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{XI}{Reconstruction and Resilience}
4 
5 Persistence can be maintained by direct transport, but it can also be restored
6 from distributed traces after a disruption. Reconstruction theory quantifies
7 when a coalition contains enough information to recover the abstract core and
8 how noise is amplified by that recovery.
9 
10 \topic{34}{Core Encoding Maps}
11 
12 Let \(C_*\) be a finite-dimensional abstract core. The exact static theory
13 provides a complete local chart
14 \[
15 \overline E_i
16 =
17 \ev_i\circ\Psi:C_*\longrightarrow C_i,
18 \]
19 where \(\Psi:C_*\to\Gamma(H)\) is the root-to-section isomorphism. A participant
20 or artifact need not expose that complete chart. Let
21 \[
22 O_i:C_i\longrightarrow\mathsf Y_i
23 \]
24 be the declared observation or access map and define the \emph{accessible
25 encoding}
26 \[
27 \boxed{
28 E_i=O_i\overline E_i:C_*\longrightarrow\mathsf Y_i.
29 }
30 \]
31 For notational economy, the direct sums below use the accessible spaces
32 \(\mathsf Y_i\). Thus injectivity of a full structural chart does not make a
33 singleton operationally reconstructive unless its access map preserves the
34 whole core.
35 
36 For a coalition \(S\subseteq V\), define the analysis operator
37 \[
38 A_S:C_*\longrightarrow\bigoplus_{i\in S}\mathsf Y_i,
39 \qquad
40 A_Sc=(E_ic)_{i\in S},
41 \]
42 and the coalition frame operator
43 \[
44 \boxed{
45 F_S=A_S^*A_S=\sum_{i\in S}E_i^*E_i.
46 }
47 \]
48 Set
49 \[
50 \alpha_S=\lambda_{\min}(F_S),
51 \qquad
52 \beta_S=\lambda_{\max}(F_S).
53 \]
54 Thus
55 \[
56 \alpha_S\|c\|^2
57 \leq
58 \|A_Sc\|^2
59 \leq
60 \beta_S\|c\|^2.
61 \]
62 
63 \begin{proposition}[Coalition reconstruction criterion]\label{prop:coalition-frame-criterion}
64 For a coalition \(S\), the following are equivalent:
65 \begin{enumerate}
66 \item \(A_S\) is injective;
67 \item \(F_S\) is positive definite;
68 \item \(\alpha_S>0\);
69 \item there exists a linear decoder \(R_S\) with \(R_SA_S=I_{C_*}\).
70 \end{enumerate}
71 When these conditions hold, the canonical decoder is
72 \[
73 \boxed{
74 R_S=A_S^\dagger=F_S^{-1}A_S^*.
75 }
76 \]
77 \end{proposition}
78 
79 \begin{proof}
80 In finite dimensions,
81 \[
82 \ker F_S=\ker A_S
83 \]
84 because
85 \[
86 \langle c,F_Sc\rangle=\|A_Sc\|^2.
87 \]
88 This proves the equivalence of the first three statements. An injective linear
89 map admits a left inverse, while a left inverse implies injectivity. The
90 formula for the Moore--Penrose left inverse follows from full column rank.
91 \end{proof}
92 
93 The lower frame bound \(\alpha_S\) measures recoverability. The upper bound
94 \(\beta_S\) measures amplification in the forward encoding. Their ratio
95 \[
96 \kappa_S=\sqrt{\frac{\beta_S}{\alpha_S}}
97 \]
98 is the coalition condition number.
99 
100 \topic{35}{Stable Reconstruction Condition}
101 
102 \subtopic{35.1 Stable reconstruction theorem \(T_7\)}
103 
104 \begin{theorem}[Stable core reconstruction]\label{thm:stable-reconstruction}
105 Assume \(\alpha_S>0\) and observe
106 \[
107 y=A_Sc+\eta.
108 \]
109 With the canonical estimate
110 \[
111 \widehat c=A_S^\dagger y,
112 \]
113 one has
114 \[
115 \boxed{
116 \|\widehat c-c\|
117 \leq
118 \frac{\|\eta\|}{\sqrt{\alpha_S}}.
119 }
120 \]
121 \end{theorem}
122 
123 \begin{proof}
124 Since \(A_S^\dagger A_S=I_{C_*}\),
125 \[
126 \widehat c-c=A_S^\dagger\eta.
127 \]
128 The singular values of \(A_S\) are the square roots of the eigenvalues of
129 \(F_S\), so
130 \[
131 \|A_S^\dagger\|
132 =
133 \frac{1}{\sigma_{\min}(A_S)}
134 =
135 \frac{1}{\sqrt{\alpha_S}}.
136 \]
137 \end{proof}
138 
139 \begin{corollary}[Encoding mismatch]\label{cor:encoding-mismatch}
140 If the true observation is
141 \[
142 y=(A_S+\Delta_S)c+\eta
143 \]
144 but the nominal decoder \(A_S^\dagger\) is used, then
145 \[
146 \boxed{
147 \|\widehat c-c\|
148 \leq
149 \frac{\|\Delta_S\|\,\|c\|+\|\eta\|}{\sqrt{\alpha_S}}.
150 }
151 \]
152 \end{corollary}
153 
154 \begin{proof}
155 Now
156 \[
157 \widehat c-c=A_S^\dagger\Delta_Sc+A_S^\dagger\eta,
158 \]
159 and the result follows from Theorem~\ref{thm:stable-reconstruction}.
160 \end{proof}
161 
162 \subtopic{35.2 Reconstruction as operational recovery}
163 
164 Geometric recovery becomes behaviorally relevant when the workspace state
165 controls a predictive kernel. Let
166 \[
167 K_w=\mathbb P(Y\in\cdot\mid W=w)
168 \]
169 and suppose
170 \[
171 \TV(K_w,K_{w'})\leq L_Kd_W(w,w').
172 \]
173 
174 \begin{proposition}[Exact reconstruction implies coalition sufficiency]
175 \label{prop:exact-reconstruction-sufficiency}
176 Let \(W\) be sufficient for \(Y\) relative to the full history \(\mathbf H\):
177 \[
178 Y\indep\mathbf H\mid W.
179 \]
180 If a coalition reconstructs the workspace exactly,
181 \[
182 W=R_S(\mathbf H^S)
183 \qquad\text{almost surely},
184 \]
185 then
186 \[
187 \boxed{
188 Y\indep\mathbf H\mid\mathbf H^S.
189 }
190 \]
191 Under logarithmic loss,
192 \[
193 \boxed{
194 \Risk_{\log}(S)=\Risk_{\log}(V).
195 }
196 \]
197 \end{proposition}
198 
199 \begin{proof}
200 Sufficiency means that the conditional target law given \(\mathbf H\) is a
201 measurable function of \(W\). Exact reconstruction makes that function
202 measurable with respect to \(\mathbf H^S\), which is the asserted conditional
203 independence. The log-risk equality follows from
204 Theorem~\ref{thm:coalition-log-risk}.
205 \end{proof}
206 
207 \begin{proposition}[Reconstruction-to-risk transfer]\label{prop:reconstruction-risk-transfer}
208 Let \(W\) be sufficient for a target \(Y\), let losses take values in
209 \([0,1]\), and let a coalition construct \(\widehat W\) from its local data.
210 Then its optimal Bayes risk satisfies
211 \[
212 \boxed{
213 0\leq
214 \Risk(S)-\Risk(W)
215 \leq
216 2L_K\,\mathbb E[d_W(W,\widehat W)].
217 }
218 \]
219 \end{proposition}
220 
221 \begin{proof}
222 For every pair \(w,\widehat w\), using the Bayes action for
223 \(K_{\widehat w}\) under the true law \(K_w\) incurs conditional regret at
224 most
225 \[
226 2\TV(K_w,K_{\widehat w}).
227 \]
228 This follows by inserting and subtracting the risk under
229 \(K_{\widehat w}\) for both the chosen action and the Bayes action for
230 \(K_w\). Average the bound and use the Lipschitz assumption. The coalition's
231 optimal rule performs at least as well as this decoder-based rule.
232 \end{proof}
233 
234 This proposition links Part~XI to the causal-state theory of Part~VI. A small
235 geometric reconstruction error is operationally meaningful only when the
236 outcome kernel changes continuously with workspace state.
237 
238 \topic{36}{k-Failure Resilience}
239 
240 For \(k\geq0\), define the worst surviving lower frame bound
241 \[
242 \boxed{
243 \alpha^{(k)}
244 =
245 \min_{\substack{A\subseteq V\\|A|\leq k}}
246 \alpha_{V\setminus A}.
247 }
248 \]
249 The sequence \(\alpha^{(k)}\) is nonincreasing in \(k\).
250 
251 \begin{theorem}[Uniform erasure resilience]\label{thm:uniform-erasure-resilience}
252 The encoding family is exactly reconstructible after every loss of at most
253 \(k\) nodes if and only if
254 \[
255 \boxed{\alpha^{(k)}>0.}
256 \]
257 If surviving observations have the form
258 \[
259 y_{V\setminus A}=A_{V\setminus A}c+\eta_A,
260 \qquad
261 |A|\leq k,
262 \]
263 then the canonical survivor decoder obeys the uniform estimate
264 \[
265 \boxed{
266 \|\widehat c_A-c\|
267 \leq
268 \frac{\|\eta_A\|}{\sqrt{\alpha^{(k)}}}.
269 }
270 \]
271 \end{theorem}
272 
273 \begin{proof}
274 Exact reconstruction for a particular survivor set is equivalent to a
275 positive lower frame bound by Proposition~\ref{prop:coalition-frame-criterion}.
276 Taking the minimum over all permitted erasures gives the first claim. The
277 uniform noise estimate follows from Theorem~\ref{thm:stable-reconstruction}.
278 \end{proof}
279 
280 \begin{proposition}[Dimensional obstruction]\label{prop:dimension-obstruction}
281 Let \(d=\dim C_*\). If
282 \[
283 \sum_{i\in S}\dim \mathsf Y_i<d,
284 \]
285 then \(\alpha_S=0\), so \(S\) cannot reconstruct the core. Consequently,
286 \(k\)-failure resilience requires
287 \[
288 \sum_{i\in V\setminus A}\dim \mathsf Y_i\geq d
289 \]
290 for every \(A\subseteq V\) with \(|A|\leq k\).
291 \end{proposition}
292 
293 \begin{proof}
294 The rank of \(A_S\) is at most the dimension of its codomain. If that
295 dimension is below \(d\), then \(A_S\) is not injective.
296 \end{proof}
297 
298 Define the accessible block support of \(c\neq0\) by
299 \[
300 \operatorname{supp}_E(c)
301 =
302 \{i\in V:E_ic\neq0\}
303 \]
304 and define the node-erasure distance
305 \[
306 \boxed{
307 d_{\mathrm{er}}
308 =
309 \min_{c\neq0}|\operatorname{supp}_E(c)|.
310 }
311 \]
312 
313 \begin{theorem}[Exact erasure-distance criterion]
314 \label{thm:erasure-distance}
315 Exact reconstruction after every loss of at most \(k\) nodes is possible if
316 and only if
317 \[
318 \boxed{
319 d_{\mathrm{er}}>k.
320 }
321 \]
322 \end{theorem}
323 
324 \begin{proof}
325 If \(d_{\mathrm{er}}\leq k\), choose \(c\neq0\) whose nonzero accessible
326 blocks are contained in a set \(A\) with \(|A|\leq k\). The surviving analysis
327 operator satisfies \(A_{V\setminus A}c=0\), so it cannot distinguish \(c\)
328 from zero. Conversely, if some erasure set \(A\) of size at most \(k\) destroys
329 injectivity, there exists \(c\neq0\) with \(E_ic=0\) for every \(i\notin A\).
330 Thus \(\operatorname{supp}_E(c)\subseteq A\), and
331 \(d_{\mathrm{er}}\leq k\).
332 \end{proof}
333 
334 The distance criterion is combinatorial. The frame quantity
335 \(\alpha^{(k)}\) is quantitative. A system may tolerate \(k\) erasures exactly
336 while amplifying noise severely because \(\alpha^{(k)}\) is close to zero.
337 
338 \subtopic{36.1 Reconstructive handoff after replacement}
339 
340 Suppose node \(r\) is removed and the surviving coalition is
341 \[
342 S=V\setminus\{r\}.
343 \]
344 A replacement node \(r'\) is to receive the encoding
345 \[
346 E_{r'}':C_*\to\mathsf Y_{r'}'.
347 \]
348 
349 \begin{corollary}[Handoff initialization]\label{cor:handoff-initialization}
350 If \(\alpha_S>0\) and the survivors observe \(y=A_Sc+\eta\), initialize the
351 replacement by
352 \[
353 x_{r'}^{\mathrm{init}}
354 =E_{r'}'A_S^\dagger y.
355 \]
356 Then
357 \[
358 \boxed{
359 \|x_{r'}^{\mathrm{init}}-E_{r'}'c\|
360 \leq
361 \frac{\|E_{r'}'\|\,\|\eta\|}{\sqrt{\alpha_S}}.
362 }
363 \]
364 \end{corollary}
365 
366 This is the basic mathematical form of artifact-mediated or
367 coalition-mediated handoff. It does not require the removed component to
368 transfer its full internal state to the replacement.
369 
370 \topic{37}{Redundancy--Irreducibility Tension}
371 
372 Define the minimum reconstructing coalition size
373 \[
374 \boxed{
375 m_{\mathrm{rec}}
376 =
377 \min\{|S|:\alpha_S>0\}
378 }
379 \]
380 and the maximum exact erasure tolerance
381 \[
382 \boxed{
383 k_{\mathrm{erase}}
384 =
385 \max\{k:\alpha^{(k)}>0\}.
386 }
387 \]
388 The condition
389 \[
390 m_{\mathrm{rec}}\geq2
391 \]
392 means that no individual encoding is sufficient for exact core recovery. The
393 condition
394 \[
395 k_{\mathrm{erase}}\geq1
396 \]
397 means that the system survives at least one node loss. These can hold
398 simultaneously because nonlocality of encoding is not the same as fragility.
399 
400 \subtopic{37.1 Three-node distributed-frame example}
401 
402 Let
403 \[
404 C_*=\mathbb R^2
405 \]
406 and define scalar node encodings
407 \[
408 E_1(c_1,c_2)=c_1,
409 \qquad
410 E_2(c_1,c_2)=c_2,
411 \qquad
412 E_3(c_1,c_2)=c_1+c_2.
413 \]
414 Every singleton has rank one, so no node reconstructs the two-dimensional
415 core. Every pair does reconstruct. For \(S=\{1,2\}\),
416 \[
417 F_S=I,
418 \qquad
419 \alpha_S=1.
420 \]
421 For \(S=\{1,3\}\) or \(S=\{2,3\}\), the frame eigenvalues are
422 \[
423 \frac{3\pm\sqrt5}{2}.
424 \]
425 Therefore
426 \[
427 \boxed{
428 m_{\mathrm{rec}}=2,
429 \qquad
430 k_{\mathrm{erase}}=1,
431 \qquad
432 d_{\mathrm{er}}=2,
433 \qquad
434 \alpha^{(1)}=\frac{3-\sqrt5}{2}.
435 }
436 \]
437 This realizes a workspace core that is irreducible to any singleton while
438 remaining reconstructible after any one node fails.
439 
440 \subtopic{37.2 Relation to predictive irreducibility}
441 
442 The geometric quantity \(m_{\mathrm{rec}}\) and the predictive quantity
443 \(k_0\) from Part~VII answer different questions. The first asks whether a
444 coalition can reconstruct a declared core coordinate. The second asks whether
445 a coalition is sufficient for a declared future target. Equality between them
446 requires structural--operational alignment and regularity such as
447 Proposition~\ref{prop:reconstruction-risk-transfer}. Neither quantity should
448 be substituted for the other without those assumptions.

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