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. v3 update (2026-04-28, b52): independent verification by Paper LXVII's lattice measurement satisfies the multi-path I-criterion of the programme's A+ pathway; the analytical (Theorems 2.1 + 3.2 + 3.3) and numerical (lattice $\alpha = 0.987 \pm 0.002$, 765$\sigma$) paths are structurally distinct and converge on the dust-like classification ($w_{\rm knot} = 0$).

Keywords: math, statistical mechanics, knot theory, dark matter, thermodynamics, equation of state, pressureless dust, Faddeev-Niemi, Hopf soliton, knot enumeration, crossing number, ropelength, Stirling approximation, Robbins bound, Kotecky-Preiss cluster expansion, ideal gas, extensivity, partition function, dilute gas, mass gap, trefoil knot, prime knot, Volovik mechanism, Gibbs-Duhem, cosmological constant, dark matter equation of state, lattice Monte Carlo, lattice gauge theory, A+ pathway, I-criterion, multi-path independent verification, Paper LXVII lattice cross-check, topological soliton, Faddeev-Skyrme, dispersion relation, non-relativistic limit, CMB temperature, Planck cosmology

Series: Paper CVI in the Hopf Soliton Programme