Strict System of Motives and Cohomologies

This work presents a rigorous and constructive 𝔹‑formalism for the Hodge conjecture, based on a fundamental ontological shift: replacing the structureless point with the right isosceles triangle (RIT) of leg 1 and hypotenuse √2 as the atomic unit of geometry and number. A 𝔹‑structure on a smooth projective manifold X consists of a triangulation into RIT simplices, a structural connection Ξ_X, and a graded complex of 𝔹‑differential forms Ω^•_𝔹(X) with a differential d_𝔹. The resulting 𝔹‑cohomology H^•𝔹(X) is canonically isomorphic to classical de Rham cohomology, and a natural Hodge filtration F^p_𝔹 is defined via the connection. The category of 𝔹‑motives is introduced, together with a realisation functor R_𝔹. An explicit formula Z_α = Σ{σ∈𝔖₃} ε(σ) (Ξ_X^{⊗p}(α))_σ is proved, which turns a 𝔹‑lift α of a Hodge class γ into an algebraic cycle Z_α whose class equals γ. A detailed algorithm (Algorithm 6.2.1) is provided for constructing the cycle. The theory is tested on projective spaces and elliptic curves, and universal manifolds X_P are constructed for arbitrary simple 𝔹‑modules. A commutative diagram of equivalences links classical motives, 𝔹‑motives and Hodge structures. The spectral gap λ₁ = 1 – √2/2 emerges as the smallest positive eigenvalue of the 𝔹‑Laplacian. Open questions and possible generalisations (mixed Hodge structures, Tate conjecture, algorithmic complexity, physical interpretation) are discussed.