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{VIII}{Causal Efficacy and Reentry}
4 
5 \topic{28}{Feasible Workspace Interventions}
6 
7 A notation such as \(\operatorname{do}(W_t=w)\) is meaningful only if the
8 candidate workspace is an endogenous variable in a structural causal model or
9 if an intervention on underlying variables is shown to realize that value.
10 Abstractly setting a projection coordinate need not correspond to any possible
11 physical or computational manipulation.
12 
13 \begin{definition}[Time-unrolled causal realization]\label{def:causal-realization}
14 A time-unrolled causal realization is a structural causal model
15 \[
16 \mathcal M
17 =
18 (\mathcal U,\mathcal V,F,\mathbb P_U)
19 \]
20 whose endogenous variables include the local states, messages, actions,
21 artifacts, target variables, and a measurable workspace extractor
22 \[
23 \pi_t:\mathsf S_t\to\mathsf W_t,
24 \qquad
25 W_t=\pi_t(S_t).
26 \]
27 The unrolled graph is acyclic even when the rolled process is recurrent.
28 \end{definition}
29 
30 \begin{definition}[Feasible implementation intervention]
31 Let \(B_t\) denote declared background variables that should remain matched.
32 A family \(\{\iota_w:w\in\mathsf W_t^\circ\}\) is a feasible implementation of
33 workspace values on the experimental domain \(\mathsf W_t^\circ\) when:
34 \begin{enumerate}
35 \item each \(\iota_w\) acts only on declared manipulable parents, channels, or
36  artifacts at or after time \(t\);
37 \item the intervened realization satisfies
38 \[
39 \pi_t(S_t^{\iota_w})=w
40 \qquad\text{almost surely};
41 \]
42 \item the background law is matched across compared interventions; and
43 \item the intervention does not directly modify \(Y_t^+\) except through
44  paths included in the declared causal design.
45 \end{enumerate}
46 \end{definition}
47 
48 Let \(D\) be a divergence or metric on probability laws. For a matched
49 background value \(b\), define
50 \[
51 \CE_{\Workspace}(w,w';b)
52 =
53 D\!\left(
54 \mathbb P^{\iota_w}(Y_t^+\in\cdot\mid B_t=b),
55 \mathbb P^{\iota_{w'}}(Y_t^+\in\cdot\mid B_t=b)
56 \right).
57 \]
58 For a declared comparison design \(\nu\) over \((w,w',b)\), define
59 \[
60 \boxed{
61 \eta_{\mathrm{causal}}^{\nu}
62 =
63 \mathbb E_{(w,w',b)\sim\nu}
64 \left[
65 \CE_{\Workspace}(w,w';b)
66 \right].
67 }
68 \]
69 The maximum
70 \[
71 \eta_{\mathrm{causal}}^{\max}
72 =
73 \sup_{w,w',b}\CE_{\Workspace}(w,w';b)
74 \]
75 may be useful for discovery, but a design-weighted effect is harder to inflate
76 by selecting a single exceptional pair after inspection.
77 
78 \begin{proposition}[Coordinate invariance of intervention effects]
79 \label{prop:causal-gauge}
80 Let \(g:\mathsf W_t\to\mathsf W_t'\) be a measurable bijection and suppose the
81 reparametrized intervention family satisfies
82 \[
83 \iota_{g(w)}'=\iota_w.
84 \]
85 Then every divergence-based causal effect is preserved:
86 \[
87 \CE_{\Workspace'}(g(w),g(w');b)
88 =
89 \CE_{\Workspace}(w,w';b).
90 \]
91 \end{proposition}
92 
93 \begin{proof}
94 The two labels designate the same underlying interventions and therefore induce
95 the same pair of target laws. Only the coordinate name has changed.
96 \end{proof}
97 
98 This is the causal analogue of gauge invariance. It also shows why effects
99 should be attached to intervention implementations rather than to arbitrary
100 numerical coordinates.
101 
102 \topic{29}{Causal Specificity and Transfer Mediation}
103 
104 A causal effect on the target is not workspace-specific if an equal-cost
105 perturbation outside the candidate workspace produces the same effect. Let
106 \(c(\iota)\) be a declared intervention cost and let
107 \[
108 \mathcal I_{\Workspace}(r),
109 \qquad
110 \mathcal I_{\perp}(r)
111 \]
112 be matched families with cost \(r\), the first changing workspace coordinates
113 and the second changing complementary coordinates while preserving the
114 workspace value as closely as the implementation permits.
115 
116 Define
117 \[
118 \eta_{\Workspace}(r)
119 =
120 \mathbb E_{\iota,\iota'\sim\mathcal I_{\Workspace}(r)}
121 D\!\left(
122 \mathbb P^{\iota}(Y_t^+),
123 \mathbb P^{\iota'}(Y_t^+)
124 \right),
125 \]
126 \[
127 \eta_{\perp}(r)
128 =
129 \mathbb E_{\jmath,\jmath'\sim\mathcal I_{\perp}(r)}
130 D\!\left(
131 \mathbb P^{\jmath}(Y_t^+),
132 \mathbb P^{\jmath'}(Y_t^+)
133 \right),
134 \]
135 and
136 \[
137 \boxed{
138 \Delta_{\mathrm{specific}}(r)
139 =
140 \eta_{\Workspace}(r)-\eta_{\perp}(r).
141 }
142 \]
143 A positive value supports causal concentration in the candidate workspace at
144 the declared perturbation scale.
145 
146 If the complement is defined by \(\Gamma_\varepsilon(H)^\perp\), the result is
147 metric dependent. Under nonunitary gauge changes, either the metric must be
148 transported with the coordinates or interventions must be matched by a
149 coordinate-independent physical cost. Otherwise specificity can change merely
150 because the representation was rescaled.
151 
152 \subtopic{Causal transfer nulls}
153 
154 Let \(\jmath_E\) randomize selected communication edges while preserving their
155 marginal message laws, rates, and timing budget. The causal transfer gap is
156 \[
157 \boxed{
158 \Delta_{\mathrm{tr}}^{\operatorname{do}}(\jmath_E)
159 =
160 \Risk_{\ell,\mathbb P^{\jmath_E}}(V)
161 -
162 \Risk_{\ell,\mathbb P}(V).
163 }
164 \]
165 Unlike an observational null law, this quantity is anchored to an implemented
166 channel intervention. A positive effect is evidence that the intact alignment
167 of messages, rather than merely their marginal content, contributes to task
168 performance.
169 
170 A stronger mediation test equalizes the workspace state after two distinct
171 channel interventions. Let \(\mu\) be a common imposed workspace distribution,
172 implemented by a feasible stochastic intervention in the sense of
173 Definition~\ref{def:causal-realization}. The notation below is shorthand for
174 that implementation. Define the residual mediation defect
175 \[
176 \delta_{\mathrm{med}}
177 =
178 \sup_{\jmath,\jmath'}
179 D\!\left(
180 \mathbb P^{\jmath,\operatorname{do}(W_t\sim\mu)}(Y_t^+),
181 \mathbb P^{\jmath',\operatorname{do}(W_t\sim\mu)}(Y_t^+)
182 \right).
183 \]
184 A small value means that once the workspace state is equalized, little of the
185 channel intervention's effect remains. This supports, but does not by itself
186 identify without the causal assumptions, the claim that \(W_t\) mediates the
187 relevant relational influence.
188 
189 \topic{30}{Causal Recurrence, Loop-Mediated Reentry, and Relationality}
190 
191 Temporal correlation is weaker than causal recurrence. Define the total
192 workspace recurrence effect at lag \(\tau\) by
193 \[
194 \boxed{
195 \eta_{\mathrm{rec}}(\tau)
196 =
197 \mathbb E_{(w,w',b)\sim\nu}
198 D\!\left(
199 \mathbb P^{\iota_w}(W_{t+\tau}\in\cdot\mid B_t=b),
200 \mathbb P^{\iota_{w'}}(W_{t+\tau}\in\cdot\mid B_t=b)
201 \right).
202 }
203 \]
204 A nonzero value shows that manipulating the present workspace changes a later
205 workspace state. It may, however, be explained by direct copying or a retained
206 memory register.
207 
208 To isolate reentry, consider the unrolled causal graph and remove every direct
209 retention edge whose source and target are both workspace variables at
210 successive times. Let
211 \[
212 \mathcal M^{\setminus\mathrm{dir}}
213 \]
214 denote the resulting path-specific model, preserving paths that leave the
215 workspace through a nonworkspace participant, action, message, or artifact and
216 later return. Define
217 \[
218 \boxed{
219 \eta_{\mathrm{loop}}(\tau)
220 =
221 \mathbb E_{(w,w',b)\sim\nu}
222 D\!\left(
223 \mathbb P_{\mathcal M^{\setminus\mathrm{dir}}}^{\iota_w}
224 (W_{t+\tau}\in\cdot\mid B_t=b),
225 \mathbb P_{\mathcal M^{\setminus\mathrm{dir}}}^{\iota_{w'}}
226 (W_{t+\tau}\in\cdot\mid B_t=b)
227 \right).
228 }
229 \]
230 Then the distinctions are:
231 \[
232 \begin{aligned}
233 I(W_t;W_{t+\tau})>0
234 &\quad\text{temporal dependence only},\\
235 \eta_{\mathrm{rec}}(\tau)>0
236 &\quad\text{causal persistence},\\
237 \eta_{\mathrm{loop}}(\tau)>0
238 &\quad\text{causal exit-and-return reentry}.
239 \end{aligned}
240 \]
241 Path-specific effects require the usual consistency and no-unmeasured-confounding
242 assumptions appropriate to the implementation. In software systems, edge
243 interventions can often be literal. In human collaboration, these assumptions
244 are substantially harder to justify.
245 
246 \begin{theorem}[Operational relationality, \(T_{13}\)]
247 \label{thm:operational-relationality}
248 Let \(\mathsf Y\) be finite, let \(P\) be the intact law at a fixed decision
249 time, and let \(W_t=\pi_t(\mathbf H_t)\) be a structurally supported candidate.
250 Let
251 \(\mathcal Q\subseteq\mathcal N_{\mathrm{pred}}(P)\) be a nonempty,
252 predeclared panel of validated relation-breaking nulls. Assume:
253 \begin{enumerate}
254 \item exact structural--operational alignment,
255 \[
256 I_P(Y_t^+;\mathbf H_t\mid W_t)=0;
257 \]
258 \item a positive node-level information gap,
259 \[
260 \delta_{\mathrm{node}}
261 =
262 \min_{i\in V}
263 I_P(Y_t^+;\mathbf H_t\mid\mathbf H_t^i)
264 >0;
265 \]
266 \item a positive robust matched-null gap,
267 \[
268 \delta_{\mathrm{null}}
269 =
270 \min_{Q\in\mathcal Q}
271 \left[
272 H_Q(Y_t^+\mid\mathbf H_t)
273 -
274 H_P(Y_t^+\mid\mathbf H_t)
275 \right]
276 >0;
277 \]
278 \item a feasible intervention design with
279 \[
280 \eta_{\mathrm{causal}}^{\nu}>0;
281 \]
282 \item positive matched causal specificity at some declared scale \(r\),
283 \[
284 \Delta_{\mathrm{specific}}(r)>0.
285 \]
286 \end{enumerate}
287 Then, relative to the declared decomposition, target, log loss, null panel, and
288 intervention family:
289 \begin{enumerate}
290 \item no statistic of any single-node history is sufficient for \(Y_t^+\), and
291  \(k_0\geq2\);
292 \item \(W_t\) cannot be reconstructed almost surely from any single-node
293  history;
294 \item every null \(Q\in\mathcal Q\) preserves all single-node predictive
295  risks but has strictly worse full-system log risk than the intact law;
296 \item the candidate workspace is causally efficacious and its effect exceeds
297  that of the declared matched off-workspace perturbations.
298 \end{enumerate}
299 If, in addition, \(\eta_{\mathrm{loop}}(\tau)>0\), the workspace is
300 loop-reentrant at lag \(\tau\).
301 \end{theorem}
302 
303 \begin{proof}
304 By Theorem~\ref{thm:coalition-log-risk}, for each node \(i\),
305 \[
306 \Risk_{\log,P}(\{i\})-
307 \Risk_{\log,P}(V)
308 =
309 I_P(Y_t^+;\mathbf H_t\mid\mathbf H_t^i)
310 \geq
311 \delta_{\mathrm{node}}>0.
312 \]
313 Hence no single-node history, and therefore no statistic of one, is sufficient;
314 this gives \(k_0\geq2\).
315 
316 Exact alignment and Theorem~\ref{thm:alignment-criterion} imply that the
317 canonical predictive state factors through \(W_t\). If
318 \(W_t=\psi_i(\mathbf H_t^i)\) almost surely for some node, then the canonical
319 state would also be a function of \(\mathbf H_t^i\), contradicting the positive
320 node gap through Proposition~\ref{prop:sufficiency-reconstruction}.
321 
322 For every \(Q\in\mathcal Q\), Proposition~\ref{prop:local-risk-null} gives
323 \[
324 \Risk_{\log,Q}(\{i\})
325 =
326 \Risk_{\log,P}(\{i\})
327 \qquad\forall i.
328 \]
329 The assumed null gap gives
330 \[
331 \Risk_{\log,Q}(V)
332 >
333 \Risk_{\log,P}(V).
334 \]
335 Thus a description determined only by the collection of single-node predictive
336 marginals cannot reproduce the intact joint capacity: it would assign the same
337 local data to \(P\) and \(Q\) while their full-system risks differ.
338 
339 Finally, the feasible intervention effect establishes causal efficacy relative
340 to the declared implementation, and the positive specificity gap shows that
341 the effect exceeds the matched complementary perturbation effect. The final
342 reentry statement is exactly the definition of \(\eta_{\mathrm{loop}}\).
343 \end{proof}
344 
345 \begin{remark}[Scope of the theorem]
346 The conclusion is operational and indexed by
347 \[
348 (V,Y_t^+,\ell,\mathcal Q,\mathcal I,\nu,r).
349 \]
350 Changing the decomposition, target, loss, nulls, or intervention design may
351 change the verdict. The theorem does not establish metaphysical
352 irreducibility, autonomous agency, selfhood, or consciousness.
353 \end{remark}
354 
355 \subtopic{Finite certification sequence}
356 
357 A reproducible dynamic-causal analysis should proceed in the following order:
358 \begin{enumerate}
359 \item Freeze the node decomposition, accessible coalition histories, target,
360  horizon, and proper scoring loss.
361 \item Estimate the structural projector on training data only and encode
362  \(W_t\) without using held-out targets.
363 \item Cross-fit conditional target laws from the full history, the workspace,
364  and every declared coalition. Report alignment and coalition risk gaps.
365 \item Construct the null panel before inspecting null performance. Verify
366  local-predictive matching and the intended destruction of transfer
367  organization separately.
368 \item Implement feasible workspace and complementary interventions with a
369  common background and cost design. Report both causal effect and
370  specificity.
371 \item Test recurrence and then cut direct retention paths to test loop-mediated
372  reentry.
373 \item Report confidence intervals or finite-sample uncertainty for every gate;
374  do not substitute training accuracy for Bayes-risk or causal estimates.
375 \end{enumerate}
376 
377 The resulting certification report is
378 \[
379 \boxed{
380 \begin{aligned}
381 \mathcal C_{\mathrm{dyn}}(W_t)
382 =\bigl(&
383 \delta_{\mathrm{align}}^{\mathcal I},
384 \{\Delta_A^\ell\}_{A\subseteq V},
385  k_\varepsilon,
386 \boldsymbol\Delta_{\mathrm{tr}}(\mathcal Q),\\
387 &\eta_{\mathrm{causal}}^\nu,
388 \Delta_{\mathrm{specific}}(r),
389 \eta_{\mathrm{rec}}(\tau),
390 \eta_{\mathrm{loop}}(\tau)
391 \bigr).
392 \end{aligned}
393 }
394 \]
395 No single coordinate certifies a relational workspace.

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