Geometric Derivation of the Hydrogen 21 cm Line and the Proton g-Factor: A k-Foam Approach

Description The hydrogen 21 cm line is a key observational signal in astronomy, widely used for probing large-scale structure and the interstellar medium. Within the standard framework of quantum mechanics, it is described as arising from hyperfine splitting, with its value determined by measured physical constants. This working paper, developed within the k-Foam framework, explores an alternative geometric interpretation of the 21 cm wavelength in terms of a discrete spatial structure. In this approach, the electron spin-flip transition is reinterpreted as a localized topological relaxation process within a k = 6 network. The study investigates whether the resulting emission scale can be related to geometric and combinatorial properties of the underlying structure, and whether similar mechanisms might also provide a geometric perspective on large-scale phenomena such as redshift. Key Observations and Hypotheses: - On the proton g-factor: A geometric construction based on the regular octahedral network suggests a value close to the experimental proton g-factor, with a relative difference of approximately 0.0015%, which may be noteworthy given the simplicity of the geometric construction. - On the structure of the 21 cm scale: By combining several geometric relations within the model, a characteristic wavelength close to the observed 21 cm scale can be obtained, suggesting a possible link between discrete topology and hyperfine structure. - On parameter dependence: The formulation attempts to minimize the introduction of free parameters, instead expressing relevant scales in terms of simple geometric and numerical relations. - On broader interpretation: The framework raises the possibility that both microscopic transitions and large-scale propagation effects could be described within a common geometric picture, though this remains speculative. This work is exploratory in nature and does not claim a definitive derivation of the 21 cm line. Rather, it proposes a geometric perspective in which known physical quantities may admit alternative interpretations in terms of discrete structure and topology.