Multi-Well Cosmology
The cosine potential is symmetric. Displacement can go either way. The result: parallel worlds that share gravity but not light.
Overview
The lattice potential V(x) = (ka²/π²)[1−cos(πx/a)] is symmetric: a node can be displaced in either direction. This means the potential has multiple wells — not just one.
In units of lattice spacing, the wells sit at positions −2, −1, 0, +1, +2. Five wells are mathematically allowed by the cosine periodicity. However, only three are stable and habitable (wells −1, 0, and +1). The ±2 wells have unstable force balance — displacements there sit at inflection points of the potential where restoring force vanishes.
The number Nc = 3 appears again: three possible occupied worlds, one for each stable well. This is the same trinary structure (+/0/−) that appears throughout GWT — from color charge to oscillator states.
Each well hosts its own matter, its own fusion, its own light. But all three wells share the same gravitational anchors. A star in well 0 gravitationally compresses the lattice at the same location where wells ±1 also feel that compression. Stars in all three wells tend to form at the same spatial coordinates — anchored by shared gravity.
Cross-Well Gravity & Dark Matter
Gravity crosses wells; electromagnetism and matter do not. Gravity is a longitudinal compression of the lattice itself — it affects all wells equally because all wells share the same underlying medium. But EM waves are transverse disturbances confined to a single well. Matter (standing waves) likewise exists only within its own well.
This means: matter in well n gravitates in well 0 but is completely invisible to well 0's photons. You cannot see it, scatter off it, or detect it electromagnetically. The only signature is gravitational.
The immediate consequence: gravitational lensing with no visible source is a possible cross-well mass signature. This is an alternative dark matter explanation — no new particles needed, no WIMPs, no axions. The “missing mass” is ordinary matter in an adjacent well.
Cross-Well Coupling
The coupling between wells follows from evanescent wave decay through the potential barrier:
The penetration depth is δ = a/π, where a is the lattice spacing. The gravitational diffusion rate is Γ ≈ 8×1041 s−1, meaning steady state is reached in approximately 10−42 s — effectively instantaneous on any observable timescale.
Soliton Dark Matter
The cosine potential also supports topological solitons (kinks) — stable field configurations that interpolate between adjacent wells. These kinks are a natural dark matter candidate:
These solitons are stable (topologically protected), EM-invisible (they are medium distortions, not standing waves), and gravitating (they carry energy). They satisfy every requirement for a dark matter particle without invoking any new physics beyond the lattice.