Phase-Volume Obstruction and the Impossibility of Local Spectral Models for the Selberg Zeta Function

Version 2. 0 — corrected proof Fixes an error in the analysis of the Legendre function Qₒ-₁. The first Born term is now analysed via the spectral decomposition of the resolvent; the simple pole at s=1/2 arises from the continuous spectrum (Eisenstein series). The main theorem remains unchanged. The previous version is kept for reference. We prove that singular potentials localized in exponentially thin tubular neighborhoods of prime geodesics on the modular surface X = PSL (2, Z) ² cannot generate the Euler product of the Selberg zeta function through Born series expansion. Using the pre-trace formula and an explicit analysis of the analytic structure near s = 1/2, we demonstrate that the first Born term possesses a simple pole at the spectral origin that cannot be cancelled by higher-order terms or absorbed into a holomorphic regular factor. This topological obstruction, which we refer to as the Artemov Phase-Volume Obstruction, establishes that one-dimensional resonator models — including thin tubes and quantum graph approximations — fundamentally fail to capture the transverse phase volume required to produce the Selberg sinh-factors. Our result explains the limitations of local spectral approaches to automorphic L-functions and necessitates a transition to global frameworks, such as the adelic noncommutative geometry developed by Connes. This paper is Part 2 of a trilogy: • Part 1: Architectural Isomorphism (DOI: 10. 5281/zenodo. 20488420) • Part 3: Null-Metric Surfaces and the Riemann Hypothesis (forthcoming)