The Infinium: A Formalization in Type Theory (Final Revised Version, May 2026)

We propose a formalization of the concept of the infinium — the minimal structural quantum of △-ontology — in the language of dependent type theory. The infinium is introduced as a signature satisfying three axioms: irreflexivity of orthogonality, the Pythagorean theorem for an internal measure, and self-similarity (cutting along the altitude). We show how finite complexes 𝒦ₙ, the limit complex 𝒦_∞, and a topos of sheaves 𝒯 on it are sequentially constructed from this signature. Inside 𝒯, the real numbers, nilpotent infinitesimals, and a smooth structure arise. The main forcing statement is formulated as a working hypothesis: the infinium forces every mathematical structure to be embeddable into 𝒯. (Brief: Three axioms — irreflexivity, Pythagoras, self-similarity — generate a topos 𝒯 that serves as a candidate universal mathematical universe.)