From Identity to Instrument: The Algebraic Fine Structure Constant as a Metrological Reference Frame

The self-consistency equation α⁻¹ + S₂·α = 4π³ + π² + π, where S₂ = 1/4! + 3!!/8! = 1683/40320, yields α⁻¹ = 137.035 999 177 from zero empirical inputs — matching the CODATA 2022 recommended value 137.035 999 177(21) to every reported digit, a residual of 0.02σ. This paper establishes three results. First, the equation is not an approximation: a round-trip computation π → α → π recovers π to the full working precision of the calculation (tested to 5,000 decimal places; limited only by allocated compute). Second, both α and π appear in Schwinger's formula for the electron anomalous magnetic moment aₑ = (1/2)(α/π) + ⋯ — and both are determined by the same equation. The quantity (1/2)(α/π) is therefore not…