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{XIV}{Scaffolded and Endogenous Formation}
4 
5 \topic{47}{Formation Systems and Scaffold Specification}
6 
7 The preceding parts begin with a transfer system that has already been chosen.
8 A formation theory must instead explain how the nodes, edges, transfer maps,
9 workspace channels, and operational boundaries are obtained. This requires a
10 strict separation between the structure supplied by the experimenter and the
11 structure selected from interaction history.
12 
13 \begin{definition}[Formation system]\label{def:formation-system}
14 A \emph{workspace formation system} is a tuple
15 \[
16 \mathfrak F
17 =
18 \bigl(
19 \Sigma,
20 \mathcal D_{\mathrm{form}},
21 \mathcal L,
22 \Omega,
23 \widehat\theta,
24 \mathcal D_{\mathrm{cert}},
25 \mathcal I_{\mathrm{form}}
26 \bigr),
27 \]
28 where:
29 \begin{itemize}
30 \item \(\Sigma\) is the externally supplied scaffold;
31 \item \(\mathcal D_{\mathrm{form}}\) is the formation or training history;
32 \item \(\mathcal L\) is the learning or selection rule;
33 \item \(\Omega\) is its internal random seed or stochastic state;
34 \item \(\widehat\theta=\mathcal L(\Sigma,\mathcal D_{\mathrm{form}},\Omega)\)
35  is the learned organization;
36 \item \(\mathcal D_{\mathrm{cert}}\) is held-out certification data; and
37 \item \(\mathcal I_{\mathrm{form}}\) is a declared family of interventions on
38  scaffold and formation history.
39 \end{itemize}
40 The learned organization is decomposed as
41 \[
42 \widehat\theta
43 =
44 \bigl(
45 \widehat V,
46 \widehat E,
47 \widehat H,
48 \widehat P,
49 \widehat{\mathcal O},
50 \widehat B
51 \bigr),
52 \]
53 where \(\widehat B\) is a learned operational boundary.
54 \end{definition}
55 
56 The scaffold is itself stratified:
57 \[
58 \boxed{
59 \Sigma
60 =
61 \bigl(
62 \Sigma_V,
63 \Sigma_E,
64 \Sigma_H,
65 \Sigma_P,
66 \Sigma_{\mathcal O},
67 \Sigma_B,
68 \Sigma_Y
69 \bigr).
70 }
71 \]
72 The final coordinate \(\Sigma_Y\) records labels, targets, privileged masks,
73 or other supervision that could directly reveal the intended workspace. This
74 coordinate is necessary because an architecture may appear to discover a
75 boundary even when the desired membership mask was encoded in its training
76 labels.
77 
78 \subtopic{47.1 Scaffolded architecture versus scaffolded solution}
79 
80 A scaffold can constrain the search space without specifying its solution.
81 For example, a graph neural architecture can fix the class of admissible local
82 message-passing rules while leaving the active edges and transfer maps to be
83 learned. Conversely, a system can use a flexible architecture while receiving
84 labels that directly encode the desired workspace. The former is
85 architecturally scaffolded but solution-open. The latter is solution-scaffolded.
86 
87 For any component \(q\) of \(\widehat\theta\), define its scaffold-only Bayes
88 risk under loss \(\ell_q\) by
89 \[
90 \mathcal R_q^{\Sigma}
91 =
92 \inf_f
93 \mathbb E
94 \left[
95 \ell_q\bigl(q,f(\Sigma,\Omega)\bigr)
96 \right].
97 \]
98 Define the full formation risk by
99 \[
100 \mathcal R_q^{\Sigma,\mathcal H}
101 =
102 \inf_g
103 \mathbb E
104 \left[
105 \ell_q\bigl(q,g(\Sigma,\mathcal D_{\mathrm{form}},\Omega)\bigr)
106 \right].
107 \]
108 The history advantage is
109 \[
110 \boxed{
111 \Delta_q^{\mathrm{hist}}
112 =
113 \mathcal R_q^{\Sigma}
114 -
115 \mathcal R_q^{\Sigma,\mathcal H}.
116 }
117 \]
118 A positive history advantage means that formation history contains information
119 about \(q\) that is not already recoverable from the declared scaffold and
120 random seed.
121 
122 \begin{proposition}[Scaffold-only impossibility]\label{prop:scaffold-only-impossibility}
123 If
124 \[
125 q=f(\Sigma,\Omega)
126 \qquad\text{almost surely},
127 \]
128 then for every formation-history intervention that leaves \((\Sigma,\Omega)\)
129 fixed,
130 \[
131 \mathcal L\!\left(q\mid
132 \operatorname{do}(\mathcal D_{\mathrm{form}}=h)
133 \right)
134 =
135 \mathcal L\!\left(q\mid
136 \operatorname{do}(\mathcal D_{\mathrm{form}}=h')
137 \right),
138 \]
139 and \(\Delta_q^{\mathrm{hist}}=0\).
140 \end{proposition}
141 
142 \begin{proof}
143 The value of \(q\) is measurable with respect to \((\Sigma,\Omega)\), so an
144 intervention on formation history cannot change its conditional law when those
145 variables are held fixed. The scaffold-only predictor \(f\) attains the same
146 risk as any predictor that also receives formation history, hence the risk gap
147 vanishes.
148 \end{proof}
149 
150 This proposition is elementary but methodologically decisive. A structure
151 that is deterministically encoded by the scaffold cannot count as discovered
152 from interaction history merely because a learned network reproduces it.
153 
154 \subtopic{47.2 Formation and certification split}
155 
156 The same sample must not be used both to select a workspace and to certify its
157 coherence, causal efficacy, or relational irreducibility. The required split is
158 \[
159 \mathcal D
160 =
161 \mathcal D_{\mathrm{form}}
162 \sqcup
163 \mathcal D_{\mathrm{cal}}
164 \sqcup
165 \mathcal D_{\mathrm{cert}}.
166 \]
167 The formation sample selects \(\widehat\theta\). The calibration sample fixes
168 thresholds, regularization strengths, and intervention magnitudes. The
169 certification sample evaluates the final gates without further structural
170 selection. In a multi-environment study, the stronger requirement is
171 \[
172 \mathcal E_{\mathrm{form}}
173 \cap
174 \mathcal E_{\mathrm{cert}}
175 =
176 \varnothing,
177 \]
178 so that the learned organization is tested under environments not used to
179 construct it.
180 
181 \begin{definition}[Formation leakage]\label{def:formation-leakage}
182 A certification variable \(Y_{\mathrm{cert}}\) leaks into formation when the
183 conditional law of \(\widehat\theta\) changes under an intervention on
184 \(Y_{\mathrm{cert}}\) that should be inaccessible during formation:
185 \[
186 \mathcal L
187 \left(
188 \widehat\theta
189 \mid
190 \operatorname{do}(Y_{\mathrm{cert}}=y)
191 \right)
192 \neq
193 \mathcal L
194 \left(
195 \widehat\theta
196 \mid
197 \operatorname{do}(Y_{\mathrm{cert}}=y')
198 \right).
199 \]
200 A valid discovery claim requires a documented information barrier or an audit
201 showing that this effect is absent up to tolerance.
202 \end{definition}
203 
204 \subtopic{47.3 Origin and reality are orthogonal}
205 
206 The formation provenance of a workspace and its operational certification are
207 separate coordinates. A fully scaffolded structure may satisfy every
208 structural and causal workspace gate. A highly endogenous structure may fail
209 to become coherent or causally useful. Symbolically,
210 \[
211 \operatorname{Endogenous}(W)
212 \centernot\Longrightarrow
213 \operatorname{SRW}_{\Theta}(W),
214 \]
215 \[
216 \operatorname{SRW}_{\Theta}(W)
217 \centernot\Longrightarrow
218 \operatorname{Endogenous}(W).
219 \]
220 The theory therefore treats endogeneity as a provenance axis rather than a
221 higher level on the workspace hierarchy.
222 
223 \topic{48}{Operational Endogeneity Signature}
224 
225 The ideal description-complexity expression from the symbolic outline is
226 retained as a conceptual limit, but raw Kolmogorov complexity is not
227 computable. A usable theory must declare a model class, a code, and an
228 intervention family.
229 
230 \begin{definition}[Code-relative history contribution]\label{def:mdl-endogeneity}
231 Let \(\mathfrak M_q\) be a declared coding or predictive model class for a
232 learned component \(q\). Let
233 \[
234 L_{\mathfrak M_q}(q\mid Z)
235 \]
236 be its held-out codelength given side information \(Z\). Define
237 \[
238 \boxed{
239  e_q^{\mathrm{MDL}}
240  =
241  \left[
242  \frac{
243  L_{\mathfrak M_q}(q\mid\Sigma)
244  -
245  L_{\mathfrak M_q}(q\mid\Sigma,\mathcal D_{\mathrm{form}})
246  }{
247  \max\{L_{\mathfrak M_q}(q\mid\Sigma),1\}
248  }
249  \right]_{0}^{1},
250 }
251 \]
252 where \([x]_0^1=\min\{1,\max\{0,x\}\}\).
253 \end{definition}
254 
255 This quantity is explicitly code-relative. It must not be presented as an
256 observer-independent metaphysical measure. Its role is to test whether the
257 formation history provides compressive information about the learned
258 organization beyond the declared scaffold.
259 
260 \begin{definition}[Causal formation sensitivity]\label{def:causal-formation-sensitivity}
261 Let \(\iota_h\) and \(\iota_{h'}\) be matched feasible interventions that alter
262 formation history while preserving the scaffold, randomization law, sample
263 size, and intervention cost. For a divergence \(D\), define
264 \[
265 \boxed{
266  e_q^{\mathrm{causal}}
267  =
268  \mathbb E_{(h,h')\sim\nu_q}
269  D\!\left(
270  \mathcal L^{\iota_h}(q),
271  \mathcal L^{\iota_{h'}}(q)
272  \right).
273 }
274 \]
275 A common null uses temporally or relationally shuffled histories that preserve
276 local marginals but destroy the candidate formation signal.
277 \end{definition}
278 
279 \begin{proposition}[Reparameterization invariance of causal provenance]
280 \label{prop:causal-provenance-invariance}
281 Let \(\phi\) be a measurable bijection on the range of \(q\), and let the
282 divergence \(D\) be invariant under measurable bijections. Then
283 \[
284  e_{\phi(q)}^{\mathrm{causal}}
285  =
286  e_q^{\mathrm{causal}}.
287 \]
288 \end{proposition}
289 
290 \begin{proof}
291 Every intervention-induced law of \(\phi(q)\) is the pushforward of the
292 corresponding law of \(q\). Bijection invariance of \(D\) preserves each
293 pairwise divergence, and therefore their design-weighted expectation.
294 \end{proof}
295 
296 The causal coordinate is therefore insensitive to a mere renaming of the
297 learned structure. The MDL coordinate need not be invariant unless the coding
298 class is transformed with the reparameterization.
299 
300 \subtopic{48.1 Scaffold dependence and cross-scaffold invariance}
301 
302 Endogeneity does not require independence from all scaffolds. Every learning
303 system has an architecture, objective, and resource budget. The relevant
304 question is whether the learned organization depends on the particular
305 solution-bearing details of the scaffold.
306 
307 For a family \(\mathfrak S\) of admissible scaffold variants, define the
308 cross-scaffold identity defect
309 \[
310 \boxed{
311 \delta_q^{\mathrm{scaf}}
312 =
313 \inf_{\{J_{\sigma\sigma'}\}}
314 \mathbb E_{\sigma,\sigma'\sim\pi}
315  d_q
316 \left(
317 J_{\sigma\sigma'}q_{\sigma},q_{\sigma'}
318 \right),
319 }
320 \]
321 where \(J_{\sigma\sigma'}\) transports the learned object between admissible
322 coordinate systems. A low value indicates that the same abstract
323 organization is recovered under multiple non-solution-bearing scaffolds.
324 
325 The complete formation signature is
326 \[
327 \boxed{
328 \mathbf E_{\mathrm{form}}(W)
329 =
330 \left(
331  e_V,e_E,e_H,e_P,e_{\mathcal O},e_B;
332  \delta^{\mathrm{scaf}},
333  \delta^{\mathrm{leak}}
334 \right),
335 }
336 \]
337 where every \(e_q\) is reported as the pair
338 \[
339  e_q
340 =
341 \left(
342  e_q^{\mathrm{MDL}},
343  e_q^{\mathrm{causal}}
344 \right).
345 \]
346 No scalar average is permitted to hide a zero history contribution for one of
347 the structures claimed to have been discovered.
348 
349 \subtopic{48.2 Formation certificate}
350 
351 For thresholds \(\Theta_{\mathrm{form}}\), define
352 \[
353 \operatorname{Form}_{\Theta_{\mathrm{form}}}(W)=1
354 \]
355 only when all applicable gates pass:
356 \[
357 \begin{aligned}
358 & e_q^{\mathrm{MDL}}\geq \tau_q^{\mathrm{MDL}},
359 && e_q^{\mathrm{causal}}\geq \tau_q^{\mathrm{causal}},\\
360 & \delta_q^{\mathrm{scaf}}\leq\varepsilon_q^{\mathrm{scaf}},
361 && \delta^{\mathrm{leak}}\leq\varepsilon_{\mathrm{leak}},\\
362 & \operatorname{SRW}_{\Theta}(W)=1,
363 && \operatorname{Pass}(N_{\mathrm{form}})=1.
364 \end{aligned}
365 \]
366 The final workspace gate is included because endogenous discovery of an
367 unstable or causally irrelevant pattern is not endogenous workspace formation.
368 
369 \topic{49}{Learned Transfer Structure and Formation Classes}
370 
371 \subtopic{49.1 Gated membership and edge learning}
372 
373 Let
374 \[
375 z_i\in\{0,1\},
376 \qquad
377 a_e\in[0,1]
378 \]
379 be node-membership and edge-activation variables satisfying
380 \[
381 a_e\leq z_{s(e)},
382 \qquad
383 a_e\leq z_{t(e)}.
384 \]
385 A weighted learned discrepancy operator is
386 \[
387 (D_{a,H}c)_e
388 =
389 \sqrt{w_ea_e}
390 \left(
391  c_{t(e)}-H_ec_{s(e)}
392 \right).
393 \]
394 One generic formation objective is
395 \[
396 \boxed{
397 \begin{aligned}
398 \widehat\theta
399 \in
400 \argmin_{z,a,H,P,\mathcal O,B}
401 \Bigl
402 \{&
403 \widehat{\mathcal R}_{\mathrm{task}}
404 +
405 \lambda_{\mathrm{coh}}\|D_{a,H}c\|^2
406 +
407 \lambda_E\|a\|_1
408 +
409 \lambda_V\|z\|_1\\
410 &+
411 \lambda_{\mathrm{stab}}\mathcal J_{\mathrm{stab}}
412 +
413 \lambda_{\mathrm{leak}}\mathcal J_{\mathrm{leak}}
414 \Bigr\}.
415 \end{aligned}
416 }
417 \]
418 The objective is schematic rather than universal. Its purpose is to expose
419 which quantities were fixed and which were optimized. If \(z\) and \(a\) are
420 fixed by \(\Sigma\), only the transfer maps or local channels can be claimed as
421 learned.
422 
423 \subtopic{49.2 Gauge-identifiable transfer learning}
424 
425 A learned transfer map is not absolutely identifiable when local coordinates
426 are latent. Under local invertible changes
427 \[
428 X_u'=g_uX_u,
429 \qquad
430 X_v'=g_vX_v,
431 \]
432 the same abstract transfer is represented by
433 \[
434 H_{uv}'=g_vH_{uv}g_u^{-1}.
435 \]
436 The identifiable object is therefore the gauge class
437 \[
438 [H]_{\mathcal G}
439 =
440 \left\{
441 (g_{t(e)}H_eg_{s(e)}^{-1})_{e\in E}
442 :
443 (g_i)_{i\in V}\in\mathcal G
444 \right\}.
445 \]
446 Any formation claim that compares raw matrices across independently trained
447 systems without gauge alignment is ill posed.
448 
449 \subtopic{49.3 Transfer identification theorem \(T_{15}\) }
450 
451 \begin{theorem}[Population transfer identification]\label{thm:transfer-identification}
452 Let \(X\in\mathbb R^{d_u}\) and \(Y\in\mathbb R^{d_v}\) satisfy
453 \[
454 Y=H_*X+\xi,
455 \qquad
456 \mathbb E[\xi X^{\mathsf T}]=0,
457 \]
458 and assume
459 \[
460 \Sigma_{XX}
461 =
462 \mathbb E[XX^{\mathsf T}]
463 \succeq
464 \kappa I
465 \]
466 for some \(\kappa>0\). Then \(H_*\) is the unique minimizer of
467 \[
468 \mathcal R(H)
469 =
470 \mathbb E\|Y-HX\|_2^2,
471 \]
472 with
473 \[
474 \boxed{
475 H_*
476 =
477 \Sigma_{YX}\Sigma_{XX}^{-1}.
478 }
479 \]
480 Moreover,
481 \[
482 \boxed{
483 \mathcal R(H)-\mathcal R(H_*)
484 =
485 \operatorname{tr}
486 \left(
487 (H-H_*)\Sigma_{XX}(H-H_*)^{\mathsf T}
488 \right)
489 \geq
490 \kappa\|H-H_*\|_F^2.
491 }
492 \]
493 \end{theorem}
494 
495 \begin{proof}
496 Substitute \(Y=H_*X+\xi\) into the risk. The cross term vanishes because
497 \(\mathbb E[\xi X^{\mathsf T}]=0\), giving
498 \[
499 \mathcal R(H)
500 =
501 \mathcal R(H_*)
502 +
503 \mathbb E\|(H-H_*)X\|_2^2.
504 \]
505 The remaining expectation is the displayed trace. Positive definiteness of
506 \(\Sigma_{XX}\) gives uniqueness and the lower bound. The normal equations
507 give the covariance formula.
508 \end{proof}
509 
510 \begin{corollary}[Deterministic empirical recovery bound]
511 \label{cor:empirical-transfer-bound}
512 For samples \((X_r,Y_r)_{r=1}^n\), write
513 \[
514 \widehat\Sigma_{XX}
515 =
516 \frac1n\sum_{r=1}^nX_rX_r^{\mathsf T},
517 \qquad
518 \widehat M
519 =
520 \frac1n\sum_{r=1}^n\xi_rX_r^{\mathsf T}.
521 \]
522 If \(\widehat\Sigma_{XX}\succeq\widehat\kappa I\), then ordinary least squares
523 satisfies
524 \[
525 \boxed{
526 \|\widehat H-H_*\|_F
527 \leq
528 \frac{\|\widehat M\|_F}{\widehat\kappa}.
529 }
530 \]
531 For ridge parameter \(\tau>0\),
532 \[
533 \boxed{
534 \|\widehat H_\tau-H_*\|_F
535 \leq
536 \frac{
537 \|\widehat M\|_F+\tau\|H_*\|_F
538 }{
539 \widehat\kappa+\tau
540 }.
541 }
542 \]
543 \end{corollary}
544 
545 \begin{proof}
546 The empirical normal equations give
547 \[
548 \widehat H-H_*
549 =
550 \widehat M\widehat\Sigma_{XX}^{-1}.
551 \]
552 For ridge regression,
553 \[
554 \widehat H_\tau-H_*
555 =
556 (\widehat M-\tau H_*)
557 (\widehat\Sigma_{XX}+\tau I)^{-1}.
558 \]
559 Taking Frobenius and operator norms yields both estimates.
560 \end{proof}
561 
562 \begin{proposition}[Gauge covariance of least squares]
563 \label{prop:ols-gauge-covariance}
564 Under invertible coordinate transformations
565 \[
566 X'=g_uX,
567 \qquad
568 Y'=g_vY,
569 \]
570 the population least-squares solution transforms as
571 \[
572 H_*'=g_vH_*g_u^{-1}.
573 \]
574 \end{proposition}
575 
576 \begin{proof}
577 The transformed covariances are
578 \[
579 \Sigma_{Y'X'}=g_v\Sigma_{YX}g_u^{\mathsf T},
580 \qquad
581 \Sigma_{X'X'}=g_u\Sigma_{XX}g_u^{\mathsf T}.
582 \]
583 Substitution into the covariance formula and inversion of the second factor
584 give the result.
585 \end{proof}
586 
587 The theorem establishes identification only under excitation and exogeneity.
588 When \(\Sigma_{XX}\) is singular, the transfer is identified only on the
589 excited source subspace. When \(\mathbb E[\xi X^{\mathsf T}]\neq0\), regression
590 recovers a predictive map rather than necessarily the intended causal
591 transfer.
592 
593 \subtopic{49.4 Held-out structure-selection theorem \(T_{16}\) }
594 
595 The same finite-margin argument applies to node membership, edge selection,
596 boundary selection, or a finite family of gauge-normalized transfer models.
597 
598 \begin{theorem}[Uniform structure-selection recovery]
599 \label{thm:uniform-structure-selection}
600 Let \(\mathfrak T\) be a finite candidate family with population score
601 \(J:\mathfrak T\to\mathbb R\). Let
602 \[
603 \theta_*
604 =
605 \argmin_{\theta\in\mathfrak T}J(\theta)
606 \]
607 be unique, with margin
608 \[
609 \Delta_J
610 =
611 \min_{\theta\neq\theta_*}
612 \bigl(J(\theta)-J(\theta_*)\bigr)>0.
613 \]
614 If a held-out estimate \(\widehat J\) satisfies
615 \[
616 \sup_{\theta\in\mathfrak T}
617 |\widehat J(\theta)-J(\theta)|
618 \leq\varepsilon,
619 \]
620 then every empirical minimizer \(\widehat\theta\) obeys
621 \[
622 \boxed{
623 J(\widehat\theta)-J(\theta_*)
624 \leq2\varepsilon.
625 }
626 \]
627 If \(2\varepsilon<\Delta_J\), then
628 \[
629 \boxed{\widehat\theta=\theta_*.}
630 \]
631 \end{theorem}
632 
633 \begin{proof}
634 By empirical optimality and uniform approximation,
635 \[
636 J(\widehat\theta)
637 \leq
638 \widehat J(\widehat\theta)+\varepsilon
639 \leq
640 \widehat J(\theta_*)+\varepsilon
641 \leq
642 J(\theta_*)+2\varepsilon.
643 \]
644 If \(\widehat\theta\neq\theta_*\), the margin would force an excess larger than
645 \(2\varepsilon\), a contradiction.
646 \end{proof}
647 
648 The theorem does not guarantee that the candidate class contains the correct
649 organization. It states exactly what follows once a finite class, score, and
650 uniform certification error are declared.
651 
652 \subtopic{49.5 Formation classes}
653 
654 The provenance classes are now defined relative to
655 \[
656 (\Sigma,\mathcal L,\mathcal I_{\mathrm{form}},\mathfrak M).
657 \]
658 They are not absolute labels.
659 
660 \begin{description}[leftmargin=2.1em,style=nextline]
661 \item[Exogenously scaffolded.]
662 The claimed nodes, edges, transfer maps, channels, operations, or boundary are
663 recoverable from the scaffold alone at the declared tolerance. Their history
664 advantages and causal formation sensitivities vanish.
665 
666 \item[Adaptively parameterized.]
667 The membership and channel topology are fixed, while transfer parameters,
668 local operators, or readouts are learned from interaction history. Typically
669 \[
670 e_H+e_P+e_{\mathcal O}>0,
671 \qquad
672 e_V\approx e_E\approx e_B\approx0.
673 \]
674 
675 \item[Structurally adaptive.]
676 At least one of membership, edges, or boundary is selected from history, but
677 the search family and objective remain externally specified.
678 
679 \item[Endogenously constituted relative to the scaffold class.]
680 Membership, transfer organization, and operational boundary have nonzero
681 history contribution and causal formation sensitivity, survive cross-scaffold
682 transport, pass leakage and matched-null tests, and produce a certified
683 workspace on held-out environments.
684 \end{description}
685 
686 The phrase ``relative to the scaffold class'' is mandatory. No finite
687 experiment can establish independence from every possible external influence.

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