Resonant Spacetime Hypothesis 2.0

For a universe with Poincar dodecahedral topology, eigenmodes are derived from the icosahedral symmetry group (order 120). The harmonics \( \psi_{nlm} \) satisfy:  

\[\Delta \psi_{nlm} = -\lambda_n \psi_{nlm},\]  

where \( \lambda_n \) are eigenvalues determined by the dodecahedron's geometry. These harmonics form a basis for scalar, vector, and tensor perturbations, with eigenvalues tabulated using group-theoretic methods (e.g., \( \lambda_1 \approx 20.0/R^2 \), where \( R \) is the circumradius).  

3. Mode Coupling via Perturbation Theory

Nonlinear structure growth arises from mode interactions. Expanding the density contrast \( \delta \rho \) to second order:  

\[\delta \rho(\mathbf{x}) \sim \sum_{n,m} A_n A_m \psi_n(\mathbf{x}) \psi_m(\mathbf{x}),\]  

where \( A_n \) are Gaussian amplitudes (\( \langle A_n A_m \rangle = P(k_n) \delta_{nm} \)) and \( P(k) \) is the primordial power spectrum. This coupling generates:  

- Filaments at intersections of constructive interference (\( \psi_n \psi_m > 0 \)).  

- Voids at nodes (\( \psi_n \psi_m \approx 0 \)).  

Fractal Helmholtz Equation

The Helmholtz equation on a fractal 3-manifold with Hausdorff dimension \( D \) is: