The Riemann Hypothesis as a Unitarity Theorem for the Arithmetic Field

The Euler product formula zeta (s) = prodₚ (1 - p^-s) ^-1 is the exact trace TrF (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 Eₚ = 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 L² (Aˣ/Qˣ, dˣ 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 Aˣ/Qˣ = 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 r^i*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.