Solving the Metric Problem in Synthetic Differential Geometry via Δ‑Ontology: A Step‑by‑Step Formal Proof with Verification in Lean 4

In synthetic differential geometry (SDG) developed by Anders Kock, there is a fundamental problem: the metric defined via the absolute value d(x,y) = |x − y| is not smooth at zero, creating a conflict between infinitesimal closeness (∼) and finite distance (#). The present work provides a formal solution to this problem within the framework of Infinium‑ontology (Δ‑ontology), where the primitive element is the infinium ℑ = △₁,₁ — a right isosceles triangle with legs 1 and hypotenuse √2. The proof is divided into eight consecutive steps: from the construction of dual numbers to the derivation of the Kock–Lawvere axiom from the self‑similarity of the infinium. Each step is accompanied by an explanatory comment and an explicit statement of the formal verification status. The formalization contains 17 fully proved theorems and is verified in Lean 4 without a single sorry.