A Geometric Ansatz for Fermion Hierarchies from a Conical Fibonacci Spiral

We present a geometric ansatz in which the effective Froggatt–Nielsen charges of Standard Model fermions emerge from a single formula involving the golden ratio φ = (1 + √5) / 2. The construction is motivated by a conical Fibonacci spiral characterized by the golden angle Θ ≈ 137.5°. Within this framework the three fermion sectors are encoded by a unified expression that produces the charge patterns (11,6,4), (9,6,3), and (10,4,0) without introducing adjustable charge parameters. The resulting hierarchy can be expressed in the usual Froggatt–Nielsen form mᵢ ~ cᵢ εⁿⁱ v with order-one coefficients cᵢ. A geometrically motivated parameter ε ≈ 0.309 reproduces the qualitative ordering of the observed fermion masses when combined with these charges. The proposal should be regarded as a phenomenological geometric ansatz rather than a complete ultraviolet theory. Nevertheless, the construction provides a compact and internally consistent way to organize the fermion hierarchy using only the golden ratio and discrete spiral structure. Possible phenomenological implications of the geometric interpretation include structures in the ~10–13 GeV missing-energy range in certain B-meson decay channels, although such implications remain speculative without a fully developed dynamical model.