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{II}{Static Core--Transfer Theory}
4 
5 \topic{3}{Interaction Structure}
6 
7 \begin{definition}[Interaction graph]\label{def:interaction-graph}
8 Let
9 \[
10 G=(V,E,s,t)
11 \]
12 be a finite directed multigraph. Parallel edges and self-loops are allowed.
13 The vertex set indexes strata, participants, artifacts, models, instruments,
14 or subworkspaces. An edge \(e:i\to j\) denotes an admissible directed
15 translation or influence channel from \(i\) to \(j\).
16 \end{definition}
17 
18 The directed path category is
19 \[
20 \mathcal C=\Path(G).
21 \]
22 Its objects are vertices. A path
23 \[
24 \gamma=e_m\cdots e_2e_1
25 \]
26 means that \(e_1\) is traversed first. Composition is concatenation, and the
27 empty path at \(i\) is denoted \(1_i\).
28 
29 For every \(i\in V\), assign a finite-dimensional Hilbert space
30 \[
31 X_i.
32 \]
33 For every edge, assign a linear transfer
34 \[
35 T_e:X_{s(e)}\longrightarrow X_{t(e)}.
36 \]
37 The path transfer is
38 \[
39 T_\gamma=T_{e_m}\cdots T_{e_2}T_{e_1},
40 \qquad
41 T_{1_i}=I_{X_i}.
42 \]
43 Accordingly, the static transfer system is a functor
44 \[
45 \mathcal X:\Path(G)\longrightarrow\mathbf{FinHilb}.
46 \]
47 
48 The finite-dimensional Hilbert assumption supplies adjoints and spectral
49 tools. The exact compatible-section definitions require less and may later be
50 formulated in categories such as
51 \[
52 \mathbf{Ban},\qquad
53 \mathbf{Meas},\qquad
54 \mathbf{Prob},\qquad
55 \mathbf{Top},\qquad
56 \mathbf{Alg}.
57 \]
58 
59 \subtopic{3.1 Directed paths versus invertible transport}
60 
61 Global compatible sections make sense even when transfers are noninvertible.
62 Holonomy, however, requires transport around reversible loops. To state the
63 exact holonomy theorem without ambiguity, let \(|G|\) be the underlying
64 undirected graph and let
65 \[
66 \Pi_1(G)
67 \]
68 denote its fundamental path groupoid. Every oriented edge is then supplied
69 with a formal reverse. The core transfers extend to \(\Pi_1(G)\) only when
70 the relevant maps \(H_e\) are invertible, with
71 \[
72 H_{e^{-1}}=H_e^{-1}.
73 \]
74 
75 Hence two regimes will be distinguished:
76 \begin{enumerate}
77 \item the \emph{diagram regime}, in which arbitrary directed maps define
78 compatible sections; and
79 \item the \emph{local-system regime}, in which invertible core maps define a
80 functor
81 \[
82 H:\Pi_1(G)\to\mathbf{FinHilb}^{\times}.
83 \]
84 \end{enumerate}
85 Theorems based on holonomy will explicitly assume the second regime.
86 
87 \subtopic{3.2 Connectivity convention}
88 
89 Unless stated otherwise, connectedness means that \(|G|\) is connected. A
90 root \(o\in V\) may then be joined to every vertex by a path in the underlying
91 graph. If \(G\) has several connected components, the theory applies to each
92 component and
93 \[
94 \Gamma(H)
95 \cong
96 \bigoplus_{\alpha}\Gamma(H|_{G_\alpha}).
97 \]
98 
99 \subtopic{3.3 Edge weights}
100 
101 For approximate theory, assign strictly positive edge weights
102 \[
103 w_e>0.
104 \]
105 They may encode channel reliability, frequency, precision, experimental
106 importance, or inverse noise variance. Exact compatibility is independent of
107 positive rescaling of these weights, while approximate spectra are not.
108 
109 \begin{example}[Human--model--artifact graph]\label{ex:hma-graph}
110 The motivating three-node system is represented by
111 \[
112 V=\{h,m,a\},
113 \qquad
114 E=\{h\to m,m\to a,a\to h\}.
115 \]
116 Additional edges may encode direct retrieval \(a\to m\), human revision
117 \(h\to a\), or independent verification. The graph records permitted
118 translations, not merely physical communication links.
119 \end{example}
120 
121 \topic{4}{Core--Residual Decomposition}
122 
123 \begin{definition}[Admissible local decomposition]\label{def:core-residual}
124 For every vertex \(i\), choose an orthogonal decomposition
125 \[
126 X_i=C_i\oplus R_i,
127 \]
128 with projectors
129 \[
130 P_i:X_i\to C_i,
131 \qquad
132 Q_i=I-P_i:X_i\to R_i.
133 \]
134 The subspace \(C_i\) is the candidate locally realized transferable core. The
135 subspace \(R_i\) contains substrate-specific, context-specific, or otherwise
136 noncore degrees of freedom relative to the present analysis.
137 \end{definition}
138 
139 The decomposition is part of the model and is not presumed unique. Calling a
140 subspace ``core'' is therefore a hypothesis requiring structural and causal
141 validation. Later null tests must determine whether the chosen projectors
142 identify a genuine shared channel rather than an arbitrary low-dimensional
143 summary.
144 
145 Relative to these decompositions, every edge map has the block form
146 \[
147 T_e=
148 \begin{pmatrix}
149 H_e & U_e\\
150 V_e & K_e
151 \end{pmatrix},
152 \]
153 where
154 \[
155 H_e=P_{t(e)}T_eP_{s(e)}:C_{s(e)}\to C_{t(e)},
156 \]
157 \[
158 U_e=P_{t(e)}T_eQ_{s(e)}:R_{s(e)}\to C_{t(e)},
159 \]
160 \[
161 V_e=Q_{t(e)}T_eP_{s(e)}:C_{s(e)}\to R_{t(e)},
162 \]
163 \[
164 K_e=Q_{t(e)}T_eQ_{s(e)}:R_{s(e)}\to R_{t(e)}.
165 \]
166 
167 \subtopic{4.1 Structural conditions}
168 
169 \textbf{Core preservation} means
170 \[
171 V_e=0.
172 \]
173 A core input cannot leak into the target residual.
174 
175 \textbf{Core autonomy} means
176 \[
177 U_e=V_e=0.
178 \]
179 Core and residual evolve as separate transfer systems.
180 
181 \textbf{Core leakage}, \textbf{residual contamination}, and \textbf{total
182 transfer defect} are measured by
183 \[
184 \lambda_e^{\mathrm{out}}=\|V_e\|,
185 \qquad
186 \lambda_e^{\mathrm{in}}=\|U_e\|,
187 \]
188 \[
189 \delta_e^{\mathrm{tr}}
190 =
191 \|U_e\|+\|V_e\|.
192 \]
193 These quantities depend on the chosen norms and projectors. Their empirical
194 meaning must therefore be tied to a declared intervention and prediction
195 protocol.
196 
197 \subtopic{4.2 Composition exposes residual mediation}
198 
199 For composable edges \(e:i\to j\) and \(f:j\to k\), direct block
200 multiplication gives
201 \[
202 T_fT_e
203 =
204 \begin{pmatrix}
205 H_fH_e+U_fV_e & H_fU_e+U_fK_e\\
206 V_fH_e+K_fV_e & V_fU_e+K_fK_e
207 \end{pmatrix}.
208 \]
209 The induced core-to-core block is therefore
210 \[
211 P_kT_fT_eP_i
212 =
213 H_fH_e+U_fV_e.
214 \]
215 The term \(U_fV_e\) is a residual-mediated correction: information leaves the
216 candidate core at the intermediate node and later re-enters it. Consequently,
217 the simple path product \(H_fH_e\) faithfully represents full core transport
218 only under a preservation condition such as \(V_e=0\), or under an explicit
219 model that includes these mediated corrections.
220 
221 \begin{proposition}[Pathwise core preservation]\label{prop:pathwise-preservation}
222 Suppose \(V_e=0\) for every edge in a directed path
223 \(\gamma=e_m\cdots e_1\). Then
224 \[
225 T_\gamma(C_{s(\gamma)})\subseteq C_{t(\gamma)}
226 \]
227 and
228 \[
229 P_{t(\gamma)}T_\gamma P_{s(\gamma)}
230 =
231 H_{e_m}\cdots H_{e_1}.
232 \]
233 \end{proposition}
234 
235 \begin{proof}
236 For a single edge, \(V_e=0\) means
237 \[
238 T_e(c,0)=(H_ec,0),
239 \]
240 so core states remain in the core. Inductively, if the first \(r\) transfers
241 map \(c\) to \((H_{e_r}\cdots H_{e_1}c,0)\), applying \(T_{e_{r+1}}\) gives
242 \[
243 (H_{e_{r+1}}H_{e_r}\cdots H_{e_1}c,0).
244 \]
245 The conclusion follows after \(m\) steps.
246 \end{proof}
247 
248 \begin{corollary}[Autonomous splitting]\label{cor:autonomous-splitting}
249 If \(U_e=V_e=0\) for every edge, then the complete diagram decomposes as the
250 direct sum of two diagrams,
251 \[
252 \mathcal X\cong H\oplus K,
253 \]
254 and every path transfer satisfies
255 \[
256 T_\gamma
257 =
258 \begin{pmatrix}
259 H_\gamma&0\\
260 0&K_\gamma
261 \end{pmatrix}.
262 \]
263 \end{corollary}
264 
265 \subtopic{4.3 Exact core-transfer systems}
266 
267 \begin{definition}[Exact core-transfer system]\label{def:exact-core-transfer}
268 An exact core-transfer system on \(G\) consists of:
269 \begin{enumerate}
270 \item decompositions \(X_i=C_i\oplus R_i\);
271 \item core-preserving edge maps \(V_e=0\); and
272 \item invertible core blocks \(H_e\in GL(C_{s(e)},C_{t(e)})\) whenever
273 holonomy is invoked.
274 \end{enumerate}
275 \end{definition}
276 
277 The requirement \(V_e=0\) concerns the relation between the selected core and
278 the full transfer. The requirement that \(H_e\) be invertible concerns the
279 ability to compare core representations along reversed paths. They answer
280 different questions and should not be conflated.
281 
282 \begin{remark}[Approximate systems]
283 In applications neither condition need hold exactly. The approximate theory
284 will replace strict invariance by low transfer energy, small leakage, and a
285 spectral gap separating workspace modes from residual modes.
286 \end{remark}
287 
288 \topic{5}{Abstract Operation Algebra}
289 
290 \begin{definition}[Operation algebra]\label{def:operation-algebra}
291 Let \(\mathbb F\in\{\mathbb R,\mathbb C\}\). An operation algebra
292 \(\mathcal A\) is a unital associative \(\mathbb F\)-algebra whose elements
293 represent abstract transformations, inferential rules, questions, or
294 procedures that can be realized in multiple strata.
295 \end{definition}
296 
297 Every core space carries a unital representation
298 \[
299 \rho_i:\mathcal A\longrightarrow\End(C_i).
300 \]
301 For \(a\in\mathcal A\), the operator \(\rho_i(a)\) is the realization of the
302 same abstract operation in local coordinates.
303 
304 \begin{definition}[Exact operation-compatible transfer]\label{def:intertwiner}
305 A core transfer \(H_e:C_i\to C_j\) is operation compatible when it is an
306 intertwiner:
307 \[
308 \boxed{
309 H_e\rho_i(a)=\rho_j(a)H_e
310 }
311 \qquad
312 \forall a\in\mathcal A.
313 \]
314 Equivalently, the diagram
315 \[
316 \begin{array}{ccc}
317 C_i&\xrightarrow{\rho_i(a)}&C_i\\
318 \downarrow H_e&&\downarrow H_e\\
319 C_j&\xrightarrow{\rho_j(a)}&C_j
320 \end{array}
321 \]
322 commutes for every \(a\).
323 \end{definition}
324 
325 This condition states that translating and then operating gives the same
326 result as operating and then translating. Agreement of static encodings alone
327 is weaker: two nodes may store corresponding vectors while implementing
328 incompatible transformations on them.
329 
330 \begin{proposition}[Path composition of intertwiners]\label{prop:path-intertwiner}
331 If every edge of a path \(\gamma:i\to j\) intertwines the local
332 \(\mathcal A\)-representations, then
333 \[
334 H_\gamma\rho_i(a)=\rho_j(a)H_\gamma
335 \qquad
336 \forall a\in\mathcal A.
337 \]
338 \end{proposition}
339 
340 \begin{proof}
341 For composable intertwiners \(H_e:C_i\to C_j\) and
342 \(H_f:C_j\to C_k\),
343 \[
344 H_fH_e\rho_i(a)
345 =H_f\rho_j(a)H_e
346 =\rho_k(a)H_fH_e.
347 \]
348 Induction over path length proves the claim.
349 \end{proof}
350 
351 \begin{corollary}[Holonomy commutes with abstract operations]\label{cor:holonomy-commutes}
352 For every loop \(\gamma:o\to o\),
353 \[
354 H_\gamma\rho_o(a)=\rho_o(a)H_\gamma.
355 \]
356 Hence
357 \[
358 \rho_o(\mathcal A)
359 \subseteq
360 \operatorname{Comm}(\Hol_o),
361 \]
362 where \(\operatorname{Comm}(\Hol_o)\) is the commutant of the holonomy
363 representation.
364 \end{corollary}
365 
366 \subtopic{5.1 Approximate operation compatibility}
367 
368 The edgewise defect is
369 \[
370 \Xi_e(a)
371 =
372 H_e\rho_{s(e)}(a)-\rho_{t(e)}(a)H_e.
373 \]
374 For a bounded generating family \(\mathcal G\subseteq\mathcal A\), define
375 \[
376 \eta_{\mathrm{int}}(\mathcal G)
377 =
378 \sup_{a\in\mathcal G}
379 \sum_{e\in E}w_e\|\Xi_e(a)\|^2.
380 \]
381 When \(\mathcal A\) is normed and its unit ball is appropriate, one may take
382 \[
383 \mathcal G=\{a:\|a\|_{\mathcal A}\leq1\}.
384 \]
385 Exact representational universality means
386 \[
387 \eta_{\mathrm{int}}=0,
388 \]
389 while approximate universality means
390 \[
391 \eta_{\mathrm{int}}\leq\varepsilon_{\mathrm{int}}.
392 \]
393 The use of a generating family prevents the definition from depending on an
394 unbounded rescaling of algebra elements.
395 
396 \begin{example}[Equivalent operations in different coordinates]\label{ex:coordinate-intertwiner}
397 Let \(C_1=C_2=\mathbb R^2\), choose \(S\in GL_2(\mathbb R)\), and set
398 \[
399 H_{12}=S,
400 \qquad
401 \rho_2(a)=S\rho_1(a)S^{-1}.
402 \]
403 Then
404 \[
405 H_{12}\rho_1(a)
406 =S\rho_1(a)
407 =\rho_2(a)S
408 =\rho_2(a)H_{12}.
409 \]
410 The local matrices can look different while realizing the same abstract
411 operation. This is the elementary model of representation change used
412 throughout the theory.
413 \end{example}

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