A Geometric Origin of a GeV-Scale Triplet in Conical Fibonacci Hierarchies

This work explores a geometric mechanism generating hierarchical mass spectra using a conical Fibonacci spiral.

The construction introduces two independent hierarchical directions: a radial hierarchy governed by a small parameter ε and an angular hierarchy governed by the golden ratio φ.

The resulting angular hierarchy naturally produces a GeV-scale triplet

(1.5, 2.4, 3.9) GeV

with approximately logarithmic spacing.

Interestingly, the central scale lies close to the region currently explored in searches for invisible particles in B⁺ → K⁺ + invisible decays at the Belle II experiment. Recent analyses suggest a possible excess compatible with a resonance near

mₓ ≈ 2.1⁺⁰·²₋₀·¹ GeV.

Within the geometric framework considered here, the scale around 2.4 GeV appears as the geometric center of the hierarchy.

Future Belle II statistics will clarify whether such a geometric hierarchy may be present in the data.