A Fixed Point of a Self-Coupled Clifford Algebra: The Fine Structure Constant From Algebraic Structure

We derive a distinguished real number from the algebraic structure of the Clifford algebra Cl(1, 6). Starting from a first-order nonlinear equation (∇Ψ)B = ½(Ψ − Ψ₀)e₀(Ψ̃ − Ψ*) with B = e₁e₂ a fixed bivector, we show that the equation's self-consistent coupling constant satisfies α⁻¹ = 128 + π² − 5/6 − cα², where 128 = dim Cl(1, 6), π² is the spectral stiffness of six compact internal winding circles, 5/6 is the residual-stiffness deficit of the winding-phase mode (a would-be Goldstone of the equation's free part, lifted by the selector's non-covariance to residual stiffness 1/6 on the (2+1)-dimensional B-core of the defect's configuration space ℝP³), and c = 69143/13542 is a rational number determined by the grade-flow dimensions of the algebra. The unique weak-coupling positive root is α⁻¹ = 137.035999175982235340655171482…, agreeing with the CODATA 2022 recommended value α⁻¹_exp = 137.035999177(21) at 0.007 ppb (0.05σ). Every coefficient in the formula is traced to a specific combinatorial, geometric, or spectral invariant of Cl(1, 6). No free parameters appear.