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 \(\vert{}\lambda_a\vert{}^{-s}\) with the complex chiral weight \(\lambda _{a}^{-s}\), 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 \((n+1) \times (n+1)\) matrix of entire functions, whose spectrum is rigidly locked to the postsingular orbit \(0 \mapsto 1 \mapsto e \mapsto e^e \mapsto \cdots\). Furthermore, we show that the chiral traces are global Grothendieck residues belonging to the postsingular ring \(\mathbb{Q}[e^{-1}, e^{-e}, \dots]\) 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.