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-rhobar) — valid regardless of their position, including off the critical line — we show that f₁* (a) = Re (zeta (a+ib*) ) has at most one zero on (0, 1). (2) Tope operator: uniqueness legitimizes the Tope theorem, giving G (f₁*) = a* - 1/2. The functional equation then forces G (f₁*) = 1/2 - a*, hence G (f₁*) = 0. (3) Contradiction: a* = 1/2. The proof applies simultaneously to f₁* and g₁*, 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, 1² for all tested zeros.
Cheikh Ahmadou Bamba Tope (Fri,) studied this question.