This interpretation allows a direct mapping from specific mode configurations to features in the cosmic web, providing a blueprint-like genesis rather than a purely stochastic evolution.
4.2 Analytical Solutions for Idealized Geometries
To make this framework tractable, we begin with analytical derivations on idealized compact manifolds, such as:
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3-Torus : Periodic boundary conditions lead to discrete, sinusoidal eigenmodes:
 \psi_{n}(x, y, z) = \sin\left(\frac{2\pi n_x x}{L_x}\right)\sin\left(\frac{2\pi n_y y}{L_y}\right)\sin\left(\frac{2\pi n_z z}{L_z}\right)
These produce a grid-like standing wave structure whose nodes and anti-nodes map onto an early crystalline cosmic structure.
3-Sphere : Eigenmodes become spherical harmonics generalized to higher dimensions. The radial and angular modes generate nested shells of resonant nodes.
Poincar Dodecahedral Space: Allows matching with the low- anomalies in CMB; the eigenmodes here are less intuitive but generate non-trivial closed-path symmetries.
Each geometry supports a distinct mode spectrum, defining how structure emerges spatially. This variety could be used to test different "cosmic templates" through simulations and CMB analysis.
4.3 Mode Coupling and Nested Resonant Shells: The Nautilus Analogy
Unlike traditional Fourier expansions, the mode structure in a compact resonant spacetime allows for nonlinear coupling and nested resonance. When certain modes and interact constructively via resonance conditions:
