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 ΞX, 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 Fᵖ_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 = Σσ∈𝔖₃ ε (σ) (ΞX^⊗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 XP 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.