The exact order has positive lower type
The single-channel result extends to finite positive feedback banks. Here a bank has weights \(h_j>0\) with \(\sum_jh_j=1\), contractions \(0<q_j,r_j<1\), and normalized solution
Write \(M(R)=M_F(R)\) and let \(q_j^{\rho_j}+r_j^{\rho_j}=1\). Then, with \(\rho=\max_j\rho_j\),
Thus logarithmic growth is trapped between two positive multiples of \(R^\rho\). In particular,
The theorem does not assert that \(\log M(R)/R^\rho\) has a limit.
A lower bound at the exact exponent
Choose a maximizing channel \((h,q,r)\) and integrate over a fixed-width terminal interval. Let
The result is the explicit all-radius bound
Unlike a fixed fractional window, the terminal window loses only an additive radius. The shift \(C\) absorbs that loss without changing the exponent.
A translated power comparison
Write \(H(R)=\log M(R)\). Positivity gives
The choice of \(C\) ensures
On the seed interval \([2,B]\), \(H(R)\ge\log R\ge\log2\ge c_-(R+C)^\rho\). If this comparison holds at the two child radii, then
Let \(Q_* =\max(q,r)\) and \(d=(1-Q_*)B+Q_*L>0\). Each child radius is at most \(R-d\) and at least two. Induction on successive intervals of length \(d\) propagates the seed comparison to every \(R\ge B\). The argument uses monotonicity of positive powers and works also when \(\rho<1\).
Explicit upper constants are available too
Set \(Q=\max_j\max(q_j,r_j)\), \(m=\min_j\min(q_j,r_j)\), \(\Delta=\log(1/m)\), and \(B_+=\max_j\log(1/(q_jr_j))+\log2\). Choose an integer \(K\) with \(Q^{-K}\ge3/m\) and put
Then \(\log M(R)\le c_+R^\rho-\log R-B_+\) for \(R\ge e\). The tangent majorant \(M(1)<2\), followed by the integral recurrence on radii \(Q^{-k}\), bounds the initial seed by \(V_K\). The logarithmic-radius comparison then supplies the displayed upper constant. The full induction is in Appendix A.
These constants are constructive and conservative. Their purpose is to establish two-sided growth, not to estimate the optimal type.