The fine-structure constant αα is traditionally a free parameter of quantum electrodynamics. Here we propose a different view: α−1α−1 is the expectation value of a quantum system whose Hilbert space consists of all finite continued fractions with partial quotients bounded above by 45. We have found an expression α−1=A(π)−1/(24A(π))−1/(A(π)2π2K)α−1=A(π)−1/(24A(π))−1/(A(π)2π2K), where A(π)=4π3+π2+πA(π)=4π3+π2+π. For the CODATA 2022 value, the self-consistency equation yields K≈9.9327912864K≈9.9327912864, whose continued fraction expansion is K=10−1/(14+1/(1+1/(7+1/(3+1/(1+1/3)))))K=10−1/(14+1/(1+1/(7+1/(3+1/(1+1/3))))). We elevate KK to an operator K^K^ acting on states of finite continued fractions. The dimensionless structure operator is S^=A(π)−1/(24A(π))−1/(A(π)2π2K^)S^=A(π)−1/(24A(π))−1/(A(π)2π2K^), where A′′′(π)=24A′′′(π)=24 is the third derivative of the polynomial. The ground state is a superposition of sequences with distribution P(q)∝e−q/10P(q)∝e−q/10. Numerical simulations (1 million collapses, depth 20, 200-digit precision) yield ⟨S^⟩=137.03599916781⟨S^⟩=137.03599916781, differing from CODATA 2022 by −9.19×10−9−9.19×10−9. The standard deviation is 1.40×10−81.40×10−8, matching historical fluctuations. The sequence [14,1,7,3,1,3][14,1,7,3,1,3] never appeared, consistent with its theoretical probability ∼1.2×10−10∼1.2×10−10. The model is falsifiable: a future measurement requiring a quotient q>45q>45 would invalidate it. The code is publicly available for independent verification.