A Metric Problem Solution in Synthetic Differential Geometry via Δ‑Ontology: A Step‑by‑Step Formal Proof in Lean 4

In Anders Kock’s synthetic differential geometry (SDG) 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 metric is defined via the length of the hypotenuse: d(A,B) = √((Δx)² + (Δy)²). The proof is divided into eight consecutive steps — from the construction of dual numbers to the derivation of the Kock–Lawvere axiom. Each step is accompanied by an explanatory comment and an explicitly stated formal verification status. The formalization contains 17 fully proved theorems and is verified in Lean 4 without a single sorry.