Discrete Geometric Origins of Standard Model Mass Ratios in the Jeandel-Rao Aperiodic Tiling

• 1. Massachusetts Institute of Technology

Description

This paper proposes a theoretical isomorphism between the Jeandel-Rao Minimal Aperiodic
Wang Shift and the structure of the Dirac Spinor. By analyzing the tiling as a dynam-
ical spacetime propagator, we demonstrate that fundamental constants of the Standard
Model emerge as combinatorial counting arguments within the phase space of the tiling.
We successfully derive the Fine Structure Constant (α−1 ≈ 137), the Proton Mass
(μ ≈ 1836), the Muon Mass (≈ 207), the Weak Mixing Angle (sin2 θW ≈ 0.231), the
Cabibbo Angle (sin θc ≈ 0.225), and the Higgs Boson Mass (mH ≈ 125 GeV) using
only the invariant integers of the lattice (N = 11 tiles, V = 4 vectors). We demonstrate
that fractional deviations in these constants are not arbitrary, but correspond precisely to
the geometric cross-sections of “Sturmian worms” (topological defects) required to enforce
aperiodicity.

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