Resulting in quantized eigenfrequencies consistent with resonant shells in an expanding universe.
A.2. Embedding Fractal Spiral Boundary Conditions
A fractal-harmonic boundary constraint is introduced using a logarithmic spiral embedding:
r(\theta) = r_0 e^{b \theta}
This spiral constraint modulates the angular component of the eigenfunctions, leading to mode nesting and shell-like structures consistent with Nautilus-like geometry.
A.3. Mode Coupling Formalism
Second-order perturbation theory is used to compute nonlinear mode coupling effects:
\Phi = \Phi^{(0)} + \epsilon \Phi^{(1)} + \epsilon^2 \Phi^{(2)} + \dots
Cross-terms in the Lagrangian are retained to model interaction between eigenmodes, yielding predictions of density contrasts and filament alignments.
Appendix B: Numerical Code and Simulation Parameters
B.1 Simulation Setup
