Theory of Relational Differentials, Part II: Topos‑Theoretic, Univalent and Quantum Extensions (A Sketch)

The present work continues the categorical formulation of Δ‑ontology and constructs a series of advanced mathematical structures based on the relational complex 𝒦_∞ and elementary relational triples (ERT). We introduce: the topos of sheaves Sh(𝒦_∞) with an internal intuitionistic logic; univalent foundations in the spirit of Homotopy Type Theory (HoTT) for relational structures; synthetic differential geometry with infinitesimal distinctions; relational motives and zeta‑functions; a set theory with non‑well‑founded membership; 𝒦_∞‑enriched categories; quantum C*-algebras of distinctions and a measure of entanglement; meta‑theorems on consistency; computational aspects (the ℛ‑Turing machine); and finally an outline of a unified field theory as a hierarchy of categories. The paper demonstrates that Δ‑ontology serves as a generative grammar for many areas of modern mathematics.