Three Exact Particle Masses from only π and ϕ: A First Principles Proof of the Pi-Phi Light Framework

This paper presents a mathematical proof from first principles, demonstrating that the exact masses of the electron, muon, and proton can be derived using only the constants π (pi) and ϕ (phi), with no fitting, no approximation, and no free parameters. This establishes a foundational result for the Pi–Phi Light Framework, strongly suggesting that mass is governed by an underlying natural law rooted in geometric constants. This work supports and extends the main Pi–Phi Light Framework manuscript, which explores the role of emergent vacuum structures and self-similar scalar interactions in particle mass formation. This mini paper focuses solely on three particles to deliver a precise and unassailable result. The mathematical proof is exact, minimal, and derived from known constants—demonstrating that these masses are not arbitrary but are deeply connected through a geometric structure that predates existing theoretical formulations. Supporting references and uploaded scripts are included to ensure full transparency and reproducibility. This paper also supports ongoing research into scalar resonance observations and theoretical models extending the Standard Model: Observation of Scalar Resonances at 227.78 and 250.00 GeV: Interpretation within a ϕ-Extended Effective Field Theory FrameworkDOI: https://doi.org/10.5281/zenodo.15353023 Observation of Novel Scalar Resonances at 227.8 and 250.0 GeV: Analysis Within an Extended SMEFT Framework and Prediction of a Third ResonanceDOI: https://doi.org/10.5281/zenodo.15341059 If you would like more information about the ENSO Framework, please contact: Eric Needham: [email protected]