We examine the consistency between the standard nonrelativistic Schrödinger description of the hydrogen atom and the Projective Dynamic Logo (PDL), a relational framework in which elementary entities are represented as minimal logical closures on signed graphs. Without attempting to derive the Schrödinger equation from the axioms of PDL, we demonstrate that, once Schrödinger dynamics for an effective degree of freedom is postulated, both the Coulomb coupling and the fine-structure constant can be reinterpreted as emergent quantities arising from the combinatorial structure of the proton and the elementary (4,6) block associated with the electron. Within controlled uncertainties, the PDL-based expression for the fine-structure constant reproduces its empirical value, and the resulting effective Coulomb potential yields the correct order of magnitude for the hydrogen ground-state binding energy and the Bohr radius. The main contribution of this work lies in providing a structural origin and interpretation of the coupling, rather than proposing any modification of the established Schrödinger phenomenology.