Arithmetic Discretization without Boundary Conditions and a Heat-Flow Program for Zeta Zeros

This paper proposes an operator-theoretic framework for the Riemann hypothesis based on arithmetic compression and heat-flow zero-mode extraction. The first main idea is that the relevant discrete arithmetic sector should be associated with the logarithmic embedding of the integers, \(n ↦ log n\), rather than with boundary conditions as the primary source of discretization. The second is that the zeta-zero problem may be reformulated as a kernel-selection problem for a nonnegative self-adjoint arithmetic constraint Laplacian, whose heat flow would isolate the zero sector. The paper develops the logarithmic arithmetic background, formulates arithmetic compression onto the logarithmic lattice, studies formal zero-mode models based on \(ζ\), \(ζ\), and the Hardy \(Z\)-function, and includes a weighted Gaussian-regularized model yielding bounded operators on \(^2( N)\). The work is programmatic rather than conclusive: it does not prove the Riemann hypothesis, but clarifies a possible route connecting logarithmic arithmetic structure, self-adjoint operator theory, constraint Laplacians, and heat-flow extraction of zeta-zero sectors.