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{XXV}{Unified Formulation}
4 
5 The complete static-to-agentive construction is
6 \[
7 \boxed{
8 \begin{aligned}
9 &
10 (\Omega,\mathfrak D,\mathfrak S,\Sigma,\mathcal D_{\mathrm{form}},\mathcal L)
11 \\[2mm]
12 &\Downarrow
13 \qquad
14 (\mathcal D,\widehat V,\widehat E,\widehat H,\widehat P,\widehat B)
15 \\[2mm]
16 &\Downarrow
17 \qquad
18 (G,\{X_i\},\{T_e\})
19 \\[2mm]
20 &\Downarrow
21 \qquad
22 X_i=C_i\oplus R_i
23 \\[2mm]
24 &
25 (\{C_i\},\{H_e\},\mathcal A,\{\rho_i\})
26 \\[2mm]
27 &\Downarrow
28 \qquad
29 \Gamma(H)=\varprojlim C_i
30 \\[2mm]
31 &\Downarrow
32 \qquad
33 L_H=D_H^*D_H
34 \\[2mm]
35 &\Downarrow
36 \qquad
37 \Gamma_\varepsilon(H)
38 =
39 \ran\,\mathbf 1_{[0,\varepsilon]}(L_H)
40 \\[2mm]
41 &\Downarrow
42 \qquad
43 W_t^{\mathrm{str}}
44 \\[2mm]
45 &\xrightarrow{\text{alignment}}
46 Z_t=\mathbb P(Y_t^+\in\cdot\mid\mathbf H_t)
47 \\[2mm]
48 &\Downarrow
49 \qquad
50 \SRW_\Theta(W_t;\mathcal D)
51 \\[2mm]
52 &\Downarrow
53 \qquad
54 (W_t,G_t,\widehat Q_t,M_t,B_t)
55 \\[2mm]
56 &\xrightarrow{\text{goal and value use}}
57 (A_t,R_t)
58 \\[2mm]
59 &\xrightarrow{\text{causal repair}}
60 e_t\leq e_{\max}
61 \\[2mm]
62 &\Downarrow
63 \qquad
64 \Workspace
65 =
66 [\mathfrak R]_{\text{gauge}+\text{causal equivalence}}.
67 \end{aligned}
68 }
69 \]
70 Compactly:
71 \[
72 \boxed{
73 \begin{aligned}
74 \text{scaffold and history}
75 &\to
76 \text{learned organization and boundary}
77 \\
78 &\to
79 \text{local core--transfer realization}
80 \\
81 &\to
82 \text{compatible sections and global descent}
83 \\
84 &\to
85 \text{causal operation and persistent identity}
86 \\
87 &\to
88 \text{cue-controlled goal retention and counterfactual action}
89 \\
90 &\to
91 \text{monitor-mediated repair and bounded maintenance deficit}
92 \\
93 &\to
94 \text{decomposition- and specification-robust identified certification}.
95 \end{aligned}
96 }
97 \]
98 The final two arrows are optional extensions rather than consequences of the
99 earlier ones. A relational workspace can exist without being an agent, and an
100 agentive controller can fail to maintain the workspace that realizes it.

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