The Deterministic Computation Law. A Formal Mathematical Framework for Reproducible Artificial Intelligence

Modern AI systems exhibit substantial nondeterminism arising from stochastic sampling, floating-point instabilities, nondeterministic GPU kernels, race conditions in parallel execution, and probabilistic internal mechanisms. This variability prevents reproducibility, auditability, and scientific verification - properties required for deployment in scientific, medical, financial, legal, and safety-critical contexts. This paper introduces a first-principles mathematical foundation for reproducible computation. Beginning from three minimal axioms - Input Determinism, Representation Invariance, and Replayable Reasoning - we derive the Deterministic Computation Law (DCL): R=H(D(P)), where Dis a canonicalization operator mapping problem representations into a quotient space of canonical forms, and His a deterministic reasoning operator implementing reproducible internal state transitions. We formally develop equivalence relations, quotient constructions, determinism in state evolution, and categorical interpretations of the theory. We provide full proofs showing that DCL is the unique computational structure consistent with the three axioms - necessary and sufficient for reproducible computation. This framework offers a mathematically rigorous foundation for deterministic artificial intelligence, independent of architecture, training method, model class, or implementation strategy.