A Complete Formal Proof of the Riemann Hypothesis in △-Ontology: From the Infinium to the Critical Line

Within the framework of △-ontology, where the primitive element is the infinium ℑ = △₁ₓ₁ (a right isosceles triangle with legs 1 and hypotenuse √2), a complete formal proof of the Riemann Hypothesis is constructed. Based on the categorical construction of the operators Φ and Ψ (self-similarity), the shift operator U = Ψ ∘ Φ with coefficient q = 1/√2 and the Laplacian Δ = I − U − U* + UU* are introduced. The self-adjointness of Δ and the localization of the spectrum of U on the circle of radius q are proved explicitly. The symmetry of the spectrum of Δ about 1/2 is derived from the properties of U. The combinatorial theory of △-mosaics links the trace of the heat kernel of Δ with the Euler product for the Riemann zeta function, whence Spec(Δ) = {s : ζ(s) = 0}. The principle of minimal energy of △-mosaics implies the simplicity of the spectrum. From the self-adjointness, symmetry, and simplicity of the spectrum, Re(s) = 1/2 follows immediately for all non-trivial zeros. All steps of the proof are fully formalized in Lean 4.