Special Functions

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1 \FrontChapter{The proof architecture}
2 The illustrated chapters develop the definitions, decisive proof steps, and worked examples. The appendices provide the complete derivations, explicit constants, computational contracts, source comparisons, and unresolved questions. Editable source files accompany the monograph.
3 
4 \begingroup\small\renewcommand{\arraystretch}{1.15}
5 \begin{tabular}{@{}P{0.13\linewidth}P{0.35\linewidth}P{0.45\linewidth}@{}}
6 \toprule
7 Appendix & Proof chapter & Main dependencies\\\midrule
8 A & Dilation type and continuity & Positive coefficient recurrence, radius bootstrap, translated-radius comparison\\
9 B & Arithmetic inverse and character resolution & Summable positive atoms, squarefree counting, Euler factors, sector partial integration\\
10 C & Multiple-prime geometry and detectors & Exact moments, isolated-node polynomials, definite matrix pencils\\
11 D & Orbit continuation and shifts & Genus-zero product, branch separation, exponential sums, two-end Mellin subtraction\\
12 E & Integer sequences and polyphase laws & Factorial recurrence, ordered trees, root filtering, carry arithmetic\\
13 F & Coupled entire growth and type & Positive coupled recurrence, characteristic matrix, translated powers, left eigenvector\\
14 G & Component reduction and exact inverse & Cubic and fifth coefficients, factorization, seventh-coefficient separation\\
15 H & Constant linear operator structure & Projection fibers, polynomial interpolation, monomial tests\\
16 I & Quantitative resolution atlas & Tap type bounds, removed-prime mass, exact interpolation and tail certificates\\
17 J & Coupled exact laboratory & Derivative recurrence, denominator clearing, positive disk majorants\\
18 \bottomrule\end{tabular}\endgroup
19 
20 \clearpage\appendix\RCAppendixtrue
21 \input{sections/dilation_full.tex}
22 \clearpage\input{sections/arithmetic_full.tex}
23 \clearpage\input{sections/sieve_full.tex}
24 \clearpage\input{sections/spectral_full.tex}
25 \clearpage\input{sections/sequence_full.tex}

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