Hypergraph Lumpability and Bayesian Refinement

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32   pdftitle={Hypergraph Lumpability and Bayesian Refinement for Dynamical Systems with Projector Resolutions},
33   pdfauthor={Komorebi P.},
34   pdfsubject={Interaction quotients, Bayesian refinement, and certified coarse filtering},
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44 \fancyhead[R]{Komorebi P.}
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114 
115 \title{\textbf{Hypergraph Lumpability and Bayesian Refinement\\
116 for Dynamical Systems with Projector Resolutions}\\[0.4em]
117 \large Exact Interaction Quotients, Filtering Cocycles,\\
118 and Certified Coarse Models}
119 \author{Komorebi P.}
120 \date{Internal completion release, version 0.1.4\\September 2026}
121 
122 \begin{document}
123 \maketitle
124 
125 \begin{abstract}
126 We study when a coarse description of a nonlinear system is exact rather than fitted after the fact.  The fine and coarse state spaces are linked by a surjective linear map $C$.  Each homogeneous term in a local analytic vector field is resolved into packet interactions.  Linear terms give ordinary directed edges, while higher order terms give directed hyperedges.  We prove that a multilinear interaction of order $m$ descends exactly when its coarse output vanishes whenever any input lies in $\Ker C$.  For block sum aggregations, this becomes a direct coefficient test.  Once the fine output coefficients are summed within a coarse fiber, the resulting value must depend only on the coarse input fibers.
127 
128 The quotient condition survives passage to the full analytic field, all of its derivatives, every rooted interaction tree, and the locally unique flow.  We represent a Bayesian observation as positive likelihood transfer followed by normalization.  This normalized update descends precisely when the fine likelihood is constant within each aggregation fiber, up to a single positive scale.  Under those two conditions, the coarse filter agrees with the aggregate of the fine filter for every finite observation sequence, and the evidence factors differ only by the same scales.
129 
130 When exact descent fails, we keep the sources of error separate.  A Grönwall argument turns interaction mismatch into a bound on the flow, while variation of the likelihood inside a fiber creates its own normalization error.  A recursive estimate carries both contributions to the final posterior.  In the Hilbert Schmidt setting, an orthogonal residual identity also gives the best approximation that retains a prescribed number of hyperedges.  Deterministic examples and the reference implementation check the finite formulas, the perturbation bounds, and negative controls.  Exact theorems, numerical checks, and claims about historical priority remain distinct throughout.
131 \end{abstract}
132 
133 \begin{statusbox}
134 \textbf{Status and scope.}
135 This manuscript is complete for internal use.  It contains the finite theorem chain, proofs, examples, algorithms, certificate code, and reproducibility files.  Before submission, it still needs independent proof review and a focused literature review.  The algebraic results concern spaces of finite dimension, and the flow results are local in time under the assumptions stated in the text.  The numerical checks verify the examples and the implementation, not the proofs.  We do not assume that useful packet resolutions always exist, that an optimal quotient is easy to find, that local flow theory guarantees global existence, or that this presentation settles priority across neighboring work.
136 \end{statusbox}
137 
138 \tableofcontents
139 \newpage
140 
141 \section{Introduction}
142 
143 \subsection{The quotient problem}
144 Let $X_{\fine}$ be a fine state space, let $X_{\coarse}$ be a coarse state space, and let
145 \[
146   C:X_{\fine}\twoheadrightarrow X_{\coarse}
147 \]
148 be a surjective linear map.  A fine trajectory satisfies
149 \begin{equation}\label{eq:fine-flow-intro}
150   \dot x(t)=F_{\fine}(x(t)),
151   \qquad x(0)=x_0,
152 \end{equation}
153 and the coarse observable of interest is
154 \[
155   z(t)=Cx(t).
156 \]
157 The first question is whether $z(t)$ obeys a closed equation of its own.  Equivalently, can we define a vector field $F_{\coarse}$ such that
158 \begin{equation}\label{eq:vector-field-quotient-intro}
159   C F_{\fine}=F_{\coarse}C?
160 \end{equation}
161 When this identity holds and both initial value problems have unique solutions, aggregation commutes with evolution:
162 \begin{equation}\label{eq:flow-quotient-intro}
163   C\Phi_t^{\fine}=\Phi_t^{\coarse}C.
164 \end{equation}
165 The coarse trajectory is then an exact quotient of the fine one.  No fitting step is involved.
166 
167 The linear versions of this question are classical.  For finite Markov chains, they appear as lumpability and Markov function criteria \cite{KemenySnell1960,RogersPitman1981,Buchholz1994}.  Nearly decomposable systems lead instead to perturbative aggregation \cite{SimonAndo1961,Meyer1989}.  Polynomial systems and reaction networks admit exact differential equivalences and reductions based on bisimulation \cite{Cardelli2015,Cardelli2017}.  We isolate a quotient test for each order of a multilinear interaction and then follow it through nonlinear composition, continuous evolution, and Bayesian observation.
168 
169 \subsection{Why hypergraphs appear}
170 Suppose a local analytic vector field has the expansion
171 \begin{equation}\label{eq:analytic-expansion-intro}
172   F_{\fine}(x)=\sum_{m\geq0}A_m^{\fine}[x,\ldots,x].
173 \end{equation}
174 The linear coefficient $A_1^{\fine}$ moves one input packet into one output packet.  The quadratic coefficient $A_2^{\fine}$ combines two input packets before producing an output.  In general, $A_m^{\fine}$ combines $m$ inputs.  We record this incidence by the directed hyperedge
175 \[
176   (j_1,\ldots,j_m)\longrightarrow i.
177 \]
178 
179 Nonlinear composition is not ordinary path concatenation.  It inserts the output of one multilinear interaction into an input of another.  Repeated insertion produces rooted trees and is governed by the Fa\`a di Bruno formula \cite{ConstantineSavits1996}.  The same tree structure appears in nonlinear evolution and in numerical integration \cite{Butcher1972,HairerLubichWanner2006}.
180 
181 At order $m$, the central question is
182 \begin{equation}\label{eq:arity-question}
183   \exists A_m^{\coarse}
184   \quad\text{such that}\quad
185   C A_m^{\fine}=A_m^{\coarse}C^{\times m}?
186 \end{equation}
187 If this question has an answer at every order, the quotient relation can be carried from individual interactions to the full field, its derivatives, its interaction trees, and its flow.
188 
189 \subsection{Why Bayesian observations add another nonlinear operation}
190 Prediction is only one part of a filter.  Given a probability vector $\mu$ and a positive likelihood $\lambda$, Bayesian conditioning is
191 \begin{equation}\label{eq:bayes-intro}
192   \Bup_{\lambda}(\mu)
193   =
194   \frac{\lambda\odot\mu}{\lambda^T\mu}.
195 \end{equation}
196 Likelihood multiplication is linear, but the evidence in the denominator couples all packets.  The full update is therefore nonlinear even when the likelihood acts separately on each state.
197 
198 Classical sufficiency theory asks whether a statistic preserves all information relevant to a chosen observation family \cite{HalmosSavage1949,Blackwell1953}.  Here the operator question is
199 \begin{equation}\label{eq:bayes-question}
200   C\Bup_{\lambda^{\fine}}
201   =
202   \Bup_{\lambda^{\coarse}}C.
203 \end{equation}
204 When both the flow identity \eqref{eq:flow-quotient-intro} and the Bayesian identity \eqref{eq:bayes-question} hold at each observation time, the whole filtering sequence commutes with aggregation.  Hidden Markov filtering is the linear unary case \cite{BaumPetrie1966}.  The same argument applies when the prediction step comes from a nonlinear quotient flow.
205 
206 \subsection{Main results}
207 Eight results form the mathematical spine of the paper.
208 
209 \begin{enumerate}[label=\textbf{C\arabic*.},leftmargin=3.5em]
210 \item \textbf{A quotient criterion for every interaction order.}
211 For a surjective map $C$, a multilinear fine interaction descends exactly when $CA_m^{\fine}$ vanishes whenever one input lies in $\Ker C$.  Equivalently, its linearization on $X_{\fine}^{\otimes m}$ factors through $C^{\otimes m}$.
212 
213 \item \textbf{A concrete packet test.}
214 For block sum aggregation, exact descent is equivalent to constancy of the aggregated fine coefficients over each product of input fibers.  These common values determine the coarse interaction uniquely.
215 
216 \item \textbf{Compatibility across several resolutions.}
217 Exact descent is transitive.  Approximate descent obeys an exact defect identity that can be iterated along a refinement tower.
218 
219 \item \textbf{Naturality of trees and flows.}
220 Descent of the coefficients passes to local analytic fields, all derivatives, every rooted interaction tree, and every locally unique flow.
221 
222 \item \textbf{Naturality after normalization.}
223 If a positive observation transfer respects the quotient up to a scalar, normalization removes that scalar and preserves the quotient.  For diagonal likelihoods, this happens exactly when the fine likelihood is constant on each fiber, up to one global scale.
224 
225 \item \textbf{Exact filtering cocycles.}
226 When both dynamics and observations descend, every finite sequence of predictions and updates agrees after aggregation.  The evidence factors obey the same quotient law.
227 
228 \item \textbf{A certificate for approximate filtering.}
229 Interaction mismatch first appears as flow error, while observation mismatch appears at normalization.  A closed recurrence carries both contributions to the final posterior and to the accumulated log evidence.
230 
231 \item \textbf{A separate pruning certificate.}
232 Discarded hyperedges remain in a named residual tensor.  Orthogonal packet blocks give an exact residual identity and an optimal retention theorem for a fixed number of hyperedges.
233 \end{enumerate}
234 
235 The proof strategy can be summarized by the chain
236 \[
237 \boxed{
238 \begin{gathered}
239   \text{multilinear quotient}
240   \Longrightarrow
241   \text{interaction tree quotient}
242   \Longrightarrow
243   \text{flow quotient}
244   \\
245   \Longrightarrow
246   \text{Bayesian quotient}
247   \Longrightarrow
248   \text{filtering cocycle quotient}
249   \Longrightarrow
250   \text{posterior certificate}.
251 \end{gathered}}
252 \]
253 
254 \subsection{Four distinctions}
255 The same partition can succeed at one stage of the problem and fail at another.  Four distinctions prevent those cases from being conflated.
256 
257 \begin{enumerate}[label=(\roman*),leftmargin=2.5em]
258 \item A \emph{resolution} states which packets are being distinguished.
259 \item An \emph{exact quotient} proves that the coarse dynamics do not depend on hidden coordinates.
260 \item A \emph{compression} deletes or approximates interaction blocks, so the omitted part must remain visible as a residual.
261 \item \emph{Statistical sufficiency} concerns the information retained for a chosen observation family.  Here it is expressed by naturality of the normalized transfer.
262 \end{enumerate}
263 
264 A partition may be exact for the dynamics yet lose information carried by the observations.  It may preserve the observation model while failing to close the dynamics.  It may also work through one interaction order and fail at the next.  Each failure receives its own defect.
265 
266 \subsection{Organization}
267 \Cref{sec:resolved} introduces packet resolutions and interaction tensors.  \Cref{sec:allarity} proves the quotient criteria.  \Cref{sec:analytic} carries the result to analytic fields, rooted trees, and flows.  \Cref{sec:bayes} studies normalized positive transfer and Bayesian refinement.  \Cref{sec:cocycles} combines prediction and observation into an exact filtering cocycle.  \Cref{sec:approx-dyn,sec:approx-bayes,sec:filter-cert} develop the error bounds.  \Cref{sec:pruning} keeps track of residuals created by hyperedge removal.  \Cref{sec:algorithms,sec:examples} describe the finite algorithms, numerical checks, and worked models.  The final sections compare the framework with nearby theories, then state the limitations and open problems.
268 
269 \section{Resolved interaction systems}\label{sec:resolved}
270 
271 \subsection{Finite packet resolutions}
272 The main algebraic results are stated for vector spaces of finite dimension over $\R$ or $\C$.  The quotient and tensor factorizations are then purely algebraic.  The later analytic results add the stated convergence conditions and the usual assumptions for local uniqueness.
273 
274 \begin{definition}[Packet resolution]\label{def:packet-resolution}
275 A packet resolution of a vector space $X$ is a finite family of projections $(P_i)_{i\in I}$ satisfying
276 \[
277   P_iP_j=\delta_{ij}P_i,
278   \qquad
279   \sum_{i\in I}P_i=I_X.
280 \]
281 Thus
282 \[
283   X=\bigoplus_{i\in I}X_i,
284   \qquad X_i=P_iX.
285 \]
286 When $X$ is a Hilbert space, the principal norm identities will assume that the $P_i$ are orthogonal.
287 \end{definition}
288 
289 Let $(P_j)_{j\in J}$ resolve a source $X$ and $(Q_i)_{i\in I}$ resolve a target $Y$.
290 
291 \begin{definition}[Resolved multilinear block and hyperedge]\label{def:resolved-block}
292 For a multilinear map $A:X^m\to Y$ of order $m$, define
293 \begin{equation}\label{eq:resolved-block}
294   A_{i\mid j_1,\ldots,j_m}
295   =
296   Q_iA(P_{j_1}\,\cdot,\ldots,P_{j_m}\,\cdot).
297 \end{equation}
298 The ordered incidence
299 \[
300   (j_1,\ldots,j_m)\longrightarrow i
301 \]
302 is the corresponding directed interaction hyperedge.  We call the hyperedge active when the block in \eqref{eq:resolved-block} is nonzero.
303 \end{definition}
304 
305 When $m=1$, the active blocks form an ordinary directed transfer graph.  When $m=2$, each active block joins a pair of input packets.  For general $m$, the block records one interaction with $m$ inputs.
306 
307 \begin{theorem}[Exact multilinear packet reconstruction]\label{thm:packet-reconstruction}
308 Every multilinear map $A:X^m\to Y$ of order $m$ satisfies
309 \begin{equation}\label{eq:packet-reconstruction}
310   A(x_1,\ldots,x_m)
311   =
312   \sum_{i,j_1,\ldots,j_m}
313   A_{i\mid j_1,\ldots,j_m}
314   (P_{j_1}x_1,\ldots,P_{j_m}x_m).
315 \end{equation}
316 \end{theorem}
317 
318 \begin{proof}
319 Insert $I_X=\sum_jP_j$ in every input slot and $I_Y=\sum_iQ_i$ after applying $A$.  Multilinearity distributes all finite sums and gives \eqref{eq:packet-reconstruction}.
320 \end{proof}
321 
322 \subsection{Coarse maps, sections, and hidden directions}
323 Let
324 \[
325   C:X_{\fine}\twoheadrightarrow X_{\coarse}
326 \]
327 be a surjective linear coarse map.  Finite dimensionality guarantees a linear right inverse
328 \[
329   E:X_{\coarse}\to X_{\fine},
330   \qquad CE=I_{X_{\coarse}}.
331 \]
332 Define
333 \begin{equation}\label{eq:representative-hidden-proj}
334   P_C=EC,
335   \qquad
336   Q_C=I_{X_{\fine}}-EC.
337 \end{equation}
338 Then
339 \[
340   P_C^2=P_C,
341   \qquad
342   Q_C^2=Q_C,
343   \qquad
344   \Ran Q_C=\Ker C.
345 \]
346 Every fine state has the exact decomposition
347 \begin{equation}\label{eq:representative-hidden-decomposition}
348   x=ECx+Q_Cx.
349 \end{equation}
350 The first term is the chosen fine representative of the coarse state.  The second lies in $\Ker C$, the hidden subspace.  Exact quotienting means that motion inside this subspace cannot change the aggregated interaction output.
351 
352 \begin{definition}[Compatible packet aggregation]\label{def:packet-aggregation}
353 Suppose $(P_i)_{i\in I_{\fine}}$ and $(\overline P_a)_{a\in I_{\coarse}}$ resolve the fine and coarse spaces.  A surjection
354 \[
355   \pi:I_{\fine}\twoheadrightarrow I_{\coarse}
356 \]
357 is compatible with $C$ when
358 \begin{equation}\label{eq:packet-aggregation-compatibility}
359   \overline P_aC
360   =
361   C\sum_{\pi(i)=a}P_i
362   \qquad(a\in I_{\coarse}).
363 \end{equation}
364 \end{definition}
365 
366 In the scalar coordinate model used later, each $P_i$ selects one coordinate and $C$ sums all coordinates in the same fiber of $\pi$.
367 
368 \subsection{Multilinear descent}
369 
370 \begin{definition}[Exact interaction quotient]\label{def:exact-interaction-quotient}
371 Let $A_m^{\fine}:X_{\fine}^m\to X_{\fine}$ and $A_m^{\coarse}:X_{\coarse}^m\to X_{\coarse}$ be multilinear maps of order $m$.  We say that $A_m^{\fine}$ \emph{descends through $C$} to $A_m^{\coarse}$, and write
372 \[
373   A_m^{\fine}\Downarrow_C A_m^{\coarse},
374 \]
375 when
376 \begin{equation}\label{eq:exact-interaction-quotient}
377   C A_m^{\fine}
378   =
379   A_m^{\coarse}C^{\times m}.
380 \end{equation}
381 For $m=0$, descent means $CA_0^{\fine}=A_0^{\coarse}$.
382 \end{definition}
383 
384 Here
385 \[
386   C^{\times m}(x_1,\ldots,x_m)
387   =(Cx_1,\ldots,Cx_m).
388 \]
389 We write $C^{\otimes m}$ for the induced linear map between algebraic tensor powers.
390 
391 \begin{definition}[Interaction defect]\label{def:interaction-defect}
392 For a proposed coarse tensor $B_m$, define
393 \begin{equation}\label{eq:interaction-defect}
394   \Delta_m[B_m]
395   =
396   CA_m^{\fine}-B_mC^{\times m}.
397 \end{equation}
398 Exact descent is the condition $\Delta_m[B_m]=0$.
399 \end{definition}
400 
401 \subsection{Norm conventions}
402 For a multilinear map $A:X^m\to Y$ of order $m$, let
403 \[
404   \norm{A}
405   =
406   \sup_{\norm{x_r}\leq1}
407   \norm{A(x_1,\ldots,x_m)}.
408 \]
409 If $E$ is a fixed right inverse of $C$, then any exact coarse tensor necessarily satisfies
410 \begin{equation}\label{eq:coarse-tensor-bound-preview}
411   A_m^{\coarse}(z_1,\ldots,z_m)
412   =
413   CA_m^{\fine}(Ez_1,\ldots,Ez_m),
414 \end{equation}
415 so
416 \[
417   \norm{A_m^{\coarse}}
418   \leq
419   \norm{C}\norm{A_m^{\fine}}\norm{E}^m.
420 \]
421 The formula also makes the later uniqueness argument immediate.
422 
423 \section{Hypergraph lumpability at every interaction order}\label{sec:allarity}
424 
425 \subsection{Invariant quotient criterion}
426 The quotient problem has a simple algebraic answer.
427 
428 \begin{theorem}[Universal multilinear quotient criterion]\label{thm:universal-quotient}
429 Let $C:X_{\fine}\twoheadrightarrow X_{\coarse}$ be surjective and let
430 \[
431   A_m^{\fine}:X_{\fine}^m\to X_{\fine}
432 \]
433 be a multilinear map of order $m$, where $m\geq1$.  The following statements are equivalent.
434 
435 \begin{enumerate}[label=(\roman*),leftmargin=2.8em]
436 \item There exists a multilinear map $A_m^{\coarse}:X_{\coarse}^m\to X_{\coarse}$ of order $m$ such that
437 \[
438   CA_m^{\fine}=A_m^{\coarse}C^{\times m}.
439 \]
440 
441 \item For every slot $r\in\{1,\ldots,m\}$,
442 \begin{equation}\label{eq:slot-kernel-condition}
443   CA_m^{\fine}(x_1,\ldots,x_{r-1},k,x_{r+1},\ldots,x_m)=0
444 \end{equation}
445 whenever $k\in\Ker C$.
446 
447 \item The map $CA_m^{\fine}$ is constant on products of fibers: if $Cx_r=Cx_r'$ for all $r$, then
448 \begin{equation}\label{eq:fiber-constancy-abstract}
449   CA_m^{\fine}(x_1,\ldots,x_m)
450   =
451   CA_m^{\fine}(x_1',\ldots,x_m').
452 \end{equation}
453 
454 \item Let $\widehat A_m^{\fine}:X_{\fine}^{\otimes m}\to X_{\coarse}$ be the linearization of $CA_m^{\fine}$.  Then
455 \begin{equation}\label{eq:tensor-kernel-condition}
456   \Ker(C^{\otimes m})
457   \subseteq
458   \Ker(\widehat A_m^{\fine}).
459 \end{equation}
460 \end{enumerate}
461 
462 When these conditions hold, the coarse tensor is unique and, for every right inverse $E$ of $C$,
463 \begin{equation}\label{eq:coarse-via-section}
464   A_m^{\coarse}(z_1,\ldots,z_m)
465   =
466   CA_m^{\fine}(Ez_1,\ldots,Ez_m).
467 \end{equation}
468 The right side is independent of the chosen right inverse.
469 \end{theorem}
470 
471 \begin{proof}
472 Assume (i).  If $k\in\Ker C$, then inserting $k$ into any input slot gives
473 \[
474   CA_m^{\fine}(x_1,\ldots,k,\ldots,x_m)
475   =
476   A_m^{\coarse}(Cx_1,\ldots,0,\ldots,Cx_m)=0,
477 \]
478 so (ii) holds.
479 
480 Assume (ii).  If $Cx_r=Cx_r'$, then $x_r'-x_r\in\Ker C$.  Replace the inputs one at a time.  Multilinearity and \eqref{eq:slot-kernel-condition} show that each replacement leaves $CA_m^{\fine}$ unchanged.  Hence (iii) holds.
481 
482 Assume (iii).  For $z_r\in X_{\coarse}$ choose arbitrary lifts $x_r$ with $Cx_r=z_r$ and set
483 \[
484   A_m^{\coarse}(z_1,\ldots,z_m)
485   =
486   CA_m^{\fine}(x_1,\ldots,x_m).
487 \]
488 Fiber constancy makes the definition independent of the lifts.  Multilinearity is inherited from $A_m^{\fine}$, and the defining identity gives (i).  Surjectivity of $C$ also gives uniqueness: every coarse input tuple has a fine lift.
489 
490 To compare (ii) and (iv), identify a multilinear map of order $m$ with its linearization on the algebraic tensor product.  Since $C$ is surjective,
491 \[
492   \Ker(C^{\otimes m})
493   =
494   \sum_{r=1}^m
495   X_{\fine}^{\otimes(r-1)}\otimes\Ker C\otimes
496   X_{\fine}^{\otimes(m-r)}.
497 \]
498 Choose a splitting $X_{\fine}=E(X_{\coarse})\oplus\Ker C$ and expand the tensor power.  Vanishing of $\widehat A_m^{\fine}$ on this sum is exactly the slot condition.  Thus (ii) and (iv) are equivalent.  Formula \eqref{eq:coarse-via-section} follows by choosing $x_r=Ez_r$.
499 \end{proof}
500 
501 \begin{remark}[Nonlinear congruence interpretation]\label{rem:nonlinear-congruence}
502 For a general map $F:X_{\fine}\to X_{\fine}$, descent through $C$ is equivalent to
503 \[
504   Cx=Cy
505   \quad\Longrightarrow\quad
506   CF(x)=CF(y).
507 \]
508 Thus the relation $x\sim_Cy$, defined by $Cx=Cy$, must be a congruence for the coarse output of $F$.  \Cref{thm:universal-quotient} turns that statement into a condition on each multilinear coefficient.
509 \end{remark}
510 
511 \begin{corollary}[Bounded coarse tensor]\label{cor:bounded-coarse-tensor}
512 Under the hypotheses of \Cref{thm:universal-quotient}, for every bounded right inverse $E$,
513 \[
514   \norm{A_m^{\coarse}}
515   \leq
516   \norm{C}\norm{A_m^{\fine}}\norm{E}^m.
517 \]
518 \end{corollary}
519 
520 \begin{proof}
521 Apply \eqref{eq:coarse-via-section} to unit coarse inputs.
522 \end{proof}
523 
524 \subsection{Criterion in packet coordinates}
525 For a block sum aggregation, the kernel condition can be read directly from the tensor coefficients.
526 
527 Let
528 \[
529   I_{\fine}=\{1,\ldots,n\},
530   \qquad
531   I_{\coarse}=\{1,\ldots,q\},
532 \]
533 and let
534 \[
535   \pi:I_{\fine}\twoheadrightarrow I_{\coarse}.
536 \]
537 Write $(e_i)$ and $(f_a)$ for the standard bases of $\R^n$ and $\R^q$.  Define
538 \begin{equation}\label{eq:block-sum-C}
539   Ce_i=f_{\pi(i)},
540   \qquad
541   (Cx)_a=\sum_{\pi(i)=a}x_i.
542 \end{equation}
543 Suppose
544 \begin{equation}\label{eq:fine-coordinates}
545   A_m^{\fine}(e_{j_1},\ldots,e_{j_m})
546   =
547   \sum_{i=1}^n
548   a^{\fine,(m)}_{i;j_1,\ldots,j_m}e_i.
549 \end{equation}
550 For each coarse output packet $a$, sum the fine coefficients over its output fiber and define
551 \begin{equation}\label{eq:aggregated-coefficient}
552   g^{(m)}_{a;j_1,\ldots,j_m}
553   =
554   \sum_{i:\pi(i)=a}
555   a^{\fine,(m)}_{i;j_1,\ldots,j_m}.
556 \end{equation}
557 
558 \begin{theorem}[Coefficient criterion for every interaction order]\label{thm:packet-coordinate-criterion}
559 The fine tensor $A_m^{\fine}$ descends through the block sum map \eqref{eq:block-sum-C} if and only if, for every coarse output $a$,
560 \begin{equation}\label{eq:packet-fiber-constancy}
561   g^{(m)}_{a;j_1,\ldots,j_m}
562   =
563   g^{(m)}_{a;j_1',\ldots,j_m'}
564 \end{equation}
565 whenever
566 \[
567   \pi(j_r)=\pi(j_r')
568   \qquad(r=1,\ldots,m).
569 \]
570 When this condition holds, the unique coarse coefficients are
571 \begin{equation}\label{eq:coarse-coordinate-formula}
572   a^{\coarse,(m)}_{a;b_1,\ldots,b_m}
573   =
574   g^{(m)}_{a;j_1,\ldots,j_m},
575   \qquad
576   \pi(j_r)=b_r.
577 \end{equation}
578 \end{theorem}
579 
580 \begin{proof}
581 Applying $C$ to \eqref{eq:fine-coordinates} gives
582 \[
583   CA_m^{\fine}(e_{j_1},\ldots,e_{j_m})
584   =
585   \sum_{a=1}^q
586   g^{(m)}_{a;j_1,\ldots,j_m}f_a.
587 \]
588 If a coarse tensor exists, this vector equals
589 \[
590   A_m^{\coarse}(f_{\pi(j_1)},\ldots,f_{\pi(j_m)}),
591 \]
592 which depends only on the coarse input labels.  This proves necessity.  Conversely, constancy on each fiber makes \eqref{eq:coarse-coordinate-formula} well defined.  Equality on every tuple of fine basis vectors then gives
593 \[
594   CA_m^{\fine}=A_m^{\coarse}C^{\times m}
595 \]
596 by multilinearity.
597 \end{proof}
598 
599 For $m=2$, the criterion has two useful views.  The hypergraph records which input fibers interact, while the coefficient matrix shows whether those interactions are constant on each product fiber.  The exact synthetic tensor used in the certificate run has the required block structure to floating point precision.
600 
601 \begin{figure}[H]
602 \centering
603 \begin{subfigure}[t]{0.57\linewidth}
604 \centering
605 \includegraphics[width=\linewidth]{hypergraph_quotient.pdf}
606 \caption{Fine interactions descend to one coarse hyperedge.}
607 \label{fig:hypergraph-quotient}
608 \end{subfigure}
609 \hfill
610 \begin{subfigure}[t]{0.39\linewidth}
611 \centering
612 \includegraphics[width=\linewidth]{fiber_constancy_map.pdf}
613 \caption{Aggregated coefficients are constant on every product fiber.}
614 \label{fig:fiber-constancy-map}
615 \end{subfigure}
616 \caption{Two views of the quadratic quotient criterion.  The incidence view identifies the descended hyperedge.  The coefficient view displays the fiber constancy condition from \Cref{thm:packet-coordinate-criterion}.}
617 \label{fig:quadratic-quotient-views}
618 \end{figure}
619 
620 \subsection{Unary and quadratic cases}
621 
622 \begin{corollary}[Unary aggregate rate criterion]\label{cor:unary-criterion}
623 For $m=1$, exact descent is equivalent to
624 \begin{equation}\label{eq:unary-aggregate-rate}
625   \sum_{i:\pi(i)=a}a^{\fine,(1)}_{i;j}
626   =
627   \sum_{i:\pi(i)=a}a^{\fine,(1)}_{i;j'}
628 \end{equation}
629 for all coarse outputs $a$ whenever $\pi(j)=\pi(j')$.
630 \end{corollary}
631 
632 \begin{proof}
633 Set $m=1$ in \Cref{thm:packet-coordinate-criterion}.  The condition then involves one input fiber and becomes \eqref{eq:unary-aggregate-rate}.
634 \end{proof}
635 
636 Let $Q_{\fine}$ be the generator of a continuous time Markov chain, using the column vector convention.  The left side of \eqref{eq:unary-aggregate-rate} is the total transition rate from fine state $j$ into coarse packet $a$.  Corollary~\ref{cor:unary-criterion} is therefore the standard criterion for strong lumpability.
637 
638 \begin{corollary}[Quadratic hypergraph criterion]\label{cor:quadratic-criterion}
639 For $m=2$, exact descent is equivalent to
640 \[
641   \sum_{i:\pi(i)=a}a^{\fine,(2)}_{i;j,k}
642   =
643   \sum_{i:\pi(i)=a}a^{\fine,(2)}_{i;j',k'}
644 \]
645 whenever
646 \[
647   \pi(j)=\pi(j'),
648   \qquad
649   \pi(k)=\pi(k').
650 \]
651 The resulting coarse interaction is the directed quadratic hyperedge
652 \[
653   (\pi(j),\pi(k))\longrightarrow a.
654 \]
655 \end{corollary}
656 
657 \begin{proof}
658 Set $m=2$ in \Cref{thm:packet-coordinate-criterion}.  Fiber constancy in the two input slots gives the displayed equality, and \eqref{eq:coarse-coordinate-formula} assigns its common value to the coarse hyperedge.
659 \end{proof}
660 
661 \begin{example}[A quotient with invisible internal interactions]\label{ex:invisible-interactions}
662 Let $X_{\coarse}=\R^q$, $X_{\fine}=\R^{2q}$, and
663 \[
664   C=\begin{bmatrix}I_q&I_q\end{bmatrix},
665   \qquad
666   E=\frac12\begin{bmatrix}I_q\\I_q\end{bmatrix}.
667 \]
668 Let $A_m^{\coarse}$ be arbitrary and let $K_m:X_{\fine}^m\to X_{\fine}$ satisfy
669 \[
670   CK_m=0.
671 \]
672 Then
673 \begin{equation}\label{eq:exact-lift-construction}
674   A_m^{\fine}
675   =
676   EA_m^{\coarse}C^{\times m}+K_m
677 \end{equation}
678 satisfies
679 \[
680   CA_m^{\fine}=A_m^{\coarse}C^{\times m}.
681 \]
682 The tensor $K_m$ may contain many fine hyperedges, so the fine model can remain complicated.  Exact quotienting asks only that every hidden contribution disappear after applying $C$.
683 \end{example}
684 
685 \subsection{Structural consequences}
686 
687 \begin{proposition}[Symmetry descends]\label{prop:symmetry-descends}
688 If $A_m^{\fine}$ is symmetric in its input slots and descends through $C$, then the unique coarse tensor $A_m^{\coarse}$ is symmetric.
689 \end{proposition}
690 
691 \begin{proof}
692 Choose lifts of the coarse inputs.  Permuting the coarse inputs permutes their lifts, and symmetry of $A_m^{\fine}$ leaves \eqref{eq:coarse-via-section} unchanged.
693 \end{proof}
694 
695 \begin{proposition}[Quotient support]\label{prop:quotient-support}
696 Under the coordinate criterion, every active coarse hyperedge
697 \[
698   (b_1,\ldots,b_m)\longrightarrow a
699 \]
700 is supported by at least one fine hyperedge whose input and output labels map to that coarse incidence.  The converse can fail because several active fine hyperedges may cancel after aggregation.
701 \end{proposition}
702 
703 \begin{proof}
704 If every fine coefficient in the corresponding incidence fiber vanishes, then the aggregate coefficient \eqref{eq:aggregated-coefficient} vanishes.  Conversely, nonzero fine coefficients can sum to zero after output aggregation.
705 \end{proof}
706 
707 \subsection{Composition across refinement levels}
708 Let
709 \[
710   X_2\xrightarrow{C_{21}}X_1\xrightarrow{C_{10}}X_0,
711   \qquad
712   C_{20}=C_{10}C_{21}.
713 \]
714 
715 \begin{proposition}[Transitivity of exact interaction quotients]\label{prop:quotient-transitivity}
716 If
717 \[
718   A_m^{(2)}\Downarrow_{C_{21}}A_m^{(1)}
719   \qquad\text{and}\qquad
720   A_m^{(1)}\Downarrow_{C_{10}}A_m^{(0)},
721 \]
722 then
723 \[
724   A_m^{(2)}\Downarrow_{C_{20}}A_m^{(0)}.
725 \]
726 \end{proposition}
727 
728 \begin{proof}
729 Compose the two identities:
730 \[
731   C_{20}A_m^{(2)}
732   =C_{10}A_m^{(1)}C_{21}^{\times m}
733   =A_m^{(0)}C_{10}^{\times m}C_{21}^{\times m}
734   =A_m^{(0)}C_{20}^{\times m}.
735 \]
736 \end{proof}
737 
738 Approximate descent has an exact composition law.
739 
740 \begin{theorem}[Interaction defect across two levels]\label{thm:multilevel-defect}
741 Define
742 \[
743   \Delta_m^{21}
744   =C_{21}A_m^{(2)}-A_m^{(1)}C_{21}^{\times m},
745 \]
746 \[
747   \Delta_m^{10}
748   =C_{10}A_m^{(1)}-A_m^{(0)}C_{10}^{\times m},
749 \]
750 and
751 \[
752   \Delta_m^{20}
753   =C_{20}A_m^{(2)}-A_m^{(0)}C_{20}^{\times m}.
754 \]
755 Then
756 \begin{equation}\label{eq:multilevel-defect}
757   \boxed{
758   \Delta_m^{20}
759   =
760   C_{10}\Delta_m^{21}
761   +
762   \Delta_m^{10}C_{21}^{\times m}.}
763 \end{equation}
764 Consequently,
765 \begin{equation}\label{eq:multilevel-defect-norm}
766   \norm{\Delta_m^{20}}
767   \leq
768   \norm{C_{10}}\norm{\Delta_m^{21}}
769   +
770   \norm{\Delta_m^{10}}\norm{C_{21}}^m.
771 \end{equation}
772 \end{theorem}
773 
774 \begin{proof}
775 Add and subtract $C_{10}A_m^{(1)}C_{21}^{\times m}$:
776 \begin{align*}
777 \Delta_m^{20}
778 &=C_{10}C_{21}A_m^{(2)}-A_m^{(0)}C_{10}^{\times m}C_{21}^{\times m}\\
779 &=C_{10}\bigl(C_{21}A_m^{(2)}-A_m^{(1)}C_{21}^{\times m}\bigr)\\
780 &\quad+
781 \bigl(C_{10}A_m^{(1)}-A_m^{(0)}C_{10}^{\times m}\bigr)C_{21}^{\times m}.
782 \end{align*}
783 This proves \eqref{eq:multilevel-defect}.  Taking norms and using submultiplicativity yields the estimate.
784 \end{proof}
785 
786 \section{Analytic fields, rooted trees, and quotient flows}\label{sec:analytic}
787 
788 \subsection{Descent of polynomial and analytic vector fields}
789 Let
790 \begin{equation}\label{eq:fine-analytic-field}
791   F_{\fine}(x)
792   =
793   \sum_{m=0}^{\infty}A_m^{\fine}[x^{\otimes m}]
794 \end{equation}
795 be locally analytic, where $A_m^{\fine}$ is a symmetric multilinear map of order $m$ for $m\geq1$ and $A_0^{\fine}\in X_{\fine}$.  Throughout,
796 \[
797   A_m[x^{\otimes m}]
798   =A_m(x,\ldots,x).
799 \]
800 
801 \begin{theorem}[Analytic interaction descent]\label{thm:analytic-descent}
802 Let $C:X_{\fine}\twoheadrightarrow X_{\coarse}$ have a bounded right inverse $E$.  Suppose the coefficient series converges normally on every smaller ball,
803 \begin{equation}\label{eq:fine-normal-summability}
804   \sum_{m=0}^{\infty}\norm{A_m^{\fine}}r^m<\infty
805   \qquad(0\leq r<R),
806 \end{equation}
807 and, for every $m\geq0$, there is a coarse coefficient $A_m^{\coarse}$ satisfying
808 \begin{equation}\label{eq:coefficientwise-descent}
809   CA_m^{\fine}=A_m^{\coarse}C^{\times m}.
810 \end{equation}
811 Then
812 \begin{equation}\label{eq:coarse-analytic-field}
813   F_{\coarse}(z)
814   =
815   \sum_{m=0}^{\infty}A_m^{\coarse}[z^{\otimes m}]
816 \end{equation}
817 converges absolutely whenever $\norm{E}\norm{z}<R$, and
818 \begin{equation}\label{eq:analytic-field-intertwining}
819   \boxed{CF_{\fine}(x)=F_{\coarse}(Cx)}
820 \end{equation}
821 for every $x$ in the fine convergence domain whose coarse image lies in the coarse domain.
822 \end{theorem}
823 
824 \begin{proof}
825 For $m\geq1$, Corollary~\ref{cor:bounded-coarse-tensor} gives
826 \[
827   \norm{A_m^{\coarse}}
828   \leq
829   \norm{C}\norm{A_m^{\fine}}\norm{E}^m.
830 \]
831 For $m=0$, the identity $A_0^{\coarse}=CA_0^{\fine}$ gives the same bound with $\norm{E}^0=1$.
832 Therefore
833 \[
834   \sum_{m\geq0}\norm{A_m^{\coarse}}\norm{z}^m
835   \leq
836   \norm{C}
837   \sum_{m\geq0}
838   \norm{A_m^{\fine}}
839   (\norm{E}\norm{z})^m,
840 \]
841 which converges when $\norm{E}\norm{z}<R$.  Coefficientwise descent and absolute convergence permit termwise aggregation:
842 \[
843   CF_{\fine}(x)
844   =\sum_{m\geq0}CA_m^{\fine}[x^{\otimes m}]
845   =\sum_{m\geq0}A_m^{\coarse}[(Cx)^{\otimes m}]
846   =F_{\coarse}(Cx).
847 \]
848 \end{proof}
849 
850 \begin{remark}[Coefficientwise descent is sufficient, not always necessary]
851 Descent of each displayed coefficient is enough to make the whole series descend.  The converse can fail for a noncanonical expansion that splits one homogeneous degree into several terms whose coarse contributions cancel.  For the Taylor expansion at a fixed base point, the derivatives are unique, so the converse does hold.  Differentiating $CF_{\fine}=F_{\coarse}C$ then recovers every descended jet.
852 \end{remark}
853 
854 \begin{theorem}[Coefficientwise characterization of polynomial descent]\label{thm:polynomial-coefficientwise}
855 Let
856 \[
857   F_{\fine}(x)=\sum_{m=0}^{M}A_m^{\fine}[x^{\otimes m}],
858   \qquad
859   F_{\coarse}(z)=\sum_{m=0}^{M}A_m^{\coarse}[z^{\otimes m}],
860 \]
861 where every $A_m^{\fine}$ and $A_m^{\coarse}$ is symmetric.  Then
862 \[
863   CF_{\fine}=F_{\coarse}C
864 \]
865 if and only if
866 \[
867   CA_m^{\fine}=A_m^{\coarse}C^{\times m}
868   \qquad(0\leq m\leq M).
869 \]
870 \end{theorem}
871 
872 \begin{proof}
873 The reverse implication follows from \Cref{thm:analytic-descent}, applied to a finite sum.  Conversely, fix $x\in X_{\fine}$ and evaluate the field identity at $tx$:
874 \[
875   \sum_{m=0}^{M}t^mCA_m^{\fine}[x^{\otimes m}]
876   =
877   \sum_{m=0}^{M}t^mA_m^{\coarse}[(Cx)^{\otimes m}].
878 \]
879 Equality of the coefficients of this polynomial with values in $X_{\coarse}$ gives equality on the diagonal for the two symmetric multilinear maps of order $m$.  The polarization identity recovers each multilinear map from its diagonal polynomial, so
880 \[
881   CA_m^{\fine}=A_m^{\coarse}C^{\times m}
882 \]
883 for every degree.
884 \end{proof}
885 
886 \subsection{Naturality of derivatives}
887 
888 \begin{proposition}[Jet naturality]\label{prop:jet-naturality}
889 Let $F_{\fine}$ and $F_{\coarse}$ be $C^r$ maps satisfying
890 \[
891   CF_{\fine}=F_{\coarse}C.
892 \]
893 Then, for every $1\leq k\leq r$,
894 \begin{equation}\label{eq:jet-naturality}
895   \boxed{
896   C D^kF_{\fine}(x)[h_1,\ldots,h_k]
897   =
898   D^kF_{\coarse}(Cx)[Ch_1,\ldots,Ch_k].}
899 \end{equation}
900 \end{proposition}
901 
902 \begin{proof}
903 Differentiate the identity $CF_{\fine}(x)=F_{\coarse}(Cx)$.  Since $C$ is linear, its derivatives of order greater than one vanish and the ordinary chain rule gives the first order statement.  Repeated differentiation gives \eqref{eq:jet-naturality}.
904 \end{proof}
905 
906 Thus the quotient preserves more than the value of the vector field.  At each base point, every fine derivative block maps to its coarse counterpart.  The same identity controls local perturbations, stability calculations, and higher order response.
907 
908 \subsection{Rooted interaction trees}
909 Repeated substitution of nonlinear interactions produces rooted trees.  Only a simple recursive definition is needed for the quotient argument.
910 
911 \begin{definition}[Interaction tree evaluation]\label{def:tree-evaluation}
912 Let $\cA=(A_m)_{m\geq1}$ be a family of multilinear maps $A_m:X^m\to X$.  A rooted interaction tree is defined recursively.  A leaf $\bullet$ has evaluation
913 \[
914   \cV_{\bullet}=I_X.
915 \]
916 If
917 \[
918   \tau=B_+(\tau_1,\ldots,\tau_m)
919 \]
920 is formed by adjoining a root of arity $m$, define
921 \begin{equation}\label{eq:tree-evaluation}
922   \cV_{\tau}(x_1,\ldots,x_{|\tau|})
923   =
924   A_m\bigl(
925   \cV_{\tau_1}(\mathbf x_1),\ldots,
926   \cV_{\tau_m}(\mathbf x_m)
927   \bigr),
928 \end{equation}
929 where $|\tau|$ denotes the number of leaves and the input list is partitioned among the subtrees in their fixed order.
930 \end{definition}
931 
932 The operation $B_+$ inserts several rooted trees beneath a new root.  In packet coordinates, every edge carries a packet label and every internal vertex must match an active resolved hyperedge.
933 
934 \begin{theorem}[Treewise quotient naturality]\label{thm:treewise-quotient}
935 Suppose every operation in a fine family descends:
936 \[
937   CA_m^{\fine}=A_m^{\coarse}C^{\times m}
938   \qquad(m\geq1).
939 \]
940 Then, for every rooted interaction tree $\tau$,
941 \begin{equation}\label{eq:treewise-quotient}
942   \boxed{
943   C\cV_{\tau}^{\fine}
944   =
945   \cV_{\tau}^{\coarse}C^{\times |\tau|}.}
946 \end{equation}
947 \end{theorem}
948 
949 \begin{proof}
950 Proceed by induction on the number of internal vertices.  The leaf case is $CI=IC$.  Suppose
951 \[
952   \tau=B_+(\tau_1,\ldots,\tau_m).
953 \]
954 Using interaction descent at the root and the inductive hypothesis on every subtree,
955 \begin{align*}
956 C\cV_{\tau}^{\fine}
957 &=
958 CA_m^{\fine}
959 \bigl(
960 \cV_{\tau_1}^{\fine},\ldots,
961 \cV_{\tau_m}^{\fine}
962 \bigr)\\
963 &=
964 A_m^{\coarse}
965 \bigl(
966 C\cV_{\tau_1}^{\fine},\ldots,
967 C\cV_{\tau_m}^{\fine}
968 \bigr)\\
969 &=
970 A_m^{\coarse}
971 \bigl(
972 \cV_{\tau_1}^{\coarse}C^{\times|\tau_1|},\ldots,
973 \cV_{\tau_m}^{\coarse}C^{\times|\tau_m|}
974 \bigr)\\
975 &=
976 \cV_{\tau}^{\coarse}C^{\times|\tau|}.
977 \end{align*}
978 \end{proof}
979 
980 \begin{corollary}[Compatibility with Fa\`a di Bruno composition]\label{cor:faa-quotient}
981 Let $F_{\fine}$ and $F_{\coarse}$ be smooth maps with $CF_{\fine}=F_{\coarse}C$, and let $G_{\fine}$ and $G_{\coarse}$ satisfy the same relation.  Then each rooted tree in the Fa\`a di Bruno expansion, written in packet coordinates, descends to the corresponding coarse tree.  Moreover,
982 \[
983   C(G_{\fine}\circ F_{\fine})
984   =(G_{\coarse}\circ F_{\coarse})C.
985 \]
986 \end{corollary}
987 
988 \begin{proof}
989 The map identity follows directly by composition.  The termwise statement follows from \Cref{thm:treewise-quotient}, because the Fa\`a di Bruno expansion is a finite sum of rooted derivative trees at each derivative order.
990 \end{proof}
991 
992 \subsection{Exact quotient flows}
993 
994 \begin{theorem}[Exact quotient flow]\label{thm:exact-quotient-flow}
995 Let $U_{\fine}\subseteq X_{\fine}$ and $U_{\coarse}\subseteq X_{\coarse}$ be open, with $C(U_{\fine})\subseteq U_{\coarse}$.  Suppose $F_{\fine}$ and $F_{\coarse}$ are locally Lipschitz and satisfy
996 \[
997   CF_{\fine}=F_{\coarse}C
998 \]
999 on $U_{\fine}$.  Let $\Phi_t^{\fine}(x)$ and $\Phi_t^{\coarse}(z)$ denote the unique maximal local flows.  Whenever both sides are defined,
1000 \begin{equation}\label{eq:exact-flow-identity}
1001   \boxed{
1002   C\Phi_t^{\fine}(x)
1003   =
1004   \Phi_t^{\coarse}(Cx).}
1005 \end{equation}
1006 \end{theorem}
1007 
1008 \begin{proof}
1009 Let $x(t)=\Phi_t^{\fine}(x_0)$ and set $z(t)=Cx(t)$.  Then
1010 \[
1011   \dot z(t)
1012   =C\dot x(t)
1013   =CF_{\fine}(x(t))
1014   =F_{\coarse}(Cx(t))
1015   =F_{\coarse}(z(t)),
1016 \]
1017 with $z(0)=Cx_0$.  Uniqueness for the coarse initial value problem gives $z(t)=\Phi_t^{\coarse}(Cx_0)$.
1018 \end{proof}
1019 
1020 \begin{corollary}[Linear semigroup naturality]\label{cor:linear-semigroup-naturality}
1021 If $CA_{\fine}=A_{\coarse}C$, then
1022 \begin{equation}\label{eq:linear-semigroup-naturality}
1023   Ce^{tA_{\fine}}
1024   =
1025   e^{tA_{\coarse}}C
1026 \end{equation}
1027 for every $t$.
1028 \end{corollary}
1029 
1030 \begin{proof}
1031 Either apply \Cref{thm:exact-quotient-flow} to the linear systems or sum the exponential series using $CA_{\fine}^k=A_{\coarse}^kC$.
1032 \end{proof}
1033 
1034 \begin{corollary}[Quotients of discrete dynamics]\label{cor:discrete-quotient}
1035 If $CF_{\fine}=F_{\coarse}C$, then
1036 \[
1037   CF_{\fine}^{\circ n}
1038   =
1039   F_{\coarse}^{\circ n}C
1040 \]
1041 for every integer $n\geq0$.
1042 \end{corollary}
1043 
1044 \begin{proof}
1045 Induct on $n$.
1046 \end{proof}
1047 
1048 \subsection{Positivity and simplex preservation}
1049 Let
1050 \[
1051   \Delta_{\fine}
1052   =\{\mu\in\R_+^n:\one^T\mu=1\},
1053   \qquad
1054   \Delta_{\coarse}
1055   =\{\nu\in\R_+^q:\one^T\nu=1\}.
1056 \]
1057 For block sum aggregation, $C\Delta_{\fine}=\Delta_{\coarse}$ and
1058 \[
1059   \one_{\coarse}^TC=\one_{\fine}^T.
1060 \]
1061 
1062 \begin{proposition}[Simplex preservation descends]\label{prop:simplex-descends}
1063 Suppose $\Phi_t^{\fine}(\Delta_{\fine})\subseteq\Delta_{\fine}$ and exact flow naturality holds.  Then
1064 \[
1065   \Phi_t^{\coarse}(\Delta_{\coarse})\subseteq\Delta_{\coarse}.
1066 \]
1067 \end{proposition}
1068 
1069 \begin{proof}
1070 For any $\nu\in\Delta_{\coarse}$, choose $\mu\in\Delta_{\fine}$ with $C\mu=\nu$.  Then
1071 \[
1072   \Phi_t^{\coarse}(\nu)
1073   =C\Phi_t^{\fine}(\mu)
1074   \in C\Delta_{\fine}
1075   =\Delta_{\coarse}.
1076 \]
1077 \end{proof}
1078 
1079 \begin{boundary}[Global existence of the flow]
1080 Exact descent does not guarantee global solutions.  A fine trajectory may blow up in finite time, leave the domain, or reach a region where the vector field is no longer Lipschitz.  \Cref{thm:exact-quotient-flow} applies only while both solutions exist and remain unique.
1081 \end{boundary}
1082 
1083 \section{Normalized positive transfer and Bayesian refinement}\label{sec:bayes}
1084 
1085 \subsection{Positive state spaces}
1086 The Bayesian result rests on a simple fact about normalizing positive operators.
1087 
1088 Let $X_{\fine}$ and $X_{\coarse}$ be finite dimensional ordered vector spaces with positive cones $K_{\fine}$ and $K_{\coarse}$.  Let
1089 \[
1090   \ell_{\fine}:X_{\fine}\to\R,
1091   \qquad
1092   \ell_{\coarse}:X_{\coarse}\to\R
1093 \]
1094 be positive mass functionals.  Define normalized state spaces
1095 \[
1096   \cS_{\fine}
1097   =\{x\in K_{\fine}:\ell_{\fine}(x)=1\},
1098   \qquad
1099   \cS_{\coarse}
1100   =\{z\in K_{\coarse}:\ell_{\coarse}(z)=1\}.
1101 \]
1102 Assume $C$ is positive and preserves mass:
1103 \begin{equation}\label{eq:mass-preserving-C}
1104   \ell_{\coarse}C=\ell_{\fine}.
1105 \end{equation}
1106 
1107 \begin{definition}[Normalized positive transfer]\label{def:normalized-transfer}
1108 For a positive linear operator $L:X\to X$, define
1109 \begin{equation}\label{eq:normalized-transfer}
1110   \Nrm_{\ell}(L)(x)
1111   =
1112   \frac{Lx}{\ell(Lx)}
1113 \end{equation}
1114 whenever $\ell(Lx)>0$.
1115 \end{definition}
1116 
1117 \subsection{Normalization naturality}
1118 
1119 \begin{theorem}[Naturality of normalized transfer]\label{thm:normalized-transfer-naturality}
1120 Let $L_{\fine}:X_{\fine}\to X_{\fine}$ and $L_{\coarse}:X_{\coarse}\to X_{\coarse}$ be positive linear operators.  Suppose
1121 \begin{equation}\label{eq:positive-intertwining}
1122   CL_{\fine}=sL_{\coarse}C
1123 \end{equation}
1124 for some $s>0$, and suppose \eqref{eq:mass-preserving-C} holds.  Then, on the common domain of normalization,
1125 \begin{equation}\label{eq:normalized-transfer-naturality}
1126   \boxed{
1127   C\Nrm_{\ell_{\fine}}(L_{\fine})
1128   =
1129   \Nrm_{\ell_{\coarse}}(L_{\coarse})C.}
1130 \end{equation}
1131 \end{theorem}
1132 
1133 \begin{proof}
1134 For a positive state $x$,
1135 \begin{align*}
1136 C\Nrm_{\ell_{\fine}}(L_{\fine})(x)
1137 &=
1138 \frac{CL_{\fine}x}{\ell_{\fine}(L_{\fine}x)}\\
1139 &=
1140 \frac{sL_{\coarse}Cx}{\ell_{\coarse}(CL_{\fine}x)}\\
1141 &=
1142 \frac{sL_{\coarse}Cx}{s\ell_{\coarse}(L_{\coarse}Cx)}\\
1143 &=
1144 \Nrm_{\ell_{\coarse}}(L_{\coarse})(Cx).
1145 \end{align*}
1146 \end{proof}
1147 
1148 \begin{remark}[Gauge invariance]
1149 Normalization removes the scale $s$.  At the operator level, this is the familiar fact that a positive rescaling of a likelihood leaves the posterior unchanged.
1150 \end{remark}
1151 
1152 \begin{proposition}[Composition law for normalized transfers]\label{prop:normalized-composition}
1153 For positive operators $L_1,L_2$ on the same ordered state space,
1154 \begin{equation}\label{eq:normalized-composition}
1155   \Nrm_{\ell}(L_2)
1156   \circ
1157   \Nrm_{\ell}(L_1)
1158   =
1159   \Nrm_{\ell}(L_2L_1)
1160 \end{equation}
1161 wherever both sides are defined.
1162 \end{proposition}
1163 
1164 \begin{proof}
1165 The first normalization contributes the same scalar to the numerator and denominator of the second update, so it cancels.
1166 \end{proof}
1167 
1168 \subsection{Classical Bayesian update}
1169 On the finite simplex, let
1170 \[
1171   L_{\lambda}=\diag(\lambda_1,\ldots,\lambda_n),
1172   \qquad
1173   \lambda_i>0.
1174 \]
1175 With $\ell(x)=\one^Tx$,
1176 \[
1177   \Nrm_{\ell}(L_{\lambda})(\mu)
1178   =
1179   \frac{\lambda\odot\mu}{\lambda^T\mu}
1180   =\Bup_{\lambda}(\mu).
1181 \]
1182 The evidence is
1183 \[
1184   Z_{\lambda}(\mu)=\lambda^T\mu.
1185 \]
1186 The composition law becomes
1187 \begin{equation}\label{eq:bayes-composition}
1188   \Bup_{\lambda_2}\circ\Bup_{\lambda_1}
1189   =
1190   \Bup_{\lambda_1\odot\lambda_2}.
1191 \end{equation}
1192 
1193 \subsection{Exact Bayesian refinement}
1194 Let $C$ be the block sum aggregation associated with a surjection $\pi:\{1,\ldots,n\}\twoheadrightarrow\{1,\ldots,q\}$.
1195 
1196 \begin{theorem}[Sufficient condition for Bayesian refinement]\label{thm:bayes-sufficient}
1197 Let $\lambda^{\fine}\in\R_{>0}^n$ and $\lambda^{\coarse}\in\R_{>0}^q$.  If
1198 \begin{equation}\label{eq:fiber-likelihood-sufficient}
1199   \lambda^{\fine}=sC^T\lambda^{\coarse}
1200 \end{equation}
1201 for some $s>0$, then
1202 \begin{equation}\label{eq:exact-bayes-refinement}
1203   \boxed{
1204   C\Bup_{\lambda^{\fine}}
1205   =
1206   \Bup_{\lambda^{\coarse}}C.}
1207 \end{equation}
1208 Moreover,
1209 \begin{equation}\label{eq:evidence-scaling}
1210   Z_{\lambda^{\fine}}(\mu)
1211   =
1212   sZ_{\lambda^{\coarse}}(C\mu).
1213 \end{equation}
1214 \end{theorem}
1215 
1216 \begin{proof}
1217 Equation \eqref{eq:fiber-likelihood-sufficient} is equivalent to
1218 \[
1219   CL_{\lambda^{\fine}}
1220   =sL_{\lambda^{\coarse}}C.
1221 \]
1222 Apply \Cref{thm:normalized-transfer-naturality}.  The evidence identity follows from
1223 \[
1224   (\lambda^{\fine})^T\mu
1225   =s(\lambda^{\coarse})^TC\mu.
1226 \]
1227 \end{proof}
1228 
1229 Condition \eqref{eq:fiber-likelihood-sufficient} says that the fine likelihood is constant on each aggregation fiber, apart from one global rescaling.  When the coarse space has at least two packets, the next theorem shows that this condition is also necessary if commutation is required for every interior prior.
1230 
1231 \begin{theorem}[Necessary condition for Bayesian refinement]\label{thm:bayes-necessity}
1232 Assume $q\geq2$, every fiber of $\pi$ is nonempty, and $\lambda^{\fine},\lambda^{\coarse}$ are strictly positive.  If
1233 \begin{equation}\label{eq:bayes-equality-all-priors}
1234   C\Bup_{\lambda^{\fine}}(\mu)
1235   =
1236   \Bup_{\lambda^{\coarse}}(C\mu)
1237 \end{equation}
1238 for every $\mu$ in the interior of $\Delta_n$, then there exists $s>0$ such that
1239 \[
1240   \lambda^{\fine}=sC^T\lambda^{\coarse}.
1241 \]
1242 \end{theorem}
1243 
1244 \begin{proof}
1245 Both sides of \eqref{eq:bayes-equality-all-priors} extend continuously to any prior with nonzero evidence.  Strict positivity guarantees nonzero evidence throughout the closed simplex.  Fix two distinct coarse packets $a$ and $b$.  Choose $i\in\pi^{-1}(a)$ and $k\in\pi^{-1}(b)$, then consider a prior supported on $\{i,k\}$ with masses $u,v>0$.  On the fine side, the ratio of the posterior masses of packets $a$ and $b$ is
1246 \[
1247   \frac{\lambda_i^{\fine}u}{\lambda_k^{\fine}v},
1248 \]
1249 whereas the coarse posterior ratio is
1250 \[
1251   \frac{\lambda_a^{\coarse}u}{\lambda_b^{\coarse}v}.
1252 \]
1253 Equality for all $u,v>0$ gives
1254 \begin{equation}\label{eq:likelihood-ratio-necessity}
1255   \frac{\lambda_i^{\fine}}{\lambda_k^{\fine}}
1256   =
1257   \frac{\lambda_a^{\coarse}}{\lambda_b^{\coarse}}.
1258 \end{equation}
1259 Take another state $j$ in the same fiber as $i$.  Applying the same argument to $j$ and $k$ gives $\lambda_j^{\fine}=\lambda_i^{\fine}$.  The fine likelihood is therefore constant on each fiber.  Write its value on fiber $a$ as $\overline\lambda_a$.  Equation \eqref{eq:likelihood-ratio-necessity} now gives
1260 \[
1261   \frac{\overline\lambda_a}{\overline\lambda_b}
1262   =
1263   \frac{\lambda_a^{\coarse}}{\lambda_b^{\coarse}}
1264 \]
1265 for every pair $a,b$.  Hence $\overline\lambda=s\lambda^{\coarse}$ for one $s>0$, proving the result.
1266 \end{proof}
1267 
1268 \begin{remark}[The case of one coarse packet]
1269 If $q=1$, both sides of \eqref{eq:bayes-equality-all-priors} equal the unique coarse state, regardless of the fine likelihood.  Necessity therefore requires a nontrivial coarse state space.
1270 \end{remark}
1271 
1272 \subsection{Differential structure of Bayesian normalization}
1273 
1274 \begin{proposition}[Bayesian differential]\label{prop:bayes-differential}
1275 At an interior prior $\mu$,
1276 \begin{equation}\label{eq:bayes-differential}
1277   D\Bup_{\lambda}(\mu)[h]
1278   =
1279   \frac{\lambda\odot h}{Z_{\lambda}(\mu)}
1280   -
1281   \Bup_{\lambda}(\mu)
1282   \frac{\lambda^Th}{Z_{\lambda}(\mu)}.
1283 \end{equation}
1284 \end{proposition}
1285 
1286 \begin{proof}
1287 Differentiate the quotient $L_{\lambda}\mu/(\one^TL_{\lambda}\mu)$.
1288 \end{proof}
1289 
1290 \begin{corollary}[Bayesian jet naturality]\label{cor:bayes-jet-naturality}
1291 Under the hypotheses of \Cref{thm:bayes-sufficient}, for every $r\geq1$ and every interior fine prior,
1292 \begin{equation}\label{eq:bayes-jet-naturality}
1293   C D^r\Bup_{\lambda^{\fine}}(\mu)
1294   [h_1,\ldots,h_r]
1295   =
1296   D^r\Bup_{\lambda^{\coarse}}(C\mu)
1297   [Ch_1,\ldots,Ch_r].
1298 \end{equation}
1299 \end{corollary}
1300 
1301 \begin{proof}
1302 Differentiate the exact map identity \eqref{eq:exact-bayes-refinement} and apply \Cref{prop:jet-naturality}.
1303 \end{proof}
1304 
1305 Thus the normalization map has a descended interaction structure of its own.  Its first derivative couples all evidence packets through a rank one term.  Higher derivatives describe interactions among larger groups of packets.
1306 
1307 \section{Filtering cocycles}\label{sec:cocycles}
1308 
1309 \subsection{Finite observation sequences}
1310 Let $0=t_0<t_1<\cdots<t_N$ and define prediction maps
1311 \[
1312   \Phi_k^{\fine}=\Phi_{t_k-t_{k-1}}^{\fine},
1313   \qquad
1314   \Phi_k^{\coarse}=\Phi_{t_k-t_{k-1}}^{\coarse}.
1315 \]
1316 For observations $y_1,\ldots,y_N$, let
1317 \[
1318   \Bup_k^{\fine}=\Bup_{\lambda_{y_k}^{\fine}},
1319   \qquad
1320   \Bup_k^{\coarse}=\Bup_{\lambda_{y_k}^{\coarse}}.
1321 \]
1322 Pair each prediction with the observation that follows it:
1323 \[
1324   \Gamma_k^{\fine}
1325   =\Bup_k^{\fine}\circ\Phi_k^{\fine},
1326   \qquad
1327   \Gamma_k^{\coarse}
1328   =\Bup_k^{\coarse}\circ\Phi_k^{\coarse},
1329 \]
1330 and define the resulting filtering cocycles
1331 \begin{equation}\label{eq:filter-cocycles}
1332   \Psi_{1:N}^{\fine}
1333   =\Gamma_N^{\fine}\circ\cdots\circ\Gamma_1^{\fine},
1334   \qquad
1335   \Psi_{1:N}^{\coarse}
1336   =\Gamma_N^{\coarse}\circ\cdots\circ\Gamma_1^{\coarse}.
1337 \end{equation}
1338 
1339 \begin{theorem}[Exact naturality of the filtering cocycle]\label{thm:exact-cocycle}
1340 Suppose, for every $k$,
1341 \begin{equation}\label{eq:step-flow-naturality}
1342   C\Phi_k^{\fine}=\Phi_k^{\coarse}C
1343 \end{equation}
1344 and
1345 \begin{equation}\label{eq:step-bayes-naturality}
1346   C\Bup_k^{\fine}=\Bup_k^{\coarse}C.
1347 \end{equation}
1348 Then
1349 \begin{equation}\label{eq:exact-cocycle-naturality}
1350   \boxed{
1351   C\Psi_{1:N}^{\fine}
1352   =
1353   \Psi_{1:N}^{\coarse}C.}
1354 \end{equation}
1355 \end{theorem}
1356 
1357 \begin{proof}
1358 For one step,
1359 \[
1360   C\Gamma_k^{\fine}
1361   =C\Bup_k^{\fine}\Phi_k^{\fine}
1362   =\Bup_k^{\coarse}C\Phi_k^{\fine}
1363   =\Bup_k^{\coarse}\Phi_k^{\coarse}C
1364   =\Gamma_k^{\coarse}C.
1365 \]
1366 Iterating this identity through \eqref{eq:filter-cocycles} proves the result.
1367 \end{proof}
1368 
1369 \begin{corollary}[Exact posterior aggregation]\label{cor:exact-posterior}
1370 If $\mu_0^{\coarse}=C\mu_0^{\fine}$, then
1371 \[
1372   \boxed{C\mu_N^{\fine}=\mu_N^{\coarse}}
1373 \]
1374 for every finite observation sequence, where
1375 \[
1376   \mu_N^{\fine}=\Psi_{1:N}^{\fine}(\mu_0^{\fine}),
1377   \qquad
1378   \mu_N^{\coarse}=\Psi_{1:N}^{\coarse}(\mu_0^{\coarse}).
1379 \]
1380 \end{corollary}
1381 
1382 \begin{proof}
1383 Apply \eqref{eq:exact-cocycle-naturality} to $\mu_0^{\fine}$ and substitute $\mu_0^{\coarse}=C\mu_0^{\fine}$.
1384 \end{proof}
1385 
1386 \subsection{Sequential evidence}
1387 Let
1388 \[
1389   \widehat\mu_k^{\fine}
1390   =\Phi_k^{\fine}(\mu_{k-1}^{\fine}),
1391   \qquad
1392   \widehat\mu_k^{\coarse}
1393   =\Phi_k^{\coarse}(\mu_{k-1}^{\coarse}),
1394 \]
1395 and define the step evidence
1396 \[
1397   z_k^{\fine}
1398   =(\lambda_{y_k}^{\fine})^T\widehat\mu_k^{\fine},
1399   \qquad
1400   z_k^{\coarse}
1401   =(\lambda_{y_k}^{\coarse})^T\widehat\mu_k^{\coarse}.
1402 \]
1403 Set
1404 \[
1405   Z_{1:N}^{\fine}=\prod_{k=1}^N z_k^{\fine},
1406   \qquad
1407   Z_{1:N}^{\coarse}=\prod_{k=1}^N z_k^{\coarse}.
1408 \]
1409 
1410 \begin{theorem}[Evidence factorization under exact refinement]\label{thm:evidence-factorization}
1411 Assume the hypotheses of \Cref{thm:exact-cocycle}, let
1412 \[
1413   \mu_0^{\coarse}=C\mu_0^{\fine},
1414 \]
1415 and suppose
1416 \[
1417   \lambda_{y_k}^{\fine}
1418   =s_kC^T\lambda_{y_k}^{\coarse}
1419   \qquad(s_k>0).
1420 \]
1421 Then
1422 \begin{equation}\label{eq:step-evidence-factor}
1423   z_k^{\fine}=s_kz_k^{\coarse}
1424 \end{equation}
1425 for every step, and the likelihood of the observation sequence satisfies
1426 \begin{equation}\label{eq:total-evidence-factor}
1427   \boxed{
1428   Z_{1:N}^{\fine}
1429   =
1430   \left(\prod_{k=1}^Ns_k\right)
1431   Z_{1:N}^{\coarse}.}
1432 \end{equation}
1433 Under the gauge $s_k=1$, the fine and coarse sequential evidences agree exactly.
1434 \end{theorem}
1435 
1436 \begin{proof}
1437 Initial compatibility gives $C\mu_0^{\fine}=\mu_0^{\coarse}$.  For later steps, \Cref{cor:exact-posterior} gives the same identity after every update.  Flow naturality therefore yields
1438 \[
1439   C\widehat\mu_k^{\fine}=\widehat\mu_k^{\coarse}.
1440 \]
1441 Therefore
1442 \[
1443   z_k^{\fine}
1444   =s_k(\lambda_{y_k}^{\coarse})^TC\widehat\mu_k^{\fine}
1445   =s_kz_k^{\coarse}.
1446 \]
1447 The likelihood of the complete observation sequence is the product of the conditional evidence terms.  Multiplying the identities gives \eqref{eq:total-evidence-factor}.
1448 \end{proof}
1449 
1450 \subsection{Hidden Markov and nonlinear specializations}
1451 
1452 \begin{corollary}[Hidden Markov lumpability]\label{cor:hmm-lumpability}
1453 Let $M_{\fine}$ and $M_{\coarse}$ be column stochastic matrices satisfying
1454 \[
1455   CM_{\fine}=M_{\coarse}C,
1456 \]
1457 and let every observation likelihood satisfy
1458 \[
1459   \lambda_y^{\fine}=s_yC^T\lambda_y^{\coarse}.
1460 \]
1461 Then, for every observation sequence $y_{1:N}$,
1462 \begin{align*}
1463 &C\bigl(
1464 \Bup_{\lambda_{y_N}^{\fine}}\circ M_{\fine}
1465 \circ\cdots\circ
1466 \Bup_{\lambda_{y_1}^{\fine}}\circ M_{\fine}
1467 \bigr)\\
1468 &\qquad=
1469 \bigl(
1470 \Bup_{\lambda_{y_N}^{\coarse}}\circ M_{\coarse}
1471 \circ\cdots\circ
1472 \Bup_{\lambda_{y_1}^{\coarse}}\circ M_{\coarse}
1473 \bigr)C.
1474 \end{align*}
1475 \end{corollary}
1476 
1477 \begin{proof}
1478 Apply \Cref{thm:exact-cocycle} with the linear prediction maps $M_{\fine}$ and $M_{\coarse}$.
1479 \end{proof}
1480 
1481 \begin{corollary}[Nonlinear quotient filtering]\label{cor:nonlinear-filtering}
1482 Suppose the fine and coarse prediction maps are flow segments of locally Lipschitz vector fields satisfying $CF_{\fine}=F_{\coarse}C$, and suppose all observation likelihoods are constant on each fiber up to scale.  Then the nonlinear filtering cocycle commutes with $C$ exactly.
1483 \end{corollary}
1484 
1485 \begin{proof}
1486 Intertwining of the vector fields gives $C\Phi_k^{\fine}=\Phi_k^{\coarse}C$ by \Cref{thm:exact-quotient-flow}.  Likelihoods that are constant on each fiber give $C\Bup_k^{\fine}=\Bup_k^{\coarse}C$ by \Cref{thm:bayes-sufficient}.  Apply \Cref{thm:exact-cocycle}.
1487 \end{proof}
1488 
1489 \begin{remark}[Two independent closure requirements]\label{rem:two-closure-requirements}
1490 The dynamical and observational conditions are independent.  Exact interaction lumpability says nothing about whether the likelihood distinguishes states inside a packet.  Conversely, an observation may ignore those distinctions even when the dynamics depend on them.  Exact coarse filtering needs both commutative squares.
1491 \end{remark}
1492 
1493 \begin{boundary}[Sufficiency for the chosen observation family]
1494 The condition $\lambda_y^{\fine}=s_yC^T\lambda_y^{\coarse}$ is sufficient for the specified finite likelihood family.  It does not assert that $C$ is sufficient for every possible future observation model.  Adding a likelihood that distinguishes two states inside one fiber can invalidate the coarse filter without changing the dynamics.
1495 \end{boundary}
1496 
1497 \section{Approximate interaction quotients}\label{sec:approx-dyn}
1498 
1499 Exact quotienting is an algebraic yes or no question.  A useful model also needs to say how far it misses when the answer is no.  We separate three sources of error:
1500 \[
1501   \text{interaction mismatch},
1502   \qquad
1503   \text{dependence on hidden directions},
1504   \qquad
1505   \text{dynamical amplification}.
1506 \]
1507 
1508 \subsection{Best coarse interaction and quotient defect}
1509 Let
1510 \[
1511   \mathfrak Q_m(C)
1512   =
1513   \{B_mC^{\times m}:B_m\in\cL^m(X_{\coarse};X_{\coarse})\}
1514 \]
1515 be the subspace of outputs from multilinear maps of order $m$ that factor through the coarse space.
1516 
1517 \begin{definition}[Best quotient error]\label{def:best-quotient-error}
1518 For a fine interaction $A_m^{\fine}$, define
1519 \begin{equation}\label{eq:best-quotient-error}
1520   \varepsilon_m^\star
1521   =
1522   \inf_{B_m\in\cL^m(X_{\coarse};X_{\coarse})}
1523   \norm{CA_m^{\fine}-B_mC^{\times m}}.
1524 \end{equation}
1525 \end{definition}
1526 
1527 \begin{proposition}[Attainment in finite dimension]\label{prop:best-quotient-attained}
1528 The infimum in \eqref{eq:best-quotient-error} is attained.
1529 \end{proposition}
1530 
1531 \begin{proof}
1532 The map
1533 \[
1534   B_m\longmapsto B_mC^{\times m}
1535 \]
1536 is injective because $C$ is surjective.  Its range is a finite dimensional linear subspace of $\cL^m(X_{\fine};X_{\coarse})$, hence it is closed.  In a finite dimensional normed space, the distance to a nonempty closed subspace attains its minimum.  A nearest point in this range therefore exists.  Any such point has a unique preimage under the displayed map, and that preimage is a minimizing coarse tensor.
1537 \end{proof}
1538 
1539 The minimum value depends on the chosen norm.  A minimizing tensor need not be unique when that norm is not strictly convex.  Exact lumpability is independent of the norm because it is the condition $\varepsilon_m^\star=0$.
1540 
1541 \subsection{Approximation induced by a section}
1542 A chosen right inverse supplies a concrete coarse tensor even when exact descent fails.
1543 
1544 \begin{definition}[Coarse tensor induced by a section]\label{def:section-coarse-tensor}
1545 Fix $E$ with $CE=I$.  Define
1546 \begin{equation}\label{eq:section-coarse-tensor}
1547   A_m^{\coarse,E}
1548   =
1549   CA_m^{\fine}E^{\times m}.
1550 \end{equation}
1551 \end{definition}
1552 
1553 Let $P_C=EC$ and $Q_C=I-EC$ as in \eqref{eq:representative-hidden-proj}.
1554 
1555 \begin{proposition}[Expansion of the section defect by hidden inputs]\label{prop:hidden-slot-expansion}
1556 The defect of \eqref{eq:section-coarse-tensor} is
1557 \begin{equation}\label{eq:section-defect}
1558   \Delta_m^E(x_1,\ldots,x_m)
1559   =
1560   CA_m^{\fine}(x_1,\ldots,x_m)
1561   -
1562   CA_m^{\fine}(P_Cx_1,\ldots,P_Cx_m),
1563 \end{equation}
1564 and admits the exact expansion
1565 \begin{equation}\label{eq:hidden-slot-sum}
1566   \boxed{
1567   \Delta_m^E(x_1,\ldots,x_m)
1568   =
1569   \sum_{\varnothing\neq S\subseteq\{1,\ldots,m\}}
1570   CA_m^{\fine}(y_1^S,\ldots,y_m^S),}
1571 \end{equation}
1572 where
1573 \[
1574   y_r^S=
1575   \begin{cases}
1576   Q_Cx_r,&r\in S,\\
1577   P_Cx_r,&r\notin S.
1578   \end{cases}
1579 \]
1580 \end{proposition}
1581 
1582 \begin{proof}
1583 Equation \eqref{eq:section-defect} follows from
1584 \[
1585   A_m^{\coarse,E}(Cx_1,\ldots,Cx_m)
1586   =CA_m^{\fine}(ECx_1,\ldots,ECx_m).
1587 \]
1588 Now expand every input as $x_r=P_Cx_r+Q_Cx_r$.  The term with no $Q_C$ factor is the term subtracted in \eqref{eq:section-defect}.  Every remaining term has at least one hidden input and appears exactly once.
1589 \end{proof}
1590 
1591 \begin{corollary}[Exactness as disappearance of hidden interactions]\label{cor:hidden-interactions-exactness}
1592 The tensor defined from the chosen section descends exactly if and only if the sum in \eqref{eq:hidden-slot-sum} vanishes for all inputs.  The stronger slotwise condition
1593 \[
1594   CA_m^{\fine}(x_1,\ldots,Q_Cx_r,\ldots,x_m)=0
1595 \]
1596 for each slot is equivalent to exact descent and makes every summand vanish separately.
1597 \end{corollary}
1598 
1599 \begin{proof}
1600 The first statement is \eqref{eq:hidden-slot-sum} with $\Delta_m^E=0$.  Since $\Ran Q_C=\Ker C$, the condition in each slot is exactly the kernel condition in \Cref{thm:universal-quotient}.  It is therefore equivalent to exact descent.  Every term indexed by a nonempty subset contains at least one hidden input, so each term vanishes.
1601 \end{proof}
1602 
1603 \subsection{Defect majorants for the full field}
1604 Suppose fine and proposed coarse fields have expansions
1605 \[
1606   F_{\fine}(x)=\sum_{m\geq0}A_m^{\fine}[x^{\otimes m}],
1607   \qquad
1608   F_{\coarse}(z)=\sum_{m\geq0}A_m^{\coarse}[z^{\otimes m}].
1609 \]
1610 Define the field defect
1611 \begin{equation}\label{eq:field-defect}
1612   \delta_F(x)
1613   =
1614   CF_{\fine}(x)-F_{\coarse}(Cx).
1615 \end{equation}
1616 Whenever both series converge absolutely,
1617 \begin{equation}\label{eq:field-defect-expansion}
1618   \delta_F(x)
1619   =
1620   \sum_{m\geq0}\Delta_m[x^{\otimes m}].
1621 \end{equation}
1622 
1623 \begin{proposition}[Analytic defect majorant]\label{prop:defect-majorant}
1624 If $\norm{\Delta_m}\leq\varepsilon_m$ and the scalar series
1625 \[
1626   \mathfrak d_F(R)=\sum_{m\geq0}\varepsilon_mR^m
1627 \]
1628 converges, then
1629 \begin{equation}\label{eq:field-defect-majorant}
1630   \boxed{
1631   \sup_{\norm{x}\leq R}\norm{\delta_F(x)}
1632   \leq
1633   \mathfrak d_F(R).}
1634 \end{equation}
1635 \end{proposition}
1636 
1637 \begin{proof}
1638 Apply the multilinear norm bound to every term in \eqref{eq:field-defect-expansion} and sum the convergent majorant.
1639 \end{proof}
1640 
1641 The sequence $(\varepsilon_m)$ shows which interaction orders obstruct the quotient.  A model may be exactly lumpable through order $r$ and fail for the first time at order $r+1$.
1642 
1643 \subsection{Propagation of dynamical defect}
1644 
1645 \begin{theorem}[Approximate quotient flow bound]\label{thm:approx-flow-bound}
1646 Let
1647 \[
1648   \dot x=F_{\fine}(x),
1649   \qquad
1650   \dot z=F_{\coarse}(z),
1651 \]
1652 and suppose both solutions remain in domains on which
1653 \[
1654   \norm{F_{\coarse}(u)-F_{\coarse}(v)}
1655   \leq L\norm{u-v}.
1656 \]
1657 Set
1658 \[
1659   e(t)=\norm{Cx(t)-z(t)}.
1660 \]
1661 Then
1662 \begin{equation}\label{eq:approx-flow-general}
1663   \boxed{
1664   e(t)
1665   \leq
1666   e^{Lt}e(0)
1667   +
1668   \int_0^t e^{L(t-s)}\norm{\delta_F(x(s))}\dd s.}
1669 \end{equation}
1670 If $e(0)=0$ and $\norm{\delta_F(x(s))}\leq\varepsilon_F$ on $[0,t]$, then
1671 \begin{equation}\label{eq:approx-flow-uniform}
1672   e(t)
1673   \leq
1674   \begin{cases}
1675   \displaystyle
1676   \varepsilon_F\frac{e^{Lt}-1}{L},&L>0,\\[1.1ex]
1677   t\varepsilon_F,&L=0.
1678   \end{cases}
1679 \end{equation}
1680 \end{theorem}
1681 
1682 \begin{proof}
1683 Let $y(t)=Cx(t)$.  From \eqref{eq:field-defect},
1684 \[
1685   \dot y(t)=F_{\coarse}(y(t))+\delta_F(x(t)).
1686 \]
1687 Subtract the integral equations for $y$ and $z$:
1688 \[
1689   y(t)-z(t)
1690   =y(0)-z(0)
1691   +\int_0^t
1692   \bigl(F_{\coarse}(y(s))-F_{\coarse}(z(s))\bigr)\dd s
1693   +\int_0^t\delta_F(x(s))\dd s.
1694 \]
1695 Taking norms and applying the Grönwall inequality yields \eqref{eq:approx-flow-general}.  The uniform cases follow by evaluating the exponential integral.
1696 \end{proof}
1697 
1698 \begin{corollary}[Coefficient certificate for a flow segment]\label{cor:coefficient-flow-certificate}
1699 If $\norm{x(s)}\leq R$ and Proposition~\ref{prop:defect-majorant} applies, then a flow interval of length $T$ satisfies
1700 \[
1701   \norm{C\Phi_T^{\fine}(x_0)-\Phi_T^{\coarse}(Cx_0)}
1702   \leq
1703   \mathfrak d_F(R)\frac{e^{LT}-1}{L}
1704 \]
1705 for $L>0$, with $T\mathfrak d_F(R)$ at $L=0$.
1706 \end{corollary}
1707 
1708 \begin{proof}
1709 By Proposition~\ref{prop:defect-majorant}, $\norm{\delta_F(x(s))}\leq\mathfrak d_F(R)$ throughout the segment.  Apply \Cref{thm:approx-flow-bound} with compatible initial data, so $e(0)=0$, and evaluate the exponential integral.
1710 \end{proof}
1711 
1712 \subsection{Refinement towers}
1713 Let
1714 \[
1715   X_L\xrightarrow{C_{L,L-1}}X_{L-1}
1716   \longrightarrow\cdots\longrightarrow
1717   X_1\xrightarrow{C_{10}}X_0.
1718 \]
1719 For $a>b$, write
1720 \[
1721   C_{ab}=C_{b+1,b}\cdots C_{a,a-1}.
1722 \]
1723 Let $\Delta_m^{ab}$ denote the interaction defect for the composite map $C_{ab}$.  Iterating \Cref{thm:multilevel-defect} yields a telescoping formula.
1724 
1725 \begin{corollary}[Telescoping interaction defect]\label{cor:tower-defect}
1726 For an interaction of order $m$ assigned at every level,
1727 \begin{equation}\label{eq:tower-defect}
1728   \Delta_m^{L0}
1729   =
1730   \sum_{\ell=0}^{L-1}
1731   C_{\ell0}\,
1732   \Delta_m^{\ell+1,\ell}
1733   C_{L,\ell+1}^{\times m},
1734 \end{equation}
1735 where $C_{00}=I$ and $C_{LL}=I$.  Consequently,
1736 \begin{equation}\label{eq:tower-defect-bound}
1737   \norm{\Delta_m^{L0}}
1738   \leq
1739   \sum_{\ell=0}^{L-1}
1740   \norm{C_{\ell0}}
1741   \norm{\Delta_m^{\ell+1,\ell}}
1742   \norm{C_{L,\ell+1}}^m.
1743 \end{equation}
1744 \end{corollary}
1745 
1746 \begin{proof}
1747 Induct on the number of levels, using \eqref{eq:multilevel-defect} at the final composition.
1748 \end{proof}
1749 
1750 The identity shows where a terminal mismatch entered the refinement tower.  Rather than storing only the final error, one can keep the local defect at each level and combine those records when needed.
1751 
1752 \section{Approximate Bayesian refinement}\label{sec:approx-bayes}
1753 
1754 \subsection{A normalization perturbation bound}
1755 
1756 \begin{lemma}[Normalization perturbation]\label{lem:normalization-perturbation}
1757 Let $u,v\in\R_+^q$ have positive total mass.  Then
1758 \begin{equation}\label{eq:normalization-perturbation}
1759   \boxed{
1760   \norm{
1761   \frac{u}{\one^Tu}
1762   -
1763   \frac{v}{\one^Tv}
1764   }_1
1765   \leq
1766   \frac{2\norm{u-v}_1}
1767   {\max\{\one^Tu,\one^Tv\}}.}
1768 \end{equation}
1769 \end{lemma}
1770 
1771 \begin{proof}
1772 Write $a=\one^Tu$ and $b=\one^Tv$.  Then
1773 \begin{align*}
1774 \norm{\frac{u}{a}-\frac{v}{b}}_1
1775 &\leq
1776 \frac{\norm{u-v}_1}{a}
1777 +
1778 \norm{v}_1\abs{\frac1a-\frac1b}\\
1779 &=
1780 \frac{\norm{u-v}_1}{a}
1781 +
1782 \frac{\abs{a-b}}{a}\\
1783 &\leq
1784 \frac{2\norm{u-v}_1}{a}.
1785 \end{align*}
1786 Interchanging $u$ and $v$ gives the same inequality with $b$ in the denominator.  Taking the smaller of the two upper bounds yields the denominator $\max\{a,b\}$.
1787 \end{proof}
1788 
1789 The factor of two need not be sharp.  It is uniform, and it makes the role of the evidence mass explicit.  No uniform estimate of this form remains useful when both masses approach zero.
1790 
1791 \subsection{Defect in the observation intertwining law}
1792 Let $L_{\fine}$ and $L_{\coarse}$ be positive linear observation operators, and choose a scale $s>0$.  On a finite simplex, we write $\Nrm(L)$ for normalization with respect to the total mass $\one^T$.  Define
1793 \begin{equation}\label{eq:observation-linear-defect}
1794   R_L
1795   =
1796   CL_{\fine}-sL_{\coarse}C.
1797 \end{equation}
1798 
1799 \begin{theorem}[Approximate normalized transfer naturality]\label{thm:approx-normalized-transfer}
1800 On finite simplices with block sum $C$, let
1801 \[
1802   u=CL_{\fine}\mu,
1803   \qquad
1804   v=sL_{\coarse}C\mu.
1805 \]
1806 If $u$ and $v$ are nonnegative with positive masses, then
1807 \begin{equation}\label{eq:approx-normalized-transfer}
1808   \boxed{
1809   \norm{
1810   C\Nrm(L_{\fine})(\mu)
1811   -
1812   \Nrm(L_{\coarse})(C\mu)
1813   }_1
1814   \leq
1815   \frac{2\norm{R_L\mu}_1}
1816   {\max\{\one^Tu,\one^Tv\}}.}
1817 \end{equation}
1818 \end{theorem}
1819 
1820 \begin{proof}
1821 Because $C$ preserves total mass,
1822 \[
1823   C\Nrm(L_{\fine})(\mu)
1824   =\frac{u}{\one^Tu}.
1825 \]
1826 Multiplication of $L_{\coarse}$ by $s$ does not change its normalized output, so
1827 \[
1828   \Nrm(L_{\coarse})(C\mu)
1829   =\frac{v}{\one^Tv}.
1830 \]
1831 Now $u-v=R_L\mu$, and Lemma~\ref{lem:normalization-perturbation} applies.
1832 \end{proof}
1833 
1834 \subsection{Likelihood variation inside a fiber}
1835 For diagonal likelihood operators, write
1836 \begin{equation}\label{eq:likelihood-residual}
1837   \lambda^{\fine}
1838   =
1839   sC^T\lambda^{\coarse}+r_{\lambda}.
1840 \end{equation}
1841 Then
1842 \[
1843   R_L\mu=C(r_{\lambda}\odot\mu).
1844 \]
1845 Since block sum aggregation is an $\ell^1$ contraction on signed vectors,
1846 \begin{equation}\label{eq:likelihood-residual-l1}
1847   \norm{R_L\mu}_1
1848   \leq
1849   \sum_i\mu_i\abs{r_{\lambda,i}}
1850   \leq
1851   \norm{r_{\lambda}}_{\infty}.
1852 \end{equation}
1853 
1854 \begin{corollary}[Bayesian refinement defect]\label{cor:bayes-refinement-defect}
1855 Let
1856 \[
1857   z_{\fine}=Z_{\lambda^{\fine}}(\mu),
1858   \qquad
1859   z_{\coarse}=sZ_{\lambda^{\coarse}}(C\mu).
1860 \]
1861 Then
1862 \begin{equation}\label{eq:bayes-refinement-defect}
1863   \boxed{
1864   \norm{
1865   C\Bup_{\lambda^{\fine}}(\mu)
1866   -
1867   \Bup_{\lambda^{\coarse}}(C\mu)
1868   }_1
1869   \leq
1870   \frac{2\sum_i\mu_i\abs{r_{\lambda,i}}}
1871   {\max\{z_{\fine},z_{\coarse}\}}.}
1872 \end{equation}
1873 If both evidences are at least $z_*>0$, then
1874 \begin{equation}\label{eq:bayes-refinement-uniform}
1875   d_{\Bup}
1876   \leq
1877   \frac{2\norm{r_{\lambda}}_{\infty}}{z_*}.
1878 \end{equation}
1879 \end{corollary}
1880 
1881 \begin{proof}
1882 Combine \Cref{thm:approx-normalized-transfer} with \eqref{eq:likelihood-residual-l1}.
1883 \end{proof}
1884 
1885 \begin{definition}[Fiber oscillation]\label{def:fiber-oscillation}
1886 For a fine likelihood, define
1887 \[
1888   \osc_a(\lambda^{\fine})
1889   =
1890   \max_{i:\pi(i)=a}\lambda_i^{\fine}
1891   -
1892   \min_{i:\pi(i)=a}\lambda_i^{\fine},
1893 \]
1894 and
1895 \[
1896   \osc_C(\lambda^{\fine})
1897   =\max_a\osc_a(\lambda^{\fine}).
1898 \]
1899 \end{definition}
1900 
1901 When $q\geq2$ and commutation is required for every interior prior, exact Bayesian refinement is equivalent to $\osc_C(\lambda^{\fine})=0$ once the coarse likelihood and global scale have been chosen consistently.  In an approximate model, $\lambda_a^{\coarse}$ may be the midpoint, the ordinary mean, or a mean chosen to reflect the task on fiber $a$.  The midpoint minimizes the largest absolute residual in that fiber and gives
1902 \[
1903   \norm{r_{\lambda}}_{\infty}
1904   =\frac12\osc_C(\lambda^{\fine})
1905 \]
1906 when $s=1$.
1907 
1908 \subsection{Lipschitz behavior of the Bayesian update}
1909 
1910 \begin{proposition}[Lipschitz bound under an evidence floor]\label{prop:bayes-lipschitz}
1911 Let $\lambda\in\R_{>0}^q$.  If
1912 \[
1913   Z_{\lambda}(\mu),Z_{\lambda}(\nu)\geq z_*>0,
1914 \]
1915 then
1916 \begin{equation}\label{eq:bayes-lipschitz}
1917   \boxed{
1918   \norm{\Bup_{\lambda}(\mu)-\Bup_{\lambda}(\nu)}_1
1919   \leq
1920   \frac{2\norm{\lambda}_{\infty}}{z_*}
1921   \norm{\mu-\nu}_1.}
1922 \end{equation}
1923 \end{proposition}
1924 
1925 \begin{proof}
1926 Apply Lemma~\ref{lem:normalization-perturbation} to
1927 \[
1928   u=\lambda\odot\mu,
1929   \qquad
1930   v=\lambda\odot\nu.
1931 \]
1932 Then
1933 \[
1934   \norm{u-v}_1
1935   \leq\norm{\lambda}_{\infty}\norm{\mu-\nu}_1.
1936 \]
1937 \end{proof}
1938 
1939 \begin{remark}[Scale invariance of the bound]
1940 Replacing $\lambda$ by $s\lambda$ multiplies both $\norm{\lambda}_{\infty}$ and $z_*$ by $s$, leaving the ratio unchanged.
1941 \end{remark}
1942 
1943 \begin{remark}[Local versus global constants]
1944 The constant in \eqref{eq:bayes-lipschitz} is valid over the whole evidence region under consideration.  Along a particular trajectory, the Jacobian in \eqref{eq:bayes-differential} may give a much smaller local factor.  The recursive theorem below can use either bound, provided the chosen constant is certified on the states being considered.
1945 \end{remark}
1946 
1947 \section{Certified approximate coarse filtering}\label{sec:filter-cert}
1948 
1949 \subsection{Prediction and update errors}
1950 Let the fine recursion be
1951 \begin{equation}\label{eq:fine-filter-recursion}
1952   \widehat\mu_k^{\fine}
1953   =\Phi_k^{\fine}(\mu_{k-1}^{\fine}),
1954   \qquad
1955   \mu_k^{\fine}
1956   =\Bup_k^{\fine}(\widehat\mu_k^{\fine}),
1957 \end{equation}
1958 and the coarse recursion be
1959 \begin{equation}\label{eq:coarse-filter-recursion}
1960   \widehat\mu_k^{\coarse}
1961   =\Phi_k^{\coarse}(\mu_{k-1}^{\coarse}),
1962   \qquad
1963   \mu_k^{\coarse}
1964   =\Bup_k^{\coarse}(\widehat\mu_k^{\coarse}).
1965 \end{equation}
1966 Define the discrepancies before and after each update by
1967 \begin{equation}\label{eq:pre-post-errors}
1968   e_k^-
1969   =\norm{C\widehat\mu_k^{\fine}-\widehat\mu_k^{\coarse}}_1,
1970   \qquad
1971   e_k^+
1972   =\norm{C\mu_k^{\fine}-\mu_k^{\coarse}}_1.
1973 \end{equation}
1974 At the initial time, set
1975 \[
1976   e_0^+=\norm{C\mu_0^{\fine}-\mu_0^{\coarse}}_1.
1977 \]
1978 
1979 Assume the coarse prediction is Lipschitz with constant $L_{\Phi,k}$ and that its quotient defect is bounded by
1980 \begin{equation}\label{eq:prediction-map-defect}
1981   \sup_{\mu}
1982   \norm{C\Phi_k^{\fine}(\mu)-\Phi_k^{\coarse}(C\mu)}_1
1983   \leq d_{\Phi,k}.
1984 \end{equation}
1985 Assume the coarse update is Lipschitz with constant $L_{\Bup,k}$ and that its direct refinement defect is bounded by
1986 \begin{equation}\label{eq:update-map-defect}
1987   \sup_{\mu}
1988   \norm{C\Bup_k^{\fine}(\mu)-\Bup_k^{\coarse}(C\mu)}_1
1989   \leq d_{\Bup,k}.
1990 \end{equation}
1991 The suprema in \eqref{eq:prediction-map-defect} and \eqref{eq:update-map-defect} are taken over a declared admissible set that contains both recursions.
1992 
1993 \begin{lemma}[Prediction bound for one step]\label{lem:prediction-step}
1994 Under \eqref{eq:prediction-map-defect},
1995 \begin{equation}\label{eq:prediction-step}
1996   e_k^-
1997   \leq
1998   L_{\Phi,k}e_{k-1}^+
1999   +d_{\Phi,k}.
2000 \end{equation}
2001 \end{lemma}
2002 
2003 \begin{proof}
2004 Insert and subtract $\Phi_k^{\coarse}(C\mu_{k-1}^{\fine})$.  The quotient defect bounds one difference, while the Lipschitz estimate bounds the other by $L_{\Phi,k}e_{k-1}^+$.
2005 \end{proof}
2006 
2007 \begin{lemma}[Update bound for one step]\label{lem:update-step}
2008 Under \eqref{eq:update-map-defect},
2009 \begin{equation}\label{eq:update-step}
2010   e_k^+
2011   \leq
2012   L_{\Bup,k}e_k^-
2013   +d_{\Bup,k}.
2014 \end{equation}
2015 \end{lemma}
2016 
2017 \begin{proof}
2018 Insert and subtract $\Bup_k^{\coarse}(C\widehat\mu_k^{\fine})$.  The direct refinement defect bounds one difference, while the Lipschitz estimate bounds the other by $L_{\Bup,k}e_k^-$.
2019 \end{proof}
2020 
2021 Combining the two inequalities gives
2022 \begin{equation}\label{eq:combined-step}
2023   e_k^+
2024   \leq
2025   \alpha_ke_{k-1}^++\beta_k,
2026 \end{equation}
2027 where
2028 \begin{equation}\label{eq:alpha-beta}
2029   \alpha_k=L_{\Bup,k}L_{\Phi,k},
2030   \qquad
2031   \beta_k=L_{\Bup,k}d_{\Phi,k}+d_{\Bup,k}.
2032 \end{equation}
2033 
2034 \subsection{Closed posterior certificate}
2035 
2036 \begin{theorem}[Certified approximate filtering]\label{thm:posterior-certificate}
2037 Under the assumptions above,
2038 \begin{equation}\label{eq:posterior-certificate}
2039   \boxed{
2040   e_N^+
2041   \leq
2042   \left(\prod_{j=1}^N\alpha_j\right)e_0^+
2043   +
2044   \sum_{k=1}^N
2045   \left(\prod_{j=k+1}^N\alpha_j\right)
2046   \beta_k.}
2047 \end{equation}
2048 Here and below, an empty product equals one.  With exact initial aggregation $e_0^+=0$,
2049 \begin{equation}\label{eq:posterior-certificate-exact-initial}
2050   e_N^+
2051   \leq
2052   \sum_{k=1}^N
2053   \left(\prod_{j=k+1}^N\alpha_j\right)
2054   \bigl(L_{\Bup,k}d_{\Phi,k}+d_{\Bup,k}\bigr).
2055 \end{equation}
2056 \end{theorem}
2057 
2058 \begin{proof}
2059 Repeated substitution in the affine recurrence \eqref{eq:combined-step} gives the stated formula.  The initial error is multiplied by every later amplification factor.  A defect introduced at step $k$ is multiplied only by the factors from steps $k+1$ through $N$.
2060 \end{proof}
2061 
2062 \begin{corollary}[Uniform stability regimes]\label{cor:filter-stability-regimes}
2063 Suppose $0\leq\alpha_k\leq\alpha$ and $0\leq\beta_k\leq\beta$.
2064 
2065 \begin{enumerate}[label=(\roman*),leftmargin=2.8em]
2066 \item If $0\leq\alpha<1$, then
2067 \begin{equation}\label{eq:stable-filter-bound}
2068   e_N^+
2069   \leq
2070   \alpha^Ne_0^+
2071   +
2072   \beta\frac{1-\alpha^N}{1-\alpha},
2073   \qquad
2074   \limsup_{N\to\infty}e_N^+
2075   \leq\frac{\beta}{1-\alpha}.
2076 \end{equation}
2077 
2078 \item If $\alpha=1$, then
2079 \[
2080   e_N^+\leq e_0^++N\beta.
2081 \]
2082 
2083 \item If $\alpha>1$, then
2084 \[
2085   e_N^+
2086   \leq
2087   \alpha^Ne_0^+
2088   +
2089   \beta\frac{\alpha^N-1}{\alpha-1}.
2090 \]
2091 \end{enumerate}
2092 \end{corollary}
2093 
2094 \begin{proof}
2095 Bound the products in \eqref{eq:posterior-certificate} by powers of $\alpha$ and sum the geometric series.
2096 \end{proof}
2097 
2098 This trichotomy describes the certificate, not necessarily the observed filter.  A stable trajectory can still come with a bound for which $\alpha>1$.  In that case, the chosen global norm estimates simply fail to certify contraction.
2099 
2100 \subsection{From interaction coefficients to posterior error}
2101 For a flow interval of length $T_k$, suppose the fine trajectory remains in a ball of radius $R_k$ and the coarse field has Lipschitz constant $L_{F,k}$.  By Corollary~\ref{cor:coefficient-flow-certificate}, one may take
2102 \begin{equation}\label{eq:dphi-coefficient}
2103   d_{\Phi,k}
2104   =
2105   \mathfrak d_{F,k}(R_k)
2106   \frac{e^{L_{F,k}T_k}-1}{L_{F,k}},
2107 \end{equation}
2108 with the continuous value $T_k\mathfrak d_{F,k}(R_k)$ at $L_{F,k}=0$.
2109 For the observation, Corollary~\ref{cor:bayes-refinement-defect} gives
2110 \begin{equation}\label{eq:dbayes-coefficient}
2111   d_{\Bup,k}
2112   \leq
2113   \frac{2\norm{r_{\lambda,k}}_{\infty}}{z_{*,k}}.
2114 \end{equation}
2115 Substituting these bounds into \eqref{eq:posterior-certificate-exact-initial} gives a certificate expressed directly in terms of interaction tensor defects and likelihood residuals within each fiber.
2116 
2117 \subsection{Observable and total variation consequences}
2118 
2119 \begin{corollary}[Coarse observable error]\label{cor:observable-error}
2120 For any coarse observable $g\in\R^q$,
2121 \[
2122   \abs{g^TC\mu_N^{\fine}-g^T\mu_N^{\coarse}}
2123   \leq
2124   \norm{g}_{\infty}e_N^+.
2125 \]
2126 In particular,
2127 \[
2128   \TV(C\mu_N^{\fine},\mu_N^{\coarse})
2129   =\frac12e_N^+.
2130 \]
2131 \end{corollary}
2132 
2133 \begin{proof}
2134 Apply the duality between $\ell^1$ and $\ell^\infty$.  The total variation identity is standard for finite probability vectors.
2135 \end{proof}
2136 
2137 \subsection{Sequential evidence error}
2138 For brevity, write $\lambda_k^{\fine}=\lambda_{y_k}^{\fine}$ and $\lambda_k^{\coarse}=\lambda_{y_k}^{\coarse}$, and decompose
2139 \begin{equation}\label{eq:step-likelihood-residual}
2140   \lambda_k^{\fine}
2141   =s_kC^T\lambda_k^{\coarse}+r_k.
2142 \end{equation}
2143 This gives
2144 \begin{align}
2145   z_k^{\fine}-s_kz_k^{\coarse}
2146   &=
2147   s_k(\lambda_k^{\coarse})^T
2148   (C\widehat\mu_k^{\fine}-\widehat\mu_k^{\coarse})
2149   +r_k^T\widehat\mu_k^{\fine}.
2150   \label{eq:evidence-error-identity}
2151 \end{align}
2152 
2153 \begin{proposition}[Evidence certificate for one step]\label{prop:evidence-error}
2154 Assume $\widehat\mu_k^{\fine}$ is a probability vector.  Under \eqref{eq:step-likelihood-residual},
2155 \begin{equation}\label{eq:evidence-error-bound}
2156   \boxed{
2157   \abs{z_k^{\fine}-s_kz_k^{\coarse}}
2158   \leq
2159   s_k\norm{\lambda_k^{\coarse}}_{\infty}e_k^-
2160   +\norm{r_k}_{\infty}.}
2161 \end{equation}
2162 If $z_k^{\fine},s_kz_k^{\coarse}\geq z_{*,k}>0$, then
2163 \begin{equation}\label{eq:log-evidence-step}
2164   \boxed{
2165   \abs{
2166   \log z_k^{\fine}
2167   -\log(s_kz_k^{\coarse})}
2168   \leq
2169   \frac{
2170   s_k\norm{\lambda_k^{\coarse}}_{\infty}e_k^-
2171   +\norm{r_k}_{\infty}
2172   }{z_{*,k}}.}
2173 \end{equation}
2174 \end{proposition}
2175 
2176 \begin{proof}
2177 Apply Hölder's inequality to \eqref{eq:evidence-error-identity}.  Since $\widehat\mu_k^{\fine}$ is a probability vector, $\norm{\widehat\mu_k^{\fine}}_1=1$.  For the logarithms, the mean value theorem gives
2178 \[
2179   \abs{\log a-\log b}
2180   \leq\frac{\abs{a-b}}{\min\{a,b\}}.
2181 \]
2182 \end{proof}
2183 
2184 \begin{corollary}[Certificate for cumulative log evidence]\label{cor:cumulative-evidence}
2185 Let $Z_{1:N}^{\fine}=\prod_kz_k^{\fine}$ and $Z_{1:N}^{\coarse}=\prod_kz_k^{\coarse}$.  Then
2186 \begin{equation}\label{eq:cumulative-evidence}
2187   \boxed{
2188   \abs{
2189   \log Z_{1:N}^{\fine}
2190   -\sum_{k=1}^N\log s_k
2191   -\log Z_{1:N}^{\coarse}}
2192   \leq
2193   \sum_{k=1}^N
2194   \frac{
2195   s_k\norm{\lambda_k^{\coarse}}_{\infty}e_k^-
2196   +\norm{r_k}_{\infty}
2197   }{z_{*,k}}.}
2198 \end{equation}
2199 \end{corollary}
2200 
2201 \begin{proof}
2202 Sum \eqref{eq:log-evidence-step} over the observation sequence.
2203 \end{proof}
2204 
2205 \section{Hyperedge pruning and certified compression}\label{sec:pruning}
2206 
2207 Exact quotienting and sparse representation are distinct.  A system may have an exact coarse quotient while its fine interaction tensor is dense.  Conversely, a sparse fine tensor can fail to descend.  This section introduces a separate residual for deleted hyperedges and then combines it with the quotient defect.
2208 
2209 \subsection{Hilbert Schmidt tensor blocks}
2210 Let $X$ and $Y$ be finite dimensional Hilbert spaces with orthogonal packet projectors $(P_j)$ and $(Q_i)$.  Identify a multilinear Hilbert Schmidt tensor of order $m$ with the Hilbert Schmidt linear map
2211 \[
2212   \widehat A:X^{\otimes m}\to Y.
2213 \]
2214 Its packet block is
2215 \[
2216   \widehat A_{i\mid j_1,\ldots,j_m}
2217   =
2218   Q_i\widehat A(P_{j_1}\otimes\cdots\otimes P_{j_m}).
2219 \]
2220 
2221 \begin{theorem}[Orthogonal hyperedge Parseval identity]\label{thm:hyperedge-parseval}
2222 Distinct packet blocks are orthogonal in Hilbert Schmidt inner product, and
2223 \begin{equation}\label{eq:hyperedge-parseval}
2224   \boxed{
2225   \norm{\widehat A}_{\mathrm{HS}}^2
2226   =
2227   \sum_{i,j_1,\ldots,j_m}
2228   \norm{\widehat A_{i\mid j_1,\ldots,j_m}}_{\mathrm{HS}}^2.}
2229 \end{equation}
2230 \end{theorem}
2231 
2232 \begin{proof}
2233 The tensor product input projectors
2234 \[
2235   P_{j_1}\otimes\cdots\otimes P_{j_m}
2236 \]
2237 are mutually orthogonal, as are the output projectors $Q_i$.  For two distinct block labels, orthogonality on at least one side forces the Hilbert Schmidt inner product to vanish.  Exact reconstruction is therefore an orthogonal sum, and Pythagoras gives \eqref{eq:hyperedge-parseval}.
2238 \end{proof}
2239 
2240 \subsection{Retained field and named residual}
2241 Choose a retained hyperedge set $\cE_m^{\mathrm{keep}}$.  Define
2242 \begin{equation}\label{eq:pruned-tensor}
2243   \widehat A_m^{\mathrm{keep}}
2244   =
2245   \sum_{h\in\cE_m^{\mathrm{keep}}}
2246   \widehat A_{m,h},
2247   \qquad
2248   R_m
2249   =
2250   \widehat A_m-\widehat A_m^{\mathrm{keep}}.
2251 \end{equation}
2252 
2253 \begin{corollary}[Exact pruning residual]\label{cor:pruning-residual}
2254 Under the hypotheses of \Cref{thm:hyperedge-parseval},
2255 \begin{equation}\label{eq:pruning-residual}
2256   \boxed{
2257   \norm{R_m}_{\mathrm{HS}}^2
2258   =
2259   \sum_{h\notin\cE_m^{\mathrm{keep}}}
2260   \norm{\widehat A_{m,h}}_{\mathrm{HS}}^2.}
2261 \end{equation}
2262 \end{corollary}
2263 
2264 \begin{proof}
2265 The retained and omitted block families are orthogonal by \Cref{thm:hyperedge-parseval}.  Since $R_m$ is exactly the sum of the omitted blocks, Pythagoras gives \eqref{eq:pruning-residual}.
2266 \end{proof}
2267 
2268 The residual is a tensor identity, not merely an empirical estimate.  Its induced state error is obtained by applying the relevant multilinear norm inequality.
2269 
2270 \subsection{\texorpdfstring{Retaining the $k$ largest energy hyperedges}{Retaining the k largest energy hyperedges}}
2271 Let
2272 \[
2273   e_h=\norm{\widehat A_{m,h}}_{\mathrm{HS}}^2
2274 \]
2275 and order the energies
2276 \[
2277   e_{h_1}\geq e_{h_2}\geq\cdots\geq e_{h_M}.
2278 \]
2279 
2280 \begin{theorem}[Optimal retention of $k$ orthogonal hyperedges]\label{thm:best-k-hyperedges}
2281 Among all retained sets of cardinality $k$, the set
2282 \[
2283   \{h_1,\ldots,h_k\}
2284 \]
2285 minimizes the Hilbert Schmidt residual.  The minimum is
2286 \begin{equation}\label{eq:best-k-residual}
2287   \boxed{
2288   \min_{|\cE|=k}
2289   \norm{R_m(\cE)}_{\mathrm{HS}}^2
2290   =
2291   \sum_{r>k}e_{h_r}.}
2292 \end{equation}
2293 \end{theorem}
2294 
2295 \begin{proof}
2296 By \eqref{eq:pruning-residual}, minimizing the omitted energy is equivalent to maximizing the retained energy.  For a fixed cardinality, this is achieved by selecting the $k$ largest nonnegative terms.
2297 \end{proof}
2298 
2299 \begin{boundary}[Overlapping blocks and structured costs]
2300 The theorem above uses orthogonal block energy and assigns the same unit cost to every retained hyperedge.  With overlapping dictionaries, shared low rank factors, communication costs, or group constraints that preserve structure, ranking individual energies need not be optimal.
2301 \end{boundary}
2302 
2303 \subsection{Hyperedge entropy}
2304 For nonzero total interaction energy, define
2305 \[
2306   p_h=\frac{e_h}{\sum_{\eta}e_{\eta}},
2307   \qquad
2308   H_m=-\sum_hp_h\log p_h,
2309 \]
2310 and the effective hyperedge count
2311 \[
2312   N_{\mathrm{eff},m}=e^{H_m}.
2313 \]
2314 Low entropy means that the interaction energy is concentrated in a small set of resolved hyperedges.  High entropy means that pruning by a fixed cutoff is less promising in the chosen resolution.  The quantity deliberately depends on the resolution.  It evaluates a representation rather than an invariant of the unresolved tensor alone.
2315 
2316 \subsection{Combined quotient and pruning defect}
2317 Let the full fine coefficient split as
2318 \[
2319   A_m^{\fine}
2320   =A_m^{\mathrm{keep}}+R_m^{\fine}.
2321 \]
2322 For a proposed coarse coefficient $A_m^{\coarse}$, define
2323 \[
2324   \Delta_m^{\mathrm{total}}
2325   =CA_m^{\fine}-A_m^{\coarse}C^{\times m}.
2326 \]
2327 Then
2328 \begin{align}
2329   \Delta_m^{\mathrm{total}}
2330   &=
2331   \underbrace{CA_m^{\mathrm{keep}}-A_m^{\coarse}C^{\times m}}
2332   _{\Delta_m^{\mathrm{keep}}}
2333   +CR_m^{\fine}.
2334   \label{eq:combined-pruning-quotient}
2335 \end{align}
2336 Therefore
2337 \begin{equation}\label{eq:combined-pruning-bound}
2338   \boxed{
2339   \norm{\Delta_m^{\mathrm{total}}}
2340   \leq
2341   \norm{\Delta_m^{\mathrm{keep}}}
2342   +\norm{C}\norm{R_m^{\fine}}.}
2343 \end{equation}
2344 
2345 For analytic fields on a ball of radius $R$,
2346 \begin{equation}\label{eq:combined-field-majorant}
2347   \mathfrak d_F(R)
2348   \leq
2349   \sum_m
2350   \norm{\Delta_m^{\mathrm{keep}}}R^m
2351   +
2352   \norm{C}\sum_m\norm{R_m^{\fine}}R^m.
2353 \end{equation}
2354 The flow and posterior bounds use this combined quantity.  A pruning decision is justified only when the omitted part remains explicit and is carried through the later estimates like any other model error.
2355 
2356 \subsection{A budgeted design problem}
2357 These terms lead to a practical design problem.  The objective below records the tradeoff without claiming that it can always be solved efficiently:
2358 \begin{equation}\label{eq:budgeted-design}
2359   \min_{C,\{A_m^{\coarse}\},\{\cE_m^{\mathrm{keep}}\}}
2360   \left[
2361   \sum_mw_m\norm{\Delta_m^{\mathrm{total}}}
2362   +\lambda\sum_m\operatorname{Cost}(\cE_m^{\mathrm{keep}})
2363   +\gamma\sum_y d_{\Bup,y}
2364   \right].
2365 \end{equation}
2366 The minimization is subject to any required positivity, mass, rank, and interpretability conditions on $C$.  The three terms measure failure of dynamical closure, computational cost, and loss of observational information.  Their weights and the admissible packet families must be chosen for the application.
2367 
2368 \section{Algorithms and finite certificates}\label{sec:algorithms}
2369 
2370 \input{tables/verification_macros.tex}
2371 
2372 Every finite criterion in the paper can be checked directly from coordinates.  The reference code uses dense arrays so that each contraction can be inspected.  A sparse implementation would loop only over the active hyperedges.
2373 
2374 \subsection{Exact test for every interaction order}
2375 For a tensor of order $m$ in scalar packet coordinates, compute the aggregated coefficient array
2376 \[
2377   g^{(m)}_{a;j_1,\ldots,j_m}
2378   =
2379   \sum_{i:\pi(i)=a}
2380   a^{\fine,(m)}_{i;j_1,\ldots,j_m}.
2381 \]
2382 For every coarse input tuple $\mathbf b=(b_1,\ldots,b_m)$, define the fine fiber product
2383 \[
2384   \cG_{\mathbf b}
2385   =
2386   \{(j_1,\ldots,j_m):\pi(j_r)=b_r\text{ for all }r\}.
2387 \]
2388 
2389 \begin{algorithm}[Exact packet coordinate lumpability test]\label{alg:exact-lumpability}
2390 For every coarse output $a$ and coarse input tuple $\mathbf b$:
2391 \begin{enumerate}[label=\arabic*.,leftmargin=2.3em]
2392 \item Evaluate $g^{(m)}_{a;\mathbf j}$ for all $\mathbf j\in\cG_{\mathbf b}$.
2393 \item In exact arithmetic, test whether all values are equal.  With floating point input, compare their spread with a declared tolerance and record that tolerance.
2394 \item When the exact test passes, assign the common value to $a^{\coarse,(m)}_{a;\mathbf b}$.  When only the numerical tolerance is met, assign a declared representative such as the mean or the midrange.
2395 \item Otherwise, return a witness $(a,\mathbf j,\mathbf j')$ with unequal aggregate coefficients.
2396 \end{enumerate}
2397 \end{algorithm}
2398 
2399 A dense implementation requires $O(n^{m+1})$ arithmetic to aggregate a tensor of order $m$ on $n$ states.  If the fine tensor is stored as $E_m$ active scalar hyperedges, the aggregation pass is $O(E_m)$, followed by grouping and consistency checks over the represented coarse incidences.
2400 
2401 \begin{proposition}[Correctness in exact arithmetic]\label{prop:algorithm-correctness}
2402 With exact arithmetic, Algorithm~\ref{alg:exact-lumpability} accepts exactly the tensors that descend through the given block sum aggregation.  On acceptance, it returns the unique coarse tensor.  A positive numerical tolerance produces an approximate certificate rather than an exact algebraic decision.
2403 \end{proposition}
2404 
2405 \begin{proof}
2406 The claim follows directly from \Cref{thm:packet-coordinate-criterion}.
2407 \end{proof}
2408 
2409 \subsection{Approximate coarse coefficients}
2410 When exact fiber constancy fails, the approximation depends on a chosen loss and, when relevant, a task distribution.  Under scalar squared loss with nonnegative weights $w_{\mathbf j}$ on a fiber product, assume their total weight is positive.  The least squares representative is
2411 \begin{equation}\label{eq:least-squares-coarse-coeff}
2412   a^{\coarse,\mathrm{LS}}_{a;\mathbf b}
2413   =
2414   \frac{
2415   \sum_{\mathbf j\in\cG_{\mathbf b}}
2416   w_{\mathbf j}g_{a;\mathbf j}
2417   }{
2418   \sum_{\mathbf j\in\cG_{\mathbf b}}w_{\mathbf j}
2419   }.
2420 \end{equation}
2421 For scalar worst case loss, a minimax representative is the midrange
2422 \begin{equation}\label{eq:minimax-coarse-coeff}
2423   a^{\coarse,\infty}_{a;\mathbf b}
2424   =
2425   \frac12
2426   \left(
2427   \max_{\mathbf j\in\cG_{\mathbf b}}g_{a;\mathbf j}
2428   +
2429   \min_{\mathbf j\in\cG_{\mathbf b}}g_{a;\mathbf j}
2430   \right),
2431 \end{equation}
2432 with optimal scalar fiber error
2433 \begin{equation}\label{eq:minimax-fiber-error}
2434   \varepsilon^{\infty}_{a;\mathbf b}
2435   =
2436   \frac12
2437   \left(
2438   \max_{\mathbf j\in\cG_{\mathbf b}}g_{a;\mathbf j}
2439   -
2440   \min_{\mathbf j\in\cG_{\mathbf b}}g_{a;\mathbf j}
2441   \right).
2442 \end{equation}
2443 These formulas are local coefficient summaries.  A global operator norm can couple several fibers and may require a joint optimization.
2444 
2445 \subsection{Certificate pipeline}
2446 The reference implementation follows the dependency chain
2447 \[
2448 \begin{array}{c}
2449 \text{interaction tensors}\\
2450 \downarrow\\
2451 \text{fiber constancy and kernel tests}\\
2452 \downarrow\\
2453 \text{coarse tensor and }\Delta_m\\
2454 \downarrow\\
2455 \mathfrak d_F(R)\text{ and }d_{\Phi,k}\\
2456 \downarrow\\
2457  r_{\lambda,k},z_{*,k},d_{\Bup,k}\\
2458 \downarrow\\
2459  e_k^-,e_k^+,\Err_N.
2460 \end{array}
2461 \]
2462 Each numerical record stores the theorem identifier, the inputs, the residual, the tolerance, and the outcome.  The suite includes examples built to satisfy the exact identities and perturbations built to violate them.
2463 
2464 \subsection{Deterministic generator for exact examples}
2465 A coarse tensor can be lifted to a fine tensor by
2466 \begin{equation}\label{eq:exact-instance-generator}
2467   A_m^{\fine}
2468   =
2469   EA_m^{\coarse}C^{\times m}+K_m,
2470   \qquad
2471   CK_m=0.
2472 \end{equation}
2473 The term $K_m$ can contain complicated hidden interactions without changing the quotient.  This makes the construction useful for testing because the correct coarse tensor is known before any numerical calculation.
2474 
2475 \begin{proposition}[Correctness of the exact example generator]\label{prop:exact-instance-generator}
2476 Every tensor constructed by \eqref{eq:exact-instance-generator} satisfies
2477 \[
2478   CA_m^{\fine}=A_m^{\coarse}C^{\times m}.
2479 \]
2480 Conversely, every exact fine tensor admits such a representation for every right inverse $E$, with
2481 \[
2482   K_m=A_m^{\fine}-EA_m^{\coarse}C^{\times m}.
2483 \]
2484 \end{proposition}
2485 
2486 \begin{proof}
2487 The forward identity follows from $CE=I$ and $CK_m=0$.  For the converse, exact descent gives
2488 \[
2489   CK_m
2490   =CA_m^{\fine}-CEA_m^{\coarse}C^{\times m}=0.
2491 \]
2492 \end{proof}
2493 
2494 \subsection{Reference implementation}
2495 The accompanying source tree contains:
2496 \begin{center}
2497 \small
2498 \begin{tabular}{@{}p{0.31\linewidth}p{0.61\linewidth}@{}}
2499 \toprule
2500 Path & Purpose\\
2501 \midrule
2502 \texttt{src/priq/core.py} & aggregation maps, tensor actions, exact lifts, quotient defects, packet coordinate recovery, Bayesian updates, and a deterministic Runge Kutta integrator\\
2503 \texttt{tests/test\_priq.py} & unit tests for the theorems and negative controls\\
2504 \texttt{examples/}\allowbreak\texttt{run\_certificates.py} & deterministic tensor, flow, filtering, perturbation, pruning, figure, and ledger generation\\
2505 \texttt{results/verification.json} & certificate summary in JSON form\\
2506 \texttt{tables/test\_results.csv} & flat test ledger\\
2507 \texttt{figures/} & vector PDF figures and large PNG figures generated from the certificate run\\
2508 \bottomrule
2509 \end{tabular}
2510 \end{center}
2511 
2512 \begin{certificate}[Executed finite suite]\label{cert:executed-suite}
2513 The deterministic reference run completed \VerificationPassCount\ checks with \VerificationFailCount\ failures.  It covers quotient tensors of orders one through three, annihilation of hidden inputs, recovery in packet coordinates, negative controls, and the defect identity across several levels.  It also covers nonlinear prediction, Bayesian refinement, evidence aggregation, approximate flow and posterior bounds, perturbation scaling, and orthogonal hyperedge residuals.  The independent unit tests also pass in the build environment.
2514 \end{certificate}
2515 
2516 \input{tables/test_results.tex}
2517 
2518 \section{Worked quotient and filtering models}\label{sec:examples}
2519 
2520 \subsection{A nonlinear fine system with four states and a quotient with two states}
2521 Let
2522 \[
2523   x=(x_1,x_2,x_3,x_4)^T\in\Delta_4,
2524   \qquad
2525   C=
2526   \begin{bmatrix}
2527   1&1&0&0\\
2528   0&0&1&1
2529   \end{bmatrix},
2530 \]
2531 so
2532 \[
2533   z=Cx=(z_1,z_2)^T,
2534   \qquad
2535   z_1=x_1+x_2,
2536   \quad
2537   z_2=x_3+x_4.
2538 \]
2539 The coarse dynamics are
2540 \begin{equation}\label{eq:coarse-two-state}
2541   \dot z_1=-az_1+bz_2,
2542   \qquad
2543   \dot z_2=az_1-bz_2,
2544   \qquad a,b>0.
2545 \end{equation}
2546 Define nonlinear internal exchange channels
2547 \begin{align}
2548   h_{12}(x)
2549   &=
2550   (r_0+r_1x_3)x_2-(s_0+s_1x_4)x_1,
2551   \label{eq:h12}\\
2552   h_{34}(x)
2553   &=
2554   (u_0+u_1x_1)x_4-(v_0+v_1x_2)x_3,
2555   \label{eq:h34}
2556 \end{align}
2557 with all coefficients nonnegative.  The fine system is
2558 \begin{equation}\label{eq:fine-four-state}
2559 \begin{aligned}
2560   \dot x_1&=-ax_1+bx_3+h_{12}(x),\\
2561   \dot x_2&=-ax_2+bx_4-h_{12}(x),\\
2562   \dot x_3&= ax_1-bx_3+h_{34}(x),\\
2563   \dot x_4&= ax_2-bx_4-h_{34}(x).
2564 \end{aligned}
2565 \end{equation}
2566 
2567 \begin{proposition}[Exact quotient with nonlinear hidden exchange]\label{prop:hidden-exchange-quotient}
2568 The system \eqref{eq:fine-four-state} preserves total mass and the nonnegative orthant.  After aggregation by $C$, it satisfies the coarse system \eqref{eq:coarse-two-state} exactly.
2569 \end{proposition}
2570 
2571 \begin{proof}
2572 Summing the four equations gives zero.  To check positivity, set one coordinate equal to zero.  Every term entering its derivative is then nonnegative.  This holds for $x_1$, $x_2$, $x_3$, and $x_4$, so the nonnegative orthant is forward invariant.  Finally,
2573 \begin{align*}
2574   \dot x_1+\dot x_2
2575   &=-a(x_1+x_2)+b(x_3+x_4),\\
2576   \dot x_3+\dot x_4
2577   &=a(x_1+x_2)-b(x_3+x_4),
2578 \end{align*}
2579 so the internal nonlinear channels cancel under $C$ and the aggregate obeys \eqref{eq:coarse-two-state}.
2580 \end{proof}
2581 
2582 The quadratic terms in \eqref{eq:h12} and \eqref{eq:h34} are genuine fine hyperedges.  Their output vectors lie in the internal directions of the two fibers,
2583 \[
2584   (1,-1,0,0)^T,
2585   \qquad
2586   (0,0,1,-1)^T,
2587 \]
2588 which are contained in $\Ker C$.  They alter the fine distribution while remaining invisible to the coarse state.  \Cref{fig:exact-quotient-flow} shows this distinction.  The two fine states inside the first fiber move differently, yet their sum follows the coarse trajectory.
2589 
2590 \begin{figure}[H]
2591 \centering
2592 \includegraphics[width=0.82\linewidth]{exact_quotient_flow.pdf}
2593 \caption{Hidden fine motion under an exact quotient.  The microstates $x_1$ and $x_2$ evolve separately.  Their sum agrees with the coarse coordinate $z_1$ throughout the trajectory.  Sparse markers on the coarse curve make the overlap visible.}
2594 \label{fig:exact-quotient-flow}
2595 \end{figure}
2596 
2597 \subsection{Exact nonlinear filtering}
2598 Choose a coarse likelihood
2599 \[
2600   \lambda^{\coarse}=(\lambda_1,\lambda_2)^T>0
2601 \]
2602 and lift it by
2603 \[
2604   \lambda^{\fine}=C^T\lambda^{\coarse}
2605   =(\lambda_1,\lambda_1,\lambda_2,\lambda_2)^T.
2606 \]
2607 By Proposition~\ref{prop:hidden-exchange-quotient} together with \Cref{thm:bayes-sufficient,thm:exact-cocycle}, every finite sequence of flow segments and such observations satisfies
2608 \[
2609   C\mu_k^{\fine}=\mu_k^{\coarse}.
2610 \]
2611 The fine filter may continue to distinguish $x_1$ from $x_2$ and $x_3$ from $x_4$, but those distinctions do not affect the exact coarse posterior.
2612 
2613 \begin{figure}[H]
2614 \centering
2615 \includegraphics[width=0.78\linewidth]{exact_filtering_residual.pdf}
2616 \caption{Aggregate posterior discrepancy after each prediction and update stage.  The residual remains at floating point scale and stays near double precision epsilon throughout the exact filtering sequence.}
2617 \label{fig:exact-filtering-residual}
2618 \end{figure}
2619 
2620 \begin{certificate}[Exact nonlinear filtering residuals]\label{cert:exact-filter-residuals}
2621 For the deterministic run with four states, the largest aggregate prediction residual was $\ExactPredictionResidual$.  The largest posterior residual after an update was $\ExactPosteriorResidual$, and the largest residual in a single evidence factor was $\ExactEvidenceResidual$.  The small nonzero values come from floating point integration and arithmetic in the finite example.
2622 \end{certificate}
2623 
2624 \subsection{A controlled dynamical violation}
2625 Add a directed transition between fine states
2626 \begin{equation}\label{eq:dynamical-perturbation}
2627   \varepsilon x_1
2628   \begin{bmatrix}
2629   -1\\0\\1\\0
2630   \end{bmatrix}
2631 \end{equation}
2632 to the fine vector field.  This perturbation preserves nonnegativity and total mass, but the added transfer between fibers is activated only by the mass in state $x_1$.  The aggregate field defect is
2633 \begin{equation}\label{eq:perturbation-field-defect}
2634   \delta_F(x)
2635   =
2636   \varepsilon x_1
2637   \begin{bmatrix}-1\\1\end{bmatrix},
2638 \end{equation}
2639 so, on the simplex,
2640 \begin{equation}\label{eq:perturbation-defect-bound}
2641   \norm{\delta_F(x)}_1
2642   =2\varepsilon x_1
2643   \leq2\varepsilon.
2644 \end{equation}
2645 The coarse Markov flow on two states is an $\ell^1$ contraction.  Over one interval, this gives
2646 \[
2647   \norm{C\Phi_T^{\fine,\varepsilon}(x)-\Phi_T^{\coarse}(Cx)}_1
2648   \leq2\varepsilon T.
2649 \]
2650 
2651 \begin{figure}[H]
2652 \centering
2653 \includegraphics[width=0.78\linewidth]{flow_perturbation_scaling.pdf}
2654 \caption{Aggregate flow error across a logarithmic perturbation sweep.  The measured discrepancy has first order scaling and remains below the certificate $2\varepsilon T$ from \eqref{eq:perturbation-defect-bound}.}
2655 \label{fig:flow-perturbation-scaling}
2656 \end{figure}
2657 
2658 \subsection{A controlled observation violation}
2659 Perturb the fine likelihood inside one fiber:
2660 \begin{equation}\label{eq:likelihood-perturbation-example}
2661   \lambda^{\fine,\eta}
2662   =
2663   C^T\lambda^{\coarse}
2664   +
2665   \eta(1,-1,0,0)^T.
2666 \end{equation}
2667 Assume $\abs{\eta}<\lambda_1^{\coarse}$, so every component of $\lambda^{\fine,\eta}$ remains strictly positive.  The fiber oscillation is $2\abs{\eta}$ and
2668 \[
2669   \norm{r_{\lambda}}_{\infty}=\abs{\eta}.
2670 \]
2671 If the evidence is bounded below by $z_*>0$, then
2672 \[
2673   d_{\Bup}
2674   \leq
2675   \frac{2\abs{\eta}}{z_*}.
2676 \]
2677 The sweep in \Cref{fig:bayesian-perturbation-scaling} varies only the observation model and isolates this effect at one interior prior.  The measured discrepancy is first order in $\eta$ and remains below the likelihood certificate.
2678 
2679 \begin{figure}[H]
2680 \centering
2681 \includegraphics[width=0.78\linewidth]{bayesian_perturbation_scaling.pdf}
2682 \caption{Posterior discrepancy caused by variation of the likelihood inside one aggregation fiber.  The error scales linearly with $\eta$ in the tested range and remains below the bound obtained from the normalization estimate.}
2683 \label{fig:bayesian-perturbation-scaling}
2684 \end{figure}
2685 
2686 Combining this observation perturbation with \eqref{eq:dynamical-perturbation} gives a posterior discrepancy that is first order in $(\varepsilon,\eta)$ over a fixed finite horizon.  \Cref{thm:posterior-certificate} supplies a conservative upper bound.
2687 
2688 \begin{figure}[H]
2689 \centering
2690 \includegraphics[width=0.78\linewidth]{posterior_error_certificate.pdf}
2691 \caption{Joint dynamical and likelihood perturbation sweep with $\varepsilon=\eta=\rho$.  For small $\rho$, the final posterior discrepancy grows linearly, while the recursive certificate remains above every measured value.}
2692 \label{fig:posterior-error-certificate}
2693 \end{figure}
2694 
2695 \begin{certificate}[Perturbation scaling]\label{cert:perturbation-scaling}
2696 The experiment used logarithmically spaced values with $\varepsilon=\eta$.  The fitted slope of the final posterior discrepancy was $\PerturbationSlope$.  Every measured error remained below its recursive certificate.
2697 \end{certificate}
2698 
2699 \subsection{Random exact tensors through order three}
2700 The implementation draws random coarse tensors of orders $m=1,2,3$, lifts them through \eqref{eq:exact-instance-generator}, and adds random hidden output tensors $K_m$ with $CK_m=0$.  It then checks:
2701 \[
2702   CA_m^{\fine}-A_m^{\coarse}C^{\times m}=0,
2703 \]
2704 annihilation of hidden inputs, constancy on packet fibers, and recovery of the original coarse tensor.  A single coefficient perturbation provides a negative control that must violate fiber constancy.
2705 
2706 This deterministic check confirms that the coordinate implementation and the invariant theorem return the same quotient.  It is not evidence for a general probabilistic law.
2707 
2708 \subsection{Concentrated hyperedge field}
2709 A synthetic interaction field with decreasing orthogonal block energies illustrates \Cref{thm:best-k-hyperedges}.  The computed hyperedge entropy is $\HyperedgeEntropyValue$, corresponding to an effective count of approximately $\EffectiveHyperedgeCount$ hyperedges despite a larger ambient support.  The ranked spectrum in \Cref{fig:hyperedge-energy-spectrum} shows why the field is compressible in this representation.
2710 
2711 \begin{figure}[H]
2712 \centering
2713 \begin{subfigure}[t]{0.48\linewidth}
2714 \centering
2715 \includegraphics[width=\linewidth]{hyperedge_energy_spectrum.pdf}
2716 \caption{Ranked normalized hyperedge energy.}
2717 \label{fig:hyperedge-energy-spectrum}
2718 \end{subfigure}
2719 \hfill
2720 \begin{subfigure}[t]{0.48\linewidth}
2721 \centering
2722 \includegraphics[width=\linewidth]{hyperedge_pruning.pdf}
2723 \caption{Residual after retaining the $k$ largest energy hyperedges.}
2724 \label{fig:hyperedge-pruning}
2725 \end{subfigure}
2726 \caption{Energy concentration and certified pruning in the synthetic interaction field.  A small leading set carries most of the energy, and the residual decreases to zero when the complete support is restored.}
2727 \label{fig:hyperedge-compression}
2728 \end{figure}
2729 
2730 \subsection{Interpretation for reaction networks}
2731 Consider a polynomial reaction system
2732 \[
2733   \dot x=S v(x),
2734   \qquad
2735   v_r(x)=k_r\prod_i x_i^{\alpha_{ir}}.
2736 \]
2737 Each reaction monomial contributes one interaction hyperedge to every species changed by the corresponding stoichiometric column.  The inputs are the reactants.  A block sum aggregation is exact precisely when the monomial coefficients, after summation over each output fiber, are constant on the product fibers specified by \Cref{thm:packet-coordinate-criterion}.  This gives a coefficient test at every interaction order.  Specialized reductions of reaction networks may also use stoichiometric, kinetic, or bisimulation structure \cite{Cardelli2015,Cardelli2017,BaezPollard2017}.
2738 
2739 \begin{boundary}[Physical interpretation]
2740 The quotient theorem is algebraic.  It does not by itself decide whether a packet represents a species class, spatial region, symmetry sector, latent state, or learned feature.  The scientific meaning of $C$ must come from the application and is not created by exact factorization alone.
2741 \end{boundary}
2742 
2743 \section{Relation to neighboring theories}\label{sec:relations}
2744 
2745 The results fit into a common finite quotient language, but each ingredient also belongs to a larger established theory.
2746 
2747 \subsection{Markov lumpability and Markov functions}
2748 For unary stochastic dynamics, the identity
2749 \[
2750   CQ_{\fine}=Q_{\coarse}C
2751 \]
2752 is a standard exact lumping condition.  It implies
2753 \[
2754   Ce^{tQ_{\fine}}=e^{tQ_{\coarse}}C
2755 \]
2756 and means that the coarse process is Markov for every fine initial distribution compatible with the quotient.  Classical treatments distinguish several forms of strong, ordinary, exact, and weak lumpability, depending on the admissible initial laws and the direction of the stochastic convention \cite{KemenySnell1960,RogersPitman1981,Buchholz1994}.  Here we fix one column vector convention and extend the coefficient criterion from unary edges to multilinear hyperedges.
2757 
2758 \subsection{Polynomial differential equivalence}
2759 Exact aggregation of polynomial vector fields has been developed through differential equivalences, bisimulations, and algorithms based on successive partition refinement, including applications to chemical reaction networks \cite{Cardelli2015,Cardelli2017}.  The universal factorization theorem here is elementary multilinear algebra once $C$ is fixed.  Its value is that it exposes every interaction order, makes the quotient condition independent of a particular polynomial syntax, and connects the same coefficient criterion to rooted tree composition and Bayesian updates.  We do not claim an algorithmic advantage over specialized methods based on partition refinement.
2760 
2761 \subsection{Hypergraphs, operads, and rooted trees}
2762 A multilinear packet block has several inputs and one output, so its incidence is naturally hypergraphic.  Rooted substitution of such blocks is operadic, and the derivative formula for composition is the multivariate Fa\`a di Bruno formula \cite{ConstantineSavits1996}.  Rooted trees also organize Butcher series and expansions for nonlinear time stepping \cite{Butcher1972,HairerLubichWanner2006}.  In this language, the quotient condition is a morphism condition.  It commutes with every generating operation and therefore with every rooted composite.
2763 
2764 \subsection{Sufficiency and comparison of experiments}
2765 A statistic is sufficient only relative to an observation family.  Classical sufficiency and comparison of experiments theory formalize when one representation preserves all information relevant to a chosen family \cite{HalmosSavage1949,Blackwell1953}.  The likelihood theorem in \Cref{sec:bayes} is a finite and explicit special case for deterministic aggregation and diagonal positive observation operators.  It characterizes exact commutation of the normalized update, not all possible notions of statistical equivalence.
2766 
2767 \subsection{Model reduction and nearly decomposable systems}
2768 Classical approaches to approximate aggregation use time scale separation, perturbation theory, balanced truncation, stochastic complementation, and related forms of model reduction \cite{SimonAndo1961,Meyer1989,Antoulas2005}.  The bounds used here follow the error through each stage.  Coefficient defects first create a field defect.  The Grönwall inequality carries that defect through the flow, and the normalization estimate carries it through each observation.  The resulting certificate can be conservative, but it keeps every source of error visible.
2769 
2770 \section{Claim boundaries and limitations}\label{sec:limitations}
2771 
2772 \begin{boundary}[Finite dimensional foundation]
2773 All quotient and coefficient theorems are proved in finite dimensional vector spaces.  The algebraic factorization extends to arbitrary vector spaces, but boundedness, convergence, measurability, domains of unbounded generators, and interchange of infinite sums require additional hypotheses in Banach, Hilbert, or measurable field settings.
2774 \end{boundary}
2775 
2776 \begin{boundary}[Local analytic dynamics]
2777 The flow theorems assume local existence and uniqueness.  The analytic coefficient theorem supplies a local coarse field, not a global invariant manifold or a global conjugacy.  Solutions may blow up, uniqueness may fail, and boundary singularities remain specific to the application.
2778 \end{boundary}
2779 
2780 \begin{boundary}[A prescribed coarse map]
2781 The principal theorems assume that $C$ is supplied.  They test and quantify whether a proposed aggregation is valid.  Discovering an optimal $C$ from data or from a large interaction tensor is a separate problem and can be combinatorial, nonconvex, or statistically nonidentifiable.
2782 \end{boundary}
2783 
2784 \begin{boundary}[Positivity and evidence floors]
2785 Bayesian normalization requires positive evidence.  The uniform perturbation bounds degrade as the evidence floor approaches zero.  This is not merely a weakness of the proof.  Small normalizing constants can amplify small absolute errors into large posterior changes.
2786 \end{boundary}
2787 
2788 \begin{boundary}[Norm dependence]
2789 Exact descent is algebraic and independent of the norm.  Best approximate coarse tensors, Lipschitz constants, pruning choices, and posterior certificates depend on the selected norms and task weights.  A useful certificate must state those choices.
2790 \end{boundary}
2791 
2792 \begin{boundary}[Numerical verification]
2793 The executed checks validate the finite implementation against the stated formulas and include negative controls.  They do not prove the theorems, establish asymptotic behavior outside the tested ranges, or demonstrate universal computational speedup.
2794 \end{boundary}
2795 
2796 \begin{boundary}[Historical priority]
2797 The central synthesis joins several established families of ideas.  The coefficient criterion at every interaction order follows from multilinear algebra on a quotient space.  Equivalent statements may already appear in neighboring theories of nonlinear aggregation under different notation.  Any claim of novelty should distinguish this elementary factorization from the combined treatment of interaction trees, Bayesian cocycles, and error certificates.  Such a claim also requires a dedicated literature review.
2798 \end{boundary}
2799 
2800 \section{Open theorem and algorithm program}\label{sec:open}
2801 
2802 \subsection{Coarsest exact joint partition}
2803 For a finite interaction family $(A_m^{\fine})_{m\leq M}$ and likelihood family $(\lambda_y^{\fine})_{y\in\cY}$, seek
2804 \begin{equation}\label{eq:coarsest-joint-partition}
2805   \min_{\pi}|I_{\coarse}|
2806 \end{equation}
2807 subject to
2808 \[
2809   CA_m^{\fine}=A_m^{\coarse}C^{\times m}
2810   \quad(m\leq M),
2811 \]
2812 and
2813 \[
2814   \lambda_y^{\fine}=s_yC^T\lambda_y^{\coarse}
2815   \quad(y\in\cY).
2816 \]
2817 The partition determined by the observations alone groups states with identical likelihood profiles.  The dynamical condition also depends on the partition because changing the partition changes the output aggregation.  A terminating refinement algorithm, together with a useful complexity bound, remains an open problem.
2818 
2819 \subsection{Approximate joint design}
2820 A finite approximate problem is
2821 \[
2822   \min_{C,\{A_m^{\coarse}\},\{\lambda_y^{\coarse}\}}
2823   \left[
2824   \sum_mw_m\norm{CA_m^{\fine}-A_m^{\coarse}C^{\times m}}
2825   +\sum_yv_yd_{\Bup,y}
2826   +\gamma\operatorname{Complexity}(C)
2827   \right].
2828 \]
2829 The main questions concern existence, tractable relaxations, and stability of the minimizers.  Another question is whether the task weights can be learned without invalidating the certificate after a distribution shift.
2830 
2831 \subsection{Posterior distributions over resolutions}
2832 Instead of selecting one $C$, place a posterior on a finite family of candidate resolutions:
2833 \[
2834   p(C\mid D)
2835   \propto
2836   p(D\mid C)p(C).
2837 \]
2838 Predictions then average both parameter and topology uncertainty:
2839 \[
2840   p(z_t\mid D)
2841   =
2842   \sum_Cp(z_t\mid C,D)p(C\mid D).
2843 \]
2844 A fuller model should distinguish uncertainty in the coefficients of a fixed hypergraph from uncertainty about the hypergraph and the quotient map themselves.
2845 
2846 \subsection{Stochastic interaction quotients}
2847 For a diffusion
2848 \[
2849   \dd X_t=b(X_t)\dd t+\sigma(X_t)\dd W_t,
2850 \]
2851 the generator is
2852 \[
2853   \cL\varphi
2854   =D\varphi[b]
2855   +\frac12\tr\bigl(\sigma\sigma^TD^2\varphi\bigr).
2856 \]
2857 Packet resolution turns drift into unary edges and quadratic variation into contracted binary hyperedges.  Let $\cC$ denote the pullback or averaging map on observables induced by the aggregation.  The natural exact condition is generator intertwining
2858 \[
2859   \cC\cL_{\fine}=\cL_{\coarse}\cC
2860 \]
2861 for the pullback or averaging map appropriate to observables.  A complete theorem would characterize closure of the drift and covariance, prove positivity of the induced semigroup, and carry generator defects into filtering error bounds.
2862 
2863 \subsection{Coarse dynamics with memory}
2864 Eliminating hidden packets generally produces memory.  The target form is
2865 \[
2866   \dot z(t)
2867   =A_{PP}z(t)
2868   +\int_0^tK(t-s)z(s)\dd s
2869   +\eta(t).
2870 \]
2871 The quotient theory should allow a fine system without memory to produce coarse dynamics with memory.  Volterra hyperedges and fractional kernels would then be treated as native coarse interactions rather than as evidence that quotienting failed.
2872 
2873 \subsection{Sharper filtering metrics}
2874 The $\ell^1$ normalization bounds are robust but conservative.  Positive operators often contract projective metrics, relative entropy, or other information geometric divergences more sharply.  One goal is to replace the factor
2875 \[
2876   \frac{2\norm{\lambda}_{\infty}}{z_*}
2877 \]
2878 by a contraction coefficient adapted to the trajectory or to the operator, while retaining an auditable finite certificate.
2879 
2880 \subsection{Pruning that preserves structure}
2881 In Hamiltonian, reversible, conservative, and monotone systems, as well as systems with detailed balance, individual hyperedges cannot always be removed independently.  The problem is to identify groups whose joint retention preserves the required structure, then solve the budget problem over those groups.
2882 
2883 \subsection{Identifiability of hidden dynamics}
2884 For fixed $C$ and coarse field $F_{\coarse}$, the exact fine realizations form the affine family
2885 \[
2886   \{F_{\fine}:CF_{\fine}=F_{\coarse}C\}.
2887 \]
2888 Any two exact fine models differ by a map into $\Ker C$.  Coarse observations alone cannot identify that hidden component.  Additional fine observations, intervention assumptions, regularity, or minimum complexity principles are required.  A statistical theory should characterize precisely which hidden hyperedges are identifiable under a given observation and intervention design.
2889 
2890 \section{Conclusion}
2891 An exact aggregation is more than a useful summary of a fine model.  It is a quotient.  Fine states with the same coarse image must produce the same coarse output.  For a multilinear interaction, this reduces to a complete finite test.  After applying $C$, the interaction must vanish whenever any input lies in $\Ker C$.  In packet coordinates, the coefficients summed over each output fiber must be constant on every product of input fibers.  The familiar unary criterion for Markov lumpability is the first order case of this general condition.
2892 
2893 The coefficient test controls the nonlinear constructions built from those interactions.  It passes to analytic vector fields, their derivatives, and every rooted interaction tree.  Under the usual local uniqueness assumptions, it also passes to the flow.  Bayesian observation has a matching quotient condition.  A normalized update commutes with aggregation exactly when the fine likelihood is constant within each fiber, apart from a positive global scale.  Once both conditions hold, every finite sequence of predictions and updates agrees after aggregation, and the evidence factors agree up to the same scales.
2894 
2895 When exactness fails, the reason matters.  A mismatch in the interaction coefficients creates a field defect, and the flow can amplify it.  Variation of the likelihood inside a fiber creates a second error at normalization.  Removing hyperedges creates a third error in the form of an explicit residual tensor.  The posterior recurrence keeps these contributions separate as it carries them forward.  An approximate coarse model is therefore auditable because one can see where each part of the final error entered.
2896 
2897 The theory does not show that an optimal quotient can always be found cheaply.  Coarse observations alone also cannot determine hidden dynamics that lie in $\Ker C$.  Those questions remain open.  The result proved here is narrower and exact:
2898 \begin{center}
2899 \fbox{\begin{minipage}{0.88\linewidth}
2900 A coarse model is exact when its interactions and observations descend through the same quotient.  When they do not, each failure can be measured at the stage where it occurs and carried forward to the final posterior or observable by an explicit certificate.
2901 \end{minipage}}
2902 \end{center}
2903 
2904 \appendix
2905 
2906 \section{Formulation on the quotient space}\label{app:quotient}
2907 Let
2908 \[
2909   q:X_{\fine}\to X_{\fine}/\Ker C
2910 \]
2911 be the canonical quotient map.  Since $C$ is surjective, there is a unique vector space isomorphism
2912 \[
2913   \widetilde C:X_{\fine}/\Ker C\to X_{\coarse}
2914 \]
2915 satisfying
2916 \[
2917   C=\widetilde Cq.
2918 \]
2919 The multilinear map $CA_m^{\fine}:X_{\fine}^m\to X_{\coarse}$ descends if and only if its value is unchanged when any one input is altered by an element of $\Ker C$.  The induced quotient tensor is
2920 \[
2921   \overline A_m
2922   ([x_1],\ldots,[x_m])
2923   =CA_m^{\fine}(x_1,\ldots,x_m),
2924 \]
2925 and
2926 \[
2927   A_m^{\coarse}
2928   =
2929   \widetilde C\,
2930   \overline A_m
2931   (\widetilde C^{-1},\ldots,\widetilde C^{-1}).
2932 \]
2933 This formulation shows that \Cref{thm:universal-quotient} is independent of coordinates and of the chosen right inverse.
2934 
2935 \section{A finite observational partition lemma}\label{app:observation-partition}
2936 Let $\cY$ be a finite observation alphabet and let each state $i$ have a strictly positive likelihood profile
2937 \[
2938   \Lambda(i)
2939   =
2940   (\lambda_y(i))_{y\in\cY}.
2941 \]
2942 Define
2943 \[
2944   i\sim_{\mathrm{obs}}j
2945   \quad\Longleftrightarrow\quad
2946   \Lambda(i)=\Lambda(j).
2947 \]
2948 
2949 \begin{proposition}[Coarsest deterministic observation partition]\label{prop:coarsest-observation-partition}
2950 The equivalence classes of $\sim_{\mathrm{obs}}$ form the coarsest partition for which every likelihood in the family is constant on every block.  Consequently, this partition is the coarsest deterministic block sum aggregation satisfying exact Bayesian refinement for all $y\in\cY$, up to one global positive scale for each observation.
2951 \end{proposition}
2952 
2953 \begin{proof}
2954 Every likelihood is constant on an equivalence class by definition.  Any partition on which every likelihood is constant on each block can place $i$ and $j$ in the same block only if all profile coordinates agree, so it refines the equivalence class partition.  The Bayesian statement follows from \Cref{thm:bayes-sufficient,thm:bayes-necessity}.
2955 \end{proof}
2956 
2957 \section{Certificate schemas and reproducibility}\label{app:schemas}
2958 A minimal interaction certificate stores
2959 \begin{equation}\label{eq:lumpability-certificate-schema}
2960   \mathsf L_m
2961   =
2962   (m,C,A_m^{\fine},A_m^{\coarse},\Delta_m,
2963   \norm{\Delta_m},\text{norm},\text{tolerance}).
2964 \end{equation}
2965 A Bayesian certificate stores
2966 \begin{equation}\label{eq:bayes-certificate-schema}
2967   \mathsf B_y
2968   =
2969   (\lambda_y^{\fine},\lambda_y^{\coarse},s_y,r_y,
2970   z_{*,y},d_{\Bup,y}).
2971 \end{equation}
2972 A filtering certificate stores
2973 \begin{equation}\label{eq:filter-certificate-schema}
2974   \mathsf F_N
2975   =
2976   (e_0^+,\{L_{\Phi,k}\},\{L_{\Bup,k}\},
2977   \{d_{\Phi,k}\},\{d_{\Bup,k}\},
2978   \{e_k^-,e_k^+\}).
2979 \end{equation}
2980 The release serializes the executed finite ledger as JSON files and CSV tables.  A certificate records a computation under specified inputs.  The ledger provides cryptographic authenticity only when it is paired with a separate signed digest layer.
2981 
2982 \subsection{Execution commands}
2983 From the source root:
2984 \begin{lstlisting}[style=code]
2985 PYTHONPATH=src python examples/run_certificates.py
2986 PYTHONPATH=src python -m pytest -q
2987 \end{lstlisting}
2988 The first command regenerates every numerical table and figure used in the manuscript.  The second runs the independent unit tests.
2989 
2990 \section{Theorem dependency ledger}\label{app:ledger}
2991 
2992 \begin{longtable}{@{}p{0.12\linewidth}p{0.55\linewidth}p{0.23\linewidth}@{}}
2993 \toprule
2994 ID & Result & Dependency\\
2995 \midrule
2996 \endfirsthead
2997 \toprule
2998 ID & Result & Dependency\\
2999 \midrule
3000 \endhead
3001 PRIQ-001 & Exact multilinear packet reconstruction & packet resolutions\\
3002 PRIQ-002 & Universal multilinear quotient criterion & factorization through a quotient space\\
3003 PRIQ-003 & Coefficient criterion at every interaction order & PRIQ-002, basis expansion\\
3004 PRIQ-004 & Symmetry descends & PRIQ-002\\
3005 PRIQ-005 & Exact quotient transitivity & intertwining algebra\\
3006 PRIQ-006 & Multilevel defect identity & algebraic decomposition\\
3007 PRIQ-007 & Analytic interaction descent & PRIQ-002, majorant convergence\\
3008 PRIQ-008 & Jet naturality & differentiation of intertwining\\
3009 PRIQ-009 & Treewise quotient naturality & PRIQ-002, rooted induction\\
3010 PRIQ-010 & Exact quotient flow & intertwining of vector fields and uniqueness\\
3011 PRIQ-011 & Normalized transfer naturality & positive intertwining, mass preservation\\
3012 PRIQ-012 & Sufficiency of likelihood constancy within fibers & PRIQ-011\\
3013 PRIQ-013 & Necessity of likelihood constancy within fibers & posterior ratios for two packets\\
3014 PRIQ-014 & Exact prediction and update cocycle & PRIQ-010, PRIQ-012\\
3015 PRIQ-015 & Sequential evidence factorization & PRIQ-014\\
3016 PRIQ-016 & Expansion by hidden inputs & decomposition through a right inverse\\
3017 PRIQ-017 & Majorant for the analytic field defect & coefficient bounds\\
3018 PRIQ-018 & Approximate quotient flow bound & Grönwall inequality\\
3019 PRIQ-019 & Normalization perturbation inequality & triangle inequality\\
3020 PRIQ-020 & Bayesian refinement defect & PRIQ-019\\
3021 PRIQ-021 & Bayesian Lipschitz certificate & PRIQ-019\\
3022 PRIQ-022 & Posterior error recurrence & prediction and update decompositions\\
3023 PRIQ-023 & Closed posterior certificate & iteration of PRIQ-022\\
3024 PRIQ-024 & Evidence error certificate & posterior error, likelihood residual\\
3025 PRIQ-025 & Hyperedge Parseval identity & orthogonal tensor blocks\\
3026 PRIQ-026 & Optimal approximation with $k$ hyperedges & PRIQ-025\\
3027 PRIQ-027 & Combined pruning and quotient defect & residual decomposition\\
3028 \bottomrule
3029 \end{longtable}
3030 
3031 \section{Notation index}\label{app:notation}
3032 \begin{longtable}{@{}p{0.22\linewidth}p{0.68\linewidth}@{}}
3033 \toprule
3034 Symbol & Meaning\\
3035 \midrule
3036 \endfirsthead
3037 \toprule
3038 Symbol & Meaning\\
3039 \midrule
3040 \endhead
3041 $X_{\fine},X_{\coarse}$ & fine and coarse state spaces\\
3042 $P_i,\overline P_a$ & fine and coarse packet projectors\\
3043 $\pi$ & map from fine packet labels to coarse labels\\
3044 $C$ & surjective aggregation or quotient map\\
3045 $E$ & right inverse selecting a fine representative of each coarse state\\
3046 $P_C=EC$ & representative projection\\
3047 $Q_C=I-EC$ & projection onto the hidden directions, with range $\Ker C$\\
3048 $A_m^{\fine},A_m^{\coarse}$ & fine and coarse interaction tensors of order $m$\\
3049 $\Delta_m$ & quotient defect at interaction order $m$\\
3050 $F_{\fine},F_{\coarse}$ & fine and coarse vector fields\\
3051 $\Phi_t^{\fine},\Phi_t^{\coarse}$ & fine and coarse flows or prediction maps\\
3052 $L_{\fine},L_{\coarse}$ & positive observation transfer operators\\
3053 $\Bup_{\lambda}$ & Bayesian normalized likelihood update\\
3054 $\Gamma_k$ & single step prediction and update map\\
3055 $\Psi_{1:N}$ & cocycle for a finite observation sequence\\
3056 $d_{\Phi,k}$ & direct prediction quotient defect at step $k$\\
3057 $d_{\Bup,k}$ & direct Bayesian refinement defect at step $k$\\
3058 $e_k^-,e_k^+$ & aggregate errors before and after an update\\
3059 $z_{*,k}$ & positive evidence floor\\
3060 $R_m$ & omitted hyperedge residual tensor\\
3061 $H_m$ & entropy of the normalized hyperedge energy distribution at order $m$\\
3062 \bottomrule
3063 \end{longtable}
3064 
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3131 
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3141 
3142 \end{thebibliography}
3143 
3144 \end{document}

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