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1 \hypertarget{positive-lower-type-and-quantitative-tap-continuity}{%
2 \chapter[Growth and tap continuity]{Positive lower type and quantitative tap
3 continuity}\label{app:growth}\label{positive-lower-type-and-quantitative-tap-continuity}}
4 
5 This chapter sharpens the entire-growth theorem for nonlinear contraction families. The symmetric definition has the antecedent identified in the reading map.
6 
7 \hypertarget{the-exact-growth-exponent-has-positive-lower-type}{%
8 \section{The exact growth exponent has positive lower
9 type}\label{the-exact-growth-exponent-has-positive-lower-type}}
10 
11 Let
12 
13 \[
14 F'(z)=1+\sum_{j=1}^{J}h_jF(q_jz)F(r_jz),\qquad F(0)=0,
15 \]
16 
17 where \(h_j>0\), \(\sum_jh_j=1\), and \(0<q_j,r_j<1\). Zero-weight
18 channels may instead be allowed if they are removed from the active bank
19 before defining its growth exponent. Put
20 
21 \[
22 \rho=\max_{j:h_j>0}\rho_j,\qquad q_j^{\rho_j}+r_j^{\rho_j}=1.
23 \]
24 
25 The inherited coefficient recurrence gives nonnegative coefficients,
26 entire extension, and \(M(R)=\max_{|z|=R}|F(z)|=F(R)\ge R\) for \(R>0\).
27 
28 \textbf{Theorem A.1.} There are explicit constants \(0<c_-\le c_+<\infty\)
29 such that
30 
31 \[
32 c_-\le\liminf_{R\to\infty}\frac{\log M(R)}{R^\rho}
33 \le\limsup_{R\to\infty}\frac{\log M(R)}{R^\rho}\le c_+.
34 \]
35 
36 In particular the bank has positive lower type and finite upper type at
37 its exact order, including when \(0<\rho<1\). Furthermore
38 
39 \[
40 \lim_{R\to\infty}\frac{\log\log M(R)}{\log R}=\rho.
41 \]
42 
43 This last radial growth-exponent limit does not assert a type limit or
44 uniform angular growth.
45 
46 \hypertarget{explicit-lower-constants-and-proof}{%
47 \subsection{Explicit lower constants and
48 proof}\label{explicit-lower-constants-and-proof}}
49 
50 Select any active maximizing channel and abbreviate it to \((h,q,r)\),
51 so \(q^\rho+r^\rho=1\). Define
52 
53 \[
54 L=\frac1h,\qquad
55 C=\max\left(\frac{qL}{1-q},\frac{rL}{1-r}\right),\qquad
56 B=L+\frac2{\min(q,r)},\qquad
57 c_-=\frac{\log2}{(B+C)^\rho}.
58 \]
59 
60 We prove the stronger bound
61 
62 \[
63 \boxed{\log M(R)\ge c_-(R+C)^\rho\quad\text{for every }R\ge2.}
64 \]
65 
66 Write \(H(R)=\log M(R)\). On the seed interval \([2,B]\),
67 \(H(R)\ge\log R\ge\log2\ge c_-(R+C)^\rho\). For \(R\ge B\), integrate
68 the positive feedback over the fixed-width interval \([R-L,R]\):
69 
70 \[
71 \begin{aligned}
72 M(R)&\ge h\int_{R-L}^{R}M(qt)M(rt)\,dt\\
73 &\ge hL M(q(R-L))M(r(R-L))\\
74 &=M(q(R-L))M(r(R-L)).
75 \end{aligned}
76 \]
77 
78 The two child radii are at least two because \(R\ge B\). Moreover
79 
80 \[
81 q(R-L)+C\ge q(R+C),\qquad
82 r(R-L)+C\ge r(R+C),
83 \]
84 
85 because respectively \((1-q)C\ge qL\) and \((1-r)C\ge rL\). If the
86 comparison bound is known at both child radii, it follows that
87 
88 \[
89 \begin{aligned}
90 H(R)&\ge H(q(R-L))+H(r(R-L))\\
91 &\ge c_-\{[q(R-L)+C]^\rho+[r(R-L)+C]^\rho\}\\
92 &\ge c_-(q^\rho+r^\rho)(R+C)^\rho
93 =c_-(R+C)^\rho.
94 \end{aligned}
95 \]
96 
97 For a rigorous propagation, put \(Q_* =\max(q,r)<1\) and
98 \(d=(1-Q_*)B+Q_*L>0\). Every child of every \(R\ge B\) is at most
99 \(R-d\). Induction on the intervals \([B+kd,B+(k+1)d]\), beginning with
100 the already controlled seed, proves the comparison for all larger radii.
101 Continuity deals with common endpoints. All power comparisons use only
102 monotonicity of \(x^\rho\) for positive \(x\); no convexity assumption
103 is made, so the argument also covers \(\rho<1\).
104 
105 \hypertarget{explicit-finite-upper-constants}{%
106 \subsection{Explicit finite upper
107 constants}\label{explicit-finite-upper-constants}}
108 
109 For all channels set
110 
111 \[
112 Q=\max_j\max(q_j,r_j),\quad
113 m=\min_j\min(q_j,r_j),\quad
114 \Delta=\log(1/m),\quad
115 B_+=\max_j\log\frac1{q_jr_j}+\log2.
116 \]
117 
118 Choose an integer \(N\ge0\) with \(Q^{-N}\ge3/m\), and put
119 
120 \[
121 V_N=(2^{N+1}-1)\log2+
122        (2^{N+1}-N-2)\log(1/Q),
123 \]
124 
125 \[
126 c_+=e^{-\rho}(V_N+1+\Delta+B_+).
127 \]
128 
129 These constants are intentionally conservative, and require no numerical
130 evaluation of an unknown solution. The tangent majorant gives
131 \(M(1)<2\). At \(R_k=Q^{-k}\), the integral equation gives
132 
133 \[
134 M(R_k)\le R_k[1+M(R_{k-1})^2].
135 \]
136 
137 Define \(U_0=2\), \(U_k=Q^{-k}(1+U_{k-1}^2)\). Then \(M(R_k)\le U_k\),
138 \(U_k\ge1\), and \(\log U_k\le k\log(1/Q)+\log2+2\log U_{k-1}\). Summing
139 this elementary recurrence gives \(\log U_N\le V_N\). Since \(e<3\), the
140 logarithmic-radius seed \(1\le t\le1+\Delta\) has
141 \(e^t\le e/m<3/m\le R_N\), and therefore \(\log M(e^t)\le V_N\).
142 
143 Let \(a_j=\log(1/q_j)\), \(b_j=\log(1/r_j)\), and
144 \(G(t)=\log M(e^t)+t+B_+\). For \(t\ge1+\Delta\), all child moduli are
145 at least one, so the positive integral and weight normalization give
146 
147 \[
148 \log M(e^t)\le t+\log2+
149 \max_j\{\log M(e^{t-a_j})+\log M(e^{t-b_j})\}.
150 \]
151 
152 The definition of \(B_+\) implies
153 
154 \[
155 G(t)\le\max_j\{G(t-a_j)+G(t-b_j)\}.
156 \]
157 
158 On the seed, \(G(t)\le V_N+1+\Delta+B_+\le c_+e^{\rho t}\). Each channel
159 satisfies \(q_j^\rho+r_j^\rho\le1\). Induction in intervals of length
160 \(\min_j\min(a_j,b_j)>0\) therefore propagates the seed comparison,
161 giving the explicit upper bound
162 
163 \[
164 \boxed{\log M(R)\le c_+R^\rho-\log R-B_+\quad(R\ge e).}
165 \]
166 
167 Dividing the two bounds by \(R^\rho\), then taking logarithms of their
168 positive logarithmic-growth bounds, proves all assertions of Theorem A.1.
169 The theorem leaves open whether \(\log M(R)/R^\rho\) has a limit.
170 Lattice resonances or logarithmic oscillations are possible research
171 directions, not results here.
172 
173 \hypertarget{joint-holomorphic-dependence-on-parameters}{%
174 \section{Joint holomorphic dependence on
175 parameters}\label{joint-holomorphic-dependence-on-parameters}}
176 
177 Fix the number \(J\) of channels. Allow complex weights \(h_j\), without
178 normalization, and complex contractions satisfying \(|q_j|,|r_j|<1\).
179 The same formal recurrence defines coefficients
180 
181 \[
182 c_n(h,q,r)\in\mathbb Q[h_1,\ldots,h_J,q_1,r_1,\ldots,q_J,r_J].
183 \]
184 
185 Each is homogeneous of degree \(n\) in the weights. If a parameter set
186 has \(|q_j|,|r_j|\le Q<1\) and \(\sum_j|h_j|\le W\), with \(W>0\),
187 coefficient induction gives
188 
189 \[
190 |c_n(h,q,r)|\le W^n d_n(Q),
191 \]
192 
193 where \(f_Q(z)=\sum d_n(Q)z^{2n+1}\) is the established diagonal
194 function. Consequently, on every disk \(|z|\le R\),
195 
196 \[
197 \sum_{n\ge0}|c_n(h,q,r)|R^{2n+1}
198 \le W^{-1/2}f_Q(\sqrt W R)<\infty.
199 \]
200 
201 Every compact subset of the complex parameter domain has such \(Q,W\).
202 The series therefore converges normally on compact sets of the joint
203 domain, and its sum is jointly holomorphic in the weights, contractions,
204 and \(z\). This proves parameter continuity at zero weights, although
205 the growth order can jump there. Complex or signed parameters do not
206 inherit the positive-bank growth theorem merely from holomorphic
207 dependence.
208 
209 \hypertarget{a-quantitative-compact-bound-for-changing-tap-weights}{%
210 \section{A quantitative compact bound for changing tap
211 weights}\label{a-quantitative-compact-bound-for-changing-tap-weights}}
212 
213 Keep fixed real contractions \(0<q_j,r_j\le Q<1\), and let \(h,g\) be
214 nonnegative normalized weight vectors. Zero weights are permitted. Let
215 \(d=\|h-g\|_1\). Then for every \(R\ge0\),
216 
217 \[
218 \boxed{\sup_{|z|\le R}|F_h(z)-F_g(z)|
219 \le\frac d4\{R f_Q'(R)-f_Q(R)\}.}
220 \]
221 
222 \textbf{Proof.} Regard \(c_n\) as a polynomial in the real weight
223 variables with the contractions fixed. Its coefficients are nonnegative.
224 Induction in the recurrence gives the coefficientwise polynomial bound
225 
226 \[
227 0\preceq c_n(h)\preceq d_n(Q)\Bigl(\sum_jh_j\Bigr)^n.
228 \]
229 
230 Here \(\preceq\) compares every monomial coefficient. Differentiation
231 preserves this comparison, so on the normalized nonnegative simplex
232 
233 \[
234 0\le\partial_{h_j}c_n\le n d_n(Q).
235 \]
236 
237 Put \(v=h-g\). Its positive and negative parts both have sum \(d/2\),
238 since \(\sum_jv_j=0\). At every point of the segment between \(g\) and
239 \(h\), all partial derivatives lie in the interval \([0,n d_n(Q)]\).
240 Hence
241 
242 \[
243 |v\cdot\nabla c_n|\le\frac d2 n d_n(Q),\qquad
244 |c_n(h)-c_n(g)|\le\frac d2 n d_n(Q).
245 \]
246 
247 Multiply by \(R^{2n+1}\) and sum. Entire convergence justifies the sum,
248 and
249 \[R f_Q'(R)-f_Q(R)=2\sum_{n\ge0}n d_n(Q)R^{2n+1},\]
250 proving the
251 claim. The same argument applies to the closed simplex and needs no
252 positive minimum weight.
253 
254 \hypertarget{a-vanishing-tap-and-discontinuous-order}{%
255 \subsection{A vanishing tap and discontinuous
256 order}\label{a-vanishing-tap-and-discontinuous-order}}
257 
258 Take the two diagonal channels \((q,r)=(1/4,1/4)\) and \((3/4,3/4)\),
259 with weights \(1-\varepsilon\) and \(\varepsilon\), respectively. For
260 \(0\le\varepsilon\le1\), write the resulting function as
261 \(F_\varepsilon\). The quantitative bound gives
262 
263 \[
264 \sup_{|z|\le R}|F_\varepsilon(z)-f_{1/4}(z)|
265 \le\frac{\varepsilon}{2}\{R f_{3/4}'(R)-f_{3/4}(R)\}.
266 \]
267 
268 Thus convergence to \(f_{1/4}\) is uniform on every fixed disk, at an
269 explicit linear rate in \(\varepsilon\). Yet
270 
271 \[
272 \rho(F_0)=\frac12,\qquad
273 \rho(F_\varepsilon)=\frac{\log2}{\log(4/3)}
274 \quad(0<\varepsilon\le1).
275 \]
276 
277 For the faster active channel the lower constants become
278 \(L=1/\varepsilon\), \(C=3/\varepsilon\), \(B=1/\varepsilon+8/3\). Its
279 certified lower-type constant is
280 
281 \[
282 c_-(\varepsilon)=\frac{\log2}{(4/\varepsilon+8/3)^{\log2/\log(4/3)}}>0.
283 \]
284 
285 It tends to zero as the faster tap vanishes. These are proved growth and
286 compact-convergence statements. They do not determine an actual
287 crossover radius or leading asymptotic constant.
288 

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