Stratified Universality

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1 \section{Admissibility change, geometric obstruction, and doctrine lifting}
2 \label{sec:block19-admissibility-lifting}
3 
4 Task strengthening changes which distinctions must be retained.  Enlarging an admissible realization class changes a different quantity: which completions or normal forms are allowed.  The two variances should not be compressed into one undifferentiated order.  This section formalizes admissibility monotonicity, separates gap from nonattainment, and places enrichment problems inside the existing bridge-passport architecture.
5 
6 \subsection{Admissible realization classes and optimized defect}
7 \label{subsec:block19-defect}
8 
9 \begin{definition}[Class-relative defect]
10 \label{def:block19-class-relative-defect}
11 Let \(X\) be a represented system, let \(\Gamma\) be a declared class of admissible auxiliary structures, and let
12 \[
13  d(X,g)\in[0,\infty]
14 \]
15 be a defect functional.  Define
16 \[
17  \delta_\Gamma(X):=\inf_{g\in\Gamma}d(X,g).
18 \]
19 The pair consisting of the infimum and its attainment status is the \emph{class-relative defect profile}.
20 \end{definition}
21 
22 \begin{theorem}[Admissibility monotonicity]
23 \label{thm:block19-admissibility-monotonicity}
24 If \(\Gamma\subseteq\Gamma'\), then
25 \[
26  \delta_{\Gamma'}(X)\leq\delta_\Gamma(X).
27 \]
28 Thus an obstruction in a restricted realization class need not remain an obstruction after the class is enlarged.
29 \end{theorem}
30 
31 \begin{proof}
32 The infimum on the left is taken over a superset of the candidates used on the right.
33 \end{proof}
34 
35 \begin{definition}[Completion, gap, and escape]
36 \label{def:block19-trichotomy}
37 Assume \(d\geq0\).  The defect profile is:
38 \begin{enumerate}[label=\textup{(\roman*)}]
39 \item a \emph{completion} when \(\delta_\Gamma(X)=0\) and the infimum is attained;
40 \item a \emph{gap obstruction} when \(\delta_\Gamma(X)>0\);
41 \item an \emph{escape obstruction} when \(\delta_\Gamma(X)=0\) but the infimum is not attained.
42 \end{enumerate}
43 \end{definition}
44 
45 The trichotomy separates approximation from realization.  A sequence of increasingly good coordinates may approach a missing completion only by leaving every compactly controlled part of the admissible parameter space.
46 
47 \begin{proposition}[Compactness certificate for escape]
48 \label{prop:block19-escape-compactness}
49 Let \(\Gamma\) be a Hausdorff parameter space, and let \(d(X,-)\) be continuous.  Suppose every sublevel set satisfying a declared conditioning bound is compact.  If \(\delta_\Gamma(X)=0\) is not attained, then every near-minimizing sequence must eventually violate every such uniform conditioning bound.
50 \end{proposition}
51 
52 \begin{proof}
53 Fix a declared conditioning bound \(c\), write \(\Gamma_c\) for the
54 parameters satisfying it, and let
55 \[
56  K_c=\{g\in\Gamma_c:d(X,g)\leq 1\}.
57 \]
58 By hypothesis \(K_c\) is compact.  If \(K_c\) is empty, every near-minimizing
59 sequence eventually has defect below \(1\) and therefore lies outside
60 \(\Gamma_c\).  Otherwise continuity gives an attained minimum
61 \[
62  a_c=\min_{g\in K_c}d(X,g).
63 \]
64 Since the zero infimum is not attained anywhere in \(\Gamma\), \(a_c>0\).
65 Every near-minimizing sequence eventually has defect below
66 \(\min\{1,a_c\}\), and therefore cannot belong to \(\Gamma_c\).  Since \(c\)
67 was arbitrary, each fixed conditioning bound is eventually violated; in
68 particular no subsequence satisfying one fixed bound can be near-minimizing.
69 \end{proof}
70 
71 \subsection{Metric packets and intrinsic distortion}
72 \label{subsec:block19-metric-distortion}
73 
74 Let \(T:(V,Q_s)\to(W,Q_t)\) be invertible between real or complex spaces of common finite dimension \(n\geq1\) with positive-definite metrics.  Choose factors \(Q_s=R_s^*R_s\) and \(Q_t=R_t^*R_t\), and put
75 \[
76  B_Q(T):=R_tTR_s^{-1}.
77 \]
78 Define
79 \[
80  D_Q(T)
81  :=
82  \frac{\sigma_{\max}(B_Q(T))}{\sigma_{\min}(B_Q(T))}
83  \geq1,
84  \qquad
85  \Delta_Q(T):=\log D_Q(T).
86 \]
87 
88 \begin{proposition}[Coordinate covariance and composition]
89 \label{prop:block19-distortion-cocycle}
90 Simultaneous coordinate transport of \(T,Q_s,Q_t\) leaves \(D_Q(T)\) invariant.  For composable invertible maps,
91 \[
92  D_Q(T_2T_1)
93  \leq
94  D_Q(T_2)D_Q(T_1),
95 \]
96 so
97 \[
98  \Delta_Q(T_2T_1)
99  \leq
100  \Delta_Q(T_2)+\Delta_Q(T_1).
101 \]
102 For a closed word \(H\),
103 \[
104  D_Q(H)
105  \geq
106  \frac{\max_{\lambda\in\operatorname{spec}(H)}|\lambda|}
107       {\min_{\lambda\in\operatorname{spec}(H)}|\lambda|}.
108 \]
109 \end{proposition}
110 
111 \begin{proof}
112 Coordinate transport changes \(B_Q(T)\) by left and right unitary factors.  Singular-value condition numbers are submultiplicative.  The spectral lower bound follows because every eigenvalue modulus lies between the smallest and largest singular values.
113 \end{proof}
114 
115 \begin{theorem}[Conformal metric-completion criterion]
116 \label{thm:block19-metric-completion}
117 Let a connected finite graph carry real or complex vector spaces of common
118 dimension \(n\geq1\) and invertible edge transports \(T_e\).  Reverse traversal
119 of an edge uses \(T_e^{-1}\).  At a chosen root, determinant-normalized closed
120 holonomy words are
121 \[
122  \widehat H:=|\det H|^{-1/n}H.
123 \]
124 They form the determinant-normalized holonomy representation.  Positive-definite
125 vertex metrics and positive edge scales satisfying
126 \[
127  T_e^*Q_{t(e)}T_e=\lambda_e^2Q_{s(e)}
128 \]
129 exist if and only if the determinant-normalized holonomy representation preserves a positive-definite metric at one root vertex.  Equivalently, the normalized holonomy is unitarizable.  In finite dimension, this is equivalent to relative compactness of its image.
130 \end{theorem}
131 
132 \begin{proof}
133 A compatible family makes every closed root holonomy \(H\) conformal for the
134 root metric \(Q_o\).  Taking determinants fixes its positive scale as
135 \(|\det H|^{1/n}\), so \(\widehat H\) preserves \(Q_o\).
136 
137 Conversely, suppose every normalized root holonomy preserves \(Q_o\).  Choose
138 spanning-tree maps \(G_v\) from the root to each vertex and put
139 \[
140  Q_v=(G_v^{-1})^*Q_oG_v^{-1}.
141 \]
142 For an edge \(e:u\to v\), the root word \(H_e=G_v^{-1}T_eG_u\) satisfies
143 \[
144  H_e^*Q_oH_e=|\det H_e|^{2/n}Q_o.
145 \]
146 Consequently
147 \[
148  T_e^*Q_vT_e=|\det H_e|^{2/n}Q_u,
149  \qquad \lambda_e=|\det H_e|^{1/n}>0.
150 \]
151 This gives the required compatible family directly.  A group preserving a
152 positive-definite metric is conjugate into a compact orthogonal or unitary
153 group.  Conversely, averaging an inner product over the compact closure of a
154 relatively compact image gives an invariant positive-definite metric.
155 \end{proof}
156 
157 \subsection{Exact convex-versus-quadratic separation}
158 \label{subsec:block19-convex-quadratic}
159 
160 Consider
161 \[
162  A_1=\begin{pmatrix}1&0\\1&0\end{pmatrix},
163  \qquad
164  A_2=\begin{pmatrix}0&1\\0&-1\end{pmatrix}.
165 \]
166 
167 \begin{theorem}[Sharp geometry-class separation]
168 \label{thm:block19-convex-quadratic-separation}
169 The norm \(\|x\|_\infty\) satisfies
170 \[
171  \max\{\|A_1x\|_\infty,\|A_2x\|_\infty\}
172  =\|x\|_\infty
173 \]
174 for every \(x\in\mathbb R^2\).  In contrast, for every positive-definite quadratic metric \(Q\),
175 \[
176  \max\{\|A_1\|_Q,\|A_2\|_Q\}\geq\sqrt2.
177 \]
178 Equality holds for scalar multiples of the Euclidean metric.  Hence the exact optimal convex/Finsler bound is \(1\), while the exact optimal quadratic bound is \(\sqrt2\).
179 \end{theorem}
180 
181 \begin{proof}
182 The two images are
183 \[
184  A_1(x_1,x_2)=(x_1,x_1),
185  \qquad
186  A_2(x_1,x_2)=(x_2,-x_2),
187 \]
188 so the first identity is immediate.
189 
190 Write
191 \[
192  Q=\begin{pmatrix}a&c\\c&b\end{pmatrix},
193  \qquad
194  d=ab-c^2>0.
195 \]
196 The squared induced norms are
197 \[
198  u:=\|A_1\|_Q^2
199  =\frac{b(a+b+2c)}{d},
200  \qquad
201  v:=\|A_2\|_Q^2
202  =\frac{a(a+b-2c)}{d}.
203 \]
204 A direct calculation gives
205 \[
206  ab\bigl((a+b)^2-4c^2\bigr)-4(ab-c^2)^2
207  =ab(a-b)^2+4c^2(ab-c^2)\geq0.
208 \]
209 Thus \(uv\geq4\), and therefore
210 \[
211  \max\{\sqrt u,\sqrt v\}\geq(uv)^{1/4}\geq\sqrt2.
212 \]
213 Equality forces \(a=b\) and \(c=0\), and then \(u=v=2\).
214 \end{proof}
215 
216 This example is a sharp instance of the established gap between general extremal norms and common quadratic Lyapunov functions \cite{AndoShih1998,John1948}.  Its role here is not a novelty claim.  It proves that obstruction status must record the admissible geometry class.
217 
218 \subsection{Escape without positive gap}
219 \label{subsec:block19-unipotent-escape}
220 
221 \begin{proposition}[Unipotent escape]
222 \label{prop:block19-unipotent-escape}
223 Let
224 \[
225  H=\begin{pmatrix}1&1\\0&1\end{pmatrix}.
226 \]
227 There is no positive-definite metric \(Q\) with \(H^*QH=Q\).  Nevertheless, for
228 \[
229  Q_s=\operatorname{diag}(s^2,1),
230 \]
231 we have
232 \[
233  D_{Q_s}(H)\longrightarrow1
234  \qquad(s\downarrow0),
235 \]
236 while
237 \[
238  \kappa_2(Q_s)=s^{-2}\longrightarrow\infty.
239 \]
240 Thus the metric-completion defect has zero infimum but is not attained.
241 \end{proposition}
242 
243 \begin{proof}
244 Writing \(Q=\begin{psmallmatrix}a&c\\c&b\end{psmallmatrix}\), the equation \(H^*QH=Q\) forces \(a=0\), contradicting positive definiteness.  For \(Q_s\), the metric-normalized matrix is
245 \[
246  \begin{pmatrix}s&0\\0&1\end{pmatrix}
247  H
248  \begin{pmatrix}s^{-1}&0\\0&1\end{pmatrix}
249  =
250  \begin{pmatrix}1&s\\0&1\end{pmatrix},
251 \]
252 whose singular-value condition number tends to one.  The conditioning formula is immediate.
253 \end{proof}
254 
255 \subsection{Forgetful descent and enrichment lifting}
256 \label{subsec:block19-doctrine-lifting}
257 
258 Suppose richer categories \(\widetilde{\mathcal C}\) and \(\widetilde{\mathcal D}\) carry forgetful functors
259 \[
260  U:\widetilde{\mathcal C}\to\mathcal C,
261  \qquad
262  V:\widetilde{\mathcal D}\to\mathcal D.
263 \]
264 
265 \begin{proposition}[Forgetful descent and enhancement lifting]
266 \label{prop:block19-forgetful-lifting}
267 A rich bridge
268 \[
269  \widetilde B:\widetilde{\mathcal C}\to\widetilde{\mathcal D}
270 \]
271 automatically determines an underlying bridge
272 \[
273  V\widetilde B:\widetilde{\mathcal C}\to\mathcal D.
274 \]
275 Given only a bare bridge \(B:\mathcal C\to\mathcal D\), constructing \(\widetilde B\) together with a comparison
276 \[
277  V\widetilde B\cong BU
278 \]
279 is a lifting problem.  Its existence, uniqueness, and residual automorphisms are additional passport fields and may carry typed obstruction objects.
280 \end{proposition}
281 
282 \begin{proof}
283 The first statement is composition.  The second is not an existence theorem but a precise reformulation of the enhancement problem: a lift is extra structure over the specified commutative square.  Any failure must therefore be recorded relative to the chosen forgetful functors and comparison data.
284 \end{proof}
285 
286 The pattern contains several earlier obstruction mechanisms:
287 \begin{center}
288 \small
289 \begin{tabularx}{.96\textwidth}{@{}L{.26\textwidth}L{.28\textwidth}Y@{}}
290 \toprule
291 Underlying structure & Requested enrichment & Typed gate or obstruction\\
292 \midrule
293 Projective semilinear cocycle & Linear semilinear descent & Relative Brauer class\\
294 Invertible transport graph & Compatible metric packet & Unitarizability of normalized holonomy\\
295 Faithful additive realization & Exact tensor fiber functor & Strong monoidal and exactness obligations\\
296 Max-plus value & Finite-temperature sum semantics & Residual partition mass\\
297 Bare quotient coordinates & Metric-decorated quotient & Compatible quotient metric and conditioning boundary\\
298 \bottomrule
299 \end{tabularx}
300 \end{center}
301 No single obstruction object is asserted for all rows.  The common statement concerns the shape of the lifting problem.
302 
303 \subsection{Two variances and the indexed architecture}
304 \label{subsec:block19-indexed-architecture}
305 
306 Task changes and admissibility changes have opposite-looking effects.  If a task is strengthened, its behavioral kernel shrinks and its quotient retains more information:
307 \[
308  N_{\mathrm{strong}}\subseteq N_{\mathrm{weak}},
309  \qquad
310  Q_{\mathrm{strong}}\twoheadrightarrow Q_{\mathrm{weak}}.
311 \]
312 If an admissible realization class is enlarged, the optimized defect can only decrease:
313 \[
314  \Gamma\subseteq\Gamma'
315  \quad\Longrightarrow\quad
316  \delta_{\Gamma'}\leq\delta_\Gamma.
317 \]
318 These are different orders and should not be merged into one scalar notion of a stronger contract.
319 
320 \begin{definition}[Doctrine-indexed task fiber]
321 \label{def:block19-task-fiber}
322 For a doctrine \(D\) and represented object \(X\), let
323 \[
324  \mathsf{Task}_D(X)
325 \]
326 denote the ordered family of continuation-stable semantic quotients admitted by the declared query universe.  Inside one fiber, kernel or congruence calculus governs task change.  A change of doctrine
327 \[
328  D\longrightarrow D'
329 \]
330 requires a bridge passport specifying carrier transport, preserved constructions, comparison maps, and any lifting obstruction.
331 \end{definition}
332 
333 \begin{theorem}[Relative stratified semantic synthesis]
334 \label{thm:block19-relative-semantic-synthesis}
335 Within the finite certified setting of this manuscript, the following statements hold.
336 \begin{enumerate}
337 \item Every declared semantic slice has a task-relative behavioral quotient whenever the required effective quotient exists.
338 \item Inside a module-valued slice, joint tasks, common coarsenings, bridge loss, essential-image failure, and task weakening are governed by the exact sequences of \cref{sec:block17-semantic-quotients,sec:block18-carrier-observer}.
339 \item Carrier changes are legal exactly on the factorization loci certified by \cref{thm:block18-carrier-factorization}; otherwise the joint semantic image is the canonical exact completion preserving both views.
340 \item Observer changes may introduce additional relation objects such as \(\Rel_K(E)\); they are not inferred from carrier compatibility alone.
341 \item Enlarging an admissible realization class weakens optimized defect but need not produce an attained completion.
342 \item Crossing to richer structure is a typed lifting problem and does not follow from the underlying bridge without an enhancement certificate.
343 \end{enumerate}
344 \end{theorem}
345 
346 \begin{proof}
347 The clauses are respectively \cref{thm:block17-universal-behavioral-coimage,thm:block17-semantic-diamond,thm:block18-bridge-loss-exact,thm:block18-carrier-factorization,thm:block18-observer-base-change,thm:block19-admissibility-monotonicity,prop:block19-forgetful-lifting}.  No clause identifies distinct carriers, observer categories, or admissibility classes without the corresponding comparison data.
348 \end{proof}
349 
350 \begin{figure}[htbp]
351 \centering
352 \includegraphics[width=.88\textwidth]{block19_contract_axes.pdf}
353 \caption{Two independent contract directions.  Horizontal task strengthening retains more semantic information.  Vertical admissibility enlargement allows more realizations and can weaken obstruction.  A commuting square is a theorem-level comparison, not an automatic identification.}
354 \label{fig:block19-contract-axes}
355 \end{figure}
356 
357 \begin{boundary}[Scope of the second-order calculus]
358 \label{bnd:block19-second-order-scope}
359 The indexed architecture does not establish a complete double category of all mathematical doctrines, nor does it prove that every task quotient or enrichment is effectively recognizable.  The fiberwise quotient and exact-sequence results are proved in the stated algebraic settings.  Cross-doctrine coherence remains conditional on explicit carrier, topology, metric, tensor, convergence, or higher-categorical passports.
360 \end{boundary}
361 
362 \begin{resultbox}
363 \textbf{Admissibility and lifting result.}
364 Obstructions are relative to both the requested enrichment and the admissible realization class.  Enlarging that class can remove a positive gap, while zero defect may still fail to be attained.  Rich structure descends under forgetting, but reconstructing it from a bare representation is a lifting problem with its own typed gate.  The resulting architecture is a family of task-quotient fibers connected only by certified doctrine bridges.
365 \end{resultbox}

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