Emergent spectral determination of the fine-structure constant in a quartic variational framework

We develop a complete and self-contained derivation of a rigid quartic variational functionaland of its coherent configuration Ψ⋆. The linearized operator obtained from thesecond variation decomposes into radial, torsional, and transverse sectors and contains aunique admissible first-order differential contribution Hloc = Vμ(Ψ⋆) ∂μ, fully determinedby the quartic structure and its projective constraints. No external parameters, scales, orphysical inputs enter the construction.A global analysis of the coherent domain shows that Hloc acts necessarily on a compactisotropic three–manifold of constant curvature, which is uniquely realized as S3.The variational constraints force Vμ to be a Killing field on this domain, implying thatHloc generates a one-parameter subgroup of isometries and possesses a discrete purelyimaginary spectrum of the form λn = inν. The norm ν is completely determined by theisotropic decomposition of the Hessian into its radial and torsional components, yieldingthe unique value ν = 1/96.These results imply that the torsional spectral shift scales as α3/2 and that the finestructureconstant is the unique positive solution of the spectral equationα − 1 −196α3/2 = 0.This establishes α−1 = 137.036 . . . as a strict spectral invariant of the quartic functional.The work is therefore structurally complete, mathematically closed, and numericallyinevitable within the variational framework.