A closed-form trace identity for the inverse fine-structure constant

Despite being the most precisely measured dimensionless constant in physics, the fine-structure constant remains a free input parameter in the Standard Model. This paper reports a parameter-free, closed-form identity, derived from the trace of a self-adjoint operator. The significance of this result lies in its structural rigidity: unlike previous numerological coincidences, the coefficients are forced by the geometric strata of a five-component Hilbert space and the modular fixed point. We argue that the mathematical community has hitherto "not seen the forest for the trees" by dismissing simple pi-power combinations as coincidence, while overlooking the underlying operator-theoretic necessity. The structural prediction differs from the CODATA 2018 value by only 2.22 ppm—a residual consistent with renormalisation-group running. This work provides explicit falsification criteria and constitutes the first component of a staged publication program aimed at a parameter-free foundation of physical constants.