Origami Proof of the Collatz Conjecture and the Unified Energy Principle of All Millennium Problems in △-ontology

This work presents an intuitive origami proof of the Collatz conjecture within the framework of △-ontology, where the foundation is the infinium △₁ₓ₁ — a right isosceles triangle. The central statement of the system: ∀ Math ≅ Topos(△₁ₓ₁). All mathematics is isomorphic to a category generated by a single triangle. It is shown that the Collatz conjecture becomes a statement about folding any triangular mosaic into a base triangle through a sequence of folds (for even numbers) and unfoldings with tripling and addition of one (for odd numbers). An energy function is introduced that monotonically decreases at each step, guaranteeing the finiteness of the process. The final part shows how the same energy principle — the tendency of any △-structure toward a global energy minimum — reformulates all seven Millennium Problems (Riemann, Goldbach, Poincaré, Hodge, Navier–Stokes, Yang–Mills, BSD) and Fermat’s Last Theorem as statements about the minimality of energy of the corresponding configurations. The logical status is fixed through forcing: △₁ₓ₁ ⊩ (all problems are true in △-ontology).