Wave Confinement Theory (WCT)
Published September 16, 2025 | Version v5
Preprint Open

Self-Emergent Fourier Cymatics: Entropic Eigenmodes out of Chaos

Authors/Creators

  • 1. Independent

Description

Self-Emergent Fourier Cymatics: Entropic Eigenmodes out of Chaos
Richard J. Reyes - September 16, 2025

GitHub Repository: github.com/rickyjreyes/geometry_of_resonance
YouTube: https://youtu.be/zn570q06gO0?si=318Glpib9h6I9fS7

This work presents a formal derivation and numerical study of self-emergent Fourier eigenmodes arising from curvature-regulated wave confinement. Starting from the Wave Confinement Theory (WCT) Lagrangian with curvature feedback and entropy suppression, we derive a Lyapunov-stable evolution equation that drives arbitrary random seeds into coherent, quantized Fourier structures. The dynamics exhibit spectral narrowing, entropy reduction, and spontaneous selection of discrete harmonic supports, without external tuning.

Established in the paper

  • Curvature feedback operator
    Θ[ψ] = −∇²ψ / (ψ + ϵe⁻ᵅ|ψ|²)

    Key results

    • Energy monotonicity is ensured by a Lyapunov functional, driving the system toward coherent states.

    • Fourier-space entropy collapses into a finite mode set K*.

    • The dynamics reveal an ontological flow: from a trivial 0D seed to structured eigenmodes via curvature–entropy coupling.

    • Coherent projection methods isolate stable modes and reconstruct observable confined fields.

Contribution
Constructively demonstrates that chaotic fields evolve toward self-organized, entropic eigenmodes governed by curvature and confinement, generalizing cymatic structures into a spectral framework and providing a rigorous bridge from random initial conditions to coherent physical forms. Extends the WCT program on curvature-regularized Lagrangians, spectral entropy functionals, and emergent harmonic quantization across physical systems. Relevant to condensed matter, plasma, and field-theoretic contexts with a reproducible formulation for spectral self-organization.

Keywords
Wave confinement; entropic eigenmodes; spectral entropy; Lyapunov descent; curvature feedback; Swift–Hohenberg; complex Ginzburg–Landau; Fourier band selection; coherence collapse; emergent order.

---

For correspondence regarding this work, please contact [Richard J. Reyes] at [reyes.ricky30@gmail.com].

ORCID iD: 0009-0005-5975-8718.

Files

Self-Emergent Fourier Cymatics Entropic Eigenmodes out of Chaos.pdf

Files (12.7 MB)

Additional details

References

  • Swift, J., & Hohenberg, P. C. (1977). Hydrodynamic fluctuations at the convective instability. Physical Review A, 15, 319.
  • Cross, M. C., & Hohenberg, P. C. (1993). Pattern formation outside of equilibrium. Reviews of Modern Physics, 65, 851–1112.
  • Aranson, I. S., & Kramer, L. (2002). The world of the complex Ginzburg–Landau equation. Reviews of Modern Physics, 74, 99–143.
  • Derrick, G. H. (1964). Comments on nonlinear wave equations as models for elementary particles. Journal of Mathematical Physics, 5, 1252.
  • Pohozaev, S. I. (1965). On the eigenfunctions of the equation ∆u + λf(u) = 0. Soviet Mathematics Doklady, 6, 1408.
  • Risken, H. (1984). The Fokker–Planck Equation: Methods of Solution and Applications. Springer Series in Synergetics, Vol. 18. Springer.
  • Cercignani, C. (1988). The Boltzmann Equation and Its Applications. Applied Mathematical Sciences, Vol. 67. Springer.
  • Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, Vol. 44. Springer.
  • Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, Vol. 840. Springer.
  • Wilczek, F., & Zee, A. (1984). Appearance of gauge structure in simple dynamical systems. Physical Review Letters, 52, 2111.
  • Reyes, R. J. (2025). The Geometry of Resonance: Wave Confinement Theory and the Emergence of Mass, Force, and Spacetime. Zenodo. https://doi.org/10.5281/zenodo.15596126