The Fine-Structure Constant from First Principles

The fine-structure constant $\alpha \approx 1/137$ governs electromagnetic interactions, yet no theory explains why it takes this value. We derive $\alpha$ from first principles within Relational Physics, a framework that bridges coalgebraic structures to physical observables. The starting point is a single categorical definition: the final coalgebra for a ternary branching functor. This algebraic structure uniquely determines a geometric realization—the srs (Laves) graph, a chiral trivalent lattice with shortest cycles of length 10. The bare coupling emerges as the dimension of the configuration space for minimal loops: 128 spinorial degrees of freedom (from binary path choices after frame-fixing) plus 9 gauge degrees of freedom (from the local unitary symmetry at each vertex), giving 137. A variational principle on loop space yields quantum corrections through a self-consistent equation, whose solution is $\alpha^{-1} = 137.035999187$. This agrees with the experimental value $137.035999177(21)$ to within $0.5\sigma$—ten significant digits with no adjustable parameters.