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
Alexey (KAMAZ) Petrov (Sat,) studied this question.