where are the eigenfunctions of the spatial Laplace--Beltrami operator on :
\Delta \psi_n + \lambda_n \psi_n = 0
and are the discrete eigenvalues determined by the boundary topology. These form a resonant spectrum of modes whose interference patterns (superpositions of ) create primordial structure templates.
3.3 Embedding Fractal Boundary Conditions and Spiral Harmonics
To reflect the observed fractal-like structure of the cosmic web and potential self-similarity in galaxy distributions, we propose that the boundary condition space itself has fractal geometry. This can be modeled using Hausdorff dimensions and recursive spatial embeddings.
The harmonic modes are no longer pure spherical harmonics , but modified to include spiral (helical) harmonics, such as:
\psi_{nkl}(r, \theta, \phi) = R_{nk}(r) \cdot \Theta_l(\theta) \cdot e^{i(m\phi + \alpha r)}
where encodes radial phase twisting, enabling logarithmic spiral features reminiscent of biological morphogenesis and galactic morphologies.
Additionally, we introduce the notion of a Fractal Helmholtz Equation, generalizing the eigenmode analysis to fractal domains:
\Delta_F \psi_n + \lambda_n \psi_n = 0
where is a fractional Laplacian operating on fractal space. This allows modeling of nested, scale-invariant harmonics akin to Nautilus shell spirals, as suggested by observational hints of logarithmic structure in galaxy distributions and cosmic voids.
