Logarithmic Curvature Flow, Filament Localization, and the Geometric Origin of the Lepton Mass Spectrum

Logarithmic Curvature Flow, Filament Localization, and the Geometric Origin of the Lepton Mass Spectrum
Richard J. Reyes - March 10, 2026

Overview

This document develops a mathematical framework within Wave Confinement Theory (WCT) in which particle masses arise from localized curvature modes of a nonlinear complex wavefield.

The work shows that the curvature–feedback operator governing the wavefield admits a logarithmic representation that transforms the dynamics into a viscous Hamilton–Jacobi equation. Through the Cole–Hopf transformation this nonlinear flow linearizes to a diffusion equation, implying that stable localized states cannot arise from local nonlinear trapping alone and instead require global topological confinement.

Within closed or periodic geometries the dynamics organize into filamentary curvature modes whose core curvature invariant determines the effective inverse length scale of the confined state and therefore its rest mass scale.

Core Result

The analysis demonstrates that curvature localization along closed filaments produces a periodic curvature spectrum. Under weak modulation this spectrum takes the form

σ(ϕ)≈(a+bcos⁡ϕ)2.\sigma(\phi) \approx (a + b \cos \phi)^2 .σ(ϕ)≈(a+bcosϕ)2.

Discrete phase offsets separated by 2π/32\pi/32π/3 generate three curvature eigenvalues whose square roots lie on a circle in m\sqrt{m}m-space.

This geometric structure reproduces the known Koide relation for the charged lepton masses.

Key Mathematical Structure

The derivation proceeds through the following chain:

curvature–feedback operator
↓
logarithmic field transform
↓
viscous Hamilton–Jacobi equation
↓
Cole–Hopf linearization
↓
diffusion dynamics
↓
topological confinement
↓
filament localization
↓
periodic curvature spectrum
↓
Koide mass geometry.

The curvature flow equation derived in this work belongs to the deterministic Kardar–Parisi–Zhang (KPZ) universality class, providing a connection between the WCT curvature dynamics and well-studied nonlinear growth equations.

Dimensional Stability

Scaling analysis of the logarithmic curvature flow identifies a critical spatial dimension

dc=3d_c = 3dc=3

at which the nonlinear gradient term becomes scale invariant. This suggests that stable curvature confinement occurs naturally in three spatial dimensions.

Scope and Interpretation

This work develops the mathematical structure linking curvature flow, topological confinement, and mass geometry. It does not claim a complete microscopic model of particle physics.

In particular, the following aspects remain open problems:

• determination of the absolute mass scale,

• selection of the phase on the Koide circle,

• derivation of discrete curvature shell eigenvalues for specific particle sectors.

The paper therefore establishes a geometric mechanism capable of generating the observed lepton mass relation, while leaving the full spectral problem for future work.

Relation to Prior Work

This manuscript builds on earlier Wave Confinement Theory papers developing curvature-based descriptions of mass, resonance, and confinement dynamics.

The present work specifically extends these ideas by:

• deriving the logarithmic curvature flow representation,

• identifying the Cole–Hopf diffusion structure,

• linking the curvature spectrum to the geometric form of the Koide relation.

Significance

If the curvature–filament framework is correct, the charged lepton mass hierarchy emerges as a geometric property of confined curvature modes rather than a set of independent fundamental parameters.

The results suggest that particle masses may reflect eigenvalues of a curvature spectrum determined by the topology and geometry of the underlying wavefield.

Keywords

wave confinement theory; curvature flow; logarithmic curvature operator; filament localization; KPZ equation; Cole–Hopf transform; curvature invariants; toroidal eigenmodes; lepton mass spectrum; Koide relation; geometric mass models; nonlinear wavefields; topological confinement.

Author & Contact

Author: Richard J. Reyes
ORCID iD: 0009-0005-5975-8718
Email: reyes.ricky30@gmail.com