The Infinite Helix Hamiltonian and the Prime Harmonic Trace Formula

We introduce a geometric Hamiltonian family HN,K constructed on a helicoidal–catenoidal background with N discrete radial sites and K helical layers. Numerical experiments indicate that, as N, K → ∞, the spectral zeta function and completed ξ-function associated to HN,K converge pointwise to the Riemann zeta and Riemann ξ-function on regions of the complex plane. Motivated by these observations, we consider the idealized infinite-limit operator H∞. Under the assumption that the one-particle spectrum of H∞ is given by {log p : p prime}, we show that the associated bosonic partition function satisfies Z(β) = Tr(e^{−βH∞}) = ζ(β) (Re β > 1),and that its logarithm admits a prime harmonic trace formula. This follows purely from the Euler product for the Riemann zeta function once the spectral assumption is granted. We present the geometric motivation, the precise trace identity, and numerical evidence for the HN,K → H∞ convergence hypothesis.