One axiom. One operation. Zero free parameters. The fine-structure constant α is the most precisely measured pure number in physics and the most conspicuously unexplained — for a century an input, never derived. This paper derives it. From a single axiom (the One) and a single operation (the fold), the inverse fine-structure constant is forced to 1/α = 2⁷ + 3²(251/250) = 34259/250 = 137.036, matching CODATA (137.035999177) to eight significant figures, six parts per billion, with zero free parameters. Each block is an independent structural count and the assembly is unique among 41,472 structural alternatives. The covering volume is itself a covered object, so the same construction continues to its own next self-similar order with no new ingredient, giving 1/α = 5995462/43751 = 137.0359991772 — level with measurement to about 0.01σ. No measured value enters the construction; this is proven as a property of the code, machine-checked by a single verifier under a static gate, and reproduces from one command. A standalone result within the Smithian Fold Theory of Everything (SFTOE). Full corpus, code, and the run-it-yourself VERIFY.md protocol: https://github.com/MettaMazza/Smithian-Fold-Theory — SFTOE record: https://doi.org/10.5281/zenodo.20775538