Cognitional Mechanics (CM) is a non‑commutative, deterministic computational framework that treats order‑sensitive operator composition as a standalone computational resource, independent of quantum physics, probabilistic interpretation, or measurement theory. While non‑commutativity has historically been monopolized by quantum mechanics, CM restores it to its natural mathematical domain and demonstrates that it can be executed directly on classical hardware. This work presents the first complete operationalization of CM as an executable computational model. It defines semantic states as vectors embedded in an abstract manifold, introduces a minimal set of non‑commutative operators, and formalizes the Logical Leap, a deterministic convergence regulator that replaces probabilistic collapse. The paper provides a reproducible reference implementation in Python and shows how CM maps naturally onto GPU/TPU batch‑parallel execution without requiring quantum simulation or physical quantization. By establishing CM as a practical, hardware‑compatible computational system, this work positions non‑commutative computation as a new domain of computer science. It offers a foundation for deterministic semantic computation, structured exploration, and hybrid symbolic‑algebraic reasoning, and clarifies the structural distinction between CM and quantum mechanics. The framework is intended as a starting point for further theoretical development and engineering‑level applications of non‑commutative computation.