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{V}{Gauge Invariance and Abstract Identity}
4 
5 Exact compatibility is a coordinate-free notion. Approximate spectra also
6 depend on the Hilbert metrics used to measure discrepancy. Gauge theory must
7 therefore distinguish a change of coordinates from a physical change of
8 metric.
9 
10 \topic{16}{Local Coordinate Transformations}
11 
12 For each \(i\in V\), let
13 \[
14 g_i\in GL(C_i),
15 \qquad
16 g=(g_i)_{i\in V}.
17 \]
18 Define the vertex and edge direct sums
19 \[
20 G_V=\bigoplus_{i\in V}g_i,
21 \qquad
22 G_E=\bigoplus_{e\in E}g_{t(e)}.
23 \]
24 The transformed transfers and local operation representations are
25 \[
26 H_e^g
27 =
28  g_{t(e)}H_eg_{s(e)}^{-1},
29 \]
30 \[
31 \rho_i^g(a)
32 =
33  g_i\rho_i(a)g_i^{-1}.
34 \]
35 The fundamental covariance identity is
36 \[
37 \boxed{
38 D_{H^g}G_V=G_ED_H.
39 }
40 \]
41 It follows directly by evaluating the \(e\)-component:
42 \[
43 \sqrt{w_e}
44 \left(
45  g_{t(e)}c_{t(e)}
46 -
47  g_{t(e)}H_ec_{s(e)}
48 \right).
49 \]
50 
51 \topic{17}{Gauge Invariance Theorem}
52 
53 \begin{theorem}[Exact gauge covariance]\label{thm:gauge-covariance}
54 The map
55 \[
56 G_V:\Gamma(H)\longrightarrow\Gamma(H^g)
57 \]
58 is an isomorphism. Therefore,
59 \[
60 \boxed{
61 \Gamma(H^g)=G_V\Gamma(H),
62 \qquad
63 \dim\Gamma(H^g)=\dim\Gamma(H).
64 }
65 \]
66 If the core transports are invertible and \(o\) is a root, then
67 \[
68 H_\gamma^g=g_oH_\gamma g_o^{-1}
69 \]
70 for every loop \(\gamma:o\to o\). Hence the holonomy-fixed core transforms by
71 \[
72 C_*^g=g_oC_*.
73 \]
74 \end{theorem}
75 
76 \begin{proof}
77 The covariance identity implies
78 \[
79 D_{H^g}(G_Vc)=G_ED_Hc.
80 \]
81 Since \(G_E\) is invertible,
82 \[
83 D_Hc=0
84 \quad\Longleftrightarrow\quad
85 D_{H^g}(G_Vc)=0.
86 \]
87 The holonomy formula follows by cancellation of adjacent factors
88 \(g_i^{-1}g_i\) along a path.
89 \end{proof}
90 
91 \begin{theorem}[Unitary gauge covariance of the Laplacian]\label{thm:unitary-gauge}
92 If every \(g_i\) is unitary, then \(G_V\) and \(G_E\) are unitary and
93 \[
94 \boxed{
95 L_{H^g}=G_VL_HG_V^{-1}.
96 }
97 \]
98 Consequently,
99 \[
100 \spec(L_{H^g})=\spec(L_H),
101 \qquad
102 P_I(L_{H^g})=G_VP_I(L_H)G_V^{-1}
103 \]
104 for every spectral interval \(I\).
105 \end{theorem}
106 
107 \begin{proof}
108 From
109 \[
110 D_{H^g}=G_ED_HG_V^{-1}
111 \]
112 and unitarity,
113 \[
114 D_{H^g}^*D_{H^g}
115 =
116 G_VD_H^*G_E^{-1}G_ED_HG_V^{-1}
117 =
118 G_VL_HG_V^{-1}.
119 \]
120 The spectral statements follow from unitary similarity.
121 \end{proof}
122 
123 \subtopic{17.1 Nonunitary gauges and metric distortion}
124 
125 For a general invertible gauge, define
126 \[
127 m_V=\sigma_{\min}(G_V),
128 \qquad
129 M_V=\|G_V\|,
130 \]
131 \[
132 m_E=\sigma_{\min}(G_E),
133 \qquad
134 M_E=\|G_E\|.
135 \]
136 The covariance identity yields
137 \[
138 m_E^2\mathcal E_H(c)
139 \leq
140 \mathcal E_{H^g}(G_Vc)
141 \leq
142 M_E^2\mathcal E_H(c),
143 \]
144 and
145 \[
146 m_V^2\|c\|^2
147 \leq
148 \|G_Vc\|^2
149 \leq
150 M_V^2\|c\|^2.
151 \]
152 Therefore the Rayleigh quotients obey
153 \[
154 \boxed{
155 \frac{m_E^2}{M_V^2}
156 \frac{\mathcal E_H(c)}{\|c\|^2}
157 \leq
158 \frac{\mathcal E_{H^g}(G_Vc)}{\|G_Vc\|^2}
159 \leq
160 \frac{M_E^2}{m_V^2}
161 \frac{\mathcal E_H(c)}{\|c\|^2}.
162 }
163 \]
164 By the min--max principle,
165 \[
166 \boxed{
167 \frac{m_E^2}{M_V^2}\lambda_k(L_H)
168 \leq
169 \lambda_k(L_{H^g})
170 \leq
171 \frac{M_E^2}{m_V^2}\lambda_k(L_H).
172 }
173 \]
174 Exact zero modes are unchanged, but fixed-metric approximate eigenvalues can be
175 strongly distorted by an ill-conditioned coordinate transformation.
176 
177 \subtopic{17.2 Metric-covariant formulation}
178 
179 A pure change of coordinates should transform the metric as well as the
180 matrices. On each transformed space, define
181 \[
182 \langle u,v\rangle_i^g
183 =
184 \left\langle g_i^{-1}u,g_i^{-1}v\right\rangle_i.
185 \]
186 Use the analogous pullback metric on every edge copy. Then \(G_V\) and
187 \(G_E\) are unitary between the original and transformed metric spaces.
188 Consequently, the complete transfer-Laplacian spectrum is exactly preserved.
189 
190 This gives two distinct interpretations:
191 \begin{itemize}
192 \item Transforming the matrices and the metrics gives the same approximate
193 workspace in new coordinates.
194 \item Transforming the matrices while holding the metrics fixed gives a
195 potentially different weighting of discrepancy.
196 \end{itemize}
197 The distinction prevents arbitrary coordinate scaling from manufacturing or
198 erasing apparent low-energy workspace directions.
199 
200 \topic{18}{Static Workspace Equivalence}
201 
202 \begin{definition}[Exact static equivalence]\label{def:static-equivalence}
203 Two static realizations are exactly gauge equivalent,
204 \[
205 \mathfrak R\sim_{\mathrm{stat}}\mathfrak R',
206 \]
207 when there exists an invertible family \(g_i\) such that
208 \[
209 H_e'=g_{t(e)}H_eg_{s(e)}^{-1},
210 \qquad
211 \rho_i'(a)=g_i\rho_i(a)g_i^{-1}.
212 \]
213 The exact static abstract workspace is
214 \[
215 \boxed{
216 \Workspace_{\mathrm{stat}}
217 =
218 [\mathfrak R]_{\sim_{\mathrm{stat}}}.
219 }
220 \]
221 \end{definition}
222 
223 For approximate theory, the realization also contains the vertex and edge
224 metrics. Two metric realizations are equivalent only when the gauge transports
225 those metrics, or when the gauge is unitary relative to the fixed metrics.
226 Accordingly:
227 \[
228 \boxed{
229 \begin{aligned}
230 \text{exact workspace identity}
231 &\text{ is invariant under }GL\text{-gauge},\\
232 \text{fixed-metric spectral geometry}
233 &\text{ is invariant under unitary gauge},\\
234 \text{metric-covariant spectral geometry}
235 &\text{ is invariant under transported }GL\text{-gauge}.
236 \end{aligned}
237 }
238 \]
239 
240 \clearpage
241 \subtopic{18.1 What the approximate theory establishes}
242 
243 Parts~IV--V now prove:
244 \[
245 \begin{aligned}
246 &\ker L_H=\Gamma(H),\\
247 &\text{low energy plus a gap implies proximity to an exact core},\\
248 &\text{a separated workspace cluster is stable under transfer noise},\\
249 &\text{approximately intertwined operations nearly preserve that cluster},\\
250 &\text{core leakage and residual reentry admit explicit pathwise bounds},\\
251 &\text{exact identity is gauge invariant while approximate geometry is metric aware}.
252 \end{aligned}
253 \]
254 These are structural and perturbative results. They still do not prove that a
255 workspace variable is causally used, predictively sufficient, irreducibly
256 distributed, or temporally self-maintaining. Those questions begin in
257 Part~VI.
258 
259 \subtopic{18.2 Finite computation protocol}
260 
261 For a declared graph, set of core dimensions, weights, and transfer maps, the
262 static analysis can be executed in the following order:
263 \begin{enumerate}
264 \item Assemble the sparse block operator \(D_H\) from the edge equations
265 \[
266 \sqrt{w_e}\bigl(c_{t(e)}-H_ec_{s(e)}\bigr).
267 \]
268 \item Compute a singular-value decomposition
269 \[
270 D_H=U\Sigma V^*,
271 \qquad
272 \lambda_k(L_H)=\sigma_k(D_H)^2.
273 \]
274 Use a declared numerical tolerance to distinguish computational zero modes
275 from positive modes.
276 \item Select a candidate rank \(d\) only at a reproducible spectral boundary
277 with \(g_d>0\). A large observed gap is evidence of stability, not evidence of
278 causal relevance.
279 \item Quantify transfer uncertainty by \(\delta_H\) and verify whether
280 \[
281 2\|D_H\|\delta_H+\delta_H^2<\frac{g_d}{2}.
282 \]
283 If this gate fails, the extracted projector has no certified perturbative
284 identity at the stated error scale.
285 \item For every proposed common operation, compute \(\beta(a)\) and the
286 operation-leakage bound in Theorem~\ref{thm:approx-operation-closure}.
287 \item For every long transfer chain used in an interpretation, report the
288 pathwise leakage certificate from Theorem~\ref{thm:residual-duhamel} rather
289 than extrapolating from single-edge defects.
290 \end{enumerate}
291 
292 A minimal structural report is therefore
293 \[
294 \boxed{
295 \left(
296  d,\lambda_d,\lambda_{d+1},g_d,
297  \delta_H,
298  \|P_d-\widetilde P_d\|_{\mathrm{bound}},
299  \beta(a),
300  \Lambda_\gamma
301 \right),
302 }
303 \]
304 where \(\Lambda_\gamma\) denotes the right-hand side of the pathwise leakage
305 bound. This structural report is intentionally incomplete as a workspace
306 certificate. Parts~VI--IX add the predictive, causal, null, and irreducibility
307 gates required to interpret the low-energy subspace operationally.

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