Special Functions

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1 \pageheading{EVIDENCE / COUPLED FAMILY}{What the coupled extension establishes}
2 \renewcommand{\arraystretch}{1.28}
3 \begin{tabular}{@{}P{.28\linewidth}P{.65\linewidth}@{}}
4 \toprule
5 Result & Scope\\\midrule
6 Coupled entire construction & Unique normalized pair for four positive strict contractions, with positive odd coefficients\\
7 Common growth law & Exact order from the characteristic matrix, positive lower type, finite upper type, and exact Perron proportions for both type vectors\\
8 Component reduction & Rescaling occurs exactly on $q=b$ or $qb=ar$\\
9 Generic exact inverse & Coefficients through degree seven identify all four ordered scales when $D\ne0$\\
10 Operator obstruction & No invertible constant linear change decouples the defining operator on all analytic inputs\\
11 Resolution certificates & Explicit compact, radius, prime-depth and polynomial-degree bounds for the three worked examples\\
12 \bottomrule
13 \end{tabular}
14 
15 \claim{Verification and reproducibility}
16 The archived coupled-system run records \textbf{5,272 exact assertions and 26 numerical
17 assertions}. Thirteen input and resource guards are recorded separately.
18 The exact inverse is checked on 256 rational parameter tuples. The coupled
19 API is compared with an independent dense Picard construction and classical
20 anchors. The resolution checks verify the exact rational certificates.
21 These counts are separate from the earlier four-family evidence of 19,871 exact
22 and 5,576 numerical assertions.
23 
24 The preserved coupled-family review reconstructs the proof implications and
25 the source files. It corrected an initial-segment quantifier in the
26 detector theorem. The worked certificate already used that initial segment.
27 The source package records the correction, the repair of a figure label, and the successful checks.
28 
29 \claim{Questions that remain open}
30 The coupled theorem does not determine an ordinary type limit, uniform angular
31 growth, or the complex zero geometry. Stable finite-precision recovery near the
32 reduction surfaces remains open. The resolution bounds are constructive and
33 can be conservative. The sieve example leaves degrees five and six undecided.
34 The literature comparison does not settle historical priority.
35 
36 The exact API and its checks run with the Python standard library. NumPy, Matplotlib, and
37 mpmath generate the figures. The full manuscript supplies the proof
38 appendices and the source archive contains the code and inputs needed for the computations.

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