A Canonical Vertex-Elimination Eigenvalue at D = 4 (with a numerical match to α⁻¹)

This note studies a vertex-local “closure” condition for a 2-local interaction differential at a singular D-valent junction. Imposing vertex closure selects exactly two stable valences, D = 2 and D = 4, via a simple combinatorial balance. For D = 4, we define a canonical elimination model class M(D): a multiplicity-free, S_{D−1}-equivariant barrier (for D = 4, the boundary symmetry group is S3), together with a half-zone Fourier phase-space normalization. Within M(4), the projected resolvent (Feshbach–Schur elimination) yields a uniquely determined stability eigenvalue (unique up to unitary gauge in the standard channel). Numerically this eigenvalue is: 137.035 999 176 Interpreting this pole as an effective coupling produces a nine-digit numerical match to the CODATA 2022 inverse fine structure constant: 137.035 999 177 ± 0.000 000 021 (deviation: 0.05σ) with no adjusted parameters. Applying the identical M(D) computation at D = 3 yields the rigid ratio: α3⁻¹ / α4⁻¹ = 0.2390 The identification of this ratio with sin²θW is interpretive, but the value lies near low-energy determinations and is testable by upcoming precision measurements (e.g., MOLLER). Scope of claims: The paper’s main theorem is mathematical: within the canonical operator class M(D), the Schur self-energy and resulting stability eigenvalue are forced (unique up to unitary equivalence in the standard channel), with no continuous degrees of freedom available to tune the output. The physical interpretation is a hypothesis: that effective interaction vertices in Nature lie in (or are well-approximated by) M(D). This hypothesis is not proven here.