𝔹‑Formalism in △‑ontology: A Sketch of a Program for a Constructive Proof of the Hodge Conjecture

This work presents a sketch of a program based on 𝔹‑formalism, a mathematical apparatus developed within the framework of △‑ontology. The program aims to give a constructive proof of the Hodge conjecture. Key concepts are introduced sequentially: the infinitum △₁ₓ₁, 𝔹‑structures, 𝔹‑cohomology, the structural Hodge filtration, the category of 𝔹‑motives, and an explicit formula for an algebraic cycle. The main result is formulated — the formula Z_α = Σσ ∈ 𝔖₃ ε(σ) · (Ξ_X⊗p(α))_σ — and a proof sketch is outlined in six steps. Special attention is given to explicit examples (ℙ¹, ℙⁿ, ℙ¹×ℙ¹, elliptic curves, ℂP²) on which the formula is tested. Key gaps that must be filled to complete the program are clearly stated, and concrete tasks for further research are proposed. Applications to the standard conjectures of Grothendieck and the Tate conjecture are also discussed. The article can serve as an “operating manual” for the 𝔹‑machine.