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{XIX}{Summary of Main Results}
4 
5 \topic{68}{Structural and Foundational Results}
6 
7 \subtopic{\(T_1\): Limit--holonomy theorem}
8 \[
9 \Gamma(H)\cong C_o^{\operatorname{Hol}_o}.
10 \]
11 
12 \subtopic{\(T_2\): Operator closure theorem}
13 \[
14 H_e\rho_i(a)=\rho_j(a)H_e
15 \Longrightarrow
16 \rho_o(a)C_*\subseteq C_*.
17 \]
18 
19 \subtopic{\(T_{2a}\): Workspace commutant theorem }
20 \[
21 \mathcal O_{\Workspace}
22 \cong
23 \operatorname{Comm}(\Hol_o).
24 \]
25 
26 \subtopic{\(T_3\): Transfer-Laplacian kernel theorem}
27 \[
28 \ker L_H=\Gamma(H).
29 \]
30 
31 \subtopic{\(T_4\): Spectral stability theorem}
32 If \(\eta=\|\widetilde L_H-L_H\|<g_d/2\), then
33 \[
34 \|\widetilde P_d-P_d\|
35 \leq
36 \frac{\eta}{g_d-\eta}
37 \leq
38 \frac{2\eta}{g_d}.
39 \]
40 
41 \subtopic{\(T_5\): Gauge invariance theorem}
42 \[
43 \Gamma(H^g)=G_V\Gamma(H),
44 \]
45 and under unitary gauge,
46 \[
47 L_{H^g}=G_VL_HG_V^{-1}.
48 \]
49 
50 \subtopic{\(T_6\): Persistence theorem}
51 
52 For the nonautonomous recurrence
53 \[
54 e_{t+1}\leq\alpha_te_t+\sigma_t,
55 \]
56 \[
57 \boxed{
58 \begin{aligned}
59 e_t
60 \leq
61 &\left(\prod_{r=0}^{t-1}\alpha_r\right)e_0\\
62 &+\sum_{j=0}^{t-1}
63 \left(\prod_{r=j+1}^{t-1}\alpha_r\right)\sigma_j.
64 \end{aligned}
65 }
66 \]
67 If \(\alpha_t\leq\bar\alpha<1\) and \(\sigma_t\leq\bar\sigma\), then
68 \[
69 \limsup_{t\to\infty}e_t
70 \leq
71 \frac{\bar\sigma}{1-\bar\alpha}.
72 \]
73 The derivation begins with a moving workspace projector and
74 an explicit system update, and adds a mean-square analogue and a non-collapse
75 gate.
76 
77 \subtopic{\(T_7\): Stable reconstruction theorem}
78 
79 For coalition analysis operator \(A_S\) with lower frame bound
80 \(\alpha_S>0\), observations
81 \[
82 y=A_Sc+\eta
83 \]
84 and canonical decoder \(A_S^\dagger\) satisfy
85 \[
86 \boxed{
87 \|A_S^\dagger y-c\|
88 \leq
89 \frac{\|\eta\|}{\sqrt{\alpha_S}}.
90 }
91 \]
92 The theorem is extended to encoding mismatch, exact support-based erasure
93 distance, uniform node erasures, reconstructive handoff, exact coalition
94 sufficiency, and operational risk under a Lipschitz predictive kernel.
95 
96 \subtopic{\(T_8\): Exact gluing theorem}
97 
98 For every finite edge-complete workspace cover,
99 \[
100 \boxed{
101 0
102 \longrightarrow
103 \mathscr W(G)
104 \xrightarrow{\ R_{\mathcal U}\ }
105 \check C^0(\mathcal U;\mathscr W)
106 \xrightarrow{\ \delta^0\ }
107 \check C^1(\mathcal U;\mathscr W)
108 }
109 \]
110 is exact. Hence
111 \[
112 \boxed{
113 \mathscr W(G)
114 \cong
115 \ker\delta^0
116 \cong
117 \operatorname{Eq}
118 \left(
119 \prod_\alpha\mathscr W_\alpha
120 \rightrightarrows
121 \prod_{\alpha<\beta}\mathscr W_{\alpha\beta}
122 \right).
123 }
124 \]
125 The same analysis also yields a cover-Laplacian distance bound, a total descent kernel
126 theorem, and a harmonic \(\check H^1\) obstruction for relative-only data.
127 
128 \subtopic{\(T_9\): Hierarchical collapse theorem}
129 
130 For every finite rooted hierarchy whose children edge-cover their parent,
131 \[
132 \boxed{
133 W_a\cong\mathscr W(G_a)
134 \quad\text{for every hierarchy node }a,
135 }
136 \]
137 and in particular
138 \[
139 \boxed{W_{\mathrm{root}}\cong\mathscr W(G).}
140 \]
141 If level \(r\) contributes defect \(\delta_r\) and downstream collapse maps
142 have Lipschitz constants \(L_s\), then
143 \[
144 \boxed{
145 \delta_{\mathrm{flat}}^{(R)}
146 \leq
147 \sum_{r=0}^{R-1}
148 \left(
149 \prod_{s=r+1}^{R-1}L_s
150 \right)\delta_r.
151 }
152 \]
153 
154 
155 \subtopic{\(T_{25}\): Faithful coarse-graining theorem}
156 
157 For a partition of the vertex set, let \(W_\Pi\) be the direct sum of the
158 internally coherent section spaces and let \(D_\Pi\) retain all boundary
159 compatibility equations. Then
160 \[
161 \boxed{
162 \Gamma(H)
163 =
164 J_\Pi\ker D_\Pi.
165 }
166 \]
167 Exact structural workspace content is therefore invariant under faithful
168 aggregation. Approximate error separates into an internal-coherence term and
169 a boundary-coherence term controlled by their respective spectral gaps.
170 
171 \subtopic{\(T_{26}\): Bayes-risk transport theorem}
172 
173 For an information-faithful coarsening \(q:V_f\to V_c\),
174 \[
175 \boxed{
176 \Risk_{\mathcal D_c}(S)
177 =
178 \Risk_{\mathcal D_f}(q^{-1}(S))
179 \qquad(S\subseteq V_c).
180 }
181 \]
182 Hence displayed coalition size can decrease under coarsening. Weighted
183 primitive support cost is a more stable localization measure and cannot
184 improve merely because several carriers were relabeled as one macro-node.
185 
186 \subtopic{\(T_{27}\): Decomposition-transport robustness theorem}
187 
188 If every gate margin at a reference decomposition exceeds its worst-case
189 transport and model-mismatch defect across \(\mathfrak D\), then the complete
190 certificate passes throughout the admissible family:
191 \[
192 \boxed{
193  m_q(W;\mathcal D_0)
194 >
195 \sup_{\mathcal D\in\mathfrak D}
196 \bigl(L_q\delta_{\mathcal D}+b_q(\mathcal D)\bigr)
197 \quad\forall q
198 \Longrightarrow
199 \operatorname{Certify}_{L,\mathfrak D}^{\mathrm{rob}}(W)=1.
200 }
201 \]
202 
203 \topic{69}{Dynamic, Causal, and Agency Results}
204 
205 \subtopic{\(T_{10}\): Approximate universality theorem }
206 
207 For the first \(d\) transfer-Laplacian modes, with
208 \(\lambda_d\leq\varepsilon<\lambda_{d+1}=\mu\),
209 \[
210 \boxed{
211 \|(I-P_d)\rho_V(a)P_d\|
212 \leq
213 \frac{\|\rho_E(a)\|\sqrt{\varepsilon}+\beta(a)}{\sqrt{\mu}}.
214 }
215 \]
216 Thus low transfer energy, a separated complement, and small intertwining defect
217 imply approximate closure under the common operation algebra.
218 
219 \subtopic{\(T_{11}\): Core--residual accumulation theorem }
220 
221 For a path \(\gamma=e_m\cdots e_1\),
222 \[
223 \boxed{
224 \begin{aligned}
225 \|Q_{t(\gamma)}T_\gamma P_{s(\gamma)}\|
226 \leq
227 \sum_{r=1}^{m}
228 &\left(
229 \prod_{\ell=r+1}^{m}\|T_{e_\ell}\|
230 \right)
231 \|V_{e_r}\|\\
232 &\times
233 \left(
234 \prod_{\ell=1}^{r-1}\|H_{e_\ell}\|
235 \right).
236 \end{aligned}
237 }
238 \]
239 The same bound controls the defect between full core-to-core path transport and
240 the pure core product. An exact Duhamel
241 identity yields both estimates.
242 
243 \subtopic{\(T_{12}\): Predictive-state minimality and alignment theorem }
244 
245 On standard Borel spaces, the canonical conditional-law state
246 \[
247 Z_t
248 =
249 \mathbb P(Y_t^+\in\cdot\mid\mathbf H_t)
250 \]
251 is sufficient and minimal:
252 \[
253 Y_t^+\indep\mathbf H_t\mid Z_t,
254 \]
255 and every sufficient statistic \(S_t=s(\mathbf H_t)\) satisfies
256 \[
257 Z_t=\phi(S_t)
258 \qquad\text{a.s.}
259 \]
260 Moreover, a structural candidate \(W_t\) is exactly operationally aligned if
261 and only if
262 \[
263 Z_t=\phi_W(W_t)
264 \qquad\text{a.s.}
265 \]
266 The countable intervention-indexed version holds for
267 \(Z_t^{\mathcal I}\).
268 
269 \subtopic{\(T_{13}\): Operational relationality theorem }
270 
271 For a finite target, exact alignment, a positive node-level conditional
272 information gap, a positive robust local-predictive-matched null gap, a feasible
273 workspace intervention effect, and positive matched specificity jointly imply
274 that:
275 \[
276 \begin{aligned}
277 &\text{no single-node statistic is sufficient},\\
278 &k_0\geq2,\\
279 &\text{no single node reconstructs the aligned workspace},\\
280 &\text{matched local nulls preserve all node risks but lose joint capacity},\\
281 &\text{and the workspace coordinate is causally efficacious at the tested scale.}
282 \end{aligned}
283 \]
284 If \(\eta_{\mathrm{loop}}(\tau)>0\), the conclusion additionally includes
285 loop-mediated reentry.
286 
287 \subtopic{\(T_{14}\): Replacement invariance theorem }
288 
289 For a unitary realization correspondence with discrepancy defect
290 \[
291 \delta_D=\|D'J_V-J_ED\|,
292 \]
293 set
294 \[
295 \eta_L=2\|D\|\delta_D+\delta_D^2.
296 \]
297 If \(\eta_L<g/2\), then the workspace projectors obey
298 \[
299 \boxed{
300 q_{\mathrm{str}}
301 \leq
302 \frac{\eta_L}{g-\eta_L}
303 \leq
304 \frac{2\eta_L}{g}.
305 }
306 \]
307 A canonical unitary correspondence between the workspace subspaces exists,
308 and its operation defect is bounded by the projector drift, the local
309 intertwining defect, and prior operation leakage. Coalition frame bounds also
310 persist under sufficiently small reconstructive transport error. With declared
311 causal-kernel and reconstructive-handoff defects,
312 \[
313 \boxed{
314 d_{\mathrm{id}}
315 \leq
316  a\,q_{\mathrm{str}}
317 +b\,\delta_{\mathrm{op}}
318 +c\,\delta_{\mathrm{causal}}
319 +d\,\delta_{\mathrm{rec}}.
320 }
321 \]
322 Under bounded norms and a uniform spectral gap, this gives the schematic
323 estimate
324 \[
325 d_{\mathrm{id}}
326 \lesssim
327 C\frac{\delta_{\mathrm{replacement}}}{g}.
328 \]
329 The causal and reconstructive terms are necessary assumptions rather than
330 consequences of structural similarity.
331 
332 
333 \subtopic{\(T_{15}\): Transfer identification theorem }
334 
335 Under the linear transfer model
336 \[
337 Y=H_*X+\xi,
338 \qquad
339 \mathbb E[\xi X^{\mathsf T}]=0,
340 \qquad
341 \Sigma_{XX}\succeq\kappa I,
342 \]
343 the unique population least-squares transfer is
344 \[
345 \boxed{H_*=\Sigma_{YX}\Sigma_{XX}^{-1},}
346 \]
347 and
348 \[
349 \mathcal R(H)-\mathcal R(H_*)
350 \geq
351 \kappa\|H-H_*\|_F^2.
352 \]
353 For empirical ordinary least squares,
354 \[
355 \|\widehat H-H_*\|_F
356 \leq
357 \frac{\|\widehat M\|_F}{\widehat\kappa},
358 \]
359 with the stated ridge extension. The identified transfer is gauge covariant
360 under independent invertible changes of source and target coordinates.
361 
362 \subtopic{\(T_{16}\): Uniform structure-selection theorem }
363 
364 For a finite candidate family with unique population optimum \(\theta_*\),
365 margin \(\Delta_J>0\), and held-out uniform score error at most \(\varepsilon\),
366 \[
367 \boxed{
368 J(\widehat\theta)-J(\theta_*)\leq2\varepsilon.
369 }
370 \]
371 If \(2\varepsilon<\Delta_J\), then \(\widehat\theta=\theta_*\). The theorem
372 applies to node membership, edge selection, gauge-normalized transfer models,
373 and finite boundary families.
374 
375 \subtopic{\(T_{17}\): Stable boundary recovery theorem }
376 
377 For a finite admissible boundary family with unique maximizer \(S_t^*\), margin
378 \(\Delta_t^B\), and uniform held-out score error \(\varepsilon_t\),
379 \[
380 \mathcal B_t(S_t^*)-
381 \mathcal B_t(\widehat S_t)
382 \leq2\varepsilon_t,
383 \]
384 and exact recovery holds when \(2\varepsilon_t<\Delta_t^B\). If the population
385 score drifts uniformly by at most \(\delta_t\), then
386 \[
387 2\delta_t<\Delta_t^B
388 \Longrightarrow
389 S_{t+1}^*=S_t^*.
390 \]
391 
392 \subtopic{\(T_{18}\): Model-specific formation transition theorem }
393 
394 For
395 \[
396 m_{t+1}=\tanh(\lambda m_t),
397 \]
398 there is a unique zero fixed point for \(0\leq\lambda\leq1\). For
399 \(\lambda>1\), the zero state is unstable and exactly two stable nonzero fixed
400 points appear. Their amplitude satisfies
401 \[
402 \boxed{
403 m_\lambda
404 =
405 \sqrt{\frac{3(\lambda-1)}{\lambda^3}}
406 +
407 O\bigl((\lambda-1)^{3/2}\bigr).
408 }
409 \]
410 This is an exact structural benchmark, not a universal law for complete
411 relational workspaces.
412 
413 \subtopic{\(T_{19}\): Cue-controlled goal-retention theorem }
414 
415 For matched future command and disturbance trajectories under
416 \[
417 G_{s+1}=R_sG_s+B_sC_{s+1}+\zeta_{s+1},
418 \]
419 two goal interventions satisfy
420 \[
421 \boxed{
422 G_{t+\tau}-G_{t+\tau}'
423 =
424 R_{t+\tau:t}(g-g').
425 }
426 \]
427 If
428 \[
429 m_s\leq\sigma_{\min}(R_s),
430 \qquad
431 \|R_s\|\leq M_s,
432 \]
433 then
434 \[
435 \boxed{
436 \left(\prod_s m_s\right)\|g-g'\|
437 \leq
438 \|G_{t+\tau}-G_{t+\tau}'\|
439 \leq
440 \left(\prod_s M_s\right)\|g-g'\|.
441 }
442 \]
443 A linear action readout gives a corresponding lower bound on the induced
444 action-mean separation. The result isolates internal retention from repeated
445 external goal rewriting.
446 
447 \subtopic{\(T_{20}\): Approximate counterfactual-policy theorem }
448 
449 If the interventional action-value estimator satisfies
450 \[
451 \sup_a|\widehat Q_t(a)-Q_t(a)|\leq\varepsilon_Q
452 \]
453 and the implemented policy has optimization error at most
454 \(\delta_{\mathrm{opt}}\) relative to \(\widehat Q_t\), then
455 \[
456 \boxed{
457 \operatorname{Reg}_t(\widehat\pi_t)
458 \leq
459 2\varepsilon_Q+\delta_{\mathrm{opt}}.
460 }
461 \]
462 For deterministic greedy action, the bound is \(2\varepsilon_Q\). A unique
463 true optimum with value gap greater than \(2\varepsilon_Q\) is recovered
464 exactly. The guarantee is relative to the declared utility, feasible action
465 support, causal identification, and horizon.
466 
467 \subtopic{\(T_{21}\): Repair-selection margin theorem }
468 
469 For a finite repair set and a repair-value estimator with uniform error
470 \(\varepsilon_R\), the selected repair \(\widehat r\) satisfies
471 \[
472 \boxed{
473 V(r^*)-V(\widehat r)
474 \leq
475 2\varepsilon_R.
476 }
477 \]
478 If the optimal repair reaches reserve \(\rho+2\varepsilon_R\), then the selected
479 repair reaches reserve \(\rho\). A unique optimal repair whose value gap
480 exceeds \(2\varepsilon_R\) is selected exactly.
481 
482 \subtopic{\(T_{22}\): Closed-loop self-maintenance theorem }
483 
484 If matched no-repair deficit obeys
485 \[
486 e_{t+1}^{(0)}\leq\alpha e_t+d_t
487 \]
488 and the implemented repair has measured causal effect
489 \[
490 e_{t+1}^{(0)}-e_{t+1}\geq\kappa e_t-\epsilon_t,
491 \]
492 set \(\beta=\alpha-\kappa\). Under
493 \[
494 0\leq\beta<1,
495 \qquad
496  d_t\leq\overline d,
497 \qquad
498  \epsilon_t\leq\overline\epsilon,
499 \]
500 \[
501 \boxed{
502 e_t
503 \leq
504 \beta^te_0
505 +
506 \frac{(\overline d+\overline\epsilon)(1-\beta^t)}{1-\beta}.
507 }
508 \]
509 Hence
510 \[
511 \limsup_{t\to\infty}e_t
512 \leq
513 \frac{\overline d+\overline\epsilon}{1-\beta}.
514 \]
515 A corresponding margin condition makes the declared bounded-deficit viability
516 set forward invariant.
517 
518 \subtopic{\(T_{23}\): Workspace-agency synthesis theorem }
519 
520 Operational relationality, cue-controlled and behaviorally operative goals,
521 calibrated counterfactual values, feasible value-to-action and action-to-outcome
522 interventions, causal closed-loop maintenance, distributed control-state
523 sufficiency, and matched local policy nulls jointly imply:
524 \[
525 \begin{aligned}
526 &\operatorname{Reg}_t\leq2\varepsilon_Q+\delta_{\mathrm{opt}},\\
527 &\limsup e_t\leq
528  \frac{\overline d+\overline\epsilon}{1-\beta},\\
529 &k_{\mathrm{ctl},\varepsilon}\geq2,
530 \end{aligned}
531 \]
532 together with causal mediation of goal-dependent action and repair through the
533 implemented internal channels. This is a sufficient, domain-relative agency
534 certificate. It does not imply consciousness, personhood, or unrestricted
535 autonomy.
536 
537 \subtopic{\(T_{24}\): Gatewise certification-validity theorem }
538 
539 Suppose a certificate requires every gate \(G_1,\ldots,G_m\) to pass, and the
540 predeclared test for gate \(j\) has false-pass probability at most \(\alpha_j\)
541 whenever that required property is absent. Under the global null that at least
542 one required property is absent,
543 \[
544 \boxed{
545 \mathbb P(\text{complete certificate passes})
546 \leq
547 \max_j\alpha_j.
548 }
549 \]
550 No independence assumption among gate tests is required. The guarantee fails
551 under adaptive gate deletion, selective reporting, or reuse of the
552 certification split.
553 
554 
555 \topic{70}{Decomposition, Specification, and Identification Results}
556 
557 \subtopic{\(T_{25}\): Faithful coarse-graining theorem}
558 For an exact partition into internally coherent macro-components,
559 \[
560 \boxed{
561 \ker D_H=J_\Pi\ker D_\Pi.
562 }
563 \]
564 Thus faithful aggregation preserves the exact shared workspace and its
565 dimension while changing the displayed node resolution.
566 
567 \subtopic{\(T_{26}\): Bayes-risk transport under aggregation}
568 For an information-faithful coarsening map \(q\),
569 \[
570 \boxed{
571 \Risk_{\mathcal D_c}(S)
572 =
573 \Risk_{\mathcal D_f}\bigl(q^{-1}(S)\bigr).
574 }
575 \]
576 Predictive content is preserved, although the visible coalition count may
577 contract. Primitive-carrier support cost is therefore preferred when claims
578 must be compared across resolutions.
579 
580 \subtopic{\(T_{27}\): Decomposition-transport robustness theorem}
581 If every required gate has reference margin larger than its worst admissible
582 decomposition-transport defect, then the complete certificate passes
583 throughout \(\mathfrak D\). When the mismatch term vanishes, the certificate
584 radius is bounded below by the smallest gate margin divided by its transport
585 sensitivity.
586 
587 \subtopic{\(T_{28}\): Specification-transport robustness theorem}
588 For a reference specification \(\varsigma_0\), if
589 \[
590 m_q(W;\varsigma_0)
591 >
592 \sup_{\varsigma\in\mathfrak S}
593 \left[
594 L_qd_{\mathfrak S}(\varsigma,\varsigma_0)+b_q(\varsigma)
595 \right]
596 \]
597 for every gate, then the transported workspace passes every admissible
598 specification. If \(b_q=0\),
599 \[
600 r_{\mathrm{spec}}(W;\varsigma_0)
601 \geq
602 \min_{q:L_q>0}\frac{m_q(W;\varsigma_0)}{L_q}.
603 \]
604 
605 \subtopic{\(T_{29}\): Experimental non-identifiability theorem}
606 If two realizations induce the same law under every declared observation and
607 intervention but have different certificate values, no decision rule based on
608 that experiment can be uniformly correct. For every such rule, the two error
609 probabilities sum to one, so the minimax error is at least one half.
610 
611 \subtopic{\(T_{30}\): Joint specification-identification theorem}
612 Let \(r_{\mathfrak S}(\mathfrak R)\) be the worst gate margin across the
613 specification family. If its infimum over the identified realization set is
614 positive, the robust certificate is identified as passing. If its supremum is
615 nonpositive, the certificate is identified as failing. Compatible realizations
616 on opposite sides of zero establish non-identifiability.
617 
618 \subtopic{\(T_{31}\): Uniform finite-sample specification theorem}
619 If all gate estimators are uniformly within their error radii across
620 \(\mathfrak S\), then requiring every empirical lower envelope to be positive
621 controls the probability of a false robust certificate by the failure
622 probability of the uniform event.
623 
624 \subtopic{\(T_{32}\): Independent selection and certification theorem}
625 A specification chosen on data independent of the certification split may be
626 evaluated by its fixed-specification test without inflating the false-pass
627 bound. Selection and certification on the same outcomes require simultaneous
628 or post-selection correction.

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