A Heat-Flow Program for Zero-Mode Extraction in the Riemann Hypothesis (Letter)

This note proposes an operator-theoretic program for considering the Riemann hypothesis as a zero-mode extraction problem. The guiding idea is that, if the zeta-zero condition can be reformulated as the kernel condition of a nonnegative self-adjoint operator, then the associated heat flow canonically projects onto the corresponding zero sector. The paper develops several formal models of this idea. It begins with diagonal constructions built from the values of the zeta function, derives the associated quadratic constraint operators, examines the limitations of a tensor-product reformulation, and then shows that on the critical line the same program admits a simpler one-component version in terms of Hardy’s Z-function. No proof of the Riemann hypothesis is claimed. Rather, the aim is to clarify the structure of such a heat-flow program and to identify its main unresolved step: the construction of an intrinsic arithmetic self-adjoint operator whose kernel encodes the zeta-zero condition.