Derivation of the Koide Relation from Coherence Shell Geometry in the Sobolev-Ozok Lattice (SOL) Framework

Koide’s relation, remains one of the most precise and unexplained empirical regularities in particle physics. Within the Standard Model, fermion masses arise from independent Yukawa couplings, and no structural mechanism exists that predicts Koide’s ratio or explains its restriction to the charged-lepton sector. In this work, Koide’s relation is derived from first principles within the Sobolev-Ozok Lattice (SOL) framework. The derivation relies only on the SOL mass primitive that particle mass arises from integrated third-order discrete Sobolev curvature energy and on a spectral admissibility condition governing the continuum limit at resolution order k=3. No Yukawa couplings, flavor symmetries, or numerical fitting are introduced. The analysis proceeds by reducing the discrete curvature functional to a radial coherence shell description, computing the exact third-order discrete difference for harmonic shell modes, and performing a controlled continuum approximation using Euler-Maclaurin summation. Koide’s ratio is then reformulated geometrically in terms of the angle between the charged lepton mass root vector and the isotropic closure direction. Spectral admissibility under lattice refinement enforces a balance between isotropic (closure) and deviatoric (excitation) curvature components, fixing this angle to 45 degrees and yielding Koide’s relation exactly. This result explains both the numerical value 2/3 and the restriction of Koide’s relation to charged leptons, which form the minimal three-state family compatible with the k=3 closure condition. The work does not modify the Standard Model Higgs mechanism but instead provides a geometric explanation for a mass pattern that remains unexplained within the Yukawa framework. The paper is intended as a focused contribution to the theoretical understanding of fermion mass regularities and as part of a broader investigation into the spectral and continuum properties of the Sobolev-Ozok Lattice framework. Declaration of Tools Used: This manuscript was prepared and typeset using LaTeX via Overleaf. Language refinement and stylistic polishing were assisted by the Overleaf AI Editor. All scientific content, mathematical derivations, conceptual development, and conclusions are original and authored by the undersigned.