Dark Matter as Fractal Solitons: Resolution of the Three Core CDM Problems

Cold Dark Matter (CDM) particle models face three persistent observational failures at galactic sub-structure scales: the core-cusp problem, the missing satellites problem, and the too-big-to-fail problem. Direct detection experiments (XENON, LUX, PandaX) have found no WIMP signal after four decades of searching. We propose that dark matter consists of fractal solitons — topologically stable, spatially extended wave packets arising from the Fibonacci harmonic cascade of the Fractal Mechanics framework. Unlike point-like WIMPs, fractal solitons span Δ𝑛 ≈ 20–30 cascade levels (∼ 1–10 kpc physical size), producing flat-core density profiles 𝜌 (𝑟) = 𝜌₀/ (1 + (𝑟/𝑟_𝑐) ²) from quantum pressure, naturally resolving the core-cusp problem. Solitons below 𝑀crit ∼ 106 𝑀⊙ evaporate via quantum tunneling, reducing the subhalo count from ∼ 500 (CDM prediction) to ∼ 50 (observed), resolving the missing satellites problem. Their extended size makes them susceptible to tidal disruption even at large masses, resolving too-big-to-fail. The effective cross-section for nucleon scattering is 𝜎ₛol ∼ 10−70 cm², explaining the absence of direct detection signals without fine-tuning. We predict universal soliton-core scaling relations testable with JWST and the Rubin Observatory, and a characteristic gravitational wave signature detectable by LISA.