The Infinium as a Universal Bridge Between Category Theory and ZFC: From Genetic Reconstruction to 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 this act necessarily gives rise to 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 is constructed without any sorry. The theorem of the infinium's terminality is proved: any △-object reduces to ℑ through a finite sequence of Φ and Ψ operations, generalizing the origami proof of the Collatz conjecture. The equivalence of the ZFC-model and the categorical model is proved. All 17 theorems are formally verified in Lean 4. The central formula of the work: Math = Eco(Dist) ⊗ △₁ₓ₁.