Categorical Formulation of the Theory of Relational Differentials (structural axiomatics of discrete self-similar space)

This work proposes an alternative mathematical foundation in which, instead of a continuum space‑time with points, a hierarchical discrete structure is introduced, built from elementary blocks — right isosceles triangles (RIT). The main tool is category theory, which makes it possible to precisely express the concepts of orthogonality, closure (hypotenuse) and self‑similarity through universal properties. A sequence of categories is constructed: distinctions, orthogonal pairs, elementary triples, complexes and the limit complex. Measures are introduced as a category of representations, and the existence of a canonical self‑similar measure is proved. The limit complex is interpreted as a space, and consistent functions on it as presheaves equivalent to sheaves on some compact metric space (Riesz theorem). Categorical versions of the Poincaré conjecture and the mass gap conjecture are formulated. Sections are added on the energy protocol (connection with Landauer's principle), on the spectral gap λ₁ = 1 – √2/2, on the geometric definition of prime numbers via mosaics, on the combinatorics of Newton's binomial, on the neurophysiological justification for choosing the right angle, as well as on categorical reformulations of the abc, Beal, Riemann and Collatz conjectures. It is shown how the monoidal structure and duality connect the theory with topological quantum field theory and with the 𝔹‑formalism. The text is supplied with detailed comments for a wide audience and contains a list of conclusions and corollaries for mathematics, physics, number theory and numerical methods.