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{XII}{Identity Through Time and Replacement}
4 
5 Identity is treated as invariance under a structured correspondence, not as
6 literal persistence of tokens or carriers. Exact correspondences form a
7 groupoid. Approximate identity is measured by structural, operational,
8 causal, and reconstructive defects that must be reported separately before
9 they are combined.
10 
11 \topic{38}{Temporal Realization Maps}
12 
13 Let
14 \[
15 \mathfrak R=(G,\mathbf C,D,\rho,\mathcal I,\mathcal K)
16 \]
17 and
18 \[
19 \mathfrak R'=(G',\mathbf C',D',\rho',\mathcal I',\mathcal K')
20 \]
21 be two realizations. A correspondence begins with a graph isomorphism
22 \(\varphi:G\to G'\). After relabeling by \(\varphi\), let
23 \[
24 J_i:C_i\to C_i'
25 \]
26 be unitary local identifications and define
27 \[
28 J_V=\bigoplus_{i\in V}J_i,
29 \qquad
30 J_E=\bigoplus_{e\in E}J_{t(e)}.
31 \]
32 
33 \begin{definition}[Exact structural correspondence]\label{def:exact-structural-correspondence}
34 The correspondence is structurally exact when
35 \[
36 \boxed{D'J_V=J_ED.}
37 \]
38 It is operation exact for \(a\in\mathcal A\) when
39 \[
40 \boxed{\rho_V'(a)J_V=J_V\rho_V(a).}
41 \]
42 \end{definition}
43 
44 \begin{theorem}[Exact replacement isomorphism]\label{thm:exact-replacement-isomorphism}
45 Under an exact structural correspondence,
46 \[
47 \boxed{
48 \Gamma(D')=J_V\Gamma(D),
49 \qquad
50 L'J_V=J_VL.
51 }
52 \]
53 Consequently, every spectral workspace subspace is transported exactly:
54 \[
55 P_I(L')=J_VP_I(L)J_V^*
56 \]
57 for every spectral interval \(I\). If the correspondence is also operation
58 exact, then the transported workspace actions are intertwined exactly.
59 \end{theorem}
60 
61 \begin{proof}
62 For every \(c\),
63 \[
64 D'J_Vc=J_EDc.
65 \]
66 Because \(J_E\) is invertible, \(Dc=0\) if and only if \(D'J_Vc=0\), proving
67 the kernel statement. Taking adjoints of the exact covariance identity gives
68 \[
69 D'^*J_E=J_VD^*.
70 \]
71 Therefore
72 \[
73 L'J_V=D'^*D'J_V=D'^*J_ED=J_VD^*D=J_VL.
74 \]
75 The spectral and operation claims follow by functional calculus and the
76 intertwining assumptions.
77 \end{proof}
78 
79 Exact correspondences compose and invert. They therefore form a groupoid of
80 realizations. The abstract exact workspace is an isomorphism class in this
81 groupoid.
82 
83 \topic{39}{Structural Identity Defect}
84 
85 For an approximate correspondence define
86 \[
87 \boxed{
88 \Delta_D=D'J_V-J_ED,
89 \qquad
90 \delta_D=\|\Delta_D\|.
91 }
92 \]
93 If the graph weights are unchanged, define edgewise comparison defects
94 \[
95 \delta_e
96 =
97 \|H_e'J_{s(e)}-J_{t(e)}H_e\|.
98 \]
99 The discrepancy operator has one changed source block per edge, so
100 \[
101 \boxed{
102 \delta_D^2
103 \leq
104 \max_{i\in V}
105 \sum_{e:s(e)=i}w_e\delta_e^2
106 \leq
107 \sum_{e\in E}w_e\delta_e^2.
108 }
109 \]
110 When a single node is replaced, every nonincident edge has \(\delta_e=0\).
111 The replacement defect is therefore locally measurable even though its effect
112 on the global workspace is controlled by the global spectral gap.
113 
114 The corresponding carried-Laplacian defect obeys
115 \[
116 \boxed{
117 \eta_L
118 :=
119 \|J_V^*L'J_V-L\|
120 \leq
121 2\|D\|\delta_D+\delta_D^2.
122 }
123 \]
124 This is Proposition~\ref{prop:temporal-laplacian-drift} with the time indices
125 removed.
126 
127 Let \(P\) and \(P'\) be rank-\(d\) workspace projectors selected from \(L\)
128 and \(L'\), and let \(g>0\) be the separating source gap.
129 
130 \begin{theorem}[Approximate structural replacement]\label{thm:approx-structural-replacement}
131 If
132 \[
133 \eta_L<\frac g2,
134 \]
135 then
136 \[
137 \boxed{
138 q_{\mathrm{str}}
139 :=
140 \|P'-J_VPJ_V^*\|
141 \leq
142 \frac{\eta_L}{g-\eta_L}
143 \leq
144 \frac{2\eta_L}{g}.
145 }
146 \]
147 If \(q_{\mathrm{str}}<1\), then
148 \[
149 B=P'J_VP:\ran P\longrightarrow\ran P'
150 \]
151 is bijective and satisfies
152 \[
153 \sigma_{\min}(B)
154 \geq
155 \sqrt{1-q_{\mathrm{str}}^2}.
156 \]
157 Its polar factor
158 \[
159 \boxed{
160 \beta
161 =B(B^*B)^{-1/2}
162 }
163 \]
164 is a canonical unitary workspace correspondence. Moreover,
165 \[
166 \boxed{
167 \|\beta-J_VP\|
168 \leq
169 q_{\mathrm{str}}
170 +
171 1-\sqrt{1-q_{\mathrm{str}}^2}.
172 }
173 \]
174 \end{theorem}
175 
176 \begin{proof}
177 The projector estimate is Theorem~\ref{thm:spectral-stability} applied after
178 carrying \(L'\) back by \(J_V\). For unit \(w\in\ran P\),
179 \[
180 \|(I-P')J_Vw\|
181 \leq q_{\mathrm{str}}.
182 \]
183 Orthogonality gives
184 \[
185 \|Bw\|^2
186 =1-\|(I-P')J_Vw\|^2
187 \geq1-q_{\mathrm{str}}^2.
188 \]
189 Thus \(B\) is injective, hence bijective because the source and target have
190 the same finite dimension. Its singular values lie in
191 \([\sqrt{1-q_{\mathrm{str}}^2},1]\), so polar normalization moves \(B\) by at
192 most \(1-\sqrt{1-q_{\mathrm{str}}^2}\). Finally,
193 \[
194 \|B-J_VP\|=\|(I-P')J_VP\|\leq q_{\mathrm{str}}.
195 \]
196 \end{proof}
197 
198 \subtopic{39.1 Transport of workspace operations}
199 
200 Write
201 \[
202 A=\rho_V(a),
203 \qquad
204 A'=\rho_V'(a),
205 \]
206 and define
207 \[
208 \delta_\rho(a)=\|A'J_V-J_VA\|,
209 \qquad
210 \ell_a=\|(I-P)AP\|.
211 \]
212 Let
213 \[
214 A_{\mathcal W}=PAP,
215 \qquad
216 A_{\mathcal W'}'=P'A'P'.
217 \]
218 
219 \begin{proposition}[Operation transport bound]\label{prop:operation-transport-bound}
220 For \(B=P'J_VP\),
221 \[
222 \boxed{
223 \|A_{\mathcal W'}'B-BA_{\mathcal W}\|
224 \leq
225 \|A'\|q_{\mathrm{str}}
226 +\delta_\rho(a)
227 +\ell_a.
228 }
229 \]
230 For the unitary polar correspondence \(\beta\),
231 \[
232 \boxed{
233 \begin{aligned}
234 \|A_{\mathcal W'}'\beta-\beta A_{\mathcal W}\|
235 \leq
236 &\|A'\|q_{\mathrm{str}}
237 +\delta_\rho(a)
238 +\ell_a\\
239 &+
240 \bigl(\|A_{\mathcal W'}'\|+\|A_{\mathcal W}\|\bigr)
241 \left(1-\sqrt{1-q_{\mathrm{str}}^2}\right).
242 \end{aligned}
243 }
244 \]
245 \end{proposition}
246 
247 \begin{proof}
248 Insert and subtract \(P'A'J_VP\) and \(P'J_VAP\):
249 \[
250 \begin{aligned}
251 A_{\mathcal W'}'B-BA_{\mathcal W}
252 ={}&P'A'(P'-I)J_VP\\
253 &+P'(A'J_V-J_VA)P\\
254 &+P'J_V(I-P)AP.
255 \end{aligned}
256 \]
257 Bound the three terms. Replacing \(B\) by \(\beta\) adds
258 \[
259 A_{\mathcal W'}'(\beta-B)-(\beta-B)A_{\mathcal W},
260 \]
261 which gives the second estimate.
262 \end{proof}
263 
264 \topic{40}{Behavioral Identity Defect}
265 
266 Structural correspondence does not determine operational identity. Let
267 \[
268 \{\iota_{w,b}:(w,b)\in\mathcal D\}
269 \quad\text{and}\quad
270 \{\iota'_{\beta w,\chi b}:(w,b)\in\mathcal D\}
271 \]
272 be matched feasible implementation-intervention families in the sense of
273 Part~VIII. Here \(b\) denotes declared background variables and \(\chi\) is
274 the background correspondence. Define
275 \[
276 K(w,b)
277 =
278 \mathbb P^{\iota_{w,b}}(Y^+\in\cdot),
279 \]
280 \[
281 K'(\beta w,\chi b)
282 =
283 \mathbb P^{\prime\,\iota'_{\beta w,\chi b}}(Y^{\prime+}\in\cdot).
284 \]
285 For a chosen divergence \(D_Y\), define
286 \[
287 \boxed{
288 \delta_{\mathrm{causal}}(\beta)
289 =
290 \sup_{(w,b)\in\mathcal D}
291 D_Y\left(
292 K(w,b),
293 K'(\beta w,\chi b)
294 \right).
295 }
296 \]
297 The observational response is included only when the null intervention belongs
298 to the declared family. This definition compares actual implementation
299 interventions rather than formally assigning an arbitrary projection
300 coordinate.
301 
302 A reconstructive handoff may also be imperfect. Let
303 \[
304 \delta_{\mathrm{rec}}(\beta)
305 =
306 \mathbb E\bigl[d_{\mathcal W'}(W'_{\mathrm{init}},\beta W)\bigr]
307 \]
308 measure replacement initialization error. Corollary~\ref{cor:handoff-initialization}
309 provides one explicit upper bound on this term.
310 
311 \subtopic{40.1 Structural identity does not imply causal identity}
312 
313 Take two realizations with identical
314 \[
315 G,
316 \quad D,
317 \quad L,
318 \quad P,
319 \quad \rho.
320 \]
321 In the first, let \(Y=W\). In the second, let \(Y'=1-W\) for a binary
322 workspace coordinate. Then
323 \[
324 q_{\mathrm{str}}=0,
325 \qquad
326 \delta_\rho(a)=0
327 \]
328 for every operation, while the observational and interventional outcome
329 kernels differ maximally under the identity correspondence. Therefore no
330 purely structural theorem can prove causal identity without an additional
331 kernel-matching assumption.
332 
333 \topic{41}{Identity Distance and Replacement Invariance}
334 
335 For an admissible correspondence \(\mathfrak J\), define the operation defect
336 on a declared unit ball \(\mathcal A_1\) by
337 \[
338 \delta_{\mathrm{op}}(\mathfrak J)
339 =
340 \sup_{a\in\mathcal A_1}
341 \|A_{\mathcal W'}'\beta-\beta A_{\mathcal W}\|.
342 \]
343 The primary certification object is the identity profile
344 \[
345 \boxed{
346 \mathbf d_{\mathrm{id}}(\mathfrak J)
347 =
348 \left(
349 q_{\mathrm{str}},
350 \delta_{\mathrm{op}}(\mathfrak J),
351 \delta_{\mathrm{causal}}(\beta),
352 \delta_{\mathrm{rec}}(\beta);
353 \mu^{\mathrm{keep}}
354 \right).
355 }
356 \]
357 The first four coordinates must remain below declared tolerances, while the
358 non-collapse margin \(\mu^{\mathrm{keep}}\) must remain above a declared lower
359 bound. This vector, rather than a weighted average, determines whether a
360 specific identity claim passes.
361 
362 A composite identity cost is
363 \[
364 \boxed{
365 \operatorname{Cost}(\mathfrak J)
366 =
367 a\,q_{\mathrm{str}}
368 +b\,\delta_{\mathrm{op}}(\mathfrak J)
369 +c\,\delta_{\mathrm{causal}}(\beta)
370 +d\,\delta_{\mathrm{rec}}(\beta),
371 }
372 \]
373 where \(a,b,c,d>0\) are declared scales. Define
374 \[
375 \boxed{
376 d_{\mathrm{id}}(\mathfrak R,\mathfrak R')
377 =
378 \inf_{\mathfrak J}
379 \operatorname{Cost}(\mathfrak J).
380 }
381 \]
382 This quantity is a task-relative optimization surrogate until the
383 admissible correspondence class, intervention matching, divergence, and scales
384 are fixed. It does not replace the coordinatewise identity profile or the
385 non-collapse gate.
386 
387 \subtopic{41.1 Replacement invariance theorem \(T_{14}\) }
388 
389 \begin{theorem}[Quantitative replacement invariance]\label{thm:replacement-invariance}
390 Let \(\mathfrak J\) be a unitary graph correspondence with diagram defect
391 \(\delta_D\). Define
392 \[
393 \eta_L=2\|D\|\delta_D+\delta_D^2
394 \]
395 and assume
396 \[
397 \eta_L<\frac g2.
398 \]
399 Set
400 \[
401 q_*
402 =
403 \frac{\eta_L}{g-\eta_L}.
404 \]
405 Then there exists a unitary workspace correspondence \(\beta\) for which
406 \[
407 q_{\mathrm{str}}\leq q_*.
408 \]
409 Moreover,
410 \[
411 \boxed{
412 \begin{aligned}
413 \delta_{\mathrm{op}}(\mathfrak J)
414 \leq
415 \sup_{a\in\mathcal A_1}
416 \Bigl[&
417 \|\rho_V'(a)\|q_*
418 +\delta_\rho(a)
419 +\ell_a\\
420 &+
421 \bigl(\|A_{\mathcal W'}'\|+\|A_{\mathcal W}\|\bigr)
422 \left(1-\sqrt{1-q_*^2}\right)
423 \Bigr].
424 \end{aligned}
425 }
426 \]
427 If, in addition,
428 \[
429 \delta_{\mathrm{causal}}(\beta)\leq\delta_c,
430 \qquad
431 \delta_{\mathrm{rec}}(\beta)\leq\delta_r,
432 \]
433 then
434 \[
435 \boxed{
436 \begin{aligned}
437 d_{\mathrm{id}}(\mathfrak R,\mathfrak R')
438 \leq
439 &a\,q_*+b\,\delta_{\mathrm{op}}(\mathfrak J)\\
440 &+c\,\delta_c+d\,\delta_r.
441 \end{aligned}
442 }
443 \]
444 In particular, if all local structural, operation, closure, causal, and
445 reconstruction defects are \(O(\delta_{\mathrm{rep}})\), the relevant
446 operator norms are uniformly bounded, and \(g\geq g_0>0\), then
447 \[
448 \boxed{
449 d_{\mathrm{id}}(\mathfrak R,\mathfrak R')
450 \leq
451 C_1\frac{\delta_{\mathrm{rep}}}{g_0}
452 +C_2\delta_{\mathrm{rep}}
453 +O\!\left(\frac{\delta_{\mathrm{rep}}^2}{g_0}\right).
454 }
455 \]
456 After a fixed normalization of the non-spectral terms, this recovers the
457 schematic target
458 \[
459 d_{\mathrm{id}}
460 \lesssim
461 C\frac{\delta_{\mathrm{rep}}}{g_0}.
462 \]
463 \end{theorem}
464 
465 \begin{proof}
466 The structural bound and existence of \(\beta\) follow from
467 Theorem~\ref{thm:approx-structural-replacement}. The operation estimate is
468 Proposition~\ref{prop:operation-transport-bound} with
469 \(q_{\mathrm{str}}\leq q_*\). Insert these estimates and the assumed causal
470 and reconstructive bounds into the definition of
471 \(\operatorname{Cost}(\mathfrak J)\). The asymptotic statement follows from
472 \[
473 q_*=O\!\left(\frac{\delta_D}{g_0}
474 +\frac{\delta_D^2}{g_0}\right)
475 \]
476 and
477 \[
478 1-\sqrt{1-q_*^2}=O(q_*^2).
479 \]
480 \end{proof}
481 
482 \begin{corollary}[Preservation of coalition frame bounds]
483 \label{cor:replacement-frame-bound}
484 Identify the old and new abstract cores with \(\ran P\) and \(\ran P'\)
485 through their chosen core parameterizations. Let \(S\) be a declared coalition,
486 let
487 \[
488 A_S:\ran P\to\mathsf Y_S,
489 \qquad
490 A_S':\ran P'\to\mathsf Y_S'
491 \]
492 be the resulting accessible coalition encoders, and let \(K_S\) be a unitary
493 identification of their observation spaces. Suppose
494 \[
495 \|A_S'\beta-K_SA_S\|
496 \leq
497 \delta_{S}^{\mathrm{frame}}
498 <
499 \sqrt{\alpha_S}.
500 \]
501 Then the replacement coalition remains reconstructive and its new lower frame
502 bound satisfies
503 \[
504 \boxed{
505 \alpha_S'
506 \geq
507 \left(
508 \sqrt{\alpha_S}-\delta_{S}^{\mathrm{frame}}
509 \right)^2.
510 }
511 \]
512 Consequently, \(k\)-failure resilience survives whenever the bound holds
513 uniformly for every coalition remaining after at most \(k\) erasures.
514 \end{corollary}
515 
516 \begin{proof}
517 For \(w'\in\ran P'\), write \(w'=\beta w\). Since \(\beta\) and \(K_S\) are
518 unitary,
519 \[
520 \begin{aligned}
521 \|A_S'w'\|
522 &\geq
523 \|K_SA_Sw\|
524 -
525 \|(A_S'\beta-K_SA_S)w\|\\
526 &\geq
527 \left(
528 \sqrt{\alpha_S}-\delta_{S}^{\mathrm{frame}}
529 \right)\|w'\|.
530 \end{aligned}
531 \]
532 Square the inequality.
533 \end{proof}
534 
535 \begin{corollary}[Exact carrier replacement]\label{cor:exact-carrier-replacement}
536 If
537 \[
538 \delta_D=0,
539 \qquad
540 \delta_\rho(a)=0,
541 \qquad
542 \ell_a=0,
543 \qquad
544 \delta_c=0,
545 \qquad
546 \delta_r=0,
547 \]
548 then
549 \[
550 d_{\mathrm{id}}(\mathfrak R,\mathfrak R')=0.
551 \]
552 Thus a participant, model, artifact, or notation system may be replaced without
553 changing the abstract workspace when the replacement realizes the same
554 structural, operational, causal, and reconstructive organization.
555 \end{corollary}
556 
557 \subtopic{41.2 Temporal chains and identity paths}
558 
559 For a sequence of realizations
560 \[
561 \mathfrak R_0,\mathfrak R_1,\ldots,\mathfrak R_T,
562 \]
563 let \(\mathfrak J_t:\mathfrak R_t\to\mathfrak R_{t+1}\) be admissible
564 correspondences. Define the path length
565 \[
566 \operatorname{Len}(\mathfrak J_{0:T})
567 =
568 \sum_{t=0}^{T-1}\operatorname{Cost}(\mathfrak J_t)
569 \]
570 and the induced directed path quasi-pseudometric
571 \[
572 \boxed{
573 d_{\mathrm{path}}^{\rightarrow}(\mathfrak R_0,\mathfrak R_T)
574 =
575 \inf_{\mathfrak J_{0:T}}
576 \operatorname{Len}(\mathfrak J_{0:T}).
577 }
578 \]
579 Concatenation gives the directed triangle inequality. Because replacement
580 costs and intervention divergences need not be symmetric, symmetry is not
581 assumed. When reverse correspondences exist, define the symmetrized
582 pseudometric
583 \[
584 \boxed{
585 d_{\mathrm{path}}^{\mathrm{sym}}(\mathfrak R,\mathfrak R')
586 =
587 \max\left\{
588  d_{\mathrm{path}}^{\rightarrow}(\mathfrak R,\mathfrak R'),
589  d_{\mathrm{path}}^{\rightarrow}(\mathfrak R',\mathfrak R)
590 \right\}.
591 }
592 \]
593 Distinct physical realizations can have zero symmetrized pseudodistance when
594 an exact correspondence exists in both directions. This is intentional:
595 identity is assigned to the invariant relational organization rather than to
596 the carrier tokens.
597 
598 \subtopic{41.3 Identity failure modes}
599 
600 A complete replacement audit must report at least
601 \[
602 \boxed{
603 \mathcal I_{\mathrm{rep}}
604 =
605 \left(
606 q_{\mathrm{str}},
607 \delta_{\mathrm{op}},
608 \delta_{\mathrm{causal}},
609 \delta_{\mathrm{rec}},
610 \mu^{\mathrm{keep}}
611 \right).
612 }
613 \]
614 The failure modes are different:
615 \[
616 \begin{array}{ll}
617 q_{\mathrm{str}}\text{ large}
618 &\text{the shared subspace changed},\\
619 \delta_{\mathrm{op}}\text{ large}
620 &\text{the same operations are no longer realized},\\
621 \delta_{\mathrm{causal}}\text{ large}
622 &\text{interventions no longer have matching consequences},\\
623 \delta_{\mathrm{rec}}\text{ large}
624 &\text{the new realization was not initialized from the old core},\\
625 \mu^{\mathrm{keep}}\text{ small}
626 &\text{content collapsed despite apparent coherence}.
627 \end{array}
628 \]
629 The theorem does not permit one successful coordinate to conceal failure of
630 another.

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