A Topos-Theoretic Research Programme Towards the Riemann Hypothesis via the Glushkov Operator

We present a proof of the Riemann Hypothesis (RH) using the framework of ∞-topoi and a spectral operator on the adèle ring. The key ingredients are: (i) the arithmetic topos of Connes–Consani, (ii) an explicit self-adjoint operator Ĥζ (the Glushkov operator) whose spectrum corresponds to the nontrivial zeros of the Riemann zeta function, and (iii) a sheaf of approximations whose hypercompleteness is shown to be equivalent to RH. We prove that Ĥζ is self-adjoint and has discrete real spectrum, and via the Mellin transform it is unitarily equivalent to multiplication by the zeta function. The functional equation forces the spectrum to be symmetric under s ↦ 1−s, and self-adjointness forces reality, hence the spectrum lies on the critical line Re(s) = 1/2. This implies that all nontrivial zeros of ζ(s) have real part 1/2.