Extensive Knotting and the Dark Matter Equation of State

Title: Extensive Knotting and the Dark Matter Equation of State

Author: Alexander Novickis (alex.novickis@gmail.com)

We prove that the knotting sector of the Faddeev--Niemi quantum field theory has equation-of-state parameter $w = 0$ (pressureless dust). The proof proceeds in three steps. First, we define the canonical partition function for a dilute gas of non-interacting knotted solitons whose energies are proportional to their minimal crossing numbers, and establish convergence of the single-particle partition function using rigorous bounds on knot enumeration. Second, we prove that the free energy is extensive (Theorem 2.1) by exploiting the ideal-gas structure inherited from exponential decay of correlations (Paper CI, Theorem 3.2). Third, we derive the equation of state from the non-relativistic dispersion relation of the massive knotted excitations: $w = \langle v^2 \rangle / (3c^2) = O(T/\Delta)$, where $\Delta > 0$ is the mass gap (Paper CI, Theorem 6.4). At the present CMB temperature $T_{\mathrm{CMB}} = 2.725\;\mathrm{K}$ and mass gap $\Delta \sim 2\;\mathrm{MeV}$, this gives $w \sim 10^{-10}$ (Theorem 3.3). The result elevates the lattice measurement of Paper LXVII ($\Delta\chi^2 = 584{,}801$ favouring $w = 0$) to a rigorous statistical-mechanical theorem. All proofs are self-contained modulo the constructive results of Paper CI.

Keywords: math, statistical mechanics, knot theory, dark matter, thermodynamics

Series: Paper CVI in the Hopf Soliton Programme