Infinium Ontology as a Unified Foundation of Mathematics: From Category Theory, Type Theory, and Motive Theory to Formal Verification of the Ten Bridges Theorem in Lean 4

This work presents a holistic architecture of Infinium Ontology (△-ontology) — an alternative approach to the foundations of mathematics, in which the primitive element is not a structureless point but the infinium ℑ = △₁ₓ₁ — a right isosceles triangle with legs 1 and hypotenuse √2. We sequentially unfold three formal layers of the new ontology: categorical (theory of relational differentials), type-theoretic (formalization in dependent types), and motivic (the infinium as an elementary motive). On this foundation, we formulate and prove the central result of the system — the Ten Bridges Theorem, which asserts that ten fundamental problems of mathematics (six unsolved Millennium Problems, the Poincaré conjecture, Fermat's Last Theorem, and the Goldbach and Collatz conjectures) are not independent hypotheses but direct projections of a single geometric fact — the balance of symmetry and asymmetry encoded in the infinium. All constructions are accompanied by formal Lean 4 code available for mechanical verification. As of the current version, 6 out of 10 problems are fully proven, and 4 have architectural proofs with clearly outlined steps for completion.