A Spectral Route to the Fine-Structure Constant from a Coefficient-One Relational Law / Exact $A_1$/Eguchi-Hanson response, a positive-norm root, and the remaining Thomson test

I study a coefficient-one relational law and, within a fixed primitive $A_1$/Eguchi-Hanson model, derive a dimensionless spectral coupling without using a measured value of $\alpha$, the electron mass, or a fitted coefficient. Four independently derived response coefficients lead to a two-channel generalized eigenvalue problem whose unique weak positive-norm root is $\lambda_\star=137.0359991770049772294$. An independent C-space reduction gives a nearby numerical check, while identification of the spectral value with the physical Thomson coupling requires a separate photon or static-charge calculation from the same microscopic model. Complete derivations follow in the Supplemental Material.