A math

We certify a dimensionless spectral constant g arising from a self– adjoint anisotropic Laplace–Beltrami operator on a three–torus with golden– ratio anisotropy, using rational–interval arithmetic and explicit tail bounds. The constant is enclosed to forty digits with a provably valid rational in- terval and a cryptographic reproduction hash. We then state a single physics identification postulate: upon normal- ising the length scale by one low–energy atomic transition (measured to ∼12 digits), the same dimensionless constant coincides numerically with the inverse fine–structure constant α−1 to all digits currently measured. No claim is made that this constitutes a pure–mathematical proof of α. Instead, this work certifies a spectral invariant and isolates its physical identification as an explicit assumption. We also enumerate open assumptions regarding scheme–independence beyond 12 digits and the absence of hidden tuning. This manuscript provides (i) rigorous convergence and uniqueness re- sults, (ii) a certified rational interval for g accurate to 10−40, (iii) a proof of regulator equality (heat–kernel vs. zeta), and (iv) a numerical appendix including SHA-256 hashes for reproducibility.