What question did this study set out to answer?

This research aims to derive six fundamental constants of particle physics using geometric and algebraic inputs without free parameters.

May 10, 2026Open Access

From the Fano 2-22 to the Lepton Masses e Lamb Shift - No Free Parameters

Key Points

This research aims to derive six fundamental constants of particle physics using geometric and algebraic inputs without free parameters.
Derivation of constants from Fano 2-22 and PEPS-5D Lagrangian structures.
Geometric relationships exploited for calculations of the fine-structure constant, electron mass, and Lamb shift.
Topological series utilized for the anomalous magnetic moment and mass ratios between leptons.
Fine-structure constant calculated as 137.03599916781, aligning with established values.
Electron anomalous magnetic moment matches experimental results within a precision of 1.33×10^-10.
Lamb shift computed as 1057.8193 MHz, differing from experimental data by only 7.3 kHz.

Abstract

We derive six fundamental constants of particle physics from the entangled topology of the Fano 2-22 (Mori-Mukai ID-69) and the PEPS-5D Lagrangian: 1. The inverse fine-structure constant: \ (₌₏ₒ^-1 = _ - = 137. 03599916781\) from a circular MPS with bond dimension \ (D=45\). 2. The electron anomalous magnetic moment: a closed topological seriesₑ^geom = R (1 - S105) + R² (T ₓ₇ 2425) - R³ (T 2425), \ (R = ₌₏ₒ/2\), yielding \ (aₑ^geom = 0. 001159652137\), agreeing with experiment to \ (1. 3310^-10\). 3. The electron mass: ₑ^phys = mₑ^bare + 110 (mₑ^bare (aₑ2 + ²2) ) = 510998. 888\ eV, \ (mₑ^bare = M₏₋ (-105-22 + S (1 - s) ) \), differing from the CODATA value \ (510998. 900\ eV\) by only \ (0. 012\ eV\) (12 meV). 4. The muon/electron mass ratio: \₌䂰 = 4 + 43 = 5. 333\ 5. The tau/muon mass ratio: \₌_ = T = 22 = 2. 8284271247461903\ 6. The Lamb shift in hydrogen: \ E₋₀₌₁^geom = fᵥ (A₅ - R\, A₆' - R² A₇) = 1057. 8193\ MHz, \ (fᵥ = ⁵ mₑ^physc²/h\), \ (A₅ = 2425ₓ₇\), \ (A₆' = T + 2 - I₀2\) with \ (I₀ = 4/ (3) \), and \ (A₇ = 4 + 1\). The difference from the experimental value \ (1057. 812\ MHz\) is \ (7. 3\ kHz\), well within the uncertainties of the proton radius puzzle. All quantities are derived from pure geometric and algebraic inputs: the bond dimension \ (D=45\) (adjoint of \ (SO (10) \) ), the Hilbert space dimension \ (105\) (adjoint of \ (SO (15) \) ), the Euler characteristic \ (=6\), the string tension \ (T=8\), the Tsirelson bound \ (T=22\), the entanglement deficit \ (S = T - 7\) from the hinge term phase \ (/6\), the Picard-Fuchs threshold correction \ (ₓ₇\), and the bosonic string parameters \ (24\) and \ (25\). No free parameters are introduced. The electron, muon and tau are the first three excitation modes of the same geometric structure. The absence of a fourth charged lepton is explained by the saturation of the Tsirelson bound. The Lamb shift emerges as a three-term topological series — impact, rotation, back-reaction — mirroring the structure of \ (aₑ\). The complete Python verification code (Derivationₗeptonsₘass+LambₛhiftfromPEPS-5DMPS-Fano₂₂2. py) is available at the Zenodo repository associated with this work (DOI: 10. 5281/zenodo. 20077021).

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