Exact All-Level Positivity of the Bosonic Veneziano Amplitude

Tree-level partial-wave unitarity of the open bosonic Veneziano amplitude follows indirectly from the no-ghost theorem, but a direct proof from the beta function has been missing at the critical dimension. We prove nonnegativity at every mass level and spin throughout the sharp range 3 < D <= 26, using a Bessel recurrence that reduces the infinite sign problem to a manifestly positive bulk and a finite exact boundary.