Wave Confinement Theory Predicts the Koide Mass Relation: A Curvature–Harmonic Origin of Fermion Mass Triplets

Wave Confinement Theory Predicts the Koide Mass Relation
A Curvature–Harmonic Origin of Fermion Mass Triplets

Richard J. Reyes - December 10, 2025

Overview

This paper demonstrates that the Koide mass relation for fermion triplets is a direct consequence of the curvature spectrum of confined toroidal eigenmodes in Wave Confinement Theory (WCT).

Dimensional stability of the WCT curvature functional imposes a universal bound on the poloidal curvature fraction of a spin-s soliton:

η² m² ≤ 2s / (2s + 1)

Stable eigenmodes saturate this inequality. This identifies a curvature harmonic:

K(s)² = 2s / (2s + 1)

which fixes the Koide mass–phase angle through the identification:

cos θ = K(s)

Combining this with Koide’s geometric identity:

Q = 1 / (3 cos² θ)

yields the spin-dependent Koide law:

Q(s) = (2s + 1) / (6s)

For s = 1/2, this gives:

Qℓ = 2/3

exactly.

Key Results

• A dimensional-stability theorem for toroidal WCT eigenmodes fixes a finite curvature budget shared between torsion (spin) and poloidal winding.

• Curvature–Koide invariant:

Q(s) · K(s)² = 1/3

• Mass–phase angle identification:

cos θ = K(s)

• Exact reproduction of the charged-lepton ratio:

Qℓ = 2/3

• Effective-spin interpretation for quark triplets:

s_eff ≈ 0.56
⇒ Q_q ≈ 0.63

• Curvature-suppressed neutrino sector consistent with:

Q_ν ≈ 1/3

Numerical Verification

The paper includes a full extraction pipeline for toroidal WCT eigenmodes.

For the m = 1, spin-1/2 soliton:

Extracted aspect ratio:
η = 0.7070 ± 0.01

Poloidal winding:
m = 1

Numerical curvature parameter:
x_num = η² m² = 0.500 ± 0.015

Theoretical prediction:
x_th = 1/2

Agreement within a few percent confirms the curvature-fraction law.

Relation to Prior WCT Work

This paper builds directly on the curvature formalism developed in:

• Rest Energy from Density-Weighted Loop Curvature (2025)
• Resonant Cavity of Vector Fields (2025)
• The Geometry of Resonance (2025)

These works introduce the Θ-operator, curvature locking, and confined toroidal eigenmodes used here.

Significance

• First derivation of Koide’s relation from a physical curvature mechanism
• Charged-lepton value emerges with no free parameters
• Quark and neutrino Koide patterns follow from the same curvature–harmonic structure

Keywords

Curvature harmonics; mass-phase geometry; Koide relation; fermion triplets; Wave Confinement Theory; toroidal eigenmodes; curvature saturation; dimensional stability; poloidal curvature; torsion budget; spin–curvature partition; effective spin; curvature-locked solitons; geometric mass ratios; curvature–Koide invariant; eigenmode spectrum; nonlinear wave confinement; Θ-operator; geometric unification.

Author & Contact

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