The Dual Soul of the Exponential Map: Chiral Collapse, Postsingular Rigidity, and the Splitting of Arithmetic

We present a groundbreaking duality in the thermodynamic formalism of the exponential branch tree. By replacing the classical conformal modulus weight \({}λ_a{}⁻ˢ\) with the complex chiral weight \(λ ₐ⁻ˢ\), the chaotic geometry of the infinite-alphabet iterated function system completely collapses into pure, rigid arithmetic. We prove that the infinite-dimensional Fredholm determinant collapses into a finite \((n+1) × (n+1)\) matrix of entire functions, whose spectrum is rigidly locked to the postsingular orbit \(0 ↦ 1 ↦ e ↦ e^e ↦ ⋯\). Furthermore, we show that the chiral traces are global Grothendieck residues belonging to the postsingular ring \(Q[e⁻¹, e⁻ᵉ, ]\) with a universal denominator of \((n!)^p\). This reveals a clean structural dichotomy: the branch tree splits into a classical modulus sector (governed by Riemann and Hurwitz zeta values) and an exact chiral sector (governed by postsingular towers), proving that chirality effectively tames the transcendental divergence of complex dynamics.