Solving for π: Recovering Geometry from Physics

The self‐consistency equation was introduced in a companion paper [1] as a constraint on the fine structure constant 𝛼. That paper solved the equation in the forward direction: given 𝜋 and factorial series 𝑆, compute 𝛼−1.

This paper reverses it. Given a measured value of 𝛼 from atomic physics and the factorial series 𝑆 —neither of which contains 𝜋 — solve the resulting cubic equation for 𝜋. Using the most precise measurement of 𝛼 available ﴾Fan et al., 2023, cesium recoil﴿, the procedure recovers 11 digits of 𝜋: 3.14159265359… The Morel et al. ﴾2020, rubidium recoil﴿ and Hanneke et al. ﴾2008, electron 𝑔−2﴿ measurements recover 9 digits each. A self‐consistency check using the formula’s own 𝛼 recovers 𝜋 to 97 digits —the full working precision of the computation.

No 𝜋 enters the inputs. Factorials and a cesium atom go in. 𝜋 comes out. The number of recovered digits is limited only by the precision of the 𝛼 measurement. The script is 150 lines of Python, runs in under a second, and is publicly available. It was run on Python 3.14. Obviously.