From the Infinitum (Right Isosceles Triangle, RIT) △₁ₓ₁ to the Hodge Conjecture: 𝔹‑Structures, Motives, and an Explicit Formula for Algebraic Cycles

This paper presents the formalism of 𝔹‑structures, based on the right isosceles triangle △₁ₓ₁ (the Infinitum, РПТ) as a fundamental geometric quantum. This formalism provides a rigorous constructive proof of the Hodge conjecture. We introduce the category of 𝔹‑manifolds, equipped with a △‑decomposition and a structural connection, construct 𝔹‑cohomology, and define on it a structural Hodge filtration. We then build the category of 𝔹‑motives, in which every morphism is automatically realized by an algebraic correspondence. The culmination is an explicit formula that expresses an algebraic cycle directly in terms of a 𝔹‑cohomology class, together with a step‑by‑step algorithm for its construction. As a result, the Hodge conjecture becomes not merely an existence theorem but a computable procedure.