Stratified Universality

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1 \begin{landscape}
2 \begin{table}[p]
3 \centering
4 \caption{Authoritative broad-stratum bridge matrix for the certified global synthesis atlas. Every decorated equivalence is restricted to the named normalized overlap; the undecorated broad categories are not thereby identified.}
5 \label{tab:block13-bridge-matrix}
6 \scriptsize
7 \setlength{\tabcolsep}{3.7pt}
8 \begin{tabular}{@{}lcccccccc@{}}
9 \toprule
10 Source $\backslash$ target & LIN & HOL & DIG & ALG & TREE & PROD & LOC & GAL\\
11 \midrule
12 LIN  & $\mathsf E$ & $\mathsf{FF}$ & $\mathsf E_{\rm per}$ & -- & -- & -- & $\mathsf E_{\rm per}$ & --\\
13 HOL  & $\mathsf P$ & $\mathsf E$ & -- & $\mathsf E_{\rm Fuss}/\mathsf P$ & $\mathsf E_{\rm Fuss}/\mathsf P$ & -- & -- & --\\
14 DIG  & $\mathsf E_{\rm per}$ & $\mathsf P$ & $\mathsf E$ & $\mathsf P$ & -- & -- & $\mathsf E_{\rm rm}$ & --\\
15 ALG  & -- & $\mathsf E_{\rm Fuss}/\mathsf D$ & $\mathsf P$ & $\mathsf E$ & $\mathsf E_{\rm Fuss}/\mathsf P$ & -- & -- & --\\
16 TREE & -- & $\mathsf E_{\rm Fuss}/\mathsf D$ & -- & $\mathsf E_{\rm Fuss}/\mathsf D$ & $\mathsf E$ & -- & -- & --\\
17 PROD & $\mathsf D$ & -- & -- & -- & -- & $\mathsf E$ & $\mathsf P$ & --\\
18 LOC  & $\mathsf E_{\rm per}$ & -- & $\mathsf E_{\rm rm}$ & -- & -- & $\mathsf P$ & $\mathsf E$ & --\\
19 GAL  & -- & -- & -- & -- & -- & -- & -- & $\mathsf E$\\
20 \bottomrule
21 \end{tabular}
22 \vspace{5pt}
23 \parbox{0.96\linewidth}{\scriptsize
24 \textbf{Legend.} $\mathsf E$: equivalence; $\mathsf{FF}$: fully faithful inclusion; $\mathsf D$: information-losing directional functor; $\mathsf P$: partial, constructor-dependent, or certificate-dependent bridge; --: no certified bridge. The subscripts $\mathrm{per}$, $\mathrm{rm}$, and $\mathrm{Fuss}$ denote the unit-periodic, rational functional-Mahler, and marked weighted Fuss overlaps. A cell of the form $\mathsf E_{\mathrm{Fuss}}/\mathsf D$ means equivalence on the marked Fuss slice and directional loss on the ambient bridge. Functional-Mahler coefficient equations and Newton--Mahler interpolation remain distinct despite sharing the LOC row.}
25 \end{table}
26 \end{landscape}

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