The Hodge Conjecture in △-Ontology: A Geometric Explanation via Infinium Self-Similarity (Most Complete Version, May 2026)

This work offers a geometric explanation of the Hodge Conjecture within the framework of △-ontology — an alternative approach to the foundations of mathematics in which the fundamental element is the infinium △₁ₓ₁ (a right isosceles triangle with legs of 1 and a hypotenuse of √2). Starting from the central postulate ∀ Math ≅ Topos(△₁ₓ₁), we show that all types of cohomology (de Rham, singular, sheaf, Hodge) are naturally expressed as constructions on △-mosaics. The key role is played by the self-similarity property of the infinium: thanks to the operators Φ and Ψ, any Hodge class of type (p,p) is representable by a finite △-mosaic, i.e., it is an algebraic cycle. A formal definition of the infinium is given as the triple ℑ = (△₁ₓ₁, D₃, √2) with the universal property of a terminal object in the cognitive topos ℰ. The motivic formalization M(ℑ) = ℚ(0) ⊕ ℚ(1)[1] ⊕ ℚ(1)[√2] allows us to express the spectral gap λ₁ through the Beilinson regulator and to establish a connection with the Langlands program. A formalization of the infinium as a higher inductive type in homotopy type theory (HoTT) is provided. The work contains explicit examples for the torus T², elliptic curves, projective spaces, and Fermat hypersurfaces, and also outlines connections with synthetic differential geometry and quantum gravity. The work is conceptual in nature and is addressed to a broad circle of mathematicians interested in the foundations of algebraic geometry.