Stratified Universality

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1 \begin{table}[htbp]
2 \centering
3 \caption{Certified connected components of the Global synthesis atlas atlas.}
4 \label{tab:block13-components}
5 \tiny
6 \begin{tabularx}{\textwidth}{@{}L{0.105\textwidth}L{0.255\textwidth}L{0.285\textwidth}Y@{}}
7 \toprule
8 Component & Semantic domain and normalized charts & Complete coordinate and reversible overlap & Directional satellites or exact boundary\\
9 \midrule
10 DRLM & fixed-prime regular sequences; canonical section module, strict shortlex record, and minimal DFAO; rational-mask/Cartier and periodic linear subcharts & strict shortlex pointed digit-linear record, refined on the rational-mask and unit-periodic full replete subcategories & integral reductions and Newton--Mahler interpolation are directional; many automatic sequences have no rational mask\\
11 LOC & continuous $\mathbb Z_p$-valued functions; Newton--Mahler coefficients, complete clopen branch packets, and affine branch coordinates & canonical Mahler coefficient sequence; complete branch analysis and synthesis at each declared depth & functional-Mahler equations require a separately declared coefficient lift and interpolation constructor\\
12 Fuss & marked $U=z+\alpha U^q$ tree system, selected algebraic branch, and compressed P-recursive coefficient chart & parameter pair $(q,\alpha)$ with its support and singularity-orbit package & forgetting the branch mark or generator grammar is information-losing\\
13 EM & fixed-basis Euler exponents, divisor ghosts, integral unit series, and prime-uniform multiplicative local factors & Euler exponent sequence on the fixed-basis, prime-uniform essential image & arbitrary product grammars and nonuniform prime-local fields remain outside the reversible overlap\\
14 Gal & subgroup, fixed field, and faithful Reynolds projector for one fixed finite Galois extension & actual subgroup $H\leq G$ with functorial core, normal-closure, and double-coset data & projective Brauer obstruction, modular or wild failure, and the tensor-fiber gate remain separately typed\\
15 \bottomrule
16 \end{tabularx}
17 \end{table}

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