The Branch Tree of the Exponential Map Sign Channels, Depletion, Stable Laws and the Arithmetic of Resonances

The inverse branches $z 7→ log z +2πik$ of the exponential map form an infinite conformal iterated function system whose limit set has Hausdorff dimension $δ∞ =1.20728 . . .$. This monograph develops the theory of that system from first principles and follows it into three areas where it produces new mathematics. First, the arithmetic: nearest-neighbour correlations of the tree are carried by a two-state sign channel whose eigenvalues are closed expressions in $ζ(s)$ and $L(s, χ−4)$; we explain how this isolates Apéry’s constant, derive a closed formula for the curvature of the dimension under rotations of the tree, and map precisely the arithmetic cost of linear forms built on the sign channel, including a trichotomy theorem that matches the symmetries of well-poised hypergeometric series with the two channels. Second, the heavy tails: we introduce the depletion function $Dδ(L)$, a deformation of the Hurwitz sums of the sign channel, show that it governs the eigenfunction and the tail of the equilibrium measure, and obtain the correct first-order law of the spectral gap of the twisted operator and the correct normalisation $p1/β/ log p$ of the stable law of winding numbers; a skew-product reformulation with fibre contraction $1/4π$ explains why the classical hypotheses fail and what replaces them. Third, the complex plane: we study the resonances of the tree, prove a zero-free half-plane and a two-term counting law, discover a beat law by which the resonances of the exponential decouple from their Hurwitz shadows at heights $t ≈ 103$, observe universality of their distribution and an anomalous shortrange rigidity, and show that the density of Fisher zeros in the half-plane of absolute convergence follows the Bohr–Jessen law for independent Lyapunov exponents while rejecting the multiplicative structure of the Hurwitz approximation at more than four standard deviations.