Universal spectrum and scaling laws for halo mass function, structure, and dark matter mass constraints

Between the linear and nonlinear regimes, we identify a universal transition range centered on a characteristic halo mass $m_h^*\propto t$, within which gravitational dynamics self-organize the matter field toward an effective spectral index n=-1. In a bottom-up hierarchy, early collapse of low-mass halos preserves imprints of the primordial spectrum, whereas prolonged assembly of halos near $m_h^*$ erases that memory and establishes universality. We formulate a scale-to-scale cascade, the redistribution of mass and energy across scales, that yields universal scaling laws for the halo mass function and internal structure. Globally, the cascade drives a random walk of halos with mass-dependent waiting time $\tau_g\propto m_h^{-\lambda}$; A Fokker-Planck equation gives mass function $f_M\propto m_h^{-\lambda}$ and $\lambda=2/3$ for the gravity-dominant transition range. Locally, a radially directed cascade governs particle migration with waiting time $\tau_{gr}\propto r^{-\gamma}$, yielding density $\rho_r\propto r^{-2\gamma}$ and $\gamma=2/3$ on scales near $m_h^*$. The cascade drives the system toward a statistically steady state that continuously releases energy and maximizes entropy, characterized by scale-independent rates, preventing mass or energy buildup at intermediate scales. Scale-dependent dominance of the primordial spectrum versus gravity implies two effective exponents, producing double-$\lambda$ mass functions and double-$\gamma$ density in excellent agreement with simulations. Using Illustris and Virgo, we measure an inverse kinetic-energy cascade from small to large scales at $\varepsilon_u \approx -10^{-7}$m$^2$/s$^3$, a direct potential-energy cascade of $-1.4\varepsilon_u$, and a net dissipation of -0.4$\varepsilon_u$ via halo mergers and particle migration. The dependence of waiting time and step length on the particle mass suggests new constraints near $10^{12}$GeV.