Mode Hierarchy and Geometric Gauge Unification

This work introduces the Konno Frequency–Thickness Law, a novel geometric formulation linking the vibration frequency (ν) of a closed surface φ to its imaginary-normal thickness Δ(iφ). The product ν·Δ(iφ) is shown to remain constant, inversely proportional to the material density ρ: This law generalizes the classic wave relation (νλ = c) by embedding it within a surface-based model of physical fields. High-frequency modes with ultra-thin φ-structures represent particle-scale fields such as electrons and photons, while low-frequency, thick φ-configurations correspond to gravitational modes. The equation provides a unified framework in which electromagnetic and gravitational phenomena are seen as dual modes of the same surface geometry, and the apparent gauge diversity across interactions arises from discrete modal indices n. The relation 1/ρ=cosθ further connects this law to Lorentz contraction and relativistic geometry. The Konno FT-Law offers a new invariant linking field vibration, spatial structure, material density, and relativistic deformation. Author: Tetsuo Konno(今野哲夫) https://orcid.org/0009-0002-1842-7865