Emergent Couplings and the 1/2 Axis: A Model Analogy to the Riemann Critical Line in the Recursive Geometry of the Bosonic Vacuum

This work develops a geometric model in which a distinguished half-axis Re S = 1/2 naturally emerges for a complexified coupling S = s + i a. Based on the Eynard–Orantin topological recursion applied to the genus-one Legendre family y² = x (x − 1)(x − λ), the model shows that modular self-duality and energy minimization uniquely select the self-dual point λ = 1/2, defining a symmetry locus structurally analogous to the Riemann critical line. The imaginary component a behaves as an axion-like phase with a periodic potential derived from the q-expansion of −log|Δ(τ)|, leading to a geometric realization of a nonperturbative Schwinger-like regime. The study is purely geometric and makes no claims regarding the zeros of the Riemann zeta function.