The Infinium as a Universal Bridge Between Category Theory and ZFC: From the Act of Distinction to Complete Formal Verification in Lean 4

This work carries out a genetic reconstruction of mathematics from a single pre-mathematical act — the act of distinction. Following the principle of minimal complexity (economy), we show that from this act there necessarily arises the right isosceles triangle △₁ₓ₁ (the infinium) with legs 1 and hypotenuse √2. This object, in turn, generates numbers, logic, geometry, analysis, and topology. In categorical language, the infinium is simultaneously an object in ZFC (a set of points) and a morphism in category theory (the hypotenuse as a universal relation). This duality makes it a universal bridge between the set-theoretic and category-theoretic approaches to the foundations of mathematics. A complete ZFC-model of △-ontology has been constructed without a single sorry. The theorem of the terminality of the infinium (any △-object reduces to ℑ) has been proved. The equivalence of the ZFC-model and the categorical model has been proved. All 20 theorems have been formally verified in Lean 4. The central formula of the work: Math = Eco(Dist) ⊗ △₁ₓ₁. The final formula of agreement: ZFC ≅ △-ontology ≅ Category Theory.