Finite Termination and Transcendence Obstructions for Exponential Orbit Sums

We study finite orbit-sum termination for the two-sided exponential iteration generated by \( f(x)=2^x-1 \) and its inverse on the unit interval. For \( w\in(0,1] \), put \( u_k(w)=f^k(w) \) for \( k\in\mathbb Z \). A binary digit sequence \( a=(a_k)_{k\in\mathbb Z} \) with $a_0=1$ is normalized by \( \sum_{k\in\mathbb Z}a_k u_k(w)=1. \) Thus the expansion scale is generated by the point being expanded, rather than by an external base or partition, and finite termination is governed by finite orbit-sum equations. We prove existence and uniqueness of normalization roots for admissible digit sequences, construct the associated greedy code, and characterize finite termination by finite orbit-sum hitting equations compatible with the greedy order. The finite terminal set is countable and has Lebesgue measure zero. On the arithmetic side, the first positive and first negative boundary roots are transcendental, and the first positive second-order boundary root is irrational. Assuming Schanuel's conjecture, we exclude all non-trivial rational finite terminal points and prove transcendence for all purely positive finite roots, for all two-term boundary roots paired by the mirror identity, and for all roots with one first negative layer and arbitrary finite positive support.