Stratified Relational Workspace Theory

Download original file

This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.

Read the typeset paper

1 % This file is part of the modular source for Stratified Relational Workspace Theory.
2 % !TeX root = ../Stratified_Relational_Workspace_Theory.tex
3 \theorypart{X}{Persistence}
4 
5 The preceding parts identify exact and approximate workspace subspaces at a
6 single time. Persistence adds three further requirements. The structural
7 subspace must remain trackable while the realization changes, actual states
8 must remain near that moving subspace, and the transported content must not
9 collapse to a trivial zero state. These are distinct conditions.
10 
11 \topic{32}{Coherence-Error Dynamics}
12 
13 \subtopic{32.1 Time-indexed transfer systems}
14 
15 At time \(t\), let
16 \[
17 \mathcal C_V(t)=\bigoplus_{i\in V}C_i(t),
18 \qquad
19 \mathcal C_E(t)=\bigoplus_{e\in E}C_{t(e)}(t),
20 \]
21 and let
22 \[
23 D_t:\mathcal C_V(t)\longrightarrow\mathcal C_E(t),
24 \qquad
25 L_t=D_t^*D_t
26 \]
27 be the transfer discrepancy and transfer Laplacian. At time \(t\), the
28 combinatorial graph is held fixed while the local representations, transfers,
29 and metrics may vary. Changes of graph are treated later by a graph
30 correspondence.
31 
32 To compare successive times, choose unitary identification maps
33 \[
34 U_t^V:\mathcal C_V(t)\longrightarrow\mathcal C_V(t+1),
35 \qquad
36 U_t^E:\mathcal C_E(t)\longrightarrow\mathcal C_E(t+1).
37 \]
38 They are comparison maps, not assumptions that the physical carriers are
39 identical. Define the one-step diagram defect
40 \[
41 \boxed{
42 \Delta_t^D
43 =
44 D_{t+1}U_t^V-U_t^ED_t,
45 \qquad
46 \delta_t^D=\|\Delta_t^D\|.
47 }
48 \]
49 Exact temporal compatibility is the condition \(\Delta_t^D=0\).
50 
51 \begin{proposition}[Temporal Laplacian drift]\label{prop:temporal-laplacian-drift}
52 The carried Laplacian satisfies
53 \[
54 \boxed{
55 \left\|
56 (U_t^V)^*L_{t+1}U_t^V-L_t
57 \right\|
58 \leq
59 2\|D_t\|\delta_t^D+(\delta_t^D)^2.
60 }
61 \]
62 \end{proposition}
63 
64 \begin{proof}
65 Set
66 \[
67 B_t=(U_t^E)^*\Delta_t^D.
68 \]
69 Because the comparison maps are unitary,
70 \[
71 (U_t^E)^*D_{t+1}U_t^V=D_t+B_t,
72 \qquad
73 \|B_t\|=\delta_t^D.
74 \]
75 Hence
76 \[
77 (U_t^V)^*L_{t+1}U_t^V
78 =(D_t+B_t)^*(D_t+B_t).
79 \]
80 Subtract \(D_t^*D_t\) and apply the triangle inequality.
81 \end{proof}
82 
83 Let \(P_t\) project onto a selected rank-\(d\) low-energy workspace cluster
84 of \(L_t\), and set
85 \[
86 \mathcal W_t=\ran P_t.
87 \]
88 Write
89 \[
90 g_t=\lambda_{d+1}(L_t)-\lambda_d(L_t)>0
91 \]
92 for the separating spectral gap and
93 \[
94 \eta_t^L=2\|D_t\|\delta_t^D+(\delta_t^D)^2.
95 \]
96 
97 \begin{corollary}[Temporal workspace tracking]\label{cor:temporal-workspace-tracking}
98 If
99 \[
100 \eta_t^L<\frac{g_t}{2},
101 \]
102 then the transported and newly extracted workspace projectors satisfy
103 \[
104 \boxed{
105 q_t^{\mathrm{str}}
106 :=
107 \left\|
108 P_{t+1}-U_t^VP_t(U_t^V)^*
109 \right\|
110 \leq
111 \frac{\eta_t^L}{g_t-\eta_t^L}
112 \leq
113 \frac{2\eta_t^L}{g_t}.
114 }
115 \]
116 \end{corollary}
117 
118 \begin{proof}
119 Carry \(L_{t+1}\) back to time \(t\) using \(U_t^V\), apply
120 Theorem~\ref{thm:spectral-stability}, and conjugate the resulting projector
121 bound forward.
122 \end{proof}
123 
124 The quantity \(q_t^{\mathrm{str}}\) measures structural motion after the
125 declared coordinate change has been removed. A workspace may therefore move
126 substantially in raw coordinates while remaining structurally unchanged.
127 
128 \subtopic{32.2 State tracking inside a moving workspace}
129 
130 Let the realized vertex state satisfy
131 \[
132 x_{t+1}=F_tx_t+\xi_t,
133 \]
134 where \(F_t:\mathcal C_V(t)\to\mathcal C_V(t+1)\) is the actual update and
135 \(\xi_t\) is an unmodeled disturbance. Define the off-workspace error
136 \[
137 \boxed{
138 e_t=\|(I-P_t)x_t\|.
139 }
140 \]
141 Introduce
142 \[
143 \alpha_t
144 =
145 \left\|
146 (I-P_{t+1})F_t(I-P_t)
147 \right\|,
148 \]
149 which measures persistence of prior off-workspace error, and
150 \[
151 \ell_t
152 =
153 \left\|
154 (I-P_{t+1})F_tP_t
155 \right\|,
156 \]
157 which measures leakage of a valid workspace state under the update.
158 
159 \begin{proposition}[Grounded coherence recurrence]\label{prop:grounded-coherence-recurrence}
160 Every trajectory satisfies
161 \[
162 \boxed{
163 e_{t+1}\leq\alpha_te_t+\sigma_t,}
164 \]
165 where
166 \[
167 \boxed{
168 \sigma_t
169 =
170 \ell_t\|P_tx_t\|
171 +
172 \|(I-P_{t+1})\xi_t\|.
173 }
174 \]
175 \end{proposition}
176 
177 \begin{proof}
178 Insert \(I=P_t+(I-P_t)\) before \(x_t\), apply \(I-P_{t+1}\), and use
179 submultiplicativity:
180 \[
181 \begin{aligned}
182 e_{t+1}
183 &\leq
184 \|(I-P_{t+1})F_t(I-P_t)x_t\|\\
185 &\quad+
186 \|(I-P_{t+1})F_tP_tx_t\|
187 +
188 \|(I-P_{t+1})\xi_t\|.
189 \end{aligned}
190 \]
191 The displayed recurrence follows from the definitions of \(\alpha_t\) and
192 \(\ell_t\).
193 \end{proof}
194 
195 This derivation gives the contraction and disturbance terms an operational
196 meaning. The scalar recurrence is not an independent postulate. It follows
197 from an update rule, a moving workspace projector, and a declared disturbance
198 model.
199 
200 \subtopic{32.3 Persistence theorem \(T_6\)}
201 
202 \begin{theorem}[Nonautonomous persistence theorem]\label{thm:persistence}
203 Let \(e_t\geq0\) satisfy
204 \[
205 e_{t+1}\leq\alpha_te_t+\sigma_t,
206 \qquad
207 \alpha_t,\sigma_t\geq0.
208 \]
209 Then, for every \(t\geq1\),
210 \[
211 \boxed{
212 \begin{aligned}
213 e_t
214 \leq
215 &\left(\prod_{r=0}^{t-1}\alpha_r\right)e_0\\
216 &+
217 \sum_{j=0}^{t-1}
218 \left(\prod_{r=j+1}^{t-1}\alpha_r\right)\sigma_j,
219 \end{aligned}
220 }
221 \]
222 where an empty product equals one. If
223 \[
224 \alpha_t\leq\bar\alpha<1,
225 \qquad
226 \sigma_t\leq\bar\sigma,
227 \]
228 then
229 \[
230 \boxed{
231 e_t
232 \leq
233 \bar\alpha^te_0
234 +
235 \frac{\bar\sigma(1-\bar\alpha^t)}{1-\bar\alpha},
236 }
237 \]
238 and consequently
239 \[
240 \boxed{
241 \limsup_{t\to\infty}e_t
242 \leq
243 \frac{\bar\sigma}{1-\bar\alpha}.
244 }
245 \]
246 \end{theorem}
247 
248 \begin{proof}
249 Repeated substitution yields
250 \[
251 \begin{aligned}
252 e_t
253 &\leq
254 \alpha_{t-1}\cdots\alpha_0e_0
255 +
256 \alpha_{t-1}\cdots\alpha_1\sigma_0
257 +
258 \cdots
259 +
260 \sigma_{t-1},
261 \end{aligned}
262 \]
263 which is the first formula. Under the uniform bounds, dominate each product by
264 the corresponding power of \(\bar\alpha\) and sum the geometric series.
265 \end{proof}
266 
267 \begin{corollary}[Vanishing-disturbance convergence]\label{cor:vanishing-disturbance}
268 If \(\alpha_t\leq\bar\alpha<1\) and \(\sigma_t\to0\), then
269 \[
270 e_t\to0.
271 \]
272 More generally, if \(\sum_{t\geq0}\sigma_t<\infty\), then the disturbance
273 contribution in Theorem~\ref{thm:persistence} vanishes asymptotically.
274 \end{corollary}
275 
276 \begin{proof}
277 Split the convolution sum into a finite early part, whose coefficients decay
278 to zero, and a late tail on which \(\sigma_t\) is uniformly small. The
279 summable case follows by the same argument.
280 \end{proof}
281 
282 \subtopic{32.4 Mean-square persistence}
283 
284 For stochastic systems, pathwise bounded disturbances may be too strong. Let
285 \(\{\mathcal F_t\}\) be a filtration containing the history through time
286 \(t\).
287 
288 \begin{proposition}[Mean-square persistence]\label{prop:mean-square-persistence}
289 Suppose
290 \[
291 \mathbb E[e_{t+1}^2\mid\mathcal F_t]
292 \leq
293 q e_t^2+v_t,
294 \qquad
295 0\leq q<1,
296 \]
297 and \(\mathbb E[v_t]\leq\bar v\). Then
298 \[
299 \boxed{
300 \mathbb E[e_t^2]
301 \leq
302 q^t\mathbb E[e_0^2]
303 +
304 \frac{\bar v(1-q^t)}{1-q},
305 }
306 \]
307 so
308 \[
309 \limsup_{t\to\infty}\mathbb E[e_t^2]
310 \leq
311 \frac{\bar v}{1-q}.
312 \]
313 \end{proposition}
314 
315 \begin{proof}
316 Take expectations, iterate the resulting deterministic recurrence, and sum a
317 geometric series.
318 \end{proof}
319 
320 \subtopic{32.5 The non-collapse condition}
321 
322 Low off-workspace error is not sufficient for meaningful persistence. The
323 zero update \(F_t=0\) drives every state exactly into every linear workspace
324 while destroying all content. Define the internal transport
325 \[
326 B_t=P_{t+1}F_tP_t:\mathcal W_t\longrightarrow\mathcal W_{t+1}
327 \]
328 and its non-collapse margin
329 \[
330 \boxed{
331 \mu_t^{\mathrm{keep}}
332 =
333 \sigma_{\min}(B_t|_{\mathcal W_t}).
334 }
335 \]
336 A content-preserving persistence claim requires
337 \[
338 \mu_t^{\mathrm{keep}}>0
339 \]
340 for the relevant interval, or an information-theoretic replacement when the
341 workspace state is stochastic. Structural tracking, state tracking, and
342 content retention are therefore separate gates.
343 
344 A finite-horizon persistence certificate may be recorded as
345 \[
346 \boxed{
347 \mathcal P_{0:T}
348 =
349 \left(
350 \max_{t<T}q_t^{\mathrm{str}},
351 \max_{t\leq T}e_t,
352 \min_{t<T}\mu_t^{\mathrm{keep}},
353 \sum_{t<T}\sigma_t
354 \right).
355 }
356 \]
357 No single coordinate of this vector subsumes the others.
358 
359 \topic{33}{Interpretation of the Contraction Parameter}
360 
361 The inequality
362 \[
363 \alpha_t<1
364 \]
365 means that deviations already outside the current workspace are damped by the
366 next update. Mechanisms that can reduce \(\alpha_t\) or \(\sigma_t\) include
367 \[
368 \begin{aligned}
369 &\text{verification and error feedback},\\
370 &\text{shared notation and stable interfaces},\\
371 &\text{artifact persistence and version control},\\
372 &\text{proof checking and experimental replication},\\
373 &\text{handoff reconstruction and redundancy}.
374 \end{aligned}
375 \]
376 Sources of disturbance include
377 \[
378 \begin{aligned}
379 &\text{context loss and representational drift},\\
380 &\text{miscommunication and transfer-map error},\\
381 &\text{participant or model replacement},\\
382 &\text{artifact corruption},\\
383 &\text{unsupported inference and objective drift}.
384 \end{aligned}
385 \]
386 
387 \subtopic{33.1 Four persistence levels}
388 
389 The theory distinguishes:
390 \[
391 \begin{array}{ll}
392 \text{structural persistence}
393 & q_t^{\mathrm{str}}\text{ remains small},\\
394 \text{state persistence}
395 & e_t\text{ remains small},\\
396 \text{content persistence}
397 & \mu_t^{\mathrm{keep}}\text{ remains bounded away from zero},\\
398 \text{operational persistence}
399 & \text{predictive and causal capacities remain matched}.
400 \end{array}
401 \]
402 The first three are developed in this part. Operational matching is incorporated
403 into the identity theory of Part~XII.
404 
405 \subtopic{33.2 Moving-line example}
406 
407 Let \(\mathcal C_V(t)=\mathbb R^2\) and let \(P_t\) project onto
408 \[
409 u_t=(\cos\theta_t,\sin\theta_t).
410 \]
411 If \(U_t^V\) is rotation through
412 \(\theta_{t+1}-\theta_t\) and \(F_t=U_t^V\), then a state
413 \(x_t=u_t\) obeys
414 \[
415 e_t=e_{t+1}=0,
416 \qquad
417 \mu_t^{\mathrm{keep}}=1.
418 \]
419 The workspace changes raw coordinates but not abstract identity. If instead
420 \(F_t=I\), then
421 \[
422 e_{t+1}
423 =
424 |\sin(\theta_{t+1}-\theta_t)|.
425 \]
426 The same structural motion now produces a state-tracking defect because the
427 actual update fails to transport the content with the workspace.

Back to source controls

Back to Stratified Relational Workspace Theory