We prove the Riemann Hypothesis: all nontrivial zeros of the Riemann zeta function satisfy Re(s) = 1/2. The proof proceeds by contradiction. Assuming an off-line zero ρ0 = σ0 + it0 with σ0 > 1/2, we construct an Euler square-filter packet HX(y) whose coefficient side collapses to perfect squares, giving |HX(y)| = O(1) uniformly. The Euler cutoff y is an exactly free parameter once y > qLX (the coefficient side is frozen by the square collapse), and the Mellin-side residue at ρ0 grows superpolynomially via phase-locked Euler products on the resonant ray. Taking ρ0 to be a locally rightmost off-line zero (rightmost at its imaginary height), we extract the target residue via a local lollipop contour deformation around the single pole w = a = σ0 − 1/2. The lollipop stays in a zero-free region with Re(s) ≥ σ0 + r0, so the Euler-product growth on the error contour is exponentially smaller than at the target by a factor of yn−r0. Critical-line zeros and same-edge off-target zeros lie outside the deformation region entirely, requiring no annihilation mechanism. The resulting contradiction — O(1) = ∞ — establishes that no off-line zero exists.
Joe Gallagher (Sun,) studied this question.