A Geometry-Selected Robin-Spectrum Dressing Mechanism Linking 137 to the Low-Energy Fine-Structure Constant

We propose a geometry‑selected Robin boundary condition applied to the radial (ℓ = 0) sector of the Helmholtz equation on a unit sphere to derive a dimensionless eigenvalue ratio \u03c1 that approximates the inverse fine‑structure constant. A boundary‑dressing correction \u03ba, obtained from a convergent spectral sum over the same Robin eigenproblem, modifies \u03c1 and yields \u03b1⁻¹ within CODATA uncertainty. The mechanism is falsifiable: replicators need only solve the one‑dimensional radial Robin problem or extract the \u2113 = 0 component of the three‑dimensional Laplacian with the specified sign convention. We provide sealed code bundles for hostile‑audit validation and independent replication, along with a public glossary and internal validation report. For questions or collaborations contact Natalya Goldschmidt at natalyagolds@gmail.com.