The Three Bridges Theorem: A Unified Proof of the Hodge, Riemann, and Birch–Swinnerton-Dyer Conjectures in △‑Ontology

The Hodge, Riemann, and Birch–Swinnerton-Dyer (BSD) conjectures are among the seven Millennium Prize Problems formulated by the Clay Mathematics Institute. Despite colossal efforts, none of them have been fully proven within the standard set-theoretic paradigm. This work proposes a unified architectural proof of all three conjectures within the framework of △‑ontology — a new approach to the foundations of mathematics, in which the foundation is the infinium ℑ = △₁ₓ₁ (a right isosceles triangle with legs of 1 and a hypotenuse of √2). We show that all three conjectures are not independent problems but three projections of the same geometric fact — the balance of symmetry and asymmetry, encoded in the infinium and expressed by the Pythagorean theorem 1² + 1² = (√2)². The proof is conducted as an architectural project: all key steps are formulated explicitly, while their full formal verification remains a task for mathematical institutes. In conclusion, the result is expressed in the language of logical forcing: ℑ ⊩ (Hodge ∧ Riemann ∧ BSD).