"Companion paper to: A Tripartite Hamiltonian Framework for the Riemann Hypothesis. " We extend the Tripartite Hamiltonian framework of 1 from a scalar operator onHw = L2 (R+, x−2) to a 2 × 2 spinor Hamiltonian on Hw ⊕ Hw. The spinor isdefined by Hregspinor = diag (H11, H11) + off-diag (ˆΩμ, regcomp), where H11 is the scalarTripartite Hamiltonian of 1 and ˆΩμ, regcomp is a new M¨obius compression operatordefined analogously to ˆΩΛcomp but with μ (n) replacing Λ (n). We resolve all four open problems identified at the outset: (A) Regularisation. The na¨ıve spinor has a negative eigenvalue −1 at n = 1. We show that excluding n = 1 and n = 2 from ˆΩμ, regcomp resolves this, at zero arithmeticcost, because μ (1) + μ (2) = 0 implies Mreg (x) = M (x) for all x ≥ 2. (B) Kato–Rellich. We prove, unconditionally, thatˆΩμ, regcompf w ≤ 1ln 6 (ˆΩexp + ˆΩΛcomp) f wfor all f ∈ D, with the sharp constant a = 1/ ln 6 ≈ 0. 558 < 1 achieved by pointmasses at n = 6 (first squarefree composite ≥ 3). This gives an unconditionalself-adjointness result for H (0) spinor. (C) Weil Denseness. The scalar class hsymf: f ∈ D from 1 consistsentirely of positive-definite functions (Bochner’s theorem), hence is not dense inWeil’s admissible class W. The spinor cross-terms introduce products f1f2 whosecosine-Mellin transforms span all of C∞c (R+) by a direct algebra argument; OpenProblem 1 of 1 is thereby resolved. (D) Deficiency Indices. The anti-unitary time-reversal Tspinor (f1, f2) = (f1, f2) maps K+ to K− unconditionally, giving n+ (Hregspinor) = n− (Hregspinor). More-over n = 0 if and only if RH, exactly as in the scalar case, with deficiency indexequal to twice the scalar deficiency. 1We close by identifying a new arithmetic tool — the Dirichlet identity (Λ ∗μ) (n) = −μ (n) log n — which introduces a joint (Λ, μ) large sieve not exploitedin the literature. We show this improves Soundararajan’s 2 bound M (x) ≪√x exp (log x) 1/2 (log log x) 14 by reducing the power of log log x.
Dipankur Bodwal (Tue,) studied this question.