Artificial general intelligence (AGI) requires long-horizon cognitive coherence, stable internal state evolution, identity continuity across time, and consistent reasoning for identical histories. Conventional AI systems-based on stochastic sampling, nondeterministic computation, floating-point variability, and approximate retrieval-provide guarantees primarily at the distributional level rather than at the level of individual cognitive histories. As a result, they cannot support strict probability-1 invariants such as reproducibility, persistent identity, synchronized multi-agent behavior, or fully interpretable reasoning. This paper introduces the Deterministic State Evolution Framework (DSEF) and formulates a Deterministic Cognitive Transition Law governing AGI cognition under strict coherence requirements. In DSEF, AGI is modeled as a deterministic dynamical system evolving over a structured cognitive state space. The framework formally defines cognitive states, histories, canonicalization operators, state-transition operators, reasoning operators, cognitive trajectories, identity invariants, self-consistency conditions, and multi-agent synchronization criteria. We prove that a broad class of core AGI invariants-including trajectory stability, identity continuity, reasoning self-consistency, perfect multi-agent synchronization, and conservation of cognitively relevant structures-are mathematically equivalent to deterministic cognitive evolution under probability-1 requirements. Conversely, the presence of any exogenous stochasticity, regardless of magnitude, violates these invariants with nonzero probability under repeated evaluation of identical histories. Together, these results establish the Deterministic Cognitive Transition Law as a universal law governing AGI systems that require strict probability-1 cognitive coherence. The framework is theoretical and non-constructive: it is architecture-agnostic, substrate-independent, and does not prescribe specific learning algorithms or implementations. Instead, it provides a rigorous foundational characterization of the necessary and sufficient conditions for reproducible, interpretable, and long-horizon-stable AGI cognition.