Stratified Relational Workspace Theory

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1 % This file is part of the modular source for Stratified Relational Workspace Theory.
2 % !TeX root = ../Stratified_Relational_Workspace_Theory.tex
3 \theorypart{VI}{Dynamic Workspace Theory}
4 
5 \topic{19}{Probabilistic Local and Relational Dynamics}
6 
7 The static theory identifies compatible directions across a declared transfer
8 network. A dynamic workspace claim requires additional probabilistic objects:
9 a history, a target future, a decision problem, and an intervention family.
10 These must be fixed before sufficiency or causal efficacy is evaluated.
11 
12 \begin{definition}[Dynamic relational realization]\label{def:dynamic-realization}
13 A dynamic relational realization is defined on a probability space
14 \((\Omega,\mathcal F,\mathbb P)\) and consists of:
15 \begin{itemize}
16 \item node states \(X_i(t)\in\mathsf X_i\) for \(i\in V\);
17 \item edge messages \(M_e(t)\in\mathsf M_e\) for \(e\in E\);
18 \item external inputs \(E_t\), local actions \(A_i(t)\), and disturbances
19  \(\xi_i(t),\zeta_e(t)\);
20 \item time-dependent transfer data \(H_e(t)\), local core vectors \(c_i(t)\),
21  and transfer Laplacians \(L_t=L_{H(t)}\); and
22 \item a future target \(Y_t^+\in\mathsf Y\).
23 \end{itemize}
24 All state, history, and target spaces are assumed standard Borel unless a
25 stronger topology is explicitly invoked.
26 \end{definition}
27 
28 A representative local update is
29 \[
30 X_i(t+1)
31 =
32 F_i\!\left(
33 X_i(t),
34 \{M_e(t):t(e)=i\},
35 A_i(t),E_t,\xi_i(t)
36 \right),
37 \]
38 with messages
39 \[
40 M_e(t)
41 =
42 K_e\!\left(
43 X_{s(e)}(t),
44 W_t,
45 E_t,\zeta_e(t)
46 \right).
47 \]
48 This notation permits, but does not require, the current workspace state to
49 modulate future transfers.
50 
51 For a coalition \(A\subseteq V\), let \(\mathbf H_t^A\) denote the random
52 history available to that coalition through time \(t\). Its generated
53 information is
54 \[
55 \mathcal F_t^A=\sigma(\mathbf H_t^A).
56 \]
57 The full relational history is \(\mathbf H_t=\mathbf H_t^V\), with
58 \[
59 \mathcal F_t^A\subseteq\mathcal F_t^B
60 \qquad
61 \text{whenever }A\subseteq B.
62 \]
63 The empty-coalition history \(\mathbf H_t^\varnothing\) is the trivial
64 constant statistic and represents prediction from the background target law
65 alone.
66 A concrete history may contain node trajectories, incident messages, actions,
67 artifacts, transfer-map estimates, and the declared graph history. The exact
68 contents must be reported because coalition sufficiency is relative to what a
69 coalition is permitted to observe.
70 
71 The target \(Y_t^+\) can be a next output, a finite future trajectory, a
72 research decision, a proof outcome, or any measurable downstream quantity.
73 Every predictive workspace claim is therefore target-relative:
74 \[
75 \operatorname{WorkspaceFor}(Y_t^+)
76 \centernot\Longrightarrow
77 \operatorname{WorkspaceFor}(\widetilde Y_t^+).
78 \]
79 
80 Let
81 \[
82 c(t)=(c_i(t))_{i\in V}
83 \in
84 \mathcal C_V(t),
85 \qquad
86 P_t=\mathbf 1_{[0,\varepsilon_t]}(L_t).
87 \]
88 The instantaneous structural workspace coordinate is
89 \[
90 \boxed{
91 W_t^{\mathrm{str}}=P_tc(t)
92 \in
93 \Gamma_{\varepsilon_t}(H(t)).
94 }
95 \]
96 When \(P_t\) varies with time, these coordinates inhabit a moving family of
97 subspaces. Parts~X--XII will supply the transport required to compare them
98 across time. In what follows, all sufficiency and intervention statements
99 are made at a fixed decision time \(t\).
100 
101 \begin{remark}[Three objects that must not be conflated]
102 The realization state \(S_t\), the complete history \(\mathbf H_t\), and the
103 structural coordinate \(W_t^{\mathrm{str}}\) are different objects. The first
104 is an instantaneous physical or computational state, the second contains all
105 admissible past information, and the third is a low-transfer-energy statistic
106 extracted from the declared core geometry.
107 \end{remark}
108 
109 \topic{20}{Canonical Predictive and Interventional Causal States}
110 
111 The quotient notation
112 \([\mathbf H_t]_{\sim_Y}\) expresses the right intuition but hides
113 measurability and null-set issues. A canonical construction is obtained from
114 regular conditional distributions.
115 
116 \begin{definition}[Predictive kernel and canonical state]\label{def:predictive-kernel}
117 Let \(\mathsf H_t\) and \(\mathsf Y\) be standard Borel spaces. A regular
118 conditional kernel is a measurable map
119 \[
120 K_t:\mathsf H_t\times\mathcal B(\mathsf Y)\to[0,1]
121 \]
122 satisfying
123 \[
124 K_t(h,A)
125 =
126 \mathbb P(Y_t^+\in A\mid\mathbf H_t=h)
127 \]
128 for \(\mathbb P_{\mathbf H_t}\)-almost every \(h\). Let
129 \(\mathcal P(\mathsf Y)\) carry its evaluation sigma-algebra. The canonical
130 predictive state is the random probability measure
131 \[
132 \boxed{
133 Z_t
134 =
135 K_t(\mathbf H_t,\cdot)
136 \in
137 \mathcal P(\mathsf Y).
138 }
139 \]
140 Two histories are predictively equivalent precisely when they induce the same
141 conditional law of \(Y_t^+\).
142 \end{definition}
143 
144 \begin{theorem}[Predictive-state sufficiency and minimality, \(T_{12}\)]
145 \label{thm:predictive-state}
146 Under Definition~\ref{def:predictive-kernel}:
147 \begin{enumerate}
148 \item \(Z_t\) is predictively sufficient:
149 \[
150 Y_t^+\indep\mathbf H_t\mid Z_t.
151 \]
152 \item If \(S_t=s(\mathbf H_t)\) is any other history statistic satisfying
153 \[
154 Y_t^+\indep\mathbf H_t\mid S_t,
155 \]
156 then there is a measurable map \(\phi\) such that
157 \[
158 Z_t=\phi(S_t)
159 \qquad\text{almost surely}.
160 \]
161 \item Consequently, any two minimal sufficient predictive states are
162 measurably isomorphic after removal of null sets.
163 \end{enumerate}
164 \end{theorem}
165 
166 \begin{proof}
167 For every bounded measurable \(f:\mathsf Y\to\mathbb R\), define
168 \[
169 Z_t(f)=\int_{\mathsf Y}f(y)\,Z_t(\mathrm dy).
170 \]
171 By construction,
172 \[
173 \mathbb E[f(Y_t^+)\mid\mathbf H_t]=Z_t(f).
174 \]
175 Since \(Z_t\) is a measurable function of \(\mathbf H_t\), the tower property
176 gives
177 \[
178 \mathbb E[f(Y_t^+)\mid Z_t]
179 =
180 \mathbb E[Z_t(f)\mid Z_t]
181 =
182 Z_t(f).
183 \]
184 Hence conditioning additionally on \(\mathbf H_t\) does not change the
185 conditional expectation, which is the stated conditional independence.
186 
187 Now let \(S_t=s(\mathbf H_t)\) be sufficient. Then, for every measurable
188 \(A\subseteq\mathsf Y\),
189 \[
190 K_t(\mathbf H_t,A)
191 =
192 \mathbb P(Y_t^+\in A\mid\mathbf H_t)
193 =
194 \mathbb P(Y_t^+\in A\mid S_t)
195 \qquad\text{a.s.}
196 \]
197 Choose a countable determining class for probability measures on the standard
198 Borel space \(\mathsf Y\). The values of \(K_t(\mathbf H_t,\cdot)\) on this
199 class are measurable functions of \(S_t\), and therefore the complete random
200 measure \(Z_t\) is a measurable function of \(S_t\). Applying this conclusion
201 in both directions to two minimal sufficient states gives the final claim.
202 \end{proof}
203 
204 \begin{remark}[Meaning of minimality]
205 The theorem does not say that \(Z_t\) has the fewest numerical coordinates in
206 a chosen parametrization. It says that every sufficient history statistic
207 must determine \(Z_t\) almost surely. This is an information order, invariant
208 under one-to-one reparametrization.
209 \end{remark}
210 
211 For an arbitrary history statistic \(S_t=s(\mathbf H_t)\), let
212 \[
213 K_t^{S}(s,\cdot)
214 =
215 \mathbb P(Y_t^+\in\cdot\mid S_t=s).
216 \]
217 Whenever the conditional relative entropy is finite, define the sufficiency
218 defect
219 \[
220 \boxed{
221 \delta_{\mathrm{suff}}(S_t)
222 =
223 \mathbb E\!\left[
224 \KL\!\left(
225 K_t(\mathbf H_t,\cdot)
226 \,\middle\|\,
227 K_t^S(S_t,\cdot)
228 \right)
229 \right].
230 }
231 \]
232 
233 \begin{proposition}[Conditional-information identity]\label{prop:conditional-information}
234 If \(S_t\) is a measurable function of \(\mathbf H_t\), then
235 \[
236 \boxed{
237 \delta_{\mathrm{suff}}(S_t)
238 =
239 I(Y_t^+;\mathbf H_t\mid S_t).
240 }
241 \]
242 In particular, the defect vanishes if and only if \(S_t\) is predictively
243 sufficient, up to the usual almost-sure qualifications.
244 \end{proposition}
245 
246 \begin{proof}
247 The conditional mutual information is the expected relative entropy between
248 the conditional law given \((\mathbf H_t,S_t)\) and the conditional law given
249 \(S_t\). Because \(S_t=s(\mathbf H_t)\), the first of these is exactly
250 \(K_t(\mathbf H_t,\cdot)\).
251 \end{proof}
252 
253 Sufficiency has a direct decision-theoretic interpretation. Use the
254 convention
255 \[
256 \TV(P,Q)=\sup_{A}|P(A)-Q(A)|.
257 \]
258 Let \(\mathsf A\) be a finite action set and
259 \(\ell:\mathsf Y\times\mathsf A\to[0,1]\) a bounded loss. Define
260 \[
261 \Risk_\ell(S_t)
262 =
263 \inf_{d}
264 \mathbb E\!\left[
265 \ell\bigl(Y_t^+,d(S_t)\bigr)
266 \right].
267 \]
268 
269 \begin{proposition}[Approximate sufficiency bounds excess Bayes risk]
270 \label{prop:risk-pinsker}
271 Under the preceding assumptions,
272 \[
273 \boxed{
274 0
275 \leq
276 \Risk_\ell(S_t)-\Risk_\ell(\mathbf H_t)
277 \leq
278 \mathbb E\!\left[
279 \TV\!\left(K_t(\mathbf H_t),K_t^S(S_t)\right)
280 \right]
281 \leq
282 \sqrt{\frac{\delta_{\mathrm{suff}}(S_t)}{2}}.
283 }
284 \]
285 \end{proposition}
286 
287 \begin{proof}
288 More information cannot increase optimal risk, giving the first inequality.
289 For any fixed action, its conditional expected loss changes by at most total
290 variation when the conditional law changes. Taking the infimum over actions
291 preserves this bound. Conditional Pinsker inequality followed by Jensen's
292 inequality gives the final estimate.
293 \end{proof}
294 
295 For a finite target alphabet and logarithmic scoring loss
296 \[
297 \ell_{\log}(y,q)=-\log q(y),
298 \qquad
299 q\in\mathcal P(\mathsf Y),
300 \]
301 the identity is exact:
302 \[
303 \boxed{
304 \Risk_{\log}(S_t)=H(Y_t^+\mid S_t),
305 \qquad
306 \Risk_{\log}(S_t)-\Risk_{\log}(\mathbf H_t)
307 =
308 I(Y_t^+;\mathbf H_t\mid S_t).
309 }
310 \]
311 Natural logarithms measure the gap in nats; base-two logarithms measure it in
312 bits.
313 
314 \subtopic{Intervention-indexed causal state}
315 
316 Observational prediction may merge histories that behave identically under the
317 current policy but respond differently to future interventions. Let
318 \(\mathcal I\) be a finite or countable family of interventions applied only
319 after time \(t\). For each \(\iota\in\mathcal I\), let
320 \[
321 K_t^{\iota}(h,\cdot)
322 =
323 \mathbb P^{\iota}(Y_t^+\in\cdot\mid\mathbf H_t=h)
324 \]
325 be a regular post-intervention kernel on the common pre-intervention history
326 space. Define
327 \[
328 \boxed{
329 Z_t^{\mathcal I}
330 =
331 \bigl(K_t^{\iota}(\mathbf H_t,\cdot)\bigr)_{\iota\in\mathcal I}
332 \in
333 \mathcal P(\mathsf Y)^{\mathcal I}.
334 }
335 \]
336 
337 \begin{corollary}[Minimal simultaneous interventional state]
338 \label{cor:interventional-state}
339 The random element \(Z_t^{\mathcal I}\) is sufficient simultaneously for all
340 post-intervention target laws. If a statistic \(S_t=s(\mathbf H_t)\)
341 determines the conditional target law for every \(\iota\in\mathcal I\), then
342 there exists a measurable \(\phi_{\mathcal I}\) such that
343 \[
344 Z_t^{\mathcal I}=\phi_{\mathcal I}(S_t)
345 \qquad\text{a.s.}
346 \]
347 \end{corollary}
348 
349 \begin{proof}
350 Apply Theorem~\ref{thm:predictive-state} componentwise. A countable product of
351 standard Borel spaces is standard Borel, so the component maps combine into a
352 single measurable factor map.
353 \end{proof}
354 
355 The observational state is the special case
356 \(\mathcal I=\{\operatorname{id}\}\). A causal workspace claim should align
357 with \(Z_t^{\mathcal I}\) for a declared intervention family rather than only
358 with the observational state.
359 
360 \topic{21}{Structural--Operational Alignment}
361 
362 A structural workspace coordinate becomes operational only when it retains the
363 canonical predictive or interventional state relevant to the declared task.
364 Let
365 \[
366 W_t=\pi_t(\mathbf H_t)
367 \]
368 be a measurable candidate workspace statistic satisfying the structural support
369 condition
370 \[
371 W_t\in\Gamma_{\varepsilon_t}(H(t))
372 \qquad\text{almost surely}.
373 \]
374 The map \(\pi_t\) may include encoding into a moving low-energy bundle; no
375 canonical operational interpretation follows from the spectral projector
376 alone.
377 
378 \begin{definition}[Exact structural--operational alignment]
379 The candidate \(W_t\) is observationally aligned when there exists a measurable
380 map \(\phi_t\) such that
381 \[
382 Z_t=\phi_t(W_t)
383 \qquad\text{a.s.}
384 \]
385 It is \(\mathcal I\)-causally aligned when
386 \[
387 Z_t^{\mathcal I}=\phi_t^{\mathcal I}(W_t)
388 \qquad\text{a.s.}
389 \]
390 \end{definition}
391 
392 \begin{theorem}[Exact alignment criterion]\label{thm:alignment-criterion}
393 For any workspace statistic \(W_t=\pi_t(\mathbf H_t)\), the following are
394 equivalent:
395 \begin{enumerate}
396 \item \(Y_t^+\indep\mathbf H_t\mid W_t\);
397 \item \(I(Y_t^+;\mathbf H_t\mid W_t)=0\);
398 \item the canonical predictive state factors through \(W_t\):
399 \[
400 Z_t=\phi_t(W_t)
401 \quad\text{a.s.}
402 \]
403 for some measurable \(\phi_t\).
404 \end{enumerate}
405 The analogous equivalence holds simultaneously over \(\mathcal I\) with
406 \(Z_t^{\mathcal I}\).
407 \end{theorem}
408 
409 \begin{proof}
410 The equivalence of the first two statements is the zero-divergence
411 characterization of conditional mutual information. The first statement and
412 Theorem~\ref{thm:predictive-state} imply the factorization in the third.
413 Conversely, if \(Z_t=\phi_t(W_t)\), then
414 \[
415 \mathbb P(Y_t^+\in A\mid\mathbf H_t)
416 =Z_t(A)
417 =\phi_t(W_t)(A),
418 \]
419 which depends on the history only through \(W_t\), proving sufficiency. The
420 interventional statement follows from
421 Corollary~\ref{cor:interventional-state}.
422 \end{proof}
423 
424 Define the observational alignment defect by
425 \[
426 \boxed{
427 \delta_{\mathrm{align}}^{\mathrm{obs}}(W_t)
428 =
429 I_{\mathbb P}(Y_t^+;\mathbf H_t\mid W_t).
430 }
431 \]
432 For a countable intervention family with declared weights
433 \(q_\iota>0\), \(\sum_\iota q_\iota=1\), define
434 \[
435 \boxed{
436 \delta_{\mathrm{align}}^{\mathcal I}(W_t)
437 =
438 \sum_{\iota\in\mathcal I}
439 q_\iota
440 I_{\mathbb P^\iota}
441 (Y_t^+;\mathbf H_t\mid W_t).
442 }
443 \]
444 For bounded decision losses, Proposition~\ref{prop:risk-pinsker} applies under
445 each intervention and gives the weighted bound
446 \[
447 \sum_{\iota}q_\iota
448 \left[
449 \Risk_\ell^{\iota}(W_t)
450 -
451 \Risk_\ell^{\iota}(\mathbf H_t)
452 \right]
453 \leq
454 \sqrt{\frac{
455 \delta_{\mathrm{align}}^{\mathcal I}(W_t)}{2}}.
456 \]
457 
458 \subtopic{Structural coherence and operational alignment are independent}
459 
460 The static defect
461 \[
462 \delta_{\mathrm{coh}}(W_t)
463 \]
464 and the operational defect
465 \[
466 \delta_{\mathrm{align}}^{\mathcal I}(W_t)
467 \]
468 answer different questions. The first measures compatibility under transfer.
469 The second measures loss of task-relevant conditional law.
470 
471 \begin{center}
472 \small
473 \begin{tabularx}{\textwidth}{@{}>{\raggedright\arraybackslash}p{0.22\textwidth}>{\raggedright\arraybackslash}p{0.34\textwidth}>{\raggedright\arraybackslash}X@{}}
474 \toprule
475 Regime & Structural status & Operational interpretation \\
476 \midrule
477 Low coherence defect, low alignment defect & Stable transported coordinate &
478 Candidate operational workspace for the declared target and interventions. \\
479 Low coherence defect, high alignment defect & Stable but task-irrelevant
480 shared structure & Archive, convention, or synchronized representation. \\
481 High coherence defect, low alignment defect & Predictively adequate but not
482 transport-stable & Localized or brittle sufficient statistic. \\
483 High coherence defect, high alignment defect & Neither stable nor sufficient &
484 Rejected candidate at the stated scales. \\
485 \bottomrule
486 \end{tabularx}
487 \end{center}
488 
489 Thus neither low transfer energy nor high predictive performance alone is
490 enough. The combined gate is
491 \[
492 \boxed{
493 \delta_{\mathrm{coh}}(W_t)
494 \leq
495 \varepsilon_{\mathrm{coh}},
496 \qquad
497 \delta_{\mathrm{align}}^{\mathcal I}(W_t)
498 \leq
499 \varepsilon_{\mathrm{align}}.
500 }
501 \]
502 
503 \begin{example}[Coherent but irrelevant archive]
504 Suppose every node perfectly copies a common random variable \(R\), so the
505 corresponding transfer energy is zero. If \(R\indep Y_t^+\), then
506 \[
507 \delta_{\mathrm{coh}}(R)=0,
508 \qquad
509 \delta_{\mathrm{align}}^{\mathrm{obs}}(R)
510 =I(Y_t^+;\mathbf H_t\mid R),
511 \]
512 which can be as large as the entire predictive information in the history.
513 Perfect agreement therefore does not imply workspace functionality.
514 \end{example}
515 
516 \begin{example}[Sufficient but localized statistic]
517 If one node contains the canonical state \(Z_t\) and all transfers from that
518 node are unstable, then \(W_t=Z_t\) is sufficient but need not lie in a stable
519 low-energy transfer subspace. This is a local controller or archive, not yet a
520 stratified shared workspace.
521 \end{example}

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