New Results Supporting the Validity of the Lindelöf Hypothesis Using the Maynard–Guth Hamiltonian

This paper develops a quantum–spectral model for prime counting in almost-short intervals, motivated by the Maynard–Guth estimate and the long-standing Riemann and Lindelöf hypotheses. We refine a perturbed Schrödinger-type Hamiltonian whose potential encodes both the main term and the exponentially decaying error term of the prime number theorem in these intervals, and we analyze its uncertainty relation, symmetry/self-adjoint structure, and spectral behavior. Numerically, we compute eigenvalues and eigenfunctions using finite-difference and finite-element discretizations on large domains and compare the unfolded spacing statistics with random-matrix predictions; we further probe Lindelöf-type boundedness by evaluating <img src=image/13439829_01.gif> at arguments derived from the computed spectrum. The resulting spectrum is real, exhibits symmetry and band structure, and shows level repulsion compatible with a Hilbert–Pólya operator candidate, while the zeta evaluations remain bounded (and frequently close to 1) in the tested ranges. The main contribution is an explicit bridge from Maynard–Guth prime-density/error terms to a concrete Hamiltonian framework that yields testable spectral and thermodynamic predictions. Limitations include finite discretization ranges and the use of conjectural inputs in parts of the prime-gap discussion; no claim of a proof is made, but the results provide quantitative guidance for future rigorous and computational work.