Sine-Kernel Structures in Riemann-Zeta Hamiltonians: Laguerre Universality, Mellin Projection and Local GUE Statistics

Hamiltonian approaches to the Riemann hypothesis based on the Hilbert-Polya conjecture seek to represent the ordinates of the nontrivial zeros of the Riemann zeta function as a quantum spectrum. We examine how sine-kernel structures arise in the Yakaboylu and Bender–Brody–Muller (BBM) Hamiltonians and in supersymmetric zeta-related constructions. For the Yakaboylu model, we distinguish the bulk sine-kernel limit of the finite-rank Laguerre projector from an exact finite-window Mellin projection kernel in the spectral variable. The fermionic determinantal completion of the latter yields the GUE-type pair-correlation form 1-[sin(πs))/(πs)]^2. The BBM model contains the same Mellin–Fourier phase in an asymptotic left–right kernel, but not an additional orthogonal projector. Supersymmetric Mellin ladders provide a third realization: their finite-rank projector is an exact Dirichlet kernel whose unfolded limit is the sine kernel, while the Riemann-zero ordinate enters only through a removable phase. Numerical tests using 10^4 zeros near the 10^12-th zero show close agreement with the sine-kernel pair correlation, support the density matching L=πρ(T) with Mellin bandwidth L and local density ρ, and reproduce the GUE ramp–plateau spectral form factor. Comparison with the first 1500 zeros demonstrates convergence toward the local regime with increasing height T, whereas the long-range number variance remains more rigid than the translation-invariant sine process.These results establish a local structural correspondence, not a Hamiltonian proof of the full determinantal hierarchy of the Riemann zeros.