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1 \pageheading{COUPLED FAMILIES}{Coupled growth and the cost of resolution}
2 The coupled family consists of two entire functions whose derivatives depend on
3 contracted values of both components. Its growth exponent is the unique
4 positive number $\rho$ for which
5 \[
6 \operatorname{spr}\begin{pmatrix}q^\rho&r^\rho\\a^\rho&b^\rho\end{pmatrix}=1,
7 \qquad 0<q,r,a,b<1.
8 \]
9 Both functions have positive lower type and finite upper type at this order.
10 The positive eigenvector fixes the proportions of both type vectors. The
11 existence of an ordinary type limit remains open.
12 
13 \claim{Local coefficients reveal the coupling}
14 Two explicit parameter surfaces describe exactly when one normalized component
15 is an amplitude and argument rescaling of the other. Outside these surfaces,
16 the coefficients through degree seven recover all four ordered scales. This
17 provides an exact local inverse for the labeled pair.
18 
19 \claim{Three ways finite observations can miss structure}
20 The resolution atlas compares a weak tap, a large excluded prime, and a negative
21 sieve atom. Each example links a structural change to a concrete observation
22 scale and a proved certificate.
23 \begin{center}
24 \renewcommand{\arraystretch}{1.35}
25 \begin{tabular}{@{}P{.22\linewidth}P{.31\linewidth}P{.39\linewidth}@{}}
26 \toprule
27 Example & Observation & What the certificate gives\\\midrule
28 Weak feedback tap & Values on a complex disk & An explicit compact error and a sufficient radius for separation\\
29 Excluded prime & Values and derivatives & Exact disk bounds and a negative-depth visibility bracket\\
30 Negative sieve atom & All polynomial tests up to a degree & Positive quadratic forms through degree four and a negative degree-seven witness\\
31 \bottomrule
32 \end{tabular}
33 \end{center}
34 
35 The computed figures follow in this chapter and Chapter~\ref{ch:resolution}.
36 Complete derivations accompany the full monograph, and the source archive
37 contains the alternative growth proof, exact laboratory, and review records.
38 The six-source literature comparison identifies classical antecedents.
39 Historical priority of the specific coupled results remains unresolved.
40 
41 \claim{How to read the figures}
42 An exact rational bound supplies a certificate. A decimal display illustrates
43 its size. Numerical curves retain their stated truncation and precision, and
44 the universal growth and recovery statements rest on their proofs.

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