Theory of Relational Differentials: A Categorical Formulation of Δ‑ontology and the Overcoming of "Point‑based" Contradictions

A rigorous categorical formulation of Δ‑ontology is proposed — a new approach to the foundations of mathematics in which the foundation is the infinium ℑ = △₁ₓ₁ (an isosceles right triangle with legs 1 and hypotenuse √2). The basic categories (Distinctions, Orthogonal Pairs, Elementary Relational Triples) and functors (orthogonality, closure, measure) are introduced. It is demonstrated that the self‑similarity of the infinium is expressed as a monoidal structure on the category of ERTs, and the measure as a contravariant functor into the category of metric spaces. The limit relational complex 𝒦_∞ is constructed as a 2‑limit, and on its basis the function space is defined as an Eilenberg–Moore category. Categorical versions of the Poincaré conjecture and the spectral gap are formulated, the latter receiving the exact value λ₁ = 1 − ½√2. A complete categorical diagram is given that encompasses all structures of Δ‑ontology. It is shown that the fundamental cause of many contradictions in mathematics and paradoxes in physics — singularities, incompatibility of quantum mechanics and relativity, “heat death” in thermodynamics — is rooted in the notion of a structureless point as the first principle. A point, having no internal structure, makes entropy a measure of chaos and leads to the conclusion of the inevitable degradation of the Universe. We replace the point with the infinium and show that entropy in this case becomes a measure of the striving for balance, and the “arrow of time” is directed not towards chaos but towards harmony and clarity. The new concept of “intropy” captures this principle: everything strives towards economy of energy. ---