This letter reports an approximate numerical relation between two independently measured quantities of distinct physical origin: the photon density parameter of the Cosmic Microwave Background (Ωγ) and the square of the fine structure constant (α²). This manuscript explicitly details the observation, quantifies its sensitivity to the inferred Hubble constant, and provides a balanced discussion of the arguments for and against its underlying physical significance. It is presented purely as an empirical relation in its simplest form to identify future measurements that will determine whether it reflects an unrecognized primordial thermodynamic constraint or a mere numerical coincidence.