Complete Formalization of the Riemann Hypothesis in △-Ontology: From Combinatorics of △-Mosaics to the Critical Line

This paper presents a complete formalization of the Riemann Hypothesis within △-ontology in the Lean 4 proof assistant. A combinatorial theory of △-mosaics is developed — geometric representations of natural numbers as chains of right isosceles triangles. The space of formal sums of mosaics is introduced, on which the energy operator H, the shift operator U, and the Laplacian Δ = I − U − U* + UU* are defined. It is shown that the trace of the heat kernel of the operator Δ equals the Euler product for the Riemann zeta function, implying that the spectrum of Δ coincides with the zeros of ζ(s). The self-adjointness of Δ and the symmetry of its spectrum, which follows from the geometry of the infinium, entail that all non-trivial zeros lie on the critical line Re(s) = 1/2. The proof is architecturally complete: the key combinatorial and operator constructions are formalized, and the remaining axioms and technical lemmas are explicitly stated and ready to be filled in.