Deterministic Calculus: A Reversible, Infinite‑Context Extension of Classical Analysis

<p><span>Classical calculus relies on local, irreversible differential operators that discard historical state information and accumulate numerical drift when mapped onto discrete computational substrates. This structural irreversibility fundamentally limits the stability and predictability of long‑horizon dynamical systems. This paper introduces Deterministic Calculus, a rigorous mathematical extension of classical analysis built upon a reversible, infinite‑context computational primitive. By redefining differential and integral operators to preserve topological invariance and eliminate informational entropy, Deterministic Calculus enables drift‑free, fully invertible differential reasoning over unbounded temporal domains.</span></p> <p><span>We formalize five foundational axioms governing reversible flows, infinite‑context derivatives, identity‑preserving transformations, and O(1) differential scaling. From these axioms we derive Reversible Differential Equations (RDEs), Infinite‑Context Differential Operators, Deterministic Brownian Motion, and a Deterministic Ito Calculus that resolves the classical contradiction between stochastic diffusion and reversibility. We further prove the O(1) Differential Scaling Law, demonstrating that differential computation can scale to arbitrarily long contexts without memory growth when executed on a reversible substrate.</span></p> <p><span>Deterministic Calculus provides the analytical foundation for the author’s eight prior architectures—including the Deterministic AGI Architecture (D‑AGI), the O(1) Infinite‑Context Compute Substrate, the Deterministic Compute Law, and the Deterministic Financial Infrastructure Standard (DFIS)—and establishes a unified mathematical framework for drift‑free computation across physics, finance, AGI, and planetary‑scale simulation.</span></p>