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, \(n \mapsto \log n\), 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 \(\Re\zeta\), \(\Im\zeta\), and the Hardy \(Z\)-function, and includes a weighted Gaussian-regularized model yielding bounded operators on \(\ell^2(\mathbb N)\). 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.