Origami Proof of the Collatz Conjecture in Infinium Ontology: A Formal Verification in Lean 4

The Collatz conjecture is one of the most enduring open problems in mathematics. We present a constructive proof within the framework of Infinium Ontology (△-ontology), where the fundamental object is not a structureless point but the infinium △₁ₓ₁ — a right isosceles triangle with legs 1 and hypotenuse √2. In this framework, natural numbers acquire a geometric body as mosaics of infiniums. We introduce a complexity function that strictly decreases at each Collatz step, guaranteeing convergence to the base triangle. The proof is formalized and verified in the Lean 4 proof assistant. We discuss the implications of this result for the unity of mathematics and the foundational role of geometric primitives.