Proof of the Riemann Hypothesis via the Tope Operator

We prove that every non-trivial zero s* = a* + ib* of the Riemann zeta function satisfies Re(s*) = 1/2, by contradiction in three steps. (1) Uniqueness: via the Hadamard product formula for xi and an exact algebraic inequality on zero pairs (rho, 1-rho_bar) — valid regardless of their position, including off the critical line — we show that fb*(a) = Re(zeta(a+ib*)) has at most one zero on (0,1). (2) Tope operator: uniqueness legitimizes the Tope theorem, giving G(fb*) = a* - 1/2. The functional equation then forces G(fb*) = 1/2 - a*, hence G(fb*) = 0. (3) Contradiction: a* = 1/2. The proof applies simultaneously to fb* and gb*, showing that 1/2 is the unique point in (0,1) where both families vanish together. Validated numerically to machine precision via the Tope kernel K(x,u) on [0,1]^2 for all tested zeros.