Stratified Universality

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1 \section{Main theorem suite and proof architecture}
2 \label{sec:block14-main-suite}
3 
4 Let
5 \[
6  \mathfrak A_{\mathrm{cert}}:=\mathfrak A_{13}
7 \]
8 denote the finite certified atlas constructed in \cref{sec:block13-global-invariant}.  Its reversible core is the disjoint union of five connected semantic components, while its directional satellites retain explicit loss or obstruction passports.
9 
10 \begin{theorem}[Stratified resolved representation, descent, and complete invariant theorem]
11 \label{thm:block14-main}
12 Let \(\mathbb K\) be the resolved theory used in this manuscript and let \(\mathfrak A_{\mathrm{cert}}\) be the declared finite atlas.  Then the following statements hold.
13 \begin{enumerate}
14 \item Every chart is a profile-preserving realization of the common resolved syntax, and every reversible edge is an equivalence on its named overlap category.
15 \item The descent category exists as the finite pseudolimit of the atlas and decomposes as the coproduct of its five connected semantic components.
16 \item Every proof whose constructor support lies in the common verified preservation profile transports to compatible local theorems and hence to the corresponding global resolved object.
17 \item The semantic normalizers induce the comparison 2-cells on all nonempty triple overlaps and force the quadruple-overlap coherence equation.
18 \item The compatible native invariant family descends to a tagged global invariant
19 \[
20  \mathcal J_{\mathrm{cert}}:
21  \Desc(\mathfrak A_{\mathrm{cert}})
22  \longrightarrow
23  \Desc(\mathcal V_{\mathrm{cert}}).
24 \]
25 \item The descended invariant is complete on the declared normalized global category:
26 \[
27  \mathcal J_{\mathrm{cert}}(X)\cong
28  \mathcal J_{\mathrm{cert}}(Y)
29  \quad\Longleftrightarrow\quad
30  X\cong Y.
31 \]
32 \item Every noninvertible construction remains outside the reversible core and carries a typed record of its preserved structure, forgotten information, named residual, or obstruction class.
33 \end{enumerate}
34 \end{theorem}
35 
36 \begin{proof}
37 The common resolved syntax and preservation-profile transport are established in \cref{sec:resolved-foundations}.  The finite atlas and its pseudolimit are supplied by \cref{thm:atlas-assembly}.  The normalizer comparison cells and their higher coherence are \cref{thm:block13-normalizer-coherence}.  Descent of compatible invariant families is \cref{thm:block13-certified-invariant-descent}.  Completeness is \cref{thm:block13-global-completeness}; its hypotheses are verified component by component in \cref{sec:block13-global-invariant}.  The bridge passports and obstruction taxonomy in \cref{sec:strata-bridges-obstructions} keep every directional construction outside the maximal subgroupoid.  Combining these results proves all seven assertions.
38 \end{proof}
39 
40 \begin{corollary}[Relative universality]
41 \label{cor:block14-relative-universality}
42 The certified atlas is universal for profile-preserving models of the declared resolved grammar on its five normalized semantic components.  This universality is relative to the chosen grammar, strata, bridges, preservation profiles, normalization targets, and descent doctrine.
43 \end{corollary}
44 
45 \begin{theorem}[Relative semantic quotient and typed contract-change theorem]
46 \label{thm:block17-main-semantic}
47 Fix a declared semantic slice in a stratum of the resolved theory.  Assume the required effective quotients exist.  Then:
48 \begin{enumerate}
49 \item the behavioral quotient is the coarsest exact representation preserving all declared future observations;
50 \item its kernel or congruence is the maximal continuation-stable invisible core;
51 \item strengthening the task induces a canonical map from the stronger quotient to the weaker quotient;
52 \item in module-valued slices, joint semantics, shared semantics, relative residuals, bridge loss, essential-image failure, and task weakening fit into canonical exact sequences;
53 \item one carrier semantics replaces another exactly on the corresponding kernel-factorization locus;
54 \item observer base change may introduce an independent relation object and therefore does not follow from carrier compatibility;
55 \item enlarging an admissible realization class can only decrease optimized defect, but zero infimum need not be attained;
56 \item lifting a bare bridge to richer metric, tensor, topological, projective, or completed structure requires a separate enhancement certificate.
57 \end{enumerate}
58 \end{theorem}
59 
60 \begin{proof}
61 The universal quotient and maximal-core statements are \cref{thm:block17-universal-behavioral-coimage,thm:block17-behavioral-core}.  Task change and the module exact sequences are \cref{thm:block17-task-change,thm:block17-semantic-diamond,thm:block18-bridge-loss-exact,thm:block18-loss-composition}.  Carrier and observer change are \cref{thm:block18-carrier-factorization,thm:block18-observer-base-change}.  Admissibility monotonicity is \cref{thm:block19-admissibility-monotonicity}, and the enrichment statement is \cref{prop:block19-forgetful-lifting}.  Each assertion retains its printed doctrine and existence hypotheses.
62 \end{proof}
63 
64 \begin{corollary}[No absolute minimal representation]
65 \label{cor:block17-no-absolute-minimum}
66 There is no observer-independent or doctrine-independent minimum asserted by this manuscript.  Minimality is relative to the carrier, continuation grammar, observation category, query family, and admissible enrichment class.
67 \end{corollary}
68 
69 \begin{boundary}[Publication boundary]
70 \label{bnd:block14-publication-boundary}
71 \Cref{thm:block14-main} does not imply a global canonical normal form for arbitrary mathematical objects, equivalence of all broad strata, decidability of arbitrary presentation equivalence, or effectivity of an unrestricted heterogeneous stack.  It also does not identify functional-Mahler equations with Newton-Mahler interpolation, forget branch data in the algebraic-to-holonomic direction, or infer Tannakian symmetry from a merely faithful additive realization.
72 \end{boundary}
73 
74 \input{generated/block14_theorem_suite_table.tex}
75 
76 \begin{figure}[htbp]
77 \centering
78 \includegraphics[width=.97\textwidth]{publication_proof_architecture.pdf}
79 \caption{Proof architecture of the journal-facing theorem.  The vertical spine contains the reversible construction and descent argument.  Directional satellites and obstruction data remain typed external inputs rather than silent equivalences.}
80 \label{fig:block14-publication-architecture}
81 \end{figure}

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