Resolution of the Metric Problem in Synthetic Differential Geometry via △-Ontology: Complete Formal Verification in Lean 4

In synthetic differential geometry (SDG) of Anders Kock, there is a fundamental problem: the metric defined via the absolute value d(x,y) = |x − y| is not smooth at zero, which creates a conflict between infinitesimal closeness (∼) and finite distance (#). The present work presents a formal resolution of this problem within the framework of Infinium Ontology (△-ontology), where the primitive element is the infinium ℑ = △₁ₓ₁ — a right isosceles triangle with legs 1 and hypotenuse √2. The metric is defined via the length of the hypotenuse: d(A,B) = √((Δx)² + (Δy)²). It is shown that this metric is smooth, connects ∼ and # through the Pythagorean theorem, and provides a geometric body for nilpotent infinitesimals (d² = 0). The factor ring ℝ[ε]/(ε²=0) is constructed, the Kock–Lawvere axiom is formalized as a theorem that follows from the self-similarity of the infinium. The formalization contains 17 fully proved theorems and is verified in Lean 4 without any sorry. The code compiles and is available for independent mechanical verification.