Positive lower type and quantitative tap continuity
This chapter sharpens the entire-growth theorem for nonlinear contraction families. The symmetric definition has the antecedent identified in the reading map.
The exact growth exponent has positive lower type
Let
where \(h_j>0\), \(\sum_jh_j=1\), and \(0<q_j,r_j<1\). Zero-weight channels may instead be allowed if they are removed from the active bank before defining its growth exponent. Put
The inherited coefficient recurrence gives nonnegative coefficients, entire extension, and \(M(R)=\max_{|z|=R}|F(z)|=F(R)\ge R\) for \(R>0\).
Theorem A.1.
There are explicit constants \(0<c_-\le c_+<\infty\) such that
In particular the bank has positive lower type and finite upper type at its exact order, including when \(0<\rho<1\). Furthermore
This last radial growth-exponent limit does not assert a type limit or uniform angular growth.
Explicit lower constants and proof
Select any active maximizing channel and abbreviate it to \((h,q,r)\), so \(q^\rho+r^\rho=1\). Define
We prove the stronger bound
Write \(H(R)=\log M(R)\). On the seed interval \([2,B]\), \(H(R)\ge\log R\ge\log2\ge c_-(R+C)^\rho\). For \(R\ge B\), integrate the positive feedback over the fixed-width interval \([R-L,R]\):
The two child radii are at least two because \(R\ge B\). Moreover
because respectively \((1-q)C\ge qL\) and \((1-r)C\ge rL\). If the comparison bound is known at both child radii, it follows that
For a rigorous propagation, put \(Q_* =\max(q,r)<1\) and \(d=(1-Q_*)B+Q_*L>0\). Every child of every \(R\ge B\) is at most \(R-d\). Induction on the intervals \([B+kd,B+(k+1)d]\), beginning with the already controlled seed, proves the comparison for all larger radii. Continuity deals with common endpoints. All power comparisons use only monotonicity of \(x^\rho\) for positive \(x\); no convexity assumption is made, so the argument also covers \(\rho<1\).
Explicit finite upper constants
For all channels set
Choose an integer \(N\ge0\) with \(Q^{-N}\ge3/m\), and put
These constants are intentionally conservative, and require no numerical evaluation of an unknown solution. The tangent majorant gives \(M(1)<2\). At \(R_k=Q^{-k}\), the integral equation gives
Define \(U_0=2\), \(U_k=Q^{-k}(1+U_{k-1}^2)\). Then \(M(R_k)\le U_k\), \(U_k\ge1\), and \(\log U_k\le k\log(1/Q)+\log2+2\log U_{k-1}\). Summing this elementary recurrence gives \(\log U_N\le V_N\). Since \(e<3\), the logarithmic-radius seed \(1\le t\le1+\Delta\) has \(e^t\le e/m<3/m\le R_N\), and therefore \(\log M(e^t)\le V_N\).
Let \(a_j=\log(1/q_j)\), \(b_j=\log(1/r_j)\), and \(G(t)=\log M(e^t)+t+B_+\). For \(t\ge1+\Delta\), all child moduli are at least one, so the positive integral and weight normalization give
The definition of \(B_+\) implies
On the seed, \(G(t)\le V_N+1+\Delta+B_+\le c_+e^{\rho t}\). Each channel satisfies \(q_j^\rho+r_j^\rho\le1\). Induction in intervals of length \(\min_j\min(a_j,b_j)>0\) therefore propagates the seed comparison, giving the explicit upper bound
Dividing the two bounds by \(R^\rho\), then taking logarithms of their positive logarithmic-growth bounds, proves all assertions of Theorem A.1. The theorem leaves open whether \(\log M(R)/R^\rho\) has a limit. Lattice resonances or logarithmic oscillations are possible research directions, not results here.
Joint holomorphic dependence on parameters
Fix the number \(J\) of channels. Allow complex weights \(h_j\), without normalization, and complex contractions satisfying \(|q_j|,|r_j|<1\). The same formal recurrence defines coefficients
Each is homogeneous of degree \(n\) in the weights. If a parameter set has \(|q_j|,|r_j|\le Q<1\) and \(\sum_j|h_j|\le W\), with \(W>0\), coefficient induction gives
where \(f_Q(z)=\sum d_n(Q)z^{2n+1}\) is the established diagonal function. Consequently, on every disk \(|z|\le R\),
Every compact subset of the complex parameter domain has such \(Q,W\). The series therefore converges normally on compact sets of the joint domain, and its sum is jointly holomorphic in the weights, contractions, and \(z\). This proves parameter continuity at zero weights, although the growth order can jump there. Complex or signed parameters do not inherit the positive-bank growth theorem merely from holomorphic dependence.
A quantitative compact bound for changing tap weights
Keep fixed real contractions \(0<q_j,r_j\le Q<1\), and let \(h,g\) be nonnegative normalized weight vectors. Zero weights are permitted. Let \(d=\|h-g\|_1\). Then for every \(R\ge0\),
Proof.
Regard \(c_n\) as a polynomial in the real weight variables with the contractions fixed. Its coefficients are nonnegative. Induction in the recurrence gives the coefficientwise polynomial bound
Here \(\preceq\) compares every monomial coefficient. Differentiation preserves this comparison, so on the normalized nonnegative simplex
Put \(v=h-g\). Its positive and negative parts both have sum \(d/2\), since \(\sum_jv_j=0\). At every point of the segment between \(g\) and \(h\), all partial derivatives lie in the interval \([0,n d_n(Q)]\). Hence
Multiply by \(R^{2n+1}\) and sum. Entire convergence justifies the sum, and
proving the claim. The same argument applies to the closed simplex and needs no positive minimum weight.
A vanishing tap and discontinuous order
Take the two diagonal channels \((q,r)=(1/4,1/4)\) and \((3/4,3/4)\), with weights \(1-\varepsilon\) and \(\varepsilon\), respectively. For \(0\le\varepsilon\le1\), write the resulting function as \(F_\varepsilon\). The quantitative bound gives
Thus convergence to \(f_{1/4}\) is uniform on every fixed disk, at an explicit linear rate in \(\varepsilon\). Yet
For the faster active channel the lower constants become \(L=1/\varepsilon\), \(C=3/\varepsilon\), \(B=1/\varepsilon+8/3\). Its certified lower-type constant is
It tends to zero as the faster tap vanishes. These are proved growth and compact-convergence statements. They do not determine an actual crossover radius or leading asymptotic constant.