Sign Channels, Dirichlet L-Values and a Non-Perturbative Hausdorff Dimension on the Branch Tree of the Exponential

Let logₖ(y)=log y + 2π ik with Im log ∈ [0, 2π), and let J∞ be the limit set of the associated conformal iterated function system on the branch tree of the exponential map. We show that the arithmetic of this system is organised by a two-state sign channel. Its transfer matrix has entries (2π)⁻ˢζ(s,5/4) and (2π)⁻ˢζ(s,7/4) and closed eigenvalues with λ₊(s)=(π/2)⁻ˢ[(1-2⁻ˢ)ζ(s)-1-3⁻ˢ] and λ₋(s)=(π/2)⁻ˢ[β(s)-1+3⁻ˢ] where β=L(·,χ₋₄) so that at $s=3$ the symmetric channel is (7ζ(3)-224/27)/π³ and the antisymmetric one is 1/4-20827π³. Apéry's constant governs the sign-blind channel and β(3)=π³/32 the sign-sensitive one. We prove a Hadamard factorisation for the fixed-point spectrum of exp giving Σ^zₖ⁻¹=1/2 and rational higher power sums that determine the Lyapunov ground state θ=1.37455701; we extend the holomorphy of the one-letter partition function to Re $s > -1$ with Z₁(0)=-3/2 and a closed finite part at $s=1$, and we obtain an exact finite-part integral for the sign-odd cubic spectral sum. We then introduce a hierarchy of cylinder transfer matrices and a Nyström discretisation in t=log n which computes the dimension of the infinite-alphabet system directly, without extrapolation in the alphabet size: δ∞=1.207283(3) and Λ∞=6.689. Twisting by the near-trivial characters χα shows that at K=∞ the spectral gap closes as α^δ∞-1 rather than quadratically, that the centred winding number obeys a symmetric stable law of index δ∞-1=0.2074, and that the leading coefficient is Γ(1-s)cos(π(s-1)/2)/π at the sign-channel level. Finally, the rotated trees of ez+iτ realise the shift parameters 1/4∓τ/2π and produce L(s,χ₋₃), L(s,χ₋₄) and L(s,χ₋₈) as one-letter partition functions of genuine dynamical systems, with δ∞(τ) increasing from 1.207284 to 1.207961.