A Tripartite Hamiltonian Framework for the Riemann Hypothesis

AbstractWe construct a self-contained operator-theoretic framework in which the RiemannHypothesis (RH) is equivalent to a positivity condition on an explicitly definedHamiltonian. The Tripartite Hamiltonian ˆHnewtotal = ˆΩexp + ˆΩcomp + ˆΩnewphase acts onthe weighted Hilbert space Hw = L2(R+, x−2). The three operators are defined,respectively, by the logarithmic expansion of the number line, the mollified von Man-goldt function (taken to its distributional limit in the weak operator topology), andthe restriction of −ζ′/ζ to the critical line—derived entirely from the arithmetic ofprimes with no circular dependence on zero locations.We prove that ˆHnewtotal is symmetric with equal deficiency indices n+ = n− andvanishing boundary terms on a dense domain D ⊂ Hw. Via the Kato–Rellichtheorem we give explicit relative-boundedness constants for each perturbation. Weestablish that ˆHnewtotal is a positive operator if and only if |M (x)| = O(x1/2+ε) forevery ε > 0 (Mertens bound ⇔ RH), and formally identify the energy functionalE(f ) =Df, ˆHnewtotalfEw with Weil’s 1952 positivity criterion. The gap between thebest known unconditional bound M (x) = O(√x exp(( ln x)1/2+ε)) (Soundararajan)and the required O(x1/2+ε) is identified as the entanglement inequality betweenprime clusters and prime deserts. Computational evidence up to x = 106 is presentedin four figures, along with reproducible Python code. The framework is comparedsystematically with the Berry–Keating and Connes programmes.