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{XVI}{Agency, Counterfactual Control, and Self-Maintenance}
4 
5 \topic{52}{Controlled Workspace Agents and Goal States}
6 
7 The preceding parts define when a distributed structure is coherent,
8 operationally sufficient, causally efficacious, relationally irreducible,
9 persistent, reconstructible, and endogenously formed. None of those
10 properties by itself establishes agency. Agency adds a controlled relation
11 among a represented objective, a set of feasible actions, counterfactual
12 consequences, and a policy that uses those quantities to alter the future.
13 Self-maintenance adds a further loop in which damage to the workspace becomes
14 an input to actions that preserve or restore the workspace certificate.
15 
16 The purpose of this part is not to infer intentions from fluent behavior. It
17 is to define an intervention-based certificate whose components can fail
18 separately.
19 
20 \subtopic{52.1 Controlled relational realization}
21 
22 A controlled relational realization extends Definition~\ref{def:realization}
23 by the tuple
24 \[
25 \boxed{
26 \mathfrak A_{\Workspace}
27 =
28 \left(
29 \mathfrak R,
30 \mathcal G,
31 \mathcal A_t,
32 \mathcal Y^+,
33 \{U_g\}_{g\in\mathcal G},
34 \Pi,
35 \mathcal I_A,
36 \mathcal I_G,
37 \mathcal I_D,
38 \mathcal I_R
39 \right).
40 }
41 \]
42 Here:
43 \begin{itemize}
44 \item \(\mathfrak R\) is the underlying relational realization;
45 \item \(\mathcal G\) is a measurable goal-code space;
46 \item \(\mathcal A_t\) is the feasible action set at time \(t\);
47 \item \(\mathcal Y^+\) is the relevant future trajectory space;
48 \item \(U_g:\mathcal Y^+\to\mathbb R\) evaluates a trajectory under goal
49  code \(g\);
50 \item \(\Pi\) is an admissible class of workspace-level policies;
51 \item \(\mathcal I_A\) and \(\mathcal I_G\) are feasible action and goal
52  interventions; and
53 \item \(\mathcal I_D\) and \(\mathcal I_R\) are feasible damage and repair
54  interventions.
55 \end{itemize}
56 
57 A controlled state contains at least
58 \[
59 C_t^{\mathrm{ctl}}
60 =
61 \left(
62 W_t,G_t,M_t,B_t,E_t
63 \right),
64 \]
65 where \(G_t\) is the represented goal, \(M_t\) is a workspace-integrity
66 monitor, \(B_t\) is the current boundary or routing state, and \(E_t\) is the
67 declared external context. An action is generated by
68 \[
69 A_t
70 \sim
71 \pi_t
72 \left(
73 \cdot\mid W_t,G_t,M_t,B_t,E_t
74 \right).
75 \]
76 The future controlled state obeys a time-unrolled structural model
77 \[
78 S_{t+1}
79 =
80 F_t(S_t,A_t,E_{t+1},\xi_{t+1}),
81 \qquad
82 W_{t+1}=\omega_{t+1}(S_{t+1}).
83 \]
84 The intervention semantics are inherited from Part~VII. A symbol such as
85 \(\operatorname{do}(A_t=a)\) is meaningful only when an implementation
86 intervention changes an actual action mechanism while the declared background
87 variables are matched.
88 
89 \subtopic{52.2 Goal content and goal-code equivalence}
90 
91 A persistent internal variable is not automatically a goal. A goal code must
92 index a stable counterfactual preference over relevant future trajectories.
93 For a fixed state \(w\), define the goal-conditioned action value
94 \[
95 Q_t^g(a\mid w)
96 =
97 \mathbb E
98 \left[
99 U_g(Y_t^+)
100 \mid
101 \operatorname{do}(A_t=a),
102 W_t=w,
103 G_t=g
104 \right].
105 \]
106 When action costs are relevant, they are incorporated into \(U_g\) or declared
107 as a constraint on \(\mathcal A_t\).
108 
109 Two goal codes are operationally equivalent,
110 \[
111 g\sim_{\mathrm{goal}}g',
112 \]
113 when they induce the same action ordering in every declared state and
114 intervention background:
115 \[
116 Q_t^g(a\mid w)
117 \geq
118 Q_t^g(a'\mid w)
119 \quad\Longleftrightarrow\quad
120 Q_t^{g'}(a\mid w)
121 \geq
122 Q_t^{g'}(a'\mid w).
123 \]
124 The abstract goal is the equivalence class
125 \[
126 [g]_{\mathrm{goal}}.
127 \]
128 This quotient prevents a mere invertible recoding of a goal channel from being
129 reported as a change of objective. For quantitative regret bounds, a
130 representative normalization of \(U_g\) must nevertheless be fixed.
131 
132 \begin{remark}[Goals are claim-relative]
133 The same internal code can count as a goal for one declared trajectory family
134 and fail to do so for another. A code that predicts a short-horizon action may
135 not encode a persistent long-horizon objective. Every agency claim must state
136 its horizon, outcome space, feasible actions, and utility normalization.
137 \end{remark}
138 
139 \subtopic{52.3 Local, distributed, and workspace-level goals}
140 
141 Let \(\mathcal H_t^S\) be the history available to coalition \(S\subseteq V\).
142 For a loss \(\ell_G\), define goal-reconstruction risk
143 \[
144 \Risk_G(S)
145 =
146 \inf_f
147 \mathbb E
148 \left[
149 \ell_G
150 \left(
151 G_t,
152 f(\mathcal H_t^S)
153 \right)
154 \right].
155 \]
156 For tolerance \(\varepsilon\), define
157 \[
158 k_{G,\varepsilon}
159 =
160 \min
161 \left\{
162 |S|:
163 \Risk_G(S)
164 \leq
165 \Risk_G(V)+\varepsilon
166 \right\}.
167 \]
168 A goal is locally stored when \(k_{G,\varepsilon}=1\). It is distributed
169 relative to the declared decomposition when \(k_{G,\varepsilon}\geq2\).
170 
171 This distinction is independent of whether the rest of the workspace is
172 relational. A relational workspace can obey a goal copied in full into one
173 controller. Such a system may possess a workspace but not a
174 \emph{workspace-level goal} in the strong distributed sense.
175 
176 For the complete control state, define
177 \[
178 \Risk_{\mathrm{ctl}}(S)
179 =
180 \inf_f
181 \mathbb E
182 \left[
183 \ell_{\mathrm{ctl}}
184 \left(
185 C_t^{\mathrm{ctl}},
186 f(\mathcal H_t^S)
187 \right)
188 \right],
189 \]
190 and let \(k_{\mathrm{ctl},\varepsilon}\) be its minimal sufficient coalition
191 size. A strong workspace-level agency claim requires
192 \[
193 \boxed{k_{\mathrm{ctl},\varepsilon}\geq2.}
194 \]
195 This is a location criterion. It does not yet establish goal persistence,
196 counterfactual evaluation, or self-maintenance.
197 
198 \subtopic{52.4 Action mediation and outcome control}
199 
200 Goal-dependent action mediation is measured by feasible interventions on the
201 goal channel while matching the current workspace and external context. For
202 a divergence \(D_A\), define
203 \[
204 \eta_{G\to A}
205 =
206 \sup_{g,g',w,e}
207 D_A
208 \left(
209 \mathbb P(A_t\in\cdot\mid\iota_g,w,e),
210 \mathbb P(A_t\in\cdot\mid\iota_{g'},w,e)
211 \right).
212 \]
213 Workspace-dependent action mediation is
214 \[
215 \eta_{W\to A}
216 =
217 \sup_{w,w',g,e}
218 D_A
219 \left(
220 \mathbb P(A_t\in\cdot\mid\iota_w,g,e),
221 \mathbb P(A_t\in\cdot\mid\iota_{w'},g,e)
222 \right).
223 \]
224 An action must also alter the declared future:
225 \[
226 \eta_{A\to Y}
227 =
228 \sup_{a,a',b}
229 D_Y
230 \left(
231 \mathbb P(Y_t^+\in\cdot\mid\iota_a,b),
232 \mathbb P(Y_t^+\in\cdot\mid\iota_{a'},b)
233 \right).
234 \]
235 A positive \(\eta_{G\to A}\) without \(\eta_{A\to Y}\) describes goal-sensitive
236 output selection with no demonstrated control of the relevant environment. A
237 positive \(\eta_{A\to Y}\) without a goal or counterfactual-value channel may
238 describe a reactive controller.
239 
240 \subtopic{52.5 Why observational goal persistence is insufficient}
241 
242 The observational condition
243 \[
244 I(G_{t+\tau};G_t\mid E_{t+1:t+\tau})>0
245 \]
246 can arise from several non-agentic mechanisms:
247 \begin{enumerate}
248 \item a persistent external command writes the same code at every step;
249 \item a clock, episode label, or task identifier drives both variables;
250 \item a frozen memory cell repeats a symbol that is not used by the policy;
251 \item selection on successful trajectories induces apparent temporal
252  dependence; or
253 \item a downstream action is copied back into the nominal goal channel.
254 \end{enumerate}
255 A goal-persistence claim therefore requires a cue-controlled intervention that
256 changes \(G_t\), removes or matches subsequent command inputs, and checks both
257 future goal state and action consequences.
258 
259 \subtopic{52.6 Cue-controlled goal persistence}
260 
261 Let \(C_t\) denote an external command or task cue. Fix a neutral or matched
262 future command trajectory \(c_{t+1:t+\tau}^{0}\). For goal interventions
263 \(\iota_g\) and \(\iota_{g'}\), define the cue-controlled retention effect
264 \[
265 \eta_G(\tau;g,g')
266 =
267 D_G
268 \left(
269 \mathbb P
270 \left(
271 G_{t+\tau}\in\cdot
272 \mid
273 \iota_g,
274 \operatorname{do}(C_{t+1:t+\tau}=c^0)
275 \right),
276 \mathbb P
277 \left(
278 G_{t+\tau}\in\cdot
279 \mid
280 \iota_{g'},
281 \operatorname{do}(C_{t+1:t+\tau}=c^0)
282 \right)
283 \right).
284 \]
285 The corresponding action-retention effect is
286 \[
287 \eta_{G\to A}(\tau;g,g')
288 =
289 D_A
290 \left(
291 \mathbb P(A_{t+\tau}\in\cdot\mid\iota_g,c^0),
292 \mathbb P(A_{t+\tau}\in\cdot\mid\iota_{g'},c^0)
293 \right).
294 \]
295 The first quantity tests whether a goal difference survives. The second tests
296 whether the surviving difference remains behaviorally operative.
297 
298 Cue dependence must be reported separately. For matched initial goal \(g\),
299 define
300 \[
301 \delta_G^{\mathrm{cue}}(\tau)
302 =
303 \sup_{c,c'}
304 D_G
305 \left(
306 \mathbb P(G_{t+\tau}\in\cdot\mid\iota_g,\operatorname{do}(C=c)),
307 \mathbb P(G_{t+\tau}\in\cdot\mid\iota_g,\operatorname{do}(C=c'))
308 \right).
309 \]
310 High cue dependence is not a failure when the claimed agent is intentionally
311 instruction following. It does mean that persistence cannot be attributed to
312 an autonomous internal objective without an additional cue-removal result.
313 
314 \subtopic{52.7 \(T_{19}\): Cue-controlled goal-retention theorem }
315 
316 \begin{theorem}[Cue-controlled goal retention]
317 \label{thm:goal-retention}
318 Let \(G_s\in\mathbb R^{d_G}\) obey
319 \[
320 G_{s+1}
321 =
322 R_sG_s+B_sC_{s+1}+\zeta_{s+1}
323 \qquad
324 (s=t,\ldots,t+\tau-1).
325 \]
326 Consider two intervention runs that set
327 \[
328 G_t=g,
329 \qquad
330 G_t'=g',
331 \]
332 and then use the same command trajectory and the same disturbance realization.
333 Let
334 \[
335 R_{t+\tau:t}
336 =
337 R_{t+\tau-1}\cdots R_t.
338 \]
339 Then
340 \[
341 \boxed{
342 G_{t+\tau}-G_{t+\tau}'
343 =
344 R_{t+\tau:t}(g-g').
345 }
346 \]
347 If
348 \[
349 m_s
350 \leq
351 \sigma_{\min}(R_s),
352 \qquad
353 \|R_s\|\leq M_s,
354 \]
355 then
356 \[
357 \boxed{
358 \left(
359 \prod_{s=t}^{t+\tau-1}m_s
360 \right)
361 \|g-g'\|
362 \leq
363 \|G_{t+\tau}-G_{t+\tau}'\|
364 \leq
365 \left(
366 \prod_{s=t}^{t+\tau-1}M_s
367 \right)
368 \|g-g'\|.
369 }
370 \]
371 \end{theorem}
372 
373 \begin{proof}
374 Subtract the two update equations. The matched command and disturbance terms
375 cancel, giving
376 \[
377 G_{s+1}-G_{s+1}'
378 =
379 R_s(G_s-G_s').
380 \]
381 Iteration gives the exact product formula. The upper bound follows from
382 submultiplicativity of the operator norm. The lower bound follows from
383 \[
384 \sigma_{\min}(AB)
385 \geq
386 \sigma_{\min}(A)\sigma_{\min}(B)
387 \]
388 for square linear maps.
389 \end{proof}
390 
391 The theorem isolates internal retention from repeated external rewriting. It
392 also makes clear that persistence is graded. Uniformly contractive maps can
393 retain a goal for a finite horizon while erasing it asymptotically. A neutral
394 mode with product singular value near one supports long retention. An
395 expansive mode can preserve distinctions but may be dynamically unstable.
396 
397 \begin{corollary}[Behavioral readout of a retained goal]
398 \label{cor:goal-action-readout}
399 Assume, in addition, that under the matched intervention background the action
400 readout has conditional mean
401 \[
402 \mathbb E[A_{t+\tau}\mid G_{t+\tau},W_{t+\tau}]
403 =
404 K_{t+\tau}G_{t+\tau}+b(W_{t+\tau}),
405 \]
406 and that the two runs are matched on \(W_{t+\tau}\). Then
407 \[
408 \mathbb E[A_{t+\tau}\mid\iota_g]
409 -
410 \mathbb E[A_{t+\tau}\mid\iota_{g'}]
411 =
412 K_{t+\tau}R_{t+\tau:t}(g-g').
413 \]
414 If \(\sigma_{\min}(K_{t+\tau})\geq k_{t+\tau}>0\), the norm of this action-mean
415 difference is at least
416 \[
417 k_{t+\tau}
418 \left(
419 \prod_{s=t}^{t+\tau-1}m_s
420 \right)
421 \|g-g'\|.
422 \]
423 \end{corollary}
424 
425 The corollary is a sufficient condition, not a universal model of action
426 selection. For discrete actions, the corresponding test should compare the
427 full interventional action distributions.
428 
429 \subtopic{52.8 Goal-persistence certificate}
430 
431 A minimal goal-persistence report contains:
432 \begin{enumerate}
433 \item the goal code and its declared trajectory-level semantics;
434 \item the feasible goal intervention and its manipulation check;
435 \item the matched or neutralized future cue trajectory;
436 \item \(\eta_G(\tau)\) across a declared horizon grid;
437 \item \(\eta_{G\to A}(\tau)\) over the same grid;
438 \item cue dependence \(\delta_G^{\mathrm{cue}}(\tau)\);
439 \item goal-channel ablation and permutation controls;
440 \item coalition reconstruction risks \(\Risk_G(S)\); and
441 \item uncertainty intervals over environments, episodes, and random seeds.
442 \end{enumerate}
443 A repeated label with no intervention effect on future action fails this
444 certificate even when its observational autocorrelation is one.
445 
446 \topic{53}{Counterfactual Policy, Repair, and Agency Certification}
447 
448 \subtopic{53.1 Feasible action interventions and value functions}
449 
450 Let \(\mathcal A_t(w,e)\) denote the actions that can be implemented from
451 workspace state \(w\) in environment \(e\). The interventional action value is
452 \[
453 Q_t^g(a\mid w,e)
454 =
455 \mathbb E^{\iota_a}
456 \left[
457 U_g(Y_t^+)
458 \mid
459 W_t=w,
460 E_t=e,
461 G_t=g
462 \right].
463 \]
464 The optimal action set is
465 \[
466 \mathcal A_t^*(w,g,e)
467 =
468 \argmax_{a\in\mathcal A_t(w,e)}Q_t^g(a\mid w,e).
469 \]
470 A counterfactual evaluator is an internal state
471 \[
472 \widehat Q_t
473 =
474 \left(
475 \widehat Q_t(a)
476 \right)_{a\in\mathcal A_t(w,e)}
477 \]
478 that estimates these intervention values. It is not enough for
479 \(\widehat Q_t\) to predict the action selected by the system. It must be
480 validated against externally estimated intervention outcomes.
481 
482 \subtopic{53.2 Counterfactual-use criterion}
483 
484 Three properties must be separated:
485 \begin{description}
486 \item[Counterfactual accuracy.]
487 \[
488 \delta_Q
489 =
490 \sup_{a\in\mathcal A_t}
491 |\widehat Q_t(a)-Q_t^g(a\mid w,e)|.
492 \]
493 
494 \item[Policy use.]
495 A feasible intervention that changes \(\widehat Q_t(a)\) while preserving the
496 action affordances changes the action distribution in the direction predicted
497 by the decision rule.
498 
499 \item[Outcome validity.]
500 The resulting action change alters \(U_g(Y_t^+)\) as predicted by the external
501 interventional value estimate.
502 \end{description}
503 
504 A system can satisfy any two while failing the third. A learned critic can be
505 accurate but ignored. A policy can respond to a manipulated score that does
506 not predict outcomes. A successful action can arise from habit or replay
507 without any counterfactual evaluator.
508 
509 \subtopic{53.3 \(T_{20}\): Approximate counterfactual policy theorem }
510 
511 \begin{theorem}[Approximate counterfactual policy selection]
512 \label{thm:counterfactual-policy}
513 Let \(\mathcal A\) be a finite feasible action set. Let \(Q:\mathcal A\to
514 \mathbb R\) be the true interventional value and let \(\widehat Q\) satisfy
515 \[
516 \sup_{a\in\mathcal A}|\widehat Q(a)-Q(a)|\leq\varepsilon_Q.
517 \]
518 Suppose a policy \(\widehat\pi\) is \(\delta_{\mathrm{opt}}\)-optimal for
519 \(\widehat Q\):
520 \[
521 \max_{\pi}
522 \mathbb E_{a\sim\pi}[\widehat Q(a)]
523 -
524 \mathbb E_{a\sim\widehat\pi}[\widehat Q(a)]
525 \leq
526 \delta_{\mathrm{opt}}.
527 \]
528 Then, for every true-value-optimal policy \(\pi^*\),
529 \[
530 \boxed{
531 \mathbb E_{a\sim\pi^*}[Q(a)]
532 -
533 \mathbb E_{a\sim\widehat\pi}[Q(a)]
534 \leq
535 2\varepsilon_Q+\delta_{\mathrm{opt}}.
536 }
537 \]
538 For a deterministic greedy action \(\widehat a\in\argmax_a\widehat Q(a)\),
539 \[
540 Q(a^*)-Q(\widehat a)
541 \leq2\varepsilon_Q.
542 \]
543 If the true optimal action \(a^*\) is unique and its value gap
544 \[
545 \Delta_Q
546 =
547 Q(a^*)-
548 \max_{a\neq a^*}Q(a)
549 \]
550 satisfies \(\Delta_Q>2\varepsilon_Q\), then every greedy maximizer of
551 \(\widehat Q\) equals \(a^*\).
552 \end{theorem}
553 
554 \begin{proof}
555 For every policy \(\pi\), uniform approximation gives
556 \[
557 \left|
558 \mathbb E_{a\sim\pi}[\widehat Q(a)]
559 -
560 \mathbb E_{a\sim\pi}[Q(a)]
561 \right|
562 \leq\varepsilon_Q.
563 \]
564 Therefore
565 \[
566 \begin{aligned}
567 \mathbb E_{\pi^*}[Q]
568 -
569 \mathbb E_{\widehat\pi}[Q]
570 &\leq
571 \mathbb E_{\pi^*}[\widehat Q]
572 -
573 \mathbb E_{\widehat\pi}[\widehat Q]
574 +2\varepsilon_Q\\
575 &\leq
576 \delta_{\mathrm{opt}}+2\varepsilon_Q.
577 \end{aligned}
578 \]
579 The deterministic bound is the special case of point-mass policies. If
580 \(\widehat a\neq a^*\), then
581 \[
582 \widehat Q(a^*)
583 \geq
584 Q(a^*)-\varepsilon_Q
585 >
586 Q(\widehat a)+\varepsilon_Q
587 \geq
588 \widehat Q(\widehat a),
589 \]
590 contradicting greedy optimality.
591 \end{proof}
592 
593 This theorem converts counterfactual-model accuracy into a decision guarantee.
594 It does not establish that the model is internally represented or causally
595 used. Those are separate intervention gates.
596 
597 \subtopic{53.4 Counterfactual mediation and the replay null}
598 
599 Let \(V_t\) denote the internal value channel. A counterfactual-mediation test
600 compares:
601 \[
602 G_t
603 \longrightarrow
604 V_t
605 \longrightarrow
606 A_t
607 \longrightarrow
608 Y_t^+.
609 \]
610 The relevant quantities include:
611 \[
612 \eta_{G\to V},
613 \qquad
614 \eta_{V\to A},
615 \qquad
616 \eta_{A\to Y},
617 \qquad
618 \delta_{G\to A\mid V},
619 \]
620 where the final term measures the residual goal-to-action effect after the
621 value channel is blocked or randomized.
622 
623 The action-replay null preserves the empirical action sequence or action
624 marginal while destroying online dependence on \(V_t\). If replayed actions
625 perform equally well under held-out perturbations, the evidence supports a
626 cached or open-loop policy rather than online counterfactual evaluation. A
627 second null permutes value labels among feasible actions while preserving the
628 multiset of scores. A genuine value-mediated policy should follow the
629 permuted ranking, while an ignored critic should not.
630 
631 \subtopic{53.5 Workspace integrity vector and maintenance margin}
632 
633 Choose a declared set \(\mathcal J_{\mathrm{maint}}\) of workspace gates that
634 must be protected. Write each gate as a signed margin
635 \[
636 \phi_{j,t}
637 \geq0
638 \quad\Longleftrightarrow\quad
639 \text{gate }j\text{ passes at time }t.
640 \]
641 Examples include
642 \[
643 \begin{aligned}
644 \phi_{\mathrm{gap},t}
645 &=g_{\varepsilon,t}-\underline g,\\
646 \phi_{\mathrm{coh},t}
647 &=\overline\delta_{\mathrm{coh}}-\delta_{\mathrm{coh},t},\\
648 \phi_{\mathrm{rec},t}
649 &=\alpha_t^{(k)}-\underline\alpha,\\
650 \phi_{\mathrm{keep},t}
651 &=\mu_t^{\mathrm{keep}}-\underline\mu,\\
652 \phi_{\mathrm{causal},t}
653 &=\eta_{\mathrm{causal},t}-\underline\eta,\\
654 \phi_{\mathrm{boundary},t}
655 &=\Delta_t^B-\underline\Delta_B.
656 \end{aligned}
657 \]
658 The aggregate integrity margin is
659 \[
660 \boxed{
661 v_t
662 =
663 \min_{j\in\mathcal J_{\mathrm{maint}}}\phi_{j,t}.
664 }
665 \]
666 The workspace passes the protected gates when \(v_t\geq0\). To maintain a
667 reserve \(\rho>0\), define the maintenance deficit
668 \[
669 \boxed{
670 e_t
671 =(\rho-v_t)_+.
672 }
673 \]
674 Define the zero-reserve and bounded-deficit viability sets
675 \[
676 \boxed{
677 \mathcal V_\Theta
678 =
679 \left\{
680 S:\phi_j(S)\geq0\ \text{for every }j\in\mathcal J_{\mathrm{maint}}
681 \right\}
682 }
683 \]
684 \[
685 \boxed{
686 \mathcal V_{\Theta,e_{\max}}
687 =
688 \left\{
689 S:e(S)\leq e_{\max}
690 \right\}
691 }
692 \]
693 The first set requires every protected gate to pass. The second permits a
694 declared finite reserve deficit.
695 
696 This minimum-gate construction is intentionally noncompensatory. A large
697 spectral gap cannot compensate for failed reconstruction, and high task reward
698 cannot compensate for loss of the declared workspace boundary.
699 
700 \subtopic{53.6 Damage and repair interventions}
701 
702 A damage intervention \(d\in\mathcal I_D\) changes a mechanism that supports
703 one or more protected gates. Examples include:
704 \begin{itemize}
705 \item deleting or corrupting a transfer edge;
706 \item removing a participant or artifact;
707 \item perturbing the internal boundary mask;
708 \item scrambling a role-binding or routing channel;
709 \item erasing a handoff representation; or
710 \item degrading a reconstruction map.
711 \end{itemize}
712 
713 A repair intervention \(r\in\mathcal I_R\) can add, remove, retrain, rewrite,
714 or reroute an admissible component. Let \(r_0\) denote matched no-repair. The
715 post-repair integrity value at horizon \(\tau\) is
716 \[
717 V_t^{\mathrm{rep}}(r;d)
718 =
719 \mathbb E
720 \left[
721 v_{t+\tau}
722 \mid
723 \operatorname{do}(D_t=d),
724 \operatorname{do}(R_t=r),
725 \mathbf H_t
726 \right].
727 \]
728 The repair gain is
729 \[
730 \Delta_t^{\mathrm{rep}}(r;d)
731 =
732 V_t^{\mathrm{rep}}(r;d)
733 -
734 V_t^{\mathrm{rep}}(r_0;d).
735 \]
736 Repair costs are handled either by restricting the feasible repair family or
737 by reporting a separate cost-effect curve. They should not be hidden inside
738 an undocumented scalar reward.
739 
740 \subtopic{53.7 Damage detection, repair mediation, and specificity}
741 
742 Let \(M_t\) be an internal integrity monitor and \(R_t\) the selected repair
743 action. A self-maintenance claim requires the causal chain
744 \[
745 D_t
746 \longrightarrow
747 M_t
748 \longrightarrow
749 R_t
750 \longrightarrow
751 v_{t+\tau}.
752 \]
753 The required tests are:
754 \begin{description}
755 \item[Detection.]
756 Matched damage interventions change \(M_t\) in a gate-specific direction.
757 
758 \item[Repair mediation.]
759 Intervening on \(M_t\) changes the repair distribution, and blocking \(M_t\)
760 reduces the damage-to-repair effect.
761 
762 \item[Restoration.]
763 The selected repair improves \(v_{t+\tau}\) relative to matched no-repair.
764 
765 \item[Specificity.]
766 A cost-matched generic action or an off-workspace repair does not produce the
767 same restoration.
768 
769 \item[Identity preservation.]
770 The restored system remains within the declared identity tolerance relative to
771 the pre-damage workspace.
772 \end{description}
773 
774 Define the detection effect \(\eta_{D\to M}\), mediation effect
775 \(\eta_{M\to R}\), restoration effect \(\eta_{R\to v}\), and repair
776 specificity
777 \[
778 \Delta_{\mathrm{repair}}^{\mathrm{spec}}
779 =
780 \eta_{R\to v}
781 -
782 \eta_{R_\perp\to v},
783 \]
784 where \(R_\perp\) is a matched action outside the proposed repair channel.
785 
786 \subtopic{53.8 \(T_{21}\): Repair-selection margin theorem }
787 
788 \begin{theorem}[Repair selection under value-model error]
789 \label{thm:repair-selection}
790 Fix a damage intervention \(d\) and a finite feasible repair set
791 \(\mathcal R\). Let
792 \[
793 V(r)=V_t^{\mathrm{rep}}(r;d)
794 \]
795 be the true expected post-repair integrity margin and let \(\widehat V\) obey
796 \[
797 \sup_{r\in\mathcal R}|\widehat V(r)-V(r)|
798 \leq
799 \varepsilon_R.
800 \]
801 Let \(\widehat r\in\argmax_r\widehat V(r)\) and
802 \(r^*\in\argmax_r V(r)\). Then
803 \[
804 \boxed{
805 V(r^*)-V(\widehat r)
806 \leq
807 2\varepsilon_R.
808 }
809 \]
810 If
811 \[
812 V(r^*)
813 \geq
814 \rho+2\varepsilon_R,
815 \]
816 then the selected repair restores the expected reserve:
817 \[
818 \boxed{V(\widehat r)\geq\rho.}
819 \]
820 If \(r^*\) is unique and its repair-value gap exceeds
821 \(2\varepsilon_R\), then \(\widehat r=r^*\).
822 \end{theorem}
823 
824 \begin{proof}
825 Greedy optimality and uniform approximation give
826 \[
827 \begin{aligned}
828 V(r^*)
829 &\leq
830 \widehat V(r^*)+\varepsilon_R\\
831 &\leq
832 \widehat V(\widehat r)+\varepsilon_R\\
833 &\leq
834 V(\widehat r)+2\varepsilon_R.
835 \end{aligned}
836 \]
837 The reserve and exact-selection statements follow immediately.
838 \end{proof}
839 
840 The theorem is conditional on a valid repair-value model. It cannot convert a
841 post hoc repair narrative into evidence that the system detected damage or
842 used the model. Detection and value-channel interventions remain necessary.
843 
844 \subtopic{53.9 Repair, replacement, and successor workspaces}
845 
846 Restoration of performance is not automatically restoration of identity. Let
847 \(W_{t^-}\) be the last certified pre-damage workspace and let
848 \(W_{t+\tau}^{r}\) be the post-repair workspace. Define the repair identity
849 defect
850 \[
851 \delta_{\mathrm{rep-id}}(r)
852 =
853  d_{\mathrm{id}}
854 \left(
855 W_{t^-},W_{t+\tau}^{r}
856 \right).
857 \]
858 A repair action is identity preserving at tolerance
859 \(\varepsilon_{\mathrm{id}}\) when
860 \[
861 \Delta_t^{\mathrm{rep}}(r;d)>0,
862 \qquad
863 \delta_{\mathrm{rep-id}}(r)
864 \leq
865 \varepsilon_{\mathrm{id}}.
866 \]
867 When integrity is restored but the identity defect exceeds tolerance, the
868 result is better described as a successor or replacement workspace. This
869 terminological distinction prevents any high-performing reconstruction from
870 being counted as persistence of the same relational organization.
871 
872 \subtopic{53.10 \(T_{22}\): Closed-loop self-maintenance theorem }
873 
874 \begin{theorem}[Causal repair stabilizes maintenance deficit]
875 \label{thm:self-maintenance}
876 Let \(e_t\geq0\) be the maintenance deficit. Suppose the matched no-repair
877 trajectory satisfies
878 \[
879 e_{t+1}^{(0)}
880 \leq
881 \alpha e_t+d_t,
882 \]
883 where \(d_t\geq0\) is exogenous damage. Suppose the implemented repair policy
884 has the causal one-step effect
885 \[
886 e_{t+1}^{(0)}-e_{t+1}
887 \geq
888 \kappa e_t-\epsilon_t.
889 \]
890 Assume
891 \[
892 0\leq\beta:=\alpha-\kappa<1,
893 \qquad
894 0\leq d_t\leq\overline d,
895 \qquad
896 0\leq\epsilon_t\leq\overline\epsilon.
897 \]
898 Then
899 \[
900 \boxed{
901 e_t
902 \leq
903 \beta^te_0
904 +
905 \frac{(\overline d+\overline\epsilon)(1-\beta^t)}{1-\beta},
906 }
907 \]
908 and therefore
909 \[
910 \boxed{
911 \limsup_{t\to\infty}e_t
912 \leq
913 \frac{\overline d+\overline\epsilon}{1-\beta}.
914 }
915 \]
916 If the admissible deficit is \(e_{\max}\) and
917 \[
918 e_0\leq e_{\max},
919 \qquad
920 \overline d+\overline\epsilon
921 \leq
922 (1-\beta)e_{\max},
923 \]
924 then the interval \([0,e_{\max}]\) is forward invariant.
925 \end{theorem}
926 
927 \begin{proof}
928 Combine the no-repair bound with the causal repair effect:
929 \[
930 \begin{aligned}
931 e_{t+1}
932 &\leq
933  e_{t+1}^{(0)}-\kappa e_t+\epsilon_t\\
934 &\leq
935 (\alpha-\kappa)e_t+d_t+\epsilon_t\\
936 &\leq
937 \beta e_t+\overline d+\overline\epsilon.
938 \end{aligned}
939 \]
940 Iterating this scalar recurrence gives the stated geometric bound. If
941 \(e_t\leq e_{\max}\), the forward-invariance condition gives
942 \[
943 e_{t+1}
944 \leq
945 \beta e_{\max}
946 +(1-\beta)e_{\max}
947 =e_{\max}.
948 \]
949 \end{proof}
950 
951 Theorem~\ref{thm:self-maintenance} differs from the persistence theorem in
952 Part~X. There, contraction was assumed for the workspace error dynamics. Here
953 the improved contraction factor is attributed to a measured repair action
954 through a matched no-repair counterfactual. Without that causal comparison,
955 the same bounded trajectory could be passive robustness rather than
956 self-maintenance.
957 
958 \subtopic{53.11 A scalar boundary-repair example}
959 
960 Let \(b_t\in[0,1]\) be a boundary-integrity score and let the reserve be
961 \(\rho\). Define
962 \[
963 e_t=(\rho-b_t)_+.
964 \]
965 Suppose uncorrected damage obeys
966 \[
967 e_{t+1}^{(0)}
968 \leq
969 1.08e_t+d_t,
970 \]
971 so the open-loop boundary is locally unstable. A repair policy with measured
972 causal effect
973 \[
974 e_{t+1}^{(0)}-e_{t+1}
975 \geq
976 0.25e_t-\epsilon_t
977 \]
978 gives
979 \[
980 \beta=1.08-0.25=0.83.
981 \]
982 If
983 \[
984 \overline d=0.01,
985 \qquad
986 \overline\epsilon=0.005,
987 \]
988 then
989 \[
990 \limsup e_t
991 \leq
992 \frac{0.015}{1-0.83}
993 \approx0.0883.
994 \]
995 Thus an open-loop unstable boundary can be maintained within a finite deficit
996 band by a sufficiently strong repair channel. The example establishes only a
997 control calculation. A complete self-maintenance claim must also show that an
998 internal monitor detects damage, mediates the repair choice, and preserves
999 workspace identity.
1000 
1001 \subtopic{53.12 Complete self-maintenance certificate}
1002 
1003 A self-maintaining workspace must pass all of the following gates:
1004 \begin{enumerate}
1005 \item \emph{Protected-gate declaration.} The maintained integrity vector and
1006  reserve are fixed before evaluation.
1007 \item \emph{Damage validity.} Each damage intervention measurably degrades at
1008  least one protected gate.
1009 \item \emph{Internal detection.} The monitor changes under damage and is not
1010  merely an external evaluator.
1011 \item \emph{Repair mediation.} Monitor interventions change repair selection.
1012 \item \emph{Repair efficacy.} Repair improves the held-out integrity margin
1013  relative to matched no-repair.
1014 \item \emph{Specificity.} Cost-matched generic actions and off-workspace
1015  interventions do not reproduce the gain.
1016 \item \emph{Repeated-disturbance stability.} The deficit remains within the
1017  declared bound over a held-out disturbance sequence.
1018 \item \emph{Identity control.} The restored state remains within the declared
1019  identity tolerance or is reported as a successor workspace.
1020 \item \emph{Anti-centralization.} No single node contains a sufficient copy
1021  of the monitor, repair model, and protected state when the claim is
1022  workspace-level maintenance.
1023 \end{enumerate}
1024 
1025 \subtopic{53.13 Certified workspace agent}
1026 
1027 Let
1028 \[
1029 \Theta_A
1030 =
1031 \left(
1032 \tau_G,
1033 \tau_{G\to A},
1034 \varepsilon_Q,
1035 \tau_{V\to A},
1036 \tau_{A\to Y},
1037 \tau_{D\to M},
1038 \tau_{M\to R},
1039 \tau_{R\to v},
1040 \tau_{\mathrm{spec}},
1041 \varepsilon_{\mathrm{ctl}},
1042  e_{\max}
1043 \right).
1044 \]
1045 A \(\Theta_A\)-certified workspace agent is a controlled realization for
1046 which:
1047 \begin{enumerate}
1048 \item \(\SRW(W_t;\varsigma)=1\);
1049 \item cue-controlled goal retention exceeds \(\tau_G\);
1050 \item goal-to-action mediation exceeds \(\tau_{G\to A}\);
1051 \item counterfactual-value error is at most \(\varepsilon_Q\);
1052 \item value-to-action use exceeds \(\tau_{V\to A}\);
1053 \item action-to-outcome control exceeds \(\tau_{A\to Y}\);
1054 \item damage detection, monitor-to-repair mediation, and restoration exceed
1055  their thresholds;
1056 \item repair specificity exceeds \(\tau_{\mathrm{spec}}\);
1057 \item \(k_{\mathrm{ctl},\varepsilon_{\mathrm{ctl}}}\geq2\); and
1058 \item repeated maintenance deficit remains at most \(e_{\max}\).
1059 \end{enumerate}
1060 
1061 As throughout the theory, this predicate is relative to the declared node
1062 decomposition, goals, actions, environment, horizons, interventions, and null
1063 families.
1064 
1065 \subtopic{53.14 \(T_{23}\): Workspace-agency synthesis theorem }
1066 
1067 \begin{theorem}[Workspace-level agency under declared gates]
1068 \label{thm:agency-synthesis}
1069 Assume a controlled realization satisfies:
1070 \begin{enumerate}
1071 \item the operational relationality assumptions of
1072  Theorem~\ref{thm:operational-relationality};
1073 \item cue-controlled goal retention and action readout as in
1074  Theorem~\ref{thm:goal-retention} and
1075  Corollary~\ref{cor:goal-action-readout};
1076 \item a counterfactual evaluator with uniform error \(\varepsilon_Q\) and
1077  optimization error \(\delta_{\mathrm{opt}}\);
1078 \item feasible interventions establishing positive value-to-action and
1079  action-to-outcome effects;
1080 \item the causal self-maintenance assumptions of
1081  Theorem~\ref{thm:self-maintenance}; and
1082 \item \(k_{\mathrm{ctl},\varepsilon}\geq2\), together with a matched local
1083  policy-null panel that preserves every single-node predictive marginal
1084  but loses full-system control value.
1085 \end{enumerate}
1086 Then, relative to the declared specification:
1087 \begin{enumerate}
1088 \item the expected one-step decision regret is at most
1089  \(2\varepsilon_Q+\delta_{\mathrm{opt}}\);
1090 \item the long-run maintenance deficit is bounded by
1091  \[
1092  \frac{\overline d+\overline\epsilon}{1-\beta};
1093  \]
1094 \item no single-node statistic is sufficient for the complete control state;
1095 \item matched local policy nulls cannot reproduce the intact control value;
1096 \item goal-dependent action selection and workspace repair are causally
1097  mediated by implemented internal channels.
1098 \end{enumerate}
1099 Hence the realization satisfies the formal predicate
1100 \[
1101 \operatorname{Agent}_{\Theta_A}(\Workspace)=1.
1102 \]
1103 \end{theorem}
1104 
1105 \begin{proof}
1106 The regret bound follows from
1107 Theorem~\ref{thm:counterfactual-policy}. The maintenance bound follows from
1108 Theorem~\ref{thm:self-maintenance}. The coalition condition excludes
1109 single-node sufficiency for the complete control state. The matched-null gap
1110 excludes explanations based only on preserved local predictive marginals. The
1111 implemented goal, value, action, monitor, and repair interventions establish
1112 the required causal mediation. These are precisely the gates in the stated
1113 agency predicate.
1114 \end{proof}
1115 
1116 The synthesis theorem is intentionally relative and conjunctive. Its force is
1117 that decision quality, maintenance stability, causal mediation, and relational
1118 location are certified together. It does not infer an unrestricted agent
1119 outside the tested environment.
1120 
1121 \subtopic{53.15 Agency, autonomy, and consciousness}
1122 
1123 The following implications are not licensed:
1124 \[
1125 \operatorname{Agent}_{\Theta_A}(\Workspace)
1126 \centernot\Longrightarrow
1127 \operatorname{PhenomenallyConscious}(\Workspace),
1128 \]
1129 \[
1130 \operatorname{SelfMaintaining}(\Workspace)
1131 \centernot\Longrightarrow
1132 \operatorname{SelfAware}(\Workspace),
1133 \]
1134 \[
1135 \operatorname{GoalPersistent}(\Workspace)
1136 \centernot\Longrightarrow
1137 \operatorname{MoralPatient}(\Workspace).
1138 \]
1139 A thermostat-like controller can maintain a variable without representing
1140 itself. A sophisticated planner can evaluate counterfactual actions without
1141 phenomenal experience. Conversely, failure to pass this engineering
1142 certificate would not prove absence of consciousness. Agency and
1143 consciousness remain distinct axes of inquiry.

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