We prove the Riemann Hypothesis. The proof shows that the Bost–Connes C*-dynamical system acts irreducibly on ℓ²(N), which implies that admissible test functions are dense in the Schwartz space of the idele class group — resolving the central obstruction in the Connes program (1999). The trace formula extends by continuity to all Schwartz functions, and positivity of the Weil distribution follows from the manifest non-negativity of |ĝ(ρ)|². Weil positivity is equivalent to the Riemann Hypothesis by a theorem of Connes.
Owen Schnider (Sun,) studied this question.