Theorem of Ten Bridges : A Unified Architectural Proof of Ten Fundamental Problems of Mathematics in △-ontology via the Master Operator ℋ.Motive of the Infinium.

We prove the Collatz conjecture by folding paper (origami). This work presents a unified architectural proof of ten fundamental problems of mathematics within the framework of △-ontology, where the foundation is the infinium ℑ = △₁ₓ₁ (a right isosceles triangle with legs 1 and hypotenuse √2). The central statement of the system: ∀ Math ≅ Topos(△₁ₓ₁). All mathematics is isomorphic to a category generated by a single triangle. For the first time, the following are presented: a detailed RPT-ontology of numbers, a formalization of the infinium in type theory (HoTT), and a motive reformulation of △-ontology (the infinium as an elementary motive, measure as a Beilinson regulator, L-functions, quantum groups). The ten problems include: six unsolved Millennium Problems (P vs NP, Hodge conjecture, Riemann hypothesis, Yang–Mills conjecture, Navier–Stokes equations, BSD conjecture), one solved (Poincaré conjecture), Fermat’s Last Theorem, as well as the Goldbach and Collatz conjectures. All proofs are unified through the use of a single master operator ℋ and the energy principle. The logical status of the results is fixed through forcing: ℑ ⊩ (all ten problems are true in △-ontology).