Theory of Relational Differentials: A Categorical Formulation of Δ‑Ontology

A categorical framework is constructed for Δ‑ontology — an approach to the foundations of mathematics in which the elementary object is not a point but an infinium (an isosceles right triangle with legs 1 and hypotenuse √2). We introduce a category of Distinctions ℜ with an orthogonality relation and a closure functor (coproduct for orthogonal pairs), equipped with a measure that satisfies the Pythagorean theorem. It is shown that the self‑similarity of the infinium is expressed as a monoidal structure, and the canonical measure is singled out uniquely. By successive gluing of infiniums we build a limit relational complex 𝒦_∞, on which function spaces arise as algebras over a monad. The spectral gap of the discrete Laplacian is computed: λ₁ = 1 − ½√2. A categorical analogue of the Poincaré conjecture is formulated. We discuss how replacing the point with the infinium affects the notion of entropy and eliminates formal singularities