On the Emergence of the Fine-Structure Constant from Minimal Null Geometry

We investigate whether the numerical value of the fine-structure constant can emerge from a minimal geometric structure rather than being introduced as an empirical parameter. Starting from a system of three null directions in Minkowski space subject to a global closure constraint, we develop a fully geometric and statistical framework in which fluctuations, phase dynamics, and correlation structure arise naturally. The geometry of fluctuations leads to a well-defined phase variation, which in turn determines an effective coupling through stability conditions. We show that the combination of constrained fluctuation geometry, phase modulation, and multi-channel correlations yields a geometric estimate of the coupling constant. Incorporating phase coherence through Gaussian averaging and a scale-invariant fluctuation spectrum, we derive a self-consistent equation for the coupling. A key result is that both the logarithmic fluctuation spectrum and the infrared scale emerge intrinsically from invariance principles and the structure of line-integral observables, rather than being externally imposed. The global coherence scale is identified with a temporal scale associated with causal propagation. The resulting fixed-point equation admits a unique and stable solution, yielding a value of the fine-structure constant in close agreement with the observed value (alpha inverse approximately 137). The derivation does not introduce new particles or interactions. Instead, the fine-structure constant appears as an emergent property of minimal null geometry, constrained correlations, and global phase coherence.