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{VII}{Relational Irreducibility}
4 
5 \topic{22}{Coalition Information and Bayes Risk}
6 
7 For \(A\subseteq V\), the coalition statistic is \(\mathbf H_t^A\). Given an
8 action space \(\mathsf A\) and loss \(\ell\), define its unrestricted Bayes
9 risk by
10 \[
11 \boxed{
12 \Risk_\ell(A)
13 =
14 \inf_d
15 \mathbb E\!\left[
16 \ell\bigl(Y_t^+,d(\mathbf H_t^A)\bigr)
17 \right].
18 }
19 \]
20 The infimum is over measurable decision rules for which the expectation is
21 defined. The full-system risk is \(\Risk_\ell(V)\).
22 
23 \begin{proposition}[Coalition monotonicity]\label{prop:coalition-monotonicity}
24 If \(A\subseteq B\subseteq V\), then
25 \[
26 \Risk_\ell(B)\leq\Risk_\ell(A).
27 \]
28 \end{proposition}
29 
30 \begin{proof}
31 Every decision rule available to coalition \(A\) is also available to coalition
32 \(B\), which can ignore its additional information.
33 \end{proof}
34 
35 For finite \(\mathsf Y\), logarithmic loss converts risk gaps into conditional
36 information exactly.
37 
38 \begin{theorem}[Coalition log-risk identity]\label{thm:coalition-log-risk}
39 Assume \(H(Y_t^+)<\infty\). If \(A\subseteq B\subseteq V\), then
40 \[
41 \boxed{
42 \Risk_{\log}(A)-\Risk_{\log}(B)
43 =
44 I\!\left(
45 Y_t^+;\mathbf H_t^B
46 \mid
47 \mathbf H_t^A
48 \right).
49 }
50 \]
51 In particular,
52 \[
53 \Risk_{\log}(A)-\Risk_{\log}(V)
54 =
55 I\!\left(
56 Y_t^+;\mathbf H_t
57 \mid
58 \mathbf H_t^A
59 \right).
60 \]
61 \end{theorem}
62 
63 \begin{proof}
64 Under logarithmic loss the optimal report is the conditional target law, so
65 \[
66 \Risk_{\log}(A)=H(Y_t^+\mid\mathbf H_t^A).
67 \]
68 Because \(\mathbf H_t^A\) is a function of \(\mathbf H_t^B\), subtracting the
69 two conditional entropies gives the stated conditional mutual information.
70 \end{proof}
71 
72 The theorem makes the decomposition explicit: irreducibility is not an
73 unqualified property of a system. It is a property relative to the chosen node
74 partition, accessible histories, target, and decision loss.
75 
76 \topic{23}{Joint Advantage and Its Scope}
77 
78 Define the coalition gap
79 \[
80 \Delta_A^{\ell}
81 =
82 \Risk_\ell(A)-\Risk_\ell(V)
83 \geq0.
84 \]
85 The node-level joint advantage is
86 \[
87 \boxed{
88 \Delta_{\mathrm{joint}}^{\ell}
89 =
90 \min_{i\in V}
91 \Delta_{\{i\}}^{\ell}.
92 }
93 \]
94 Under logarithmic loss,
95 \[
96 \boxed{
97 \Delta_{\mathrm{joint}}^{\log}
98 =
99 \min_{i\in V}
100 I\!\left(
101 Y_t^+;\mathbf H_t
102 \mid
103 \mathbf H_t^{\{i\}}
104 \right).
105 }
106 \]
107 Therefore
108 \[
109 \Delta_{\mathrm{joint}}^{\log}>0
110 \]
111 means that every individual component omits some predictive information
112 available to the complete system.
113 
114 This quantity should not automatically be identified with a particular notion
115 of partial-information synergy. A positive gap can arise from complementary
116 information distributed across components, from higher-order interaction, or
117 from both. The theory uses the operationally weaker and more directly
118 testable statement:
119 \[
120 \text{no individual node is sufficient for the declared target.}
121 \]
122 
123 For \(0\leq r\leq |V|\), define the order-\(r\) coalition gap profile by
124 \[
125 \boxed{
126 \Delta^{(r)}_\ell
127 =
128 \min_{\substack{A\subseteq V\\|A|\leq r}}
129 \left[
130 \Risk_\ell(A)-\Risk_\ell(V)
131 \right].
132 }
133 \]
134 The profile is nonincreasing in \(r\) and reaches zero by \(r=|V|\).
135 It records how rapidly operative capacity becomes reconstructible as larger
136 coalitions are permitted.
137 
138 \topic{24}{Minimal Sufficient Coalition}
139 
140 For tolerance \(\varepsilon\geq0\), define
141 \[
142 \boxed{
143 k_\varepsilon
144 =
145 \min\left\{
146 |A|:
147 \Risk_\ell(A)
148 \leq
149 \Risk_\ell(V)+\varepsilon
150 \right\}.
151 }
152 \]
153 The set of minimal coalitions is
154 \[
155 \mathcal K_\varepsilon
156 =
157 \left\{
158 A\subseteq V:
159 |A|=k_\varepsilon,
160 \Risk_\ell(A)
161 \leq
162 \Risk_\ell(V)+\varepsilon
163 \right\}.
164 \]
165 
166 \begin{corollary}[Information form of coalition size]
167 Under the assumptions of Theorem~\ref{thm:coalition-log-risk},
168 \[
169 \boxed{
170 k_\varepsilon
171 =
172 \min\left\{
173 |A|:
174 I(Y_t^+;\mathbf H_t\mid\mathbf H_t^A)
175 \leq\varepsilon
176 \right\}.
177 }
178 \]
179 Equivalently, \(k_\varepsilon\) is the first \(r\) for which
180 \(\Delta_{\log}^{(r)}\leq\varepsilon\).
181 \end{corollary}
182 
183 This yields a graded classification:
184 \[
185 \begin{aligned}
186 k_0=0
187 &\quad\Longrightarrow\quad
188 \text{the background law alone is sufficient},\\
189 k_0=1
190 &\quad\Longrightarrow\quad
191 \text{some individual node is sufficient},\\
192 k_0\geq2
193 &\quad\Longrightarrow\quad
194 \text{node-level relational irreducibility},\\
195 k_0=|V|
196 &\quad\Longrightarrow\quad
197 \text{every proper coalition is insufficient}.
198 \end{aligned}
199 \]
200 The last condition is much stronger and should not be inferred from the first.
201 
202 \topic{25}{Reconstructibility of the Canonical State}
203 
204 Let \(d_{\mathcal P}\) be a bounded metric generating the measurable structure
205 on \(\mathcal P(\mathsf Y)\). Define the reconstruction defect of the canonical
206 state from coalition \(A\) by
207 \[
208 \delta_A^Z
209 =
210 \inf_{\psi_A}
211 \mathbb E\!\left[
212  d_{\mathcal P}\!\left(
213  Z_t,\psi_A(\mathbf H_t^A)
214  \right)
215 \right].
216 \]
217 
218 \begin{proposition}[Sufficiency--reconstruction equivalence]
219 \label{prop:sufficiency-reconstruction}
220 For a coalition \(A\subseteq V\), the following are equivalent:
221 \begin{enumerate}
222 \item \(\mathbf H_t^A\) is sufficient for \(Y_t^+\) relative to the full
223  history;
224 \item \(Z_t\) is measurable with respect to \(\mathcal F_t^A\);
225 \item there exists a measurable \(\psi_A\) such that
226 \[
227 Z_t=\psi_A(\mathbf H_t^A)
228 \qquad\text{almost surely}.
229 \]
230 \end{enumerate}
231 Each condition implies \(\delta_A^Z=0\). Under finite-target logarithmic loss,
232 they are also equivalent to
233 \[
234 \Risk_{\log}(A)=\Risk_{\log}(V).
235 \]
236 If the reconstruction infimum is known to be attained, then
237 \(\delta_A^Z=0\) is itself equivalent to the three conditions.
238 \end{proposition}
239 
240 \begin{proof}
241 The equivalence of the first two statements is
242 Theorem~\ref{thm:predictive-state}; the second and third are two formulations
243 of measurability up to null sets. An exact reconstructor has zero expected
244 metric error. The log-risk statement follows from
245 Theorem~\ref{thm:coalition-log-risk} and the zero condition for conditional
246 mutual information. Without an attainment or compactness condition, an
247 infimum of zero alone certifies arbitrarily accurate reconstruction but need
248 not supply a single exact minimizing map.
249 \end{proof}
250 
251 For an arbitrary enriched workspace variable \(W_t\), local
252 non-reconstructibility need not imply predictive irreducibility: \(W_t\) may
253 contain task-irrelevant material. The equivalence is exact for the canonical
254 state \(Z_t\), or for a candidate \(W_t\) known to be minimally aligned with
255 it.
256 
257 \topic{26}{Matched Relational Nulls}
258 
259 A null control must preserve the local facts that the claim is not intended to
260 challenge while destroying the relational organization being tested. Merely
261 reducing model capacity or replacing the workspace by noise is not a matched
262 null.
263 
264 Let \(P\) be the intact law of
265 \((Y_t^+,\mathbf H_t^1,\ldots,\mathbf H_t^n)\). Define the local-marginal null
266 class
267 \[
268 \mathcal N_{\mathrm{marg}}(P)
269 =
270 \left\{
271 Q:
272 Q_{\mathbf H_t^i}=P_{\mathbf H_t^i}
273 \text{ for every }i
274 \right\},
275 \]
276 and the stronger local-predictive-matched class
277 \[
278 \boxed{
279 \mathcal N_{\mathrm{pred}}(P)
280 =
281 \left\{
282 Q:
283 Q_{(Y_t^+,\mathbf H_t^i)}
284 =
285 P_{(Y_t^+,\mathbf H_t^i)}
286 \text{ for every }i
287 \right\}.
288 }
289 \]
290 Because \(\mathbf H_t^i\) can include the complete local trajectory, the latter
291 class can preserve all single-component temporal and predictive behavior while
292 altering cross-component organization.
293 
294 \begin{proposition}[Local-risk invariance under predictive matching]
295 \label{prop:local-risk-null}
296 If \(Q\in\mathcal N_{\mathrm{pred}}(P)\), then for every node \(i\) and every
297 loss determined by \((Y_t^+,d(\mathbf H_t^i))\),
298 \[
299 \boxed{
300 \Risk_{\ell,Q}(\{i\})
301 =
302 \Risk_{\ell,P}(\{i\}).
303 }
304 \]
305 \end{proposition}
306 
307 \begin{proof}
308 Every such risk depends only on the joint law of
309 \((Y_t^+,\mathbf H_t^i)\), which is identical under \(P\) and \(Q\).
310 \end{proof}
311 
312 A valid null must also satisfy a declared relation-breaking condition. Examples
313 include:
314 \begin{itemize}
315 \item permuting source--target message pairings while preserving each message
316  marginal;
317 \item shuffling temporal correspondence across nodes while preserving every
318  local trajectory;
319 \item permuting role bindings while preserving the local token or state
320  distributions;
321 \item replacing an artifact by a matched-size, matched-format artifact whose
322  relation to the surrounding history has been destroyed; and
323 \item randomizing transfer channels through a physical or software
324  intervention while preserving message rate and marginal content.
325 \end{itemize}
326 
327 For a predeclared null \(Q\), define
328 \[
329 \boxed{
330 \Delta_{\mathrm{tr}}(Q)
331 =
332 \Risk_{\ell,Q}(V)
333 -
334 \Risk_{\ell,P}(V).
335 }
336 \]
337 For a finite validated panel
338 \(\mathcal Q=\{Q_1,\ldots,Q_m\}\subseteq\mathcal N_{\mathrm{pred}}(P)\), report
339 both the complete vector
340 \[
341 \boldsymbol\Delta_{\mathrm{tr}}(\mathcal Q)
342 =
343 \bigl(
344 \Delta_{\mathrm{tr}}(Q_1),\ldots,
345 \Delta_{\mathrm{tr}}(Q_m)
346 \bigr)
347 \]
348 and the robust panel minimum
349 \[
350 \Delta_{\mathrm{tr}}^{\min}(\mathcal Q)
351 =
352 \min_{Q\in\mathcal Q}\Delta_{\mathrm{tr}}(Q).
353 \]
354 The panel must be fixed independently of the observed favorable gap. Otherwise
355 null selection can manufacture apparent relational dependence.
356 
357 \topic{27}{Relational Witness Pairs}
358 
359 \begin{definition}[Relational witness pair]\label{def:relational-witness}
360 A pair of laws \((P,Q)\) is a predictive relational witness pair for the
361 decomposition \(V\) and target \(Y_t^+\) when
362 \[
363 Q\in\mathcal N_{\mathrm{pred}}(P)
364 \]
365 but
366 \[
367 \Risk_{\ell,Q}(V)
368 \neq
369 \Risk_{\ell,P}(V).
370 \]
371 The two systems agree on every single-node predictive marginal but differ in
372 full-system operative capacity.
373 \end{definition}
374 
375 \begin{example}[XOR witness pair]\label{ex:xor-witness}
376 Under the intact law \(P\), let
377 \[
378 X_1,X_2\overset{\mathrm{iid}}{\sim}\operatorname{Bernoulli}(1/2),
379 \qquad
380 Y=X_1\oplus X_2.
381 \]
382 Then each \((X_i,Y)\) pair is independent and uniform, so
383 \[
384 H_P(Y\mid X_i)=1\text{ bit},
385 \qquad
386 H_P(Y\mid X_1,X_2)=0.
387 \]
388 Let \(Q\) make \(X_1,X_2,Y\) mutually independent fair bits. Then
389 \[
390 Q_{(X_i,Y)}=P_{(X_i,Y)}
391 \qquad(i=1,2),
392 \]
393 but
394 \[
395 H_Q(Y\mid X_1,X_2)=1\text{ bit}.
396 \]
397 Therefore
398 \[
399 \Delta_{\mathrm{joint}}^{\log}=1\text{ bit},
400 \qquad
401 k_0=2,
402 \qquad
403 \Delta_{\mathrm{tr}}(Q)=1\text{ bit}.
404 \]
405 This proves predictive relational structure relative to the declared variables.
406 It does not, by itself, prove that any workspace variable is causally active.
407 \end{example}
408 
409 \begin{example}[Redundant-copy control]
410 Let \(Y\) be a fair bit and set \(X_1=X_2=Y\). The two nodes are perfectly
411 coherent and the full system predicts perfectly, but
412 \[
413 H(Y\mid X_i)=0,
414 \qquad
415 \Delta_{\mathrm{joint}}^{\log}=0,
416 \qquad
417 k_0=1.
418 \]
419 Duplication is therefore not relational irreducibility.
420 \end{example}
421 
422 The witness-pair criterion is stronger than a performance comparison against a
423 smaller model because it holds local predictive behavior fixed. It remains
424 observational. Part~VIII adds interventions and converts predictive
425 relationality into an operational causal claim.

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