We derive the fine-structure constant α from six properties that anyquantum coupling constant must satisfy: scale invariance, self-reference,cumulativity, probability normalization, self-consistency, and contractivity.These chain together with zero tunable parameters to produce a uniquefixed-point equation erfi(b)² = e^(b²), whose solution b* generates afive-letter algebraic alphabet. A convergent Riccati series makes thederivation analytic, extending α to unlimited precision.Three closure theorems force every component of the master equationα = R/(D_M − R α L^d): (I) a universal escape identity constrains theprojection ratio R; (II) Hurwitz's division algebra theorem forces d = 4;(III) Euler's ζ(2) = π²/6 organizes the master denominator D_M.Theorems II and III unify as facets of the graded real spectral tripleon S¹_{2π}, whose chirality grading forces the spectral convention uniquely(the competing convention is excluded at 134σ). The Padé form of theself-energy Σ_2 — previously the last remaining assumption — is nowderived: the fixed-point equation forces constant coupling (β = 0),which forces the Dyson self-energy series to be geometric, closing toa rational function with a single simple pole. A spectral dimension jump(Gabriel's Horn) connects the logarithmic branch point of the Riccatiseries (radius r = 2π = det'(D), exponent consistent with p = −1;extracted from 50 coefficients via Wynn acceleration) to the convergentspectral zeta value ζ(2) that organizes D_M.The result, 1/α = 137.035 999 075, agrees with CODATA 2018 at 0.4σ andwith Parker et al. (2018) at 1.1σ. The 4.9σ tension with CODATA 2022constitutes a falsifiable prediction. The same alphabet reproducesα_s(M_Z) (0.11σ), sin²θ_W (0.25σ), λ_H (0.03σ), m_μ/m_e (0.2σ),|V_us| (0.9σ), and Newton's constant (0.15σ) — seven observables froma single transcendental number with zero free parameters. No theorempostdating 1954 enters the derivation chain; the spectral-geometricinterpretation of the denominator draws on Connes (1994).