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{XIII}{Gluing and Workspaces of Workspaces}
4 
5 The preceding parts treat one interaction diagram at a time. A large
6 collaborative system is usually assembled from overlapping local laboratories,
7 subteams, artifacts, models, or task-specific workspaces. This part asks when
8 those local workspaces determine one coherent global workspace, how failed
9 gluing is detected, and whether recursive assembly changes the resulting
10 object. The exact theory is a descent theorem. The approximate theory is a
11 second transfer-Laplacian construction on the cover itself.
12 
13 \topic{42}{Local Workspace Covers}
14 
15 \subtopic{42.1 Edge-complete covers}
16 
17 Let
18 \[
19 G=(V,E)
20 \]
21 be a finite directed interaction graph with local core spaces \(C_i\), core
22 transfers \(H_e\), vertex space \(\mathcal C_V(G)\), discrepancy operator
23 \(D_G\), and exact workspace
24 \[
25 \mathscr W(G)
26 :=
27 \Gamma(H|_G)
28 =
29 \ker D_G.
30 \]
31 
32 \begin{definition}[Finite workspace cover]\label{def:workspace-cover}
33 A finite family of subgraphs
34 \[
35 \mathcal U=\{G_\alpha\}_{\alpha\in A}
36 \]
37 is an \emph{edge-complete workspace cover} of \(G\) when
38 \[
39 V=\bigcup_{\alpha\in A}V_\alpha,
40 \qquad
41 E=\bigcup_{\alpha\in A}E_\alpha.
42 \]
43 Every edge in \(E_\alpha\) carries the transfer inherited from \(G\). For a
44 nonempty finite index set
45 \[
46 I=\{\alpha_0,\ldots,\alpha_p\}\subseteq A,
47 \]
48 define the intersection subgraph
49 \[
50 G_I
51 =
52 G_{\alpha_0\cdots\alpha_p}
53 :=
54 \bigcap_{\alpha\in I}G_\alpha
55 \]
56 by intersecting both vertex and edge sets. Isolated vertices are retained.
57 \end{definition}
58 
59 Covering only the vertices is insufficient. If an edge belongs to no chart,
60 its transfer equation is absent from every local workspace and cannot be
61 recovered by overlap agreement. Edge completeness is therefore the minimal
62 cover assumption used below.
63 
64 If \(K\subseteq L\subseteq G\) are subgraphs, coordinate restriction gives
65 \[
66 \rho_K^L:\mathcal C_V(L)\longrightarrow\mathcal C_V(K).
67 \]
68 Because every edge equation on \(K\) is also an edge equation on \(L\),
69 restriction maps exact sections to exact sections:
70 \[
71 r_K^L
72 :=
73 \rho_K^L|_{\mathscr W(L)}:
74 \mathscr W(L)\longrightarrow\mathscr W(K).
75 \]
76 They obey
77 \[
78 r_L^L=I,
79 \qquad
80 r_J^K r_K^L=r_J^L
81 \]
82 whenever \(J\subseteq K\subseteq L\). Thus \(K\mapsto\mathscr W(K)\) is a
83 contravariant system of local workspaces.
84 
85 \subtopic{42.2 The cover nerve and overlap interfaces}
86 
87 The nerve \(N(\mathcal U)\) has one vertex for each chart \(G_\alpha\) and a
88 \(p\)-simplex
89 \[
90 [\alpha_0\cdots\alpha_p]
91 \]
92 whenever \(G_{\alpha_0\cdots\alpha_p}\) is nonempty. The nerve records which
93 local workspaces can compare states directly. It does not replace the
94 underlying interaction graph: \(G\) carries the internal transfer equations,
95 while \(N(\mathcal U)\) carries overlap-consistency equations between already
96 constructed local objects.
97 
98 For pairwise overlaps write
99 \[
100 \mathscr W_\alpha=\mathscr W(G_\alpha),
101 \qquad
102 \mathscr W_{\alpha\beta}=\mathscr W(G_{\alpha\beta}),
103 \]
104 and
105 \[
106 r_{\alpha\beta}^{\alpha}:
107 \mathscr W_\alpha\to\mathscr W_{\alpha\beta},
108 \qquad
109 r_{\alpha\beta}^{\beta}:
110 \mathscr W_\beta\to\mathscr W_{\alpha\beta}.
111 \]
112 The interface is therefore a cospan
113 \[
114 \mathscr W_\alpha
115 \xrightarrow{\ r_{\alpha\beta}^{\alpha}\ }
116 \mathscr W_{\alpha\beta}
117 \xleftarrow{\ r_{\alpha\beta}^{\beta}\ }
118 \mathscr W_\beta,
119 \]
120 not necessarily a direct transfer map from one complete local workspace to the
121 other. Both charts are compared through the content they expose on their
122 common domain.
123 
124 \subtopic{42.3 \v{C}ech cochains of local workspaces}
125 
126 For \(p\geq0\), define the finite cochain space
127 \[
128 \check C^p(\mathcal U;\mathscr W)
129 =
130 \bigoplus_{\alpha_0<\cdots<\alpha_p}
131 \mathscr W(G_{\alpha_0\cdots\alpha_p}),
132 \]
133 where empty intersections contribute the zero space. The coboundary
134 \[
135 \delta^p:\check C^p\longrightarrow\check C^{p+1}
136 \]
137 is
138 \[
139 \boxed{
140 (\delta^p s)_{\alpha_0\cdots\alpha_{p+1}}
141 =
142 \sum_{j=0}^{p+1}(-1)^j
143 r_{\alpha_0\cdots\alpha_{p+1}}^%
144 {\alpha_0\cdots\widehat{\alpha_j}\cdots\alpha_{p+1}}
145 %
146 s_{\alpha_0\cdots\widehat{\alpha_j}\cdots\alpha_{p+1}}.
147 }
148 \]
149 The restriction identities imply
150 \[
151 \delta^{p+1}\delta^p=0.
152 \]
153 For a local family \(w=(w_\alpha)_\alpha\),
154 \[
155 \boxed{
156 (\delta^0w)_{\alpha\beta}
157 =
158 r_{\alpha\beta}^{\beta}w_\beta
159 -
160 r_{\alpha\beta}^{\alpha}w_\alpha.
161 }
162 \]
163 Thus \(\delta^0\) is the overlap-discrepancy operator at the level of exact
164 local workspaces.
165 
166 \topic{43}{Compatibility and Gluing Obstructions}
167 
168 \subtopic{43.1 Direct overlap mismatch}
169 
170 A family \(w=(w_\alpha)_\alpha\) is exactly compatible when
171 \[
172 \delta^0w=0.
173 \]
174 Its direct mismatch is
175 \[
176 \boxed{
177 \mathfrak m_{\mathcal U}(w)=\delta^0w.
178 }
179 \]
180 After choosing inner products and positive overlap weights, the mismatch
181 energy is
182 \[
183 \boxed{
184 \mathcal E_{\mathrm{ov}}(w)
185 =
186 \|\delta^0w\|^2
187 =
188 \sum_{\alpha<\beta}
189 \nu_{\alpha\beta}
190 \left\|
191  r_{\alpha\beta}^{\beta}w_\beta
192 -r_{\alpha\beta}^{\alpha}w_\alpha
193 \right\|^2.
194 }
195 \]
196 This is the appropriate obstruction when the full local states are already
197 given. Such a family glues exactly if and only if its mismatch vector itself
198 vanishes. No cohomological quotient is needed for that direct question.
199 
200 \subtopic{43.2 Relative-only and affine descent data}
201 
202 A different problem arises when only relative transition data are supplied.
203 Let
204 \[
205 b=(b_{\alpha\beta})\in\check C^1(\mathcal U;\mathscr W)
206 \]
207 be a prescribed family of overlap offsets. The triple-overlap consistency
208 condition is
209 \[
210 \delta^1b=0.
211 \]
212 One seeks local potentials \(q=(q_\alpha)\in\check C^0\) satisfying
213 \[
214 \delta^0q=b.
215 \]
216 Define
217 \[
218 \boxed{
219 \check H^1(\mathcal U;\mathscr W)
220 =
221 \frac{\ker\delta^1}{\operatorname{im}\delta^0}.
222 }
223 \]
224 Then the affine gluing problem is solvable precisely when
225 \[
226 \boxed{[b]=0\quad\text{in}\quad\check H^1(\mathcal U;\mathscr W).}
227 \]
228 A nonzero class means that all local transition statements can be mutually
229 consistent on triple overlaps while still failing to arise from any family of
230 local workspace states. This is a cover-level analogue of nontrivial holonomy.
231 
232 \subtopic{43.3 Harmonic representative of the obstruction}
233 
234 Equip the finite cochain spaces with Hilbert inner products and define the
235 first \v{C}ech Laplacian
236 \[
237 \boxed{
238 \Delta_1
239 =
240 \delta^0(\delta^0)^*
241 +
242 (\delta^1)^*\delta^1.
243 }
244 \]
245 Let
246 \[
247 \mathcal H^1_{\mathcal U}=\ker\Delta_1.
248 \]
249 
250 \begin{proposition}[Harmonic gluing obstruction]\label{prop:harmonic-gluing-obstruction}
251 For every cocycle \(b\in\ker\delta^1\), there is an orthogonal decomposition
252 \[
253 \boxed{
254 b=\delta^0q^\dagger+h,
255 \qquad
256 h\in\mathcal H^1_{\mathcal U},
257 }
258 \]
259 where \(q^\dagger=(\delta^0)^\dagger b\) is a least-squares potential. Moreover,
260 \[
261 \boxed{
262 \inf_q\|b-\delta^0q\|
263 =
264 \|h\|
265 =
266 \|P_{\mathcal H^1_{\mathcal U}}b\|.
267 }
268 \]
269 Hence \(b\) is exactly integrable if and only if its harmonic component
270 vanishes.
271 \end{proposition}
272 
273 \begin{proof}
274 The finite-dimensional Hodge decomposition gives
275 \[
276 \check C^1
277 =
278 \operatorname{im}\delta^0
279 \oplus
280 \ker\Delta_1
281 \oplus
282 \operatorname{im}(\delta^1)^*.
283 \]
284 If \(b\in\ker\delta^1\), then \(b\) is orthogonal to
285 \(\operatorname{im}(\delta^1)^*\). Its remaining two orthogonal components are
286 therefore an exact part \(\delta^0q^\dagger\) and a harmonic part \(h\). The
287 orthogonal projection theorem gives the least-squares identity.
288 \end{proof}
289 
290 The scalar
291 \[
292 \omega_{\mathcal U}(b)
293 :=
294 \|P_{\mathcal H^1_{\mathcal U}}b\|
295 \]
296 is an irreducible overlap-frustration measure. Its vanishing is
297 invariant under admissible invertible coordinate changes, although its numerical
298 norm depends on the chosen metric. It cannot be removed by adding local
299 potentials alone.
300 
301 \subtopic{43.4 Cycle example}
302 
303 Consider three charts arranged in a cycle, with one-dimensional overlap
304 spaces and oriented offsets
305 \[
306 b_{12},\qquad b_{23},\qquad b_{31}.
307 \]
308 Local potentials would have to satisfy
309 \[
310 q_2-q_1=b_{12},
311 \qquad
312 q_3-q_2=b_{23},
313 \qquad
314 q_1-q_3=b_{31}.
315 \]
316 Adding the equations gives the necessary and sufficient condition
317 \[
318 \boxed{
319 b_{12}+b_{23}+b_{31}=0.}
320 \]
321 With the standard edge norm, the harmonic obstruction has magnitude
322 \[
323 \boxed{
324 \omega_{\mathcal U}(b)
325 =
326 \frac{|b_{12}+b_{23}+b_{31}|}{\sqrt3}.
327 }
328 \]
329 Every pairwise relation may be individually meaningful while the complete
330 cycle remains globally inconsistent.
331 
332 \topic{44}{Exact Gluing Theorem T8}
333 
334 Let
335 \[
336 R_{\mathcal U}:\mathscr W(G)
337 \longrightarrow
338 \check C^0(\mathcal U;\mathscr W)
339 \]
340 be the restriction map
341 \[
342 R_{\mathcal U}(w)
343 =
344 (w|_{G_\alpha})_{\alpha\in A}.
345 \]
346 
347 \begin{theorem}[Exact workspace gluing]\label{thm:exact-workspace-gluing}
348 For every finite edge-complete workspace cover \(\mathcal U\), the sequence
349 \[
350 \boxed{
351 0
352 \longrightarrow
353 \mathscr W(G)
354 \xrightarrow{\ R_{\mathcal U}\ }
355 \check C^0(\mathcal U;\mathscr W)
356 \xrightarrow{\ \delta^0\ }
357 \check C^1(\mathcal U;\mathscr W)
358 }
359 \]
360 is exact. Equivalently,
361 \[
362 \boxed{
363 \mathscr W(G)
364 \cong
365 \ker\delta^0
366 \cong
367 \operatorname{Eq}
368 \left(
369 \prod_\alpha\mathscr W_\alpha
370 \rightrightarrows
371 \prod_{\alpha<\beta}\mathscr W_{\alpha\beta}
372 \right).
373 }
374 \]
375 Every pairwise compatible family of exact local workspace states therefore
376 glues to a unique exact global workspace state.
377 \end{theorem}
378 
379 \begin{proof}
380 The cover contains every vertex, so \(R_{\mathcal U}\) is injective. A global
381 section has identical restrictions on every overlap, hence
382 \(\delta^0R_{\mathcal U}=0\).
383 
384 Conversely, let \(w=(w_\alpha)_\alpha\in\ker\delta^0\). For each vertex
385 \(i\in V\), choose any chart \(G_\alpha\) containing \(i\) and set
386 \[
387 c_i=(w_\alpha)_i.
388 \]
389 If another chart \(G_\beta\) contains \(i\), then \(i\in G_{\alpha\beta}\)
390 and overlap compatibility gives
391 \((w_\alpha)_i=(w_\beta)_i\). Thus \(c_i\) is well defined.
392 
393 For every edge \(e:i\to j\), edge completeness provides a chart
394 \(G_\alpha\) containing \(e\). Since \(w_\alpha\in\mathscr W(G_\alpha)\),
395 \[
396 c_j=(w_\alpha)_j=H_e(w_\alpha)_i=H_ec_i.
397 \]
398 Hence \(c=(c_i)\in\mathscr W(G)\), and by construction
399 \(R_{\mathcal U}c=w\). Uniqueness follows because the charts cover all
400 vertices.
401 \end{proof}
402 
403 \begin{corollary}[Zeroth descent cohomology]\label{cor:zeroth-descent}
404 For every finite edge-complete cover,
405 \[
406 \boxed{
407 \mathscr W(G)
408 \cong
409 \check H^0(\mathcal U;\mathscr W)
410 =
411 \ker\delta^0.
412 }
413 \]
414 In finite dimensions,
415 \[
416 \boxed{
417 \dim\mathscr W(G)
418 =
419 \sum_{\alpha\in A}\dim\mathscr W_\alpha
420 -
421 \operatorname{rank}(\delta^0).
422 }
423 \]
424 \end{corollary}
425 
426 The theorem does not require the nerve to be acyclic. Cycles in the cover are
427 allowed; their effect is encoded in the rank of \(\delta^0\) and, for
428 relative-only data, in \(\check H^1\). It also shows that the local-workspace
429 assignment is a genuine sheaf on the finite subgraph cover used here rather
430 than merely a presheaf.
431 
432 \subtopic{44.1 Local nontriviality does not imply global nontriviality}
433 
434 Let \(G\) be the directed triangle
435 \[
436 1\xrightarrow{\ a\ }2
437 \xrightarrow{\ b\ }3
438 \xrightarrow{\ c\ }1
439 \]
440 with one-dimensional vertex cores. Cover it by its three edge charts. Their
441 local workspaces are all one-dimensional:
442 \[
443 \begin{aligned}
444 \mathscr W_{12}&=\{(x,ax):x\in\mathbb R\},\\
445 \mathscr W_{23}&=\{(y,by):y\in\mathbb R\},\\
446 \mathscr W_{31}&=\{(z,cz):z\in\mathbb R\}.
447 \end{aligned}
448 \]
449 Overlap agreement requires
450 \[
451 y=ax,
452 \qquad
453 z=by,
454 \qquad
455 x=cz.
456 \]
457 Therefore
458 \[
459 (1-abc)x=0,
460 \]
461 and
462 \[
463 \boxed{
464 \dim\mathscr W(G)
465 =
466 \begin{cases}
467 1,&abc=1,\\
468 0,&abc\neq1.
469 \end{cases}
470 }
471 \]
472 The meta-level gluing obstruction is exactly the original cycle holonomy.
473 Strong local workspaces can therefore coexist with a trivial global core.
474 
475 \topic{45}{Approximate Gluing and Meta-Workspaces}
476 
477 \subtopic{45.1 The cover-level transfer Laplacian}
478 
479 First assume each local state already lies in its exact local workspace. Define
480 \[
481 B_{\mathcal U}
482 :=
483 \delta^0:
484 \check C^0(\mathcal U;\mathscr W)
485 \longrightarrow
486 \check C^1(\mathcal U;\mathscr W)
487 \]
488 and the meta-Laplacian
489 \[
490 \boxed{
491 L_{\mathcal U}^{\mathrm{meta}}
492 =
493 B_{\mathcal U}^*B_{\mathcal U}.
494 }
495 \]
496 By Theorem~\ref{thm:exact-workspace-gluing},
497 \[
498 \boxed{
499 \ker L_{\mathcal U}^{\mathrm{meta}}
500 =
501 R_{\mathcal U}\mathscr W(G).
502 }
503 \]
504 Thus the gluing problem is itself a workspace problem whose ``nodes'' are
505 local workspaces and whose interfaces are overlaps.
506 
507 Let
508 \[
509 \gamma_{\mathcal U}^{\mathrm{meta}}
510 =
511 \lambda_{\min}^+
512 \left(L_{\mathcal U}^{\mathrm{meta}}\right)
513 \]
514 be the first positive eigenvalue.
515 
516 \begin{proposition}[Approximate overlap gluing]\label{prop:approx-overlap-gluing}
517 For every \(w\in\check C^0(\mathcal U;\mathscr W)\),
518 \[
519 \boxed{
520 \operatorname{dist}
521 \left(
522  w,
523  R_{\mathcal U}\mathscr W(G)
524 \right)^2
525 \leq
526 \frac{\|B_{\mathcal U}w\|^2}
527 {\gamma_{\mathcal U}^{\mathrm{meta}}}.
528 }
529 \]
530 \end{proposition}
531 
532 \begin{proof}
533 Decompose \(w\) into its component in \(\ker L_{\mathcal U}^{\mathrm{meta}}\)
534 and its orthogonal complement. On the complement,
535 \(L_{\mathcal U}^{\mathrm{meta}}\geq
536 \gamma_{\mathcal U}^{\mathrm{meta}}I\). Therefore
537 \[
538 \|B_{\mathcal U}w\|^2
539 =
540 \langle w,L_{\mathcal U}^{\mathrm{meta}}w\rangle
541 \geq
542 \gamma_{\mathcal U}^{\mathrm{meta}}
543 \operatorname{dist}
544 \left(w,\ker L_{\mathcal U}^{\mathrm{meta}}\right)^2.
545 \]
546 \end{proof}
547 
548 The gap \(\gamma_{\mathcal U}^{\mathrm{meta}}\) measures how strongly the cover
549 penalizes disagreement. A small overlap mismatch need not imply a nearby
550 global workspace when this gap is itself very small.
551 
552 \subtopic{45.2 Simultaneous local and overlap error}
553 
554 Local candidates may also violate their internal transfer equations. Define
555 the local-copy space
556 \[
557 \widehat{\mathcal C}_{\mathcal U}
558 =
559 \bigoplus_{\alpha\in A}\mathcal C_V(G_\alpha)
560 \]
561 and the block internal discrepancy
562 \[
563 \widehat D_{\mathcal U}
564 =
565 \bigoplus_{\alpha\in A}D_{G_\alpha}.
566 \]
567 For \(u=(u_\alpha)_\alpha\in\widehat{\mathcal C}_{\mathcal U}\), define the
568 raw overlap synchronization operator
569 \[
570 (S_{\mathcal U}u)_{\alpha\beta}
571 =
572 \rho_{G_{\alpha\beta}}^{G_\beta}u_\beta
573 -
574 \rho_{G_{\alpha\beta}}^{G_\alpha}u_\alpha.
575 \]
576 For an overlap weight \(\lambda_{\mathrm{ov}}>0\), set
577 \[
578 \boxed{
579 K_{\mathcal U}
580 =
581 \begin{pmatrix}
582 \widehat D_{\mathcal U}\\[0.2em]
583 \sqrt{\lambda_{\mathrm{ov}}}\,S_{\mathcal U}
584 \end{pmatrix},
585 \qquad
586 L_{\mathcal U}^{\mathrm{tot}}
587 =
588 K_{\mathcal U}^*K_{\mathcal U}.
589 }
590 \]
591 Its energy is
592 \[
593 \boxed{
594 \begin{aligned}
595 \mathcal E_{\mathcal U}^{\mathrm{tot}}(u)
596 ={}&
597 \sum_\alpha\|D_{G_\alpha}u_\alpha\|^2\\
598 &+
599 \lambda_{\mathrm{ov}}
600 \sum_{\alpha<\beta}
601 \left\|
602 \rho^{G_\beta}_{G_{\alpha\beta}}u_\beta
603 -
604 \rho^{G_\alpha}_{G_{\alpha\beta}}u_\alpha
605 \right\|^2.
606 \end{aligned}
607 }
608 \]
609 
610 Let
611 \[
612 J_{\mathcal U}:\mathcal C_V(G)\to\widehat{\mathcal C}_{\mathcal U},
613 \qquad
614 J_{\mathcal U}c=(c|_{G_\alpha})_\alpha,
615 \]
616 be the duplication map.
617 
618 \begin{theorem}[Total descent kernel]\label{thm:total-descent-kernel}
619 For every finite edge-complete cover,
620 \[
621 \boxed{
622 \ker K_{\mathcal U}
623 =
624 J_{\mathcal U}\mathscr W(G).
625 }
626 \]
627 Consequently, if
628 \[
629 \gamma_{\mathcal U}^{\mathrm{tot}}
630 =
631 \lambda_{\min}^+
632 \left(L_{\mathcal U}^{\mathrm{tot}}\right),
633 \]
634 then
635 \[
636 \boxed{
637 \operatorname{dist}
638 \left(
639  u,
640  J_{\mathcal U}\mathscr W(G)
641 \right)^2
642 \leq
643 \frac{\mathcal E_{\mathcal U}^{\mathrm{tot}}(u)}
644 {\gamma_{\mathcal U}^{\mathrm{tot}}}.
645 }
646 \]
647 \end{theorem}
648 
649 \begin{proof}
650 If \(u=J_{\mathcal U}c\) for a global exact section, every local discrepancy
651 and every overlap discrepancy vanishes. Conversely, if \(K_{\mathcal U}u=0\),
652 then every \(u_\alpha\) is an exact local section and the family agrees on all
653 overlaps. Theorem~\ref{thm:exact-workspace-gluing} gives a unique
654 \(c\in\mathscr W(G)\) with \(u=J_{\mathcal U}c\). The distance estimate follows
655 from the spectral lower bound on the orthogonal complement of the kernel.
656 \end{proof}
657 
658 This theorem combines two distinct failure modes:
659 \[
660 \text{local incoherence}
661 \qquad\text{and}\qquad
662 \text{cross-chart disagreement}.
663 \]
664 Neither can be hidden by the other because they occupy separate blocks of
665 \(K_{\mathcal U}\).
666 
667 \subtopic{45.3 Least-squares global assembly}
668 
669 Let
670 \[
671 m_i=|\{\alpha:i\in V_\alpha\}|
672 \]
673 be the multiplicity of vertex \(i\). Then
674 \[
675 J_{\mathcal U}^*J_{\mathcal U}=M_{\mathcal U},
676 \qquad
677 (M_{\mathcal U}c)_i=m_ic_i.
678 \]
679 The pseudoinverse performs coordinatewise averaging:
680 \[
681 \boxed{
682 (J_{\mathcal U}^\dagger u)_i
683 =
684 \frac{1}{m_i}
685 \sum_{\alpha:i\in V_\alpha}(u_\alpha)_i.
686 }
687 \]
688 Let
689 \[
690 \widehat u
691 =
692 P_{\ker K_{\mathcal U}}u
693 =
694 J_{\mathcal U}\widehat c.
695 \]
696 Then
697 \[
698 \widehat c
699 =
700 J_{\mathcal U}^\dagger\widehat u
701 \]
702 and
703 \[
704 \boxed{
705 \|u-J_{\mathcal U}\widehat c\|
706 \leq
707 \frac{\|K_{\mathcal U}u\|}
708 {\sqrt{\gamma_{\mathcal U}^{\mathrm{tot}}}}.
709 }
710 \]
711 If \(m_{\min}=\min_i m_i\), then
712 \[
713 \boxed{
714 \left\|
715 \widehat c-J_{\mathcal U}^\dagger u
716 \right\|
717 \leq
718 \frac{\|K_{\mathcal U}u\|}
719 {\sqrt{m_{\min}\gamma_{\mathcal U}^{\mathrm{tot}}}}.
720 }
721 \]
722 The simple overlap average and the globally coherent least-squares assembly
723 are close only when the total descent energy is small relative to the cover
724 gap.
725 
726 \subtopic{45.4 A workspace of workspaces}
727 
728 Define the exact meta-workspace
729 \[
730 \boxed{
731 \mathscr W_{\mathrm{meta}}(\mathcal U)
732 :=
733 \ker B_{\mathcal U}
734 \subseteq
735 \bigoplus_\alpha\mathscr W_\alpha.
736 }
737 \]
738 Theorem~\ref{thm:exact-workspace-gluing} gives the canonical identification
739 \[
740 \boxed{
741 \mathscr W_{\mathrm{meta}}(\mathcal U)
742 \cong
743 \mathscr W(G).
744 }
745 \]
746 Thus exact formation of a workspace from local workspaces does not create an
747 additional mysterious carrier. It gives a second presentation of the same
748 global compatible-section object.
749 
750 Suppose local operations
751 \[
752 A_\alpha:\mathscr W_\alpha\to\mathscr W_\alpha
753 \]
754 and overlap operations
755 \[
756 A_{\alpha\beta}:\mathscr W_{\alpha\beta}\to
757 \mathscr W_{\alpha\beta}
758 \]
759 satisfy
760 \[
761 r_{\alpha\beta}^{\alpha}A_\alpha
762 =
763 A_{\alpha\beta}r_{\alpha\beta}^{\alpha},
764 \qquad
765 r_{\alpha\beta}^{\beta}A_\beta
766 =
767 A_{\alpha\beta}r_{\alpha\beta}^{\beta}.
768 \]
769 Then, with \(A^0=\bigoplus_\alpha A_\alpha\) and
770 \(A^1=\bigoplus_{\alpha<\beta}A_{\alpha\beta}\),
771 \[
772 B_{\mathcal U}A^0=A^1B_{\mathcal U}.
773 \]
774 Therefore
775 \[
776 A^0\ker B_{\mathcal U}\subseteq\ker B_{\mathcal U}.
777 \]
778 The operation-closure theorem from Part~III reappears at the meta level:
779 operations on subworkspaces descend globally precisely when they agree on
780 interfaces.
781 
782 \subtopic{45.5 Descent-faithful compression}
783 
784 A practical hierarchy rarely transmits each complete local workspace. Let
785 \[
786 Q^p:\check C^p(\mathcal U;\mathscr W)\to\overline C^p
787 \]
788 be compression maps into smaller local and overlap representations. Suppose
789 there is a compressed coboundary \(\overline\delta^p\) such that
790 \[
791 \boxed{
792 Q^{p+1}\delta^p
793 =
794 \overline\delta^p Q^p.
795 }
796 \]
797 The maps \(Q^\bullet\) then form a cochain map.
798 
799 \begin{definition}[Descent-faithful compression]\label{def:descent-faithful}
800 The compression is \emph{descent-faithful in degree zero} when the induced map
801 \[
802 \boxed{
803 H^0(Q):
804 \ker\delta^0
805 \longrightarrow
806 \ker\overline\delta^0
807 }
808 \]
809 is an isomorphism.
810 \end{definition}
811 
812 Injectivity prevents two distinct global workspace states from collapsing to
813 the same compressed meta-state. Surjectivity prevents the compressed hierarchy
814 from introducing spurious compatible states that have no full-resolution
815 lift. Under this condition,
816 \[
817 \boxed{
818 \ker\overline\delta^0
819 \cong
820 \mathscr W(G).
821 }
822 \]
823 This is the exact criterion for a compressed workspace of workspaces to retain
824 the global core.
825 
826 \subtopic{45.6 Structural and operational gluing are distinct}
827 
828 Exact or approximate overlap coherence does not by itself establish global
829 predictive sufficiency. Let \(W_t\) denote a glued structural state and let
830 \[
831 Z_t^{\mathrm{loc}}
832 =
833 (Z_{\alpha,t})_{\alpha\in A}
834 \]
835 be local predictive summaries recovered from its chart restrictions.
836 
837 \begin{proposition}[Local-to-global predictive alignment]\label{prop:local-global-predictive}
838 Assume \(W_t\) is measurable with respect to the complete history
839 \(\mathbf H_t\), every \(Z_{\alpha,t}\) is a measurable function of
840 \(W_t|_{G_\alpha}\), and
841 \[
842 I(Y_t^+;\mathbf H_t\mid Z_t^{\mathrm{loc}})
843 \leq
844 \varepsilon.
845 \]
846 Then
847 \[
848 \boxed{
849 I(Y_t^+;\mathbf H_t\mid W_t)
850 \leq
851 \varepsilon.
852 }
853 \]
854 In particular, exact sufficiency of the complete local-summary tuple implies
855 exact sufficiency of the glued workspace.
856 \end{proposition}
857 
858 \begin{proof}
859 The tuple \(Z_t^{\mathrm{loc}}\) is a measurable function of \(W_t\), while
860 \(W_t\) is itself a history statistic. Conditioning on the finer statistic
861 cannot increase the residual information that the full history carries about
862 the target. Equivalently, use the conditional-law characterization from
863 Theorem~\ref{thm:predictive-state}.
864 \end{proof}
865 
866 Causal gluing requires an additional interface gate: interventions described
867 in two overlapping charts must induce matched outcome laws on the overlap. A
868 structurally compatible family can fail this gate, just as structurally
869 identical replacement systems can fail causal identity in Part~XII.
870 
871 \subtopic{45.7 Anti-centralization gate}
872 
873 The isomorphism
874 \[
875 \mathscr W_{\mathrm{meta}}(\mathcal U)\cong\mathscr W(G)
876 \]
877 is a compositional statement, not a relational-irreducibility statement. A
878 single chart may still expose enough accessible information to reconstruct the
879 whole meta-state. A claim that the hierarchy is genuinely distributed must
880 therefore reuse the coalition gates from Parts~VII and XI. At the chart level,
881 one may require, for example,
882 \[
883 \alpha_{\{\alpha\}}^{\mathrm{meta}}=0
884 \quad\text{for every }\alpha,
885 \qquad
886 \alpha_A^{\mathrm{meta}}>0
887 \quad\text{for some }|A|\geq2,
888 \]
889 or the predictive analogue
890 \[
891 k_\varepsilon^{\mathrm{meta}}\geq2,
892 \qquad
893 \Delta_{\mathrm{joint}}^{\mathrm{meta}}>0.
894 \]
895 Capacity-matched local-memory and chart-ablation controls are still required.
896 Hierarchical notation cannot convert a centralized copy into a relational
897 workspace.
898 
899 \topic{46}{Hierarchical Collapse Theorem T9}
900 
901 \subtopic{46.1 Finite cover hierarchies}
902 
903 A finite workspace hierarchy is a rooted tree \(\mathcal T\). Each node
904 \(a\in\mathcal T\) carries a subgraph \(G_a\). The root \(r\) carries
905 \[
906 G_r=G.
907 \]
908 For every internal node \(a\), its children
909 \[
910 \operatorname{Ch}(a)
911 \]
912 form a finite edge-complete cover of \(G_a\). Leaves carry primitive local
913 workspace spaces
914 \[
915 W_\ell=\mathscr W(G_\ell).
916 \]
917 For an internal node, define recursively
918 \[
919 \boxed{
920 W_a
921 =
922 \ker
923 \left(
924 \delta_a^0:
925 \bigoplus_{b\in\operatorname{Ch}(a)}W_b
926 \longrightarrow
927 \bigoplus_{b<c}W_{bc}
928 \right),
929 }
930 \]
931 where the overlap spaces and restrictions are inherited from the child cover.
932 
933 \begin{theorem}[Exact hierarchical collapse]\label{thm:hierarchical-collapse}
934 For every finite workspace hierarchy as above, there is a canonical
935 isomorphism
936 \[
937 \boxed{
938 W_a\cong\mathscr W(G_a)
939 }
940 \]
941 for every node \(a\). In particular,
942 \[
943 \boxed{
944 W_r\cong\mathscr W(G).
945 }
946 \]
947 Therefore recursively gluing local workspaces, using any other finite
948 parenthesization built from edge-complete intermediate covers that retain all
949 required overlap objects, and solving the flattened global transfer system all
950 produce the same exact workspace up to canonical isomorphism.
951 \end{theorem}
952 
953 \begin{proof}
954 Proceed upward from the leaves. The claim holds at every leaf by definition.
955 Assume it holds for all children of an internal node \(a\). Replacing each
956 child space \(W_b\) by its canonical copy \(\mathscr W(G_b)\), the recursive
957 definition of \(W_a\) is exactly the compatible-family kernel for the
958 edge-complete child cover of \(G_a\). Theorem~\ref{thm:exact-workspace-gluing}
959 therefore gives
960 \[
961 W_a\cong\mathscr W(G_a).
962 \]
963 Finite induction reaches the root.
964 \end{proof}
965 
966 \subtopic{46.2 Flattening and limit interchange}
967 
968 Let \(L(\mathcal T)\) be the leaf set. Repeated edge completeness implies that
969 the flattened family
970 \[
971 \mathcal U_{\mathrm{flat}}
972 =
973 \{G_\ell:\ell\in L(\mathcal T)\}
974 \]
975 is itself an edge-complete cover of \(G\). Let
976 \[
977 \delta_{\mathrm{flat}}^0
978 \]
979 be its overlap coboundary and define
980 \[
981 W_{\mathrm{flat}}
982 =
983 \ker\delta_{\mathrm{flat}}^0.
984 \]
985 Theorem~\ref{thm:exact-workspace-gluing} and
986 Theorem~\ref{thm:hierarchical-collapse} give
987 \[
988 \boxed{
989 W_r
990 \cong
991 W_{\mathrm{flat}}
992 \cong
993 \mathscr W(G).
994 }
995 \]
996 Equivalently,
997 \[
998 \boxed{
999 \operatorname{Lim}_{\mathcal T}^{\mathrm{rec}}
1000 \left(
1001 \{\mathscr W(G_\ell)\}_{\ell\in L(\mathcal T)}
1002 \right)
1003 \cong
1004 \operatorname{Eq}_{\mathcal U_{\mathrm{flat}}}
1005 \left(
1006 \{\mathscr W(G_\ell)\}_{\ell\in L(\mathcal T)}
1007 \right).
1008 }
1009 \]
1010 The notation on the left means recursive finite descent along the tree; the
1011 right side means one-shot equalization over the flattened leaf cover. In
1012 finite-dimensional vector spaces, both are kernels of the same complete family
1013 of local and overlap constraints, grouped in different orders. Hierarchical
1014 collapse is therefore a finite-limit interchange property, not a claim that
1015 intermediate levels are physically absent or operationally irrelevant.
1016 
1017 \subtopic{46.3 Approximate hierarchical collapse}
1018 
1019 At level \(r\), let \(K_r\) be the corresponding total descent operator,
1020 \[
1021 \gamma_r
1022 =
1023 \lambda_{\min}^+(K_r^*K_r),
1024 \qquad
1025 \|K_ru_r\|\leq\varepsilon_r.
1026 \]
1027 The nearest exact compatible state then differs from \(u_r\) by at most
1028 \[
1029 \frac{\varepsilon_r}{\sqrt{\gamma_r}}.
1030 \]
1031 Allow an additional implementation, compression, or causal-summary defect
1032 \(\zeta_r\), and define
1033 \[
1034 \delta_r
1035 =
1036 \frac{\varepsilon_r}{\sqrt{\gamma_r}}
1037 +
1038 \zeta_r.
1039 \]
1040 
1041 \begin{theorem}[Approximate hierarchical collapse bound]\label{thm:approx-hierarchical-collapse}
1042 Let
1043 \[
1044 F_r:\mathcal H_r\to\mathcal H_{r+1},
1045 \qquad
1046 \widetilde F_r:\mathcal H_r\to\mathcal H_{r+1}
1047 \]
1048 be exact and implemented level-collapse maps for
1049 \(r=0,\ldots,R-1\). Let \(\mathcal R_r\subseteq\mathcal H_r\) contain the exact
1050 and implemented states reachable at level \(r\). Assume, on \(\mathcal R_r\),
1051 \[
1052 \|F_r(x)-F_r(y)\|
1053 \leq
1054 L_r\|x-y\|
1055 \]
1056 and
1057 \[
1058 \sup_{x\in\mathcal R_r}
1059 \|\widetilde F_r(x)-F_r(x)\|
1060 \leq
1061 \delta_r.
1062 \]
1063 Then
1064 \[
1065 \boxed{
1066 \begin{aligned}
1067 &\left\|
1068 \widetilde F_{R-1}\circ\cdots\circ\widetilde F_0(x)
1069 -
1070 F_{R-1}\circ\cdots\circ F_0(x)
1071 \right\|\\
1072 &\qquad\leq
1073 \sum_{r=0}^{R-1}
1074 \left(
1075 \prod_{s=r+1}^{R-1}L_s
1076 \right)
1077 \delta_r.
1078 \end{aligned}
1079 }
1080 \]
1081 In particular, if \(L_r\leq1\), then the total error is at most
1082 \[
1083 \sum_{r=0}^{R-1}\delta_r.
1084 \]
1085 If \(L_r\leq L<1\) and \(\delta_r\leq\delta\), then
1086 \[
1087 \boxed{
1088 \text{total error}
1089 \leq
1090 \frac{\delta(1-L^R)}{1-L}
1091 \leq
1092 \frac{\delta}{1-L}.
1093 }
1094 \]
1095 \end{theorem}
1096 
1097 \begin{proof}
1098 Let \(x_r\) and \(\widetilde x_r\) be the exact and implemented states after
1099 \(r\) levels. Then
1100 \[
1101 \begin{aligned}
1102 \|\widetilde x_{r+1}-x_{r+1}\|
1103 &\leq
1104 \|\widetilde F_r(\widetilde x_r)-F_r(\widetilde x_r)\|\\
1105 &\quad+
1106 \|F_r(\widetilde x_r)-F_r(x_r)\|\\
1107 &\leq
1108 \delta_r+L_r\|\widetilde x_r-x_r\|.
1109 \end{aligned}
1110 \]
1111 Iterating this recurrence gives the displayed product-sum estimate. The final
1112 two bounds follow by summing a finite or infinite geometric series.
1113 \end{proof}
1114 
1115 This theorem identifies three regimes:
1116 \[
1117 \begin{array}{ll}
1118 L_r<1
1119 &\text{hierarchical correction is contractive},\\
1120 L_r=1
1121 &\text{level errors accumulate at most additively},\\
1122 L_r>1
1123 &\text{deep hierarchies can amplify early mismatch}.
1124 \end{array}
1125 \]
1126 A deep hierarchy is therefore stable only when its level gaps remain separated
1127 and its assembly maps do not magnify small overlap errors faster than those
1128 errors are corrected.
1129 
1130 \subtopic{46.4 Causal-kernel accumulation}
1131 
1132 Let \(K_r\) and \(\widetilde K_r\) now denote exact and implemented Markov
1133 kernels at successive hierarchy levels, and suppose
1134 \[
1135 \sup_x
1136 \operatorname{TV}
1137 \left(
1138 \widetilde K_r(x,\cdot),
1139 K_r(x,\cdot)
1140 \right)
1141 \leq
1142 \zeta_r.
1143 \]
1144 Because Markov kernels contract total variation, a telescoping argument gives
1145 \[
1146 \boxed{
1147 \sup_x
1148 \operatorname{TV}
1149 \left(
1150 \widetilde K_0\cdots\widetilde K_{R-1}(x,\cdot),
1151 K_0\cdots K_{R-1}(x,\cdot)
1152 \right)
1153 \leq
1154 \sum_{r=0}^{R-1}\zeta_r.
1155 }
1156 \]
1157 Thus causal semantics can also be transported through a hierarchy, but only
1158 under an explicit kernel-level compatibility test. Structural gluing alone
1159 provides no such guarantee.
1160 
1161 \subtopic{46.5 Direct versus hierarchical computation}
1162 
1163 The exact theorem states
1164 \[
1165 \boxed{
1166 \text{recursive exact gluing}
1167 =
1168 \text{one-shot exact gluing}.
1169 }
1170 \]
1171 The approximate theorem instead supplies a measurable flattening defect:
1172 \[
1173 \boxed{
1174 \delta_{\mathrm{flat}}^{(R)}
1175 =
1176 \left\|
1177 \widetilde F_{R-1}\cdots\widetilde F_0(x)
1178 -
1179 F_{R-1}\cdots F_0(x)
1180 \right\|.
1181 }
1182 \]
1183 A hierarchy is empirically justified when it reduces computation,
1184 communication, or local model complexity while keeping
1185 \(\delta_{\mathrm{flat}}^{(R)}\), predictive risk, causal-kernel mismatch, and
1186 anti-centralization gates within declared tolerances. Hierarchy by itself is
1187 not evidence of emergence.
1188 
1189 \subtopic{46.6 Gluing and hierarchical synthesis}
1190 
1191 The exact and approximate constructions can be compressed as
1192 \[
1193 \boxed{
1194 \begin{aligned}
1195 \text{local transfer systems}
1196 &\xrightarrow{\ \ker D_\alpha\ }
1197 \text{local workspaces}\\
1198 &\xrightarrow{\ \ker\delta^0\ }
1199 \text{meta-workspace}\\
1200 &\xrightarrow{\ \text{finite recursive descent}\ }
1201 \text{global workspace}.
1202 \end{aligned}
1203 }
1204 \]
1205 At exact resolution, all finite parenthesizations are canonically equivalent.
1206 At approximate resolution, every level contributes a defect scaled by its
1207 spectral gap and by downstream amplification. Relative-only overlap data can
1208 carry a harmonic \(H^1\) obstruction even when every local chart is internally
1209 coherent. Operational and relational claims still require predictive, causal,
1210 coalition, and null-model gates in addition to structural descent.

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