The Riemann Hypothesis as a Unitarity Theorem for the Arithmetic Field

The Euler product formula zeta(s) = prod_p (1 - p⁻ˢ)⁻¹ is the exact trace Tr_F(N⁻ˢ) of the number operator N on the bosonic Fock space F built from one-particle states labelled by primes, in which integers are Fock states, primes are elementary quanta with single-particle energies E_p = log p, and zeta(s) is the partition function. We prove that all non-trivial zeros of zeta(s) lie on the critical line Re(s) = 1/2. The proof identifies F with L^2(A^x/Q^x, d^x a) via the Fundamental Theorem of Arithmetic and the adelic product formula. The scaling generator A = -i d/d(log|a|) is self-adjoint by Stone's theorem. Meyer's unconditional spectral realization (2005) identifies the non-trivial zeros as atoms of the trace spectral measure of A via the Weil explicit formula. The product decomposition A^x/Q^x = K x R_+^x, where K is compact, reduces the eigenvalue equation to a first-order ODE on R_+^x whose solution space is one-dimensional: each ordinate gamma admits exactly one eigenfunction ri*gamma. The Spectral-Weil identity equates the atom weight at each ordinate with the total analytic multiplicity, which therefore equals 1. The functional equation xi(s) = xi(1-s) pairs any zero rho = sigma + it with a companion (1-sigma) + it at the same ordinate; when sigma != 1/2 these are distinct, forcing multiplicity at least 2 — contradicting the spectral bound. All zeros therefore satisfy Re(s) = 1/2, and as an immediate corollary, all zeros are simple.