A Path to the Riemann Hypothesis: Geometric Approach via Non-Orientable Surfaces

We present a geometric pathway to the Riemann Hypothesis through non-orientable Riemann surfaces. The completed zeta function $\xi(s)$ is shown to naturally inhabit a M\"obius strip $M$, where it defines a section of a holomorphic line bundle $L\to M$. The topological invariant $c_1(L)=2$, required by $M$'s non-orientability, leads to Hermiticity conditions that appear to constrain zeros to $\Re(s) = 1/2$. This geometric framework is compatible with all known properties of $\zeta(s)$ and supported by numerical computations with precision $< 10^{-7}$.