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The aim is to present a new formalism for the Hodge conjecture using a novel geometric structure based on right isosceles triangles.

June 15, 2026Open Access

Strict System of Motives and Cohomologies

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The aim is to present a new formalism for the Hodge conjecture using a novel geometric structure based on right isosceles triangles.
Developed a B-structure on smooth projective manifolds through triangulation into right isosceles triangle simplices.
Introduced a B-motives category and an explicit algorithm for constructing algebraic cycles.
Demonstrated results on projective spaces and elliptic curves.
Established an isomorphism between B-cohomology H^•𝔹(X) and classical de Rham cohomology.
Identified the smallest positive eigenvalue of the B-Laplacian as the spectral gap λ₁ = 1 – √2/2.
Linked classical motives, B-motives, and Hodge structures through a commutative diagram of equivalences.

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Constructed a rigorous B-formalism for the Hodge conjecture, revealing new insights on motives and cohomologies.

Alexey (KAMAZ) Petrov (2026) studied this question.

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