Abstract
The Riemann Hypothesis can be read as a spectral-reality problem once every non-trivial zero ρ is assigned the quadratic energy E_ρ=ρ(1-ρ). In this paper we develop that observation into a precise operator-theoretic program. For a zero written as ρ=σ+it, the imaginary part of the energy is t(1-2σ). Hence, for every non-real zero in the critical strip, E_ρis real exactly on the critical line. The result shows the Riemann Hypothesis is the assertion that all quadratic zeta energies are real. The paper then formulates the Hamiltonian structure required to turn this equivalence into a proof. The proposed operator is not a finite spectral fit, instead, it is a modular-arithmetic Hamiltonian built on logarithmic space, prime translations, an arithmetic boundary domain, a prime-power trace formula, and determinant closure with the completed zeta function. The main theorem is conditional since if such a self-adjoint operator has determinant ξ(s)in the spectral coordinate s(1-s), then all non-trivial zeros lie on the critical line.
