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{XV}{Boundary Formation and Stability}
4 
5 \topic{50}{Candidate Workspace Boundaries}
6 
7 A boundary is not merely a subset of physical components. It is a subset
8 relative to a declared node decomposition, prediction target, intervention
9 family, environment distribution, and time scale.
10 
11 \begin{definition}[Boundary problem]\label{def:boundary-problem}
12 A boundary problem is a tuple
13 \[
14 \mathfrak B
15 =
16 \left(
17 V,
18 \mathfrak S,
19 \tau,
20 \mathcal Y,
21 \mathcal I,
22 \Pi,
23 \mathcal G_B
24 \right),
25 \]
26 where \(\mathfrak S\subseteq2^V\) is the admissible candidate family,
27 \(\tau\) is the temporal scale, \(\mathcal Y\) is the operational target,
28 \(\mathcal I\) is the feasible intervention family, \(\Pi\) is the
29 environment distribution, and \(\mathcal G_B\) is a symmetry group under
30 which boundaries are considered equivalent.
31 \end{definition}
32 
33 Unless the application requires otherwise, the candidate family excludes
34 \[
35 \varnothing,
36 \qquad
37 V,
38 \qquad
39 \{i\}\ \text{for }i\in V.
40 \]
41 This anti-triviality rule prevents the empty set, the entire realization, or a
42 single centralized controller from winning solely through a degenerate score.
43 
44 \subtopic{50.1 Observational recurrence, dependence, and outward influence}
45 
46 For \(S\in\mathfrak S\), define
47 \[
48 \Ret_\tau(S)
49 =
50 I\!\left(
51 X_S(t+\tau);X_S(t)
52 \mid X_{\bar S}(t),E_t
53 \right),
54 \]
55 \[
56 \Dep_\tau(S)
57 =
58 I\!\left(
59 X_S(t+\tau);X_{\bar S}(t)
60 \mid X_S(t),E_t
61 \right),
62 \]
63 \[
64 \Out_\tau(S)
65 =
66 I\!\left(
67 X_{\bar S}(t+\tau);X_S(t)
68 \mid X_{\bar S}(t),E_t
69 \right).
70 \]
71 These quantities are observational. They can be nonzero because of common
72 causes, selection, or feedback paths and therefore cannot alone establish a
73 causal boundary.
74 
75 \subtopic{50.2 Causal boundary terms}
76 
77 Let \(\iota_{S,x}\) be a feasible intervention that changes the internal
78 workspace coordinate of \(S\) to \(x\) while matching background variables and
79 intervention cost. Define outward causal efficacy
80 \[
81 \boxed{
82 \COut_\tau(S)
83 =
84 \mathbb E_{x,x'\sim\nu_S}
85 D\!\left(
86 \mathcal L^{\iota_{S,x}}
87 (X_{\bar S}(t+\tau)),
88 \mathcal L^{\iota_{S,x'}}
89 (X_{\bar S}(t+\tau))
90 \right).
91 }
92 \]
93 Define inward causal dependence analogously by intervening on \(\bar S\) and
94 measuring the future law of \(X_S\):
95 \[
96 \boxed{
97 \CIn_\tau(S)
98 =
99 \mathbb E_{u,u'\sim\nu_{\bar S}}
100 D\!\left(
101 \mathcal L^{\iota_{\bar S,u}}
102 (X_S(t+\tau)),
103 \mathcal L^{\iota_{\bar S,u'}}
104 (X_S(t+\tau))
105 \right).
106 }
107 \]
108 Neither term should be minimized to zero in general. A useful workspace may
109 receive substantial external input. The boundary objective must instead
110 express the intended balance between internal recurrence, external dependence,
111 and outward efficacy.
112 
113 \subtopic{50.3 Structural and anti-centralization gates}
114 
115 For the induced subsystem on \(S\), define
116 \[
117 \mathsf{Coh}(S)
118 =
119 \operatorname{tr}
120 \left(
121 P_{S,d}L_SP_{S,d}
122 \right),
123 \]
124 where \(P_{S,d}\) projects onto the selected low-energy workspace modes. Let
125 \[
126 \mathsf{Cut}(S)
127 =
128 \sum_{e:s(e)\in S,\ t(e)\notin S}w_e
129 +
130 \sum_{e:s(e)\notin S,\ t(e)\in S}w_e.
131 \]
132 Let \(k_\varepsilon(S)\) be the minimal sufficient coalition size inside the
133 candidate subsystem. A candidate is admissible only if
134 \[
135 \operatorname{SRW}_{\Theta_S}(W_S)=1,
136 \qquad
137 k_\varepsilon(S)\geq2,
138 \]
139 unless the claim explicitly concerns a centralized workspace.
140 
141 \topic{51}{Boundary Objective, Recovery, and Persistence}
142 
143 A general score is
144 \[
145 \boxed{
146 \begin{aligned}
147 \mathcal B_t(S)
148 ={}&
149  a_R\Ret_\tau(S)
150 +a_O\Out_\tau(S)
151 +a_C\COut_\tau(S)\\
152 &-a_D\Dep_\tau(S)
153 -a_I\CIn_\tau(S)
154 -a_L\mathsf{Coh}(S)\\
155 &-a_K\mathsf{Cut}(S)
156 -a_Z\mathsf{Size}(S)
157 -a_T\Instability_t(S).
158 \end{aligned}
159 }
160 \]
161 The coefficients and all normalizations are part of the boundary problem. No
162 single choice is universal. The selected boundary is
163 \[
164 \widehat S_t
165 \in
166 \argmax_{S\in\mathfrak S_{\mathrm{pass}}}
167 \widehat{\mathcal B}_t(S),
168 \]
169 where \(\mathfrak S_{\mathrm{pass}}\) contains only candidates that satisfy the
170 required workspace and anti-centralization gates.
171 
172 \subtopic{51.1 Boundary equivalence and identifiability}
173 
174 If a symmetry \(g\in\mathcal G_B\) leaves the full observational and
175 interventional law invariant, \(S\) and \(gS\) cannot be distinguished by the
176 declared experiment. The identifiable boundary is then the orbit
177 \[
178 [S]_{\mathcal G_B}
179 =
180 \{gS:g\in\mathcal G_B\}.
181 \]
182 A unique subset should not be claimed when only an equivalence class is
183 identifiable.
184 
185 \subtopic{51.2 Stable boundary recovery \(T_{17}\) }
186 
187 \begin{theorem}[Held-out boundary recovery]\label{thm:heldout-boundary-recovery}
188 Let \(\mathfrak S_{\mathrm{pass}}\) be finite and let
189 \[
190 S_t^*
191 =
192 \argmax_{S\in\mathfrak S_{\mathrm{pass}}}
193 \mathcal B_t(S)
194 \]
195 be unique. Define the population margin
196 \[
197 \Delta_t^B
198 =
199 \mathcal B_t(S_t^*)
200 -
201 \max_{S\neq S_t^*}\mathcal B_t(S)>0.
202 \]
203 If
204 \[
205 \sup_{S\in\mathfrak S_{\mathrm{pass}}}
206 \left|
207 \widehat{\mathcal B}_t(S)-\mathcal B_t(S)
208 \right|
209 \leq\varepsilon_t,
210 \]
211 then every empirical maximizer \(\widehat S_t\) satisfies
212 \[
213 \boxed{
214 \mathcal B_t(S_t^*)-
215 \mathcal B_t(\widehat S_t)
216 \leq2\varepsilon_t.
217 }
218 \]
219 If \(2\varepsilon_t<\Delta_t^B\), then
220 \[
221 \boxed{\widehat S_t=S_t^*.}
222 \]
223 \end{theorem}
224 
225 \begin{proof}
226 Apply Theorem~\ref{thm:uniform-structure-selection} to
227 \(J(S)=-\mathcal B_t(S)\).
228 \end{proof}
229 
230 The theorem converts uncertainty in mutual-information estimation, causal
231 interventions, structural scores, and stability penalties into an explicit
232 boundary-identification condition. It also exposes when the data are
233 insufficient: a small margin makes the boundary intrinsically fragile.
234 
235 \subtopic{51.3 Temporal boundary stability}
236 
237 \begin{theorem}[Boundary persistence under score drift]
238 \label{thm:boundary-drift-stability}
239 Assume \(S_t^*\) is the unique maximizer of \(\mathcal B_t\) with margin
240 \(\Delta_t^B\), and suppose
241 \[
242 \sup_{S\in\mathfrak S_{\mathrm{pass}}}
243 |\mathcal B_{t+1}(S)-\mathcal B_t(S)|
244 \leq\delta_t.
245 \]
246 If
247 \[
248 2\delta_t<\Delta_t^B,
249 \]
250 then
251 \[
252 \boxed{S_{t+1}^*=S_t^*.}
253 \]
254 More generally,
255 \[
256 \mathcal B_{t+1}(S_{t+1}^*)
257 -
258 \mathcal B_{t+1}(S_t^*)
259 \leq2\delta_t.
260 \]
261 \end{theorem}
262 
263 \begin{proof}
264 For every competitor \(S\neq S_t^*\),
265 \[
266 \begin{aligned}
267 \mathcal B_{t+1}(S_t^*)-
268 \mathcal B_{t+1}(S)
269 &\geq
270 \mathcal B_t(S_t^*)-
271 \mathcal B_t(S)-2\delta_t\\
272 &\geq
273 \Delta_t^B-2\delta_t>0.
274 \end{aligned}
275 \]
276 Hence the maximizer is unchanged. The approximate bound follows by comparing
277 both time-indexed scores at their respective maximizers.
278 \end{proof}
279 
280 If the score depends on a parameter \(\vartheta_t\) and satisfies
281 \[
282 \sup_S
283 |\mathcal B_{\vartheta}(S)-\mathcal B_{\vartheta'}(S)|
284 \leq
285 L_Bd(\vartheta,\vartheta'),
286 \]
287 then the sufficient stability condition becomes
288 \[
289 2L_Bd(\vartheta_t,\vartheta_{t+1})
290 <
291 \Delta_t^B.
292 \]
293 
294 \subtopic{51.4 Analyst-discovered, system-instantiated, and self-maintained boundaries}
295 
296 Three claims must remain separate.
297 
298 \begin{description}[leftmargin=2.1em,style=nextline]
299 \item[Analyst-discovered boundary.]
300 An external procedure fits \(\widehat S\) after observing the system. The
301 system need not represent or use that boundary.
302 
303 \item[System-instantiated boundary.]
304 The system contains a channel, mask, or routing variable causally equivalent to
305 \(\widehat S\), and interventions on that variable alter downstream operation.
306 
307 \item[Self-maintained boundary.]
308 The system detects degradation of its boundary certificate and takes actions
309 that restore membership, transfer, or reconstruction properties.
310 \end{description}
311 
312 Only the second and third claims concern a boundary realized within the
313 system. A post hoc partition that predicts behavior well is not by itself a
314 self-discovered boundary.
315 
316 \subtopic{51.5 Boundary formation certificate}
317 
318 A boundary \(\widehat S\) is certified as endogenously formed only if:
319 \begin{enumerate}
320 \item the membership mask was not directly encoded in \(\Sigma_B\) or
321  \(\Sigma_Y\);
322 \item \(e_B^{\mathrm{MDL}}\) and \(e_B^{\mathrm{causal}}\) exceed declared
323  thresholds;
324 \item held-out recovery satisfies Theorem~\ref{thm:heldout-boundary-recovery};
325 \item the boundary remains stable under admissible environment drift or its
326  changes are predicted by score-margin closure;
327 \item matched history-shuffle and label-leakage controls fail to reproduce
328  the result, as do full-system, empty-set, singleton, and centralized
329  boundary nulls;
330 \item the induced subsystem satisfies its workspace certificate; and
331 \item a system-instantiation intervention demonstrates that the boundary is
332  used by the system rather than only by the analyst.
333 \end{enumerate}
334 
335 The results below establish the statistical and deterministic stability components. They
336 does not prove that an arbitrary adaptive system will spontaneously learn such
337 a boundary.

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