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{III}{Exact Shared Cores}
4 
5 \topic{6}{Global Compatible Sections}
6 
7 From this point through Theorem~\ref{thm:limit-holonomy}, assume the core maps
8 define a local system on a connected graph unless a weaker hypothesis is
9 stated explicitly.
10 
11 \begin{definition}[Global compatible section]\label{def:global-section}
12 The exact structural workspace is
13 \[
14 \boxed{
15 \Gamma(H)
16 =
17 \left\{
18 (c_i)_{i\in V}\in\bigoplus_{i\in V}C_i:
19 \ c_{t(e)}=H_ec_{s(e)}\ \forall e\in E
20 \right\}.
21 }
22 \]
23 An element \(c=(c_i)\in\Gamma(H)\) is a global compatible section.
24 \end{definition}
25 
26 Equivalently,
27 \[
28 \Gamma(H)=\varprojlim_{\Path(G)}C_i.
29 \]
30 A section is not a single vector copied verbatim at every vertex. It is one
31 abstract element represented by a family of local vectors that agree under the
32 specified translation maps.
33 
34 A nontrivial exact structural workspace exists when
35 \[
36 \dim\Gamma(H)>0.
37 \]
38 Its exact dimension is
39 \[
40 d_{\Workspace}=\dim\Gamma(H).
41 \]
42 
43 Fix a root \(o\). The evaluation map is
44 \[
45 \ev_o:\Gamma(H)\to C_o,
46 \qquad
47 \ev_o((c_i))=c_o.
48 \]
49 
50 \begin{lemma}[Root determination]\label{lem:root-determination}
51 If \(|G|\) is connected and every core transfer is invertible, then
52 \(\ev_o\) is injective. Consequently,
53 \[
54 \dim\Gamma(H)\leq\dim C_o.
55 \]
56 \end{lemma}
57 
58 \begin{proof}
59 Choose for each vertex \(i\) a path \(\tau_i:o\to i\). Compatibility gives
60 \[
61 c_i=H_{\tau_i}c_o.
62 \]
63 Thus a section with \(c_o=0\) vanishes at every vertex. Hence \(\ev_o\) is
64 injective.
65 \end{proof}
66 
67 \begin{lemma}[Tree propagation]\label{lem:tree-propagation}
68 If \(|G|\) is a tree and every \(H_e\) is invertible, then every
69 \(x\in C_o\) extends uniquely to a global compatible section.
70 \end{lemma}
71 
72 \begin{proof}
73 A tree contains a unique undirected path \(\tau_i:o\to i\) for every vertex.
74 Define \(c_i=H_{\tau_i}x\). Because every edge belongs to these unique path
75 relations, the family is compatible. Uniqueness follows from
76 Lemma~\ref{lem:root-determination}.
77 \end{proof}
78 
79 The obstruction to a global core therefore appears only when distinct paths
80 must agree. Cycles encode exactly those consistency constraints.
81 
82 \topic{7}{Holonomy and Path Obstruction}
83 
84 Fix a root \(o\in V\). For every loop class
85 \[
86 [\gamma]\in\pi_1(|G|,o),
87 \]
88 define the holonomy
89 \[
90 \Hol([\gamma])=H_\gamma\in GL(C_o).
91 \]
92 Functoriality gives a group representation
93 \[
94 \operatorname{hol}_o:
95 \pi_1(|G|,o)\longrightarrow GL(C_o).
96 \]
97 Its image is the holonomy group
98 \[
99 \Hol_o=\operatorname{im}(\operatorname{hol}_o).
100 \]
101 
102 The obstruction associated with a loop is
103 \[
104 \Omega_\gamma=H_\gamma-I_{C_o}.
105 \]
106 The maximal path-independent root core is
107 \[
108 \boxed{
109 C_*
110 =
111 \bigcap_{\gamma:o\to o}\ker(H_\gamma-I).
112 }
113 \]
114 Equivalently,
115 \[
116 C_*=C_o^{\Hol_o}=\operatorname{Fix}(\Hol_o).
117 \]
118 
119 \subtopic{7.1 Spanning-tree reduction}
120 
121 Choose a spanning tree \(\mathcal T\subseteq|G|\). Let
122 \[
123 \tau_i:o\to i
124 \]
125 be the unique tree path. Every non-tree edge \(e:i\to j\) determines a
126 fundamental loop
127 \[
128 \gamma_e=\tau_j^{-1}e\tau_i
129 \]
130 with transfer
131 \[
132 H_{\gamma_e}=H_{\tau_j}^{-1}H_eH_{\tau_i}.
133 \]
134 The fundamental group of a finite graph is generated freely by these cotree
135 loops. Hence it is enough to impose fixed-point conditions for a finite set of
136 generators.
137 
138 \begin{lemma}[Finite obstruction set]\label{lem:finite-obstruction}
139 Let \(e_1,\ldots,e_r\) be the non-tree edges of a connected finite graph.
140 Then
141 \[
142 C_*
143 =
144 \bigcap_{q=1}^{r}
145 \ker(H_{\gamma_{e_q}}-I).
146 \]
147 \end{lemma}
148 
149 \begin{proof}
150 A vector fixed by every generator is fixed by each inverse generator and by
151 every finite product of generators and inverses. Those products represent all
152 elements of \(\pi_1(|G|,o)\). The reverse inclusion is immediate.
153 \end{proof}
154 
155 The number of independent graph cycles is
156 \[
157 r=|E_{\mathrm{und}}|-|V|+1
158 \]
159 for a connected underlying graph, after treating parallel edges with their
160 multiplicity.
161 
162 \topic{8}{Exact Existence Theorem}
163 
164 \begin{theorem}[Limit--holonomy equivalence]\label{thm:limit-holonomy}
165 Let \(|G|\) be connected and let
166 \[
167 H:\Pi_1(G)\to\mathbf{FinHilb}^{\times}
168 \]
169 be a finite-dimensional local system. For any root \(o\in V\), evaluation at
170 the root induces a canonical vector-space isomorphism
171 \[
172 \boxed{
173 \ev_o:\Gamma(H)
174 \xrightarrow{\ \cong\ }
175 C_o^{\Hol_o}
176 =
177 \bigcap_{\gamma:o\to o}\ker(H_\gamma-I).
178 }
179 \]
180 Consequently,
181 \[
182 \Gamma(H)\neq\{0\}
183 \]
184 if and only if the holonomy representation contains a nonzero trivial
185 subrepresentation.
186 \end{theorem}
187 
188 \begin{proof}
189 Let \(c=(c_i)\in\Gamma(H)\). For any loop \(\gamma:o\to o\), compatibility
190 along the loop gives
191 \[
192 H_\gamma c_o=c_o.
193 \]
194 Therefore \(\ev_o(c)=c_o\in C_o^{\Hol_o}\). By
195 Lemma~\ref{lem:root-determination}, \(\ev_o\) is injective.
196 
197 Conversely, let \(x\in C_o^{\Hol_o}\). Choose a path
198 \(\tau_i:o\to i\) for each vertex and define
199 \[
200 c_i=H_{\tau_i}x.
201 \]
202 This definition is independent of the chosen path. Indeed, if
203 \(\sigma_i:o\to i\) is another path, then
204 \[
205 \sigma_i^{-1}\tau_i:o\to o
206 \]
207 is a loop, and invariance of \(x\) gives
208 \[
209 H_{\sigma_i}^{-1}H_{\tau_i}x=x.
210 \]
211 Multiplying by \(H_{\sigma_i}\) yields
212 \[
213 H_{\tau_i}x=H_{\sigma_i}x.
214 \]
215 For any edge \(e:i\to j\), the paths \(e\tau_i\) and \(\tau_j\) both run
216 from \(o\) to \(j\). Path independence therefore gives
217 \[
218 H_ec_i
219 =H_eH_{\tau_i}x
220 =H_{\tau_j}x
221 =c_j.
222 \]
223 Thus \((c_i)\) is a global compatible section and \(\ev_o((c_i))=x\).
224 Hence \(\ev_o\) is surjective and therefore an isomorphism.
225 \end{proof}
226 
227 \begin{corollary}[Tree case]\label{cor:tree-core}
228 If \(|G|\) is a tree, then \(\Hol_o=\{I\}\) and
229 \[
230 \Gamma(H)\cong C_o.
231 \]
232 In particular,
233 \[
234 d_{\Workspace}=\dim C_o.
235 \]
236 \end{corollary}
237 
238 \begin{corollary}[Finite matrix test]\label{cor:finite-matrix-test}
239 Let \(\gamma_1,\ldots,\gamma_r\) be fundamental cycle generators and suppose
240 \(\dim C_o=d\). Define the stacked obstruction matrix
241 \[
242 \mathcal M
243 =
244 \begin{pmatrix}
245 H_{\gamma_1}-I\\
246 \vdots\\
247 H_{\gamma_r}-I
248 \end{pmatrix}.
249 \]
250 Then
251 \[
252 \boxed{
253 C_*=\ker\mathcal M,
254 \qquad
255 d_{\Workspace}=d-\operatorname{rank}(\mathcal M).
256 }
257 \]
258 \end{corollary}
259 
260 \begin{proof}
261 The kernel of a vertically stacked operator is the intersection of the kernels
262 of its blocks. Apply Lemma~\ref{lem:finite-obstruction} and
263 Theorem~\ref{thm:limit-holonomy}.
264 \end{proof}
265 
266 \begin{corollary}[Averaging projector for finite holonomy]\label{cor:averaging-projector}
267 If \(\Hol_o\) is finite and acts unitarily, then
268 \[
269 P_*=
270 \frac{1}{|\Hol_o|}
271 \sum_{h\in\Hol_o}h
272 \]
273 is the orthogonal projector onto \(C_*\).
274 \end{corollary}
275 
276 \begin{proof}
277 For every \(g\in\Hol_o\), left multiplication permutes the group elements, so
278 \(gP_*=P_*\). Thus \(\operatorname{ran}P_*\subseteq C_*\). If
279 \(x\in C_*\), then \(P_*x=x\), so \(C_*\subseteq\operatorname{ran}P_*\).
280 Unitarity gives \(P_*^*=P_*\), and the range characterization implies
281 \(P_*^2=P_*\).
282 \end{proof}
283 
284 \subtopic{8.1 Worked examples}
285 
286 \begin{example}[One-dimensional core surviving a contradictory cycle]\label{ex:reflection-cycle}
287 Let a three-node cycle have \(C_1=C_2=C_3=\mathbb R^2\) and
288 \[
289 H_{12}=I,
290 \qquad
291 H_{23}=I,
292 \qquad
293 H_{31}=J,
294 \qquad
295 J=
296 \begin{pmatrix}
297 1&0\\
298 0&-1
299 \end{pmatrix}.
300 \]
301 The cycle holonomy is \(J\), hence
302 \[
303 C_*=\ker(J-I)=\operatorname{span}\{(1,0)^\top\}.
304 \]
305 The first coordinate circulates consistently through all three strata. The
306 second returns with its sign reversed and therefore cannot define a nonzero
307 exact shared section. Thus
308 \[
309 d_{\Workspace}=1.
310 \]
311 \end{example}
312 
313 \begin{example}[Local compatibility without a global nonzero core]\label{ex:rotation-obstruction}
314 Let the only nontrivial cycle holonomy on \(\mathbb R^2\) be a rotation
315 \[
316 R_\theta=
317 \begin{pmatrix}
318 \cos\theta&-\sin\theta\\
319 \sin\theta&\cos\theta
320 \end{pmatrix}
321 \]
322 with \(\theta\notin2\pi\mathbb Z\). Then
323 \[
324 \ker(R_\theta-I)=\{0\}.
325 \]
326 Every edge map is individually invertible and well-defined, yet the complete
327 cycle admits no nonzero exact shared core. Pairwise translatability is
328 therefore insufficient for global path independence.
329 \end{example}
330 
331 \begin{example}[Acyclic collaboration]\label{ex:tree-workspace}
332 Suppose a root concept is translated into a proof, implementation, and report
333 along a tree of invertible exact transfers. No alternative route returns a
334 conflicting representation. The tree has trivial holonomy, so every root-core
335 direction extends uniquely. This does not yet establish causal efficacy; it
336 establishes only exact structural compatibility.
337 \end{example}
338 
339 \subtopic{8.2 Interpretive consequence}
340 
341 The theorem gives a precise meaning to ``something shared across
342 representations.'' The shared object is neither any one local vector nor an
343 extra vector outside the system. It is the equivalence class of all compatible
344 local realizations, canonically represented at a root by the subspace fixed by
345 every translation cycle.
346 
347 \topic{9}{Operator Closure Theorem}
348 
349 \begin{theorem}[Global operation closure]\label{thm:operator-closure}
350 Suppose every core transfer intertwines the representations of
351 \(\mathcal A\):
352 \[
353 H_e\rho_{s(e)}(a)=\rho_{t(e)}(a)H_e
354 \qquad
355 \forall e\in E,\ a\in\mathcal A.
356 \]
357 Then \(\Gamma(H)\) is an \(\mathcal A\)-module under the componentwise action
358 \[
359 \rho_\Gamma(a)(c_i)_{i\in V}
360 =
361 (\rho_i(a)c_i)_{i\in V}.
362 \]
363 Equivalently, the holonomy-fixed root core is invariant:
364 \[
365 \boxed{
366 \rho_o(a)C_*\subseteq C_*
367 \qquad
368 \forall a\in\mathcal A.
369 }
370 \]
371 \end{theorem}
372 
373 \begin{proof}
374 Let \(c=(c_i)\in\Gamma(H)\). For every edge \(e:i\to j\),
375 \[
376 H_e\rho_i(a)c_i
377 =
378 \rho_j(a)H_ec_i
379 =
380 \rho_j(a)c_j.
381 \]
382 Therefore the family \((\rho_i(a)c_i)\) is compatible and belongs to
383 \(\Gamma(H)\). The representation identities on each \(C_i\) hold
384 componentwise, so \(\rho_\Gamma\) is an \(\mathcal A\)-representation.
385 
386 Under the isomorphism of Theorem~\ref{thm:limit-holonomy}, this action is
387 represented at the root by \(\rho_o\). Alternatively, if \(x\in C_*\), then
388 Corollary~\ref{cor:holonomy-commutes} gives, for every loop \(\gamma\),
389 \[
390 H_\gamma\rho_o(a)x
391 =
392 \rho_o(a)H_\gamma x
393 =
394 \rho_o(a)x.
395 \]
396 Hence \(\rho_o(a)x\in C_*\).
397 \end{proof}
398 
399 \begin{corollary}[Inherited algebraic structure]\label{cor:inherited-structure}
400 Any algebraic relation satisfied by \(\rho_i(\mathcal A)\) is inherited by the
401 global action. In particular, if \(a,b\in\mathcal A\), then
402 \[
403 \rho_\Gamma(ab)=\rho_\Gamma(a)\rho_\Gamma(b),
404 \qquad
405 \rho_\Gamma(1)=I_{\Gamma(H)}.
406 \]
407 \end{corollary}
408 
409 The result distinguishes a shared \emph{operational} representation from a
410 shared static code. A core closed under the common algebra supports the same
411 abstract manipulations across strata without leaving the globally compatible
412 subspace.
413 
414 \topic{10}{Workspace Operator Algebra}
415 
416 \begin{definition}[Natural workspace operators]\label{def:workspace-operator-algebra}
417 Define
418 \[
419 \boxed{
420 \mathcal O_{\Workspace}
421 =
422 \left\{
423 (A_i)_{i\in V}:
424 A_{t(e)}H_e=H_eA_{s(e)}\ \forall e
425 \right\}.
426 }
427 \]
428 Categorically,
429 \[
430 \mathcal O_{\Workspace}=\Nat(H,H),
431 \]
432 the algebra of natural endomorphisms of the local system.
433 \end{definition}
434 
435 Pointwise addition, scalar multiplication, and composition make
436 \(\mathcal O_{\Workspace}\) a unital associative algebra. Each
437 \(A=(A_i)\) acts on global sections by
438 \[
439 \pi_{\Workspace}(A)(c_i)=(A_ic_i).
440 \]
441 Naturality guarantees that \((A_ic_i)\) is again compatible.
442 
443 \begin{theorem}[Commutant characterization]\label{thm:commutant-characterization}
444 Let \(|G|\) be connected and let \(H\) be an invertible local system. Root
445 evaluation induces an algebra isomorphism
446 \[
447 \boxed{
448 \Phi_o:\mathcal O_{\Workspace}
449 \xrightarrow{\ \cong\ }
450 \operatorname{Comm}(\Hol_o),
451 \qquad
452 \Phi_o((A_i))=A_o,
453 }
454 \]
455 where
456 \[
457 \operatorname{Comm}(\Hol_o)
458 =
459 \{B\in\End(C_o):Bh=hB\ \forall h\in\Hol_o\}.
460 \]
461 \end{theorem}
462 
463 \begin{proof}
464 If \((A_i)\in\mathcal O_{\Workspace}\), naturality along every loop gives
465 \[
466 A_oH_\gamma=H_\gamma A_o.
467 \]
468 Thus \(A_o\in\operatorname{Comm}(\Hol_o)\), and \(\Phi_o\) is a well-defined
469 algebra homomorphism.
470 
471 Choose paths \(\tau_i:o\to i\). Naturality along \(\tau_i\) gives
472 \[
473 A_iH_{\tau_i}=H_{\tau_i}A_o,
474 \]
475 so
476 \[
477 A_i=H_{\tau_i}A_oH_{\tau_i}^{-1}.
478 \]
479 Therefore \(A_o\) determines every \(A_i\), and \(\Phi_o\) is injective.
480 
481 Conversely, take \(B\in\operatorname{Comm}(\Hol_o)\) and define
482 \[
483 A_i=H_{\tau_i}BH_{\tau_i}^{-1}.
484 \]
485 This is independent of the chosen root path: changing \(\tau_i\) multiplies
486 it by a loop holonomy that commutes with \(B\). For an edge \(e:i\to j\), the
487 operator
488 \[
489 H_{\tau_j}^{-1}H_eH_{\tau_i}
490 \]
491 is a root holonomy. Commutation with \(B\) rearranges to
492 \[
493 A_jH_e=H_eA_i.
494 \]
495 Hence \((A_i)\) is natural and \(\Phi_o((A_i))=B\). Thus \(\Phi_o\) is
496 surjective.
497 \end{proof}
498 
499 The structural workspace is therefore more accurately represented by the
500 triple
501 \[
502 \boxed{
503 \Workspace_{\mathrm{static}}
504 =
505 \bigl(
506 \Gamma(H),
507 \mathcal O_{\Workspace},
508 \pi_{\Workspace}
509 \bigr),
510 }
511 \]
512 not merely by the vector space \(\Gamma(H)\).
513 
514 \subtopic{10.1 Effective operator algebra}
515 
516 Some natural operators may vanish on every global section while acting
517 nontrivially on locally available but globally inconsistent directions. Define
518 the annihilator
519 \[
520 \operatorname{Ann}(\Gamma(H))
521 =
522 \{A\in\mathcal O_{\Workspace}:
523 \pi_{\Workspace}(A)=0\}.
524 \]
525 The effective workspace algebra is
526 \[
527 \boxed{
528 \mathcal O_{\Workspace}^{\mathrm{eff}}
529 =
530 \mathcal O_{\Workspace}/\operatorname{Ann}(\Gamma(H)).
531 }
532 \]
533 Its action on \(\Gamma(H)\) is faithful by construction.
534 
535 \begin{example}[Operators in the reflection-cycle workspace]\label{ex:reflection-operators}
536 For Example~\ref{ex:reflection-cycle}, the holonomy is
537 \[
538 J=\operatorname{diag}(1,-1).
539 \]
540 Its commutant consists of diagonal matrices:
541 \[
542 \operatorname{Comm}(J)
543 =
544 \left\{
545 \begin{pmatrix}a&0\\0&b\end{pmatrix}:a,b\in\mathbb R
546 \right\}.
547 \]
548 The exact workspace core is the first coordinate. Consequently, only the
549 parameter \(a\) acts effectively on \(\Gamma(H)\); the \(b\)-direction belongs
550 to the annihilator of the global core. Hence
551 \[
552 \mathcal O_{\Workspace}^{\mathrm{eff}}\cong\mathbb R.
553 \]
554 \end{example}
555 
556 \subtopic{10.2 Scope of the exact theory}
557 
558 Parts~I--III now establish three exact facts:
559 \[
560 \begin{aligned}
561 &\text{global shared content is the holonomy-fixed core},\\
562 &\text{universally transported operations preserve that core},\\
563 &\text{natural workspace operations are the commutant of holonomy}.
564 \end{aligned}
565 \]
566 These statements do not yet prove that the shared core is robust, causally
567 used, irreducibly distributed, or temporally persistent. Those are separate
568 questions developed in the approximate and dynamic theory that follows.

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