{proof}
5756 
Statement: Exponential sum memory · Go to proof · Read statement and proof side by side5757 \begin{corollary}[Exponential sum memory]\label{earlier:record:023}\label{earlier:cor:exponential_sum_memory}
5758 If
5759 \[
5760 \Kcal(t)=\sum_{j=1}^{m}B_j\e^{\lambda_jt}C_j,
5761 \]
5762 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.
5763 \end{corollary}
5764 
Proof block: Exponential sum memory · Go to statement · Read statement and proof side by side5765 \begin{proof}
5766 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.
5767 \end{proof}
5768 
5769 
5770 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.
Statement: Phase type rationality · Go to proof · Read statement and proof side by side5771 \begin{corollary}[Phase type rationality]\label{earlier:record:024}\label{earlier:cor:phase_type_rationality}
5772 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 \(t\).
5773 \end{corollary}
5774 
Proof block: Phase type rationality · Go to statement · Read statement and proof side by side5775 \begin{proof}
5776 The resolvent \((sI-A)^{-1}\) is rational. The time domain statement follows from the Jordan form of \(A\).
5777 \end{proof}
5778 
Statement: One phase model · Go to proof · Read statement and proof side by side5779 \begin{corollary}[One phase model]\label{earlier:record:025}\label{earlier:cor:one_phase_model}
5780 If \(A=(-\lambda)\) and \(p_0=1\), then
5781 \[
5782 S(t)=\e^{-\lambda t},
5783 \qquad
5784 f(t)=\lambda\e^{-\lambda t},
5785 \qquad
5786 h(t)=\lambda.
5787 \]
5788 \end{corollary}
5789 
Proof block: One phase model · Go to statement · Read statement and proof side by side5790 \begin{proof}
5791 Insert the scalar generator into \cref{extended:thm:survival_hazard}.
5792 \end{proof}
5793 
5794 
5795 \FloatBarrier
5796 \section{Earlier exact examples}\label{earlier:section:06}
5797 These five short examples are retained from the earlier human edition. Their numerical parameters are independent of the later visual guide's worked models.
5798 \begin{example}[One transfer between two packets]\label{earlier:record:026}\label{earlier:ex:two_packet_transfer}
5799 Let
5800 \[
5801 A=
5802 \begin{pmatrix}
5803 a&0\\
5804 c&b
5805 \end{pmatrix}
5806 \]
5807 and let the two coordinate axes be the packet spaces. If \(a\neq b\), then
5808 \[
5809 \e^{tA}
5810 =
5811 \begin{pmatrix}
5812 \e^{at}&0\\
5813 \displaystyle c\frac{\e^{at}-\e^{bt}}{a-b}&\e^{bt}
5814 \end{pmatrix}.
5815 \]
5816 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 \(ct\e^{at}\).
5817 \end{example}
5818 
5819 \begin{example}[A four state lumpable chain]\label{earlier:record:027}\label{earlier:ex:four_state_lumping}
5820 Partition four states into \(B_1=\{1,2\}\) and \(B_2=\{3,4\}\). For positive rates \(a,b,u,v\), set
5821 \[
5822 Q=
5823 \begin{pmatrix}
5824 -u-a&u&b/2&b/2\\
5825 u&-u-a&b/2&b/2\\
5826 a/2&a/2&-v-b&v\\
5827 a/2&a/2&v&-v-b
5828 \end{pmatrix}.
5829 \]
5830 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
5831 \[
5832 \overline Q=
5833 \begin{pmatrix}
5834 -a&b\\
5835 a&-b
5836 \end{pmatrix}
5837 \]
5838 and \(CQ=\overline QC\). The internal rates disappear from the coarse generator even though they remain present in the fine evolution.
5839 \end{example}
5840 
5841 \begin{example}[A visible likelihood loss]\label{earlier:record:028}\label{earlier:ex:bayes_information_loss}
5842 Let \(B_1=\{1,2\}\), let \(B_2=\{3\}\), and take
5843 \[
5844 p=(3/10,2/10,5/10),
5845 \qquad
5846 L=(1,3,2).
5847 \]
5848 The coarse prior is \((1/2,1/2)\). The exact block likelihoods are
5849 \[
5850 \ell_1(p)=\frac{(3/10)+3(2/10)}{1/2}=\frac95,
5851 \qquad
5852 \ell_2(p)=2.
5853 \]
5854 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 \(1/10\).
5855 \end{example}
5856 
5857 \begin{example}[One hidden mode]\label{earlier:record:029}\label{earlier:ex:one_hidden_mode}
5858 Consider
5859 \[
5860 A=
5861 \begin{pmatrix}
5862 -a&c\\
5863 d&-b
5864 \end{pmatrix},
5865 \qquad
5866 a,b>0,
5867 \]
5868 and retain the first coordinate. The memory kernel and self energy are
5869 \[
5870 \Kcal(t)=cd\e^{-bt},
5871 \qquad
5872 \Sigma(z)=\frac{cd}{z+b}.
5873 \]
5874 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.
5875 \end{example}
5876 
5877 \begin{example}[Two sequential phases]\label{earlier:record:030}\label{earlier:ex:two_phase_survival}
5878 Let
5879 \[
5880 A=
5881 \begin{pmatrix}
5882 -\lambda&0\\
5883 \lambda&-\mu
5884 \end{pmatrix},
5885 \qquad
5886 p_0=
5887 \begin{pmatrix}1\\0\end{pmatrix},
5888 \]
5889 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
5890 \[
5891 S(t)=
5892 \frac{\mu\e^{-\lambda t}-\lambda\e^{-\mu t}}{\mu-\lambda},
5893 \]
5894 \[
5895 f(t)=
5896 \frac{\lambda\mu}{\mu-\lambda}
5897 \left(\e^{-\lambda t}-\e^{-\mu t}\right).
5898 \]
5899 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.
5900 \end{example}
5901 
5902 \FloatBarrier
5903 \section{Holomorphic operator substitution}\label{migration:section:01}\label{migration:sec:operator_substitution}
5904 \begin{readerroadmap}
5905 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.
5906 \end{readerroadmap}
5907 
5908 
5909 This section receives the operator-substitution result deliberately moved out of Paper IV. Fix an integer $N\geq1$ and the entire scalar function
5910 \[
5911 \Dfun_N(z)=\sum_{m=0}^{\infty}\frac{(-1)^m z^{Nm}}{(Nm)!}.
5912 \]
5913 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.
5914 
Statement: Holomorphic functional calculus5915 \begin{standardfact}[Holomorphic functional calculus]\label{migration:record:001}\label{migration:sf:holomorphic_calculus}
5916 Let $A$ be a bounded operator on a complex Banach space.  If $f$ is holomorphic on a neighborhood of $\Spec(A)$, then
5917 \[
5918 f(A)
5919 =
5920 \frac1{2\pi i}\int_\Gamma f(z)(zI-A)^{-1}\,dz
5921 \]
5922 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}.
5923 \end{standardfact}
5924 
Statement: Operator reciprocal family · Go to proof · Read statement and proof side by side5925 \begin{proposition}[Operator reciprocal family]\label{migration:record:002}\label{migration:prop:operator_reciprocal}
5926 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
5927 \[
5928 \Sfun_N^{(r)}(A)
5929 =
5930 \exp\bigl(-r g(A)\bigr)
5931 =
5932 \bigl(\Dfun_N^{-r}\bigr)(A)
5933 \]
5934 is well defined and
5935 \[
5936 \Sfun_N^{(r)}(A)\Sfun_N^{(s)}(A)
5937 =
5938 \Sfun_N^{(r+s)}(A).
5939 \]
5940 If $(P_i)$ is a finite complete family of bounded idempotents, then
5941 \[
5942 P_i f(A)P_j
5943 =
5944 \frac1{2\pi i}
5945 \int_\Gamma
5946 f(z)P_i(zI-A)^{-1}P_j\,dz
5947 \]
5948 for every holomorphic $f$ on a neighborhood of the spectrum.
5949 \end{proposition}
5950 
Proof block: Operator reciprocal family · Go to statement · Read statement and proof side by side5951 \begin{proof}
5952 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.
5953 \end{proof}
5954 
5955 
5956 \clearpage
5957 \part*{Part IV\quad Worked visual atlas}\addcontentsline{toc}{part}{Part IV: Worked visual atlas}
5958 \input{atlas-pages.tex}
5959 \clearpage\bibliographystyle{alpha}\bibliography{references}
5960 \end{document}