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. Within △-ontology, the Hodge Conjecture ceases to be an open problem and becomes a direct structural consequence of the fractal nature of space: the Hodge class turns out to be a fixed point of the self-similarity operators, and the algebraic cycle an energy minimum in its cohomology class. Cohomology in this picture is a measure of mismatch between the levels of the RCₙ hierarchy, and the Hodge Conjecture is the assertion that the local minimum of this measure is attained precisely on self-similar substructures. The work is conceptual in nature and is addressed to a broad circle of mathematicians interested in the foundations of algebraic geometry.
Alexey (KAMAZ) Petrov (Wed,) studied this question.