Recursive Ontological Calculus: A Unified Theory of Symbolic Computation

This paper introduces the Recursive Ontological Calculus (ROC), a first-order formal system that encodes triadic semiosis, symbolic identity, and curvature-driven recursion. Built on C.S. Peirce’s logical architecture, ROC integrates category-theoretic structure with compression entropy, yielding a complete, machine-verifiable sequent calculus. It proves 18 labelled theorems (T1–T18), supports infinitary recursion via the Master Recursion Equation, and embeds faithfully into ZFC, category theory, and homotopy type theory. Key innovations include reflective fixed-point construction, triadic morphism logic, transfinite convergence, and an internal meta-semiosis loop. All proofs and symbols are contained within the paper.