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{IX}{Formal Workspace Definition}
4 
5 \topic{31}{Certified Stratified Relational Workspace}
6 
7 Let
8 \[
9 \Theta
10 =
11 \left(
12 \varepsilon_{\mathrm{coh}},
13 \varepsilon_{\mathrm{align}},
14 \varepsilon_k,
15 \underline\eta_c,
16 \underline\eta_{\mathrm{loop}},
17 \delta_{\mathrm{node}},
18 \delta_{\mathrm{null}},
19 \delta_{\mathrm{spec}},
20 k,
21 \varepsilon_{\mathrm{pers}}
22 \right).
23 \]
24 Let
25 \[
26 \varsigma
27 =
28 (\mathcal D,Y^+,\ell,\tau,\mathcal I,\mathcal Q,\Theta)
29 \]
30 be the complete declared scientific specification, where
31 \(\mathcal Q\subseteq\mathcal N_{\mathrm{pred}}(P)\) is a validated null panel.
32 A dynamic realization contains a certified stratified relational workspace
33 when there exists a candidate \(W_t\) satisfying all gates required for the
34 claimed level under \(\varsigma\). The detailed robustness and identifiability
35 requirements for varying \(\varsigma\) are developed in Parts~XVII and XVIII.
36 
37 \subtopic{Structural coherence}
38 \[
39 \delta_{\mathrm{coh}}(W_t)
40 \leq
41 \varepsilon_{\mathrm{coh}}.
42 \]
43 
44 \subtopic{Interventional structural--operational alignment}
45 \[
46 \delta_{\mathrm{align}}^{\mathcal I}(W_t)
47 \leq
48 \varepsilon_{\mathrm{align}}.
49 \]
50 
51 \subtopic{Causal efficacy}
52 \[
53 \eta_{\mathrm{causal}}^\nu
54 \geq
55 \underline\eta_c.
56 \]
57 
58 \subtopic{Loop-mediated reentry, when claimed}
59 \[
60 \eta_{\mathrm{loop}}(\tau)
61 \geq
62 \underline\eta_{\mathrm{loop}}.
63 \]
64 
65 \subtopic{Node-level joint advantage}
66 \[
67 \Delta_{\mathrm{joint}}^{\ell}
68 \geq
69 \delta_{\mathrm{node}}.
70 \]
71 
72 \subtopic{Matched-null transfer dependence}
73 \[
74 \Delta_{\mathrm{tr}}^{\min}(\mathcal Q)
75 \geq
76 \delta_{\mathrm{null}}.
77 \]
78 
79 \subtopic{Causal specificity}
80 \[
81 \Delta_{\mathrm{specific}}(r)
82 \geq
83 \delta_{\mathrm{spec}}.
84 \]
85 
86 \subtopic{Distributedness}
87 \[
88 k_{\varepsilon_k}
89 \geq
90 k.
91 \]
92 
93 \subtopic{Persistence, when claimed}
94 \[
95 d_{\mathrm{id}}(W_t,W_{t+\tau})
96 \leq
97 \varepsilon_{\mathrm{pers}}.
98 \]
99 
100 Symbolically,
101 \[
102 \boxed{
103 \SRW(W_t;\varsigma)
104 =
105 \bigwedge_{q\in\mathcal J(\varsigma)}
106 \operatorname{Gate}_q(W_t;\varsigma).
107 }
108 \]
109 The gate set is claim-relative. A one-step operational workspace need not pass
110 persistence or reentry gates, while a persistent reentrant workspace must.
111 No scalar total score is sufficient: failure of a required gate cannot be
112 hidden by high performance on another gate.
113 
114 \begin{remark}[Structural, predictive, causal, and temporal claims]
115 The formal hierarchy now distinguishes four logically independent questions:
116 \[
117 \begin{aligned}
118 &\text{Does a stable shared coordinate exist?}\\
119 &\text{Does it retain the task-relevant conditional law?}\\
120 &\text{Does manipulating it change downstream behavior specifically?}\\
121 &\text{Does it persist or reenter through time?}
122 \end{aligned}
123 \]
124 The preceding structural and causal analysis answers the first three at the level of definitions and conditional
125 theorems. The following parts develop the fourth.
126 \end{remark}

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