△‑Ontology: A Manifesto of Geometric‑Numerical Synthesis. The Infinitum Paradigm

This manifesto presents △‑ontology — a new paradigm for the foundations of mathematics, in which the primary object is not a structureless point but a geometric quantum: the Infinitum △₁ₓ₁ — a right isosceles triangle with legs of length 1 and hypotenuse √2. The very name "Infinitum" (from Latin infimum, infinitum, infinitesimal, continuum) points to the unique properties of this object that no other figure combines: it is an energy minimum, possesses fractal self‑similarity, provides a geometric body for nilpotent infinitesimals, and serves as a bridge between the discrete and the continuous. From the three properties of the Infinitum — orthogonality, fractal self‑similarity, and the irrationality of √2 — all types of numbers and spaces, Lebesgue measure, smoothness, and a metric are sequentially derived. The Unity Formula ∀ Math ≅ Topos(△₁ₓ₁) asserts that all of mathematics is isomorphic to a topos generated by a single triangle. We show how △‑ontology constructively models ZFC, resolves the contradiction between set theory and category theory, provides a geometric explanation for the fine‑structure constant α ≈ 1/137.04, and offers architectural proofs of ten fundamental problems of mathematics, including the Riemann, Goldbach, Collatz, and P vs NP conjectures. The concept of intropy is introduced as a measure of a system's striving toward balance, harmony, and clarity. The final formula: Math = Proj(Topos(△₁ₓ₁)) — all of mathematics is a set of projections of the single topos of the Infinitum. -- -