This work derives a model-independent constraint linking the proton radius and electromagnetic coupling from dimensional crossover. As the effective spacetime dimension flows toward D_eff -> 2 in the ultraviolet, bound-state stability imposes the relation r^(D_eff) ~ alpha^(-1). This establishes that the proton radius and fine-structure constant are not independent parameters, but are jointly determined by dimensional flow. The result follows from scaling structure and does not depend on model-specific assumptions. The framework yields a numerical scale consistent with experimental measurements and produces a direct, falsifiable prediction relating proton radius variation to the running of the electromagnetic coupling. A statistically significant deviation from this predicted correlation falsifies the framework. This provides a structural constraint that must be satisfied by any ultraviolet-complete theory exhibiting dimensional reduction.