Dark Matter as a Topological Vacuum Condensate

Title: Dark Matter as a Topological Vacuum Condensate Author: Alexander Novickis ([email protected]) We propose that dark matter within the Hopf soliton framework is the topological vacuum condensate --- the diffuse background of knotted vacuum topology that the lattice Monte Carlo naturally discovers. Two lattice results motivate this picture: (1) the $H = 1$ Hopf soliton dissolves in 3D MC, with the topology spreading into a diffuse background carrying energy but no compact core (Paper XL, $E_\text{sol}/E_\text{BS} \to 0$ as $\beta \to \infty$); (2) the $H = 0$ trefoil knot dissolves under cooling, its knot topology unprotected by the Hopf invariant. We propose that the topological vacuum has two non-perturbative sectors: (i) a linking sector of virtual soliton-antisoliton pairs that produces dark energy ($\rho_\Lambda = N_\text{DE} \cdot \alpha^{16} m_e^4$, Paper XI), and (ii) a knotting sector of virtual knotted vacuum fluctuations that produces dark matter, with knotting vertex amplitude $\rho_\text{DM}^{(0)} = N_\text{DM} \cdot \alpha^{16} m_e^4$ set at the Hopf phase transition. Both sectors share the same $\alpha^{16}$ suppression from the topological vertex structure, differing only in the combinatorial multiplicity $N$. The present-day density is $\rho_\text{DM}(a_0) = \rho_\text{DM}^{(0)} (a_c/a_0)^3$, where $a_c$ is the scale factor at the phase transition. The ratio of multiplicities $N_\text{DM}/N_\text{DE}$ determines the density ratio at the epoch when both sectors are set; the observed present-day ratio $\Omega_\text{DM}/\Omega_\Lambda = 0.387$ then implies $N_\text{DM}/N_\text{DE} \approx 0.387/(a_c/a_0)^3$. The near-equality $0.387 \approx 3/8$ implies $(a_c/a_0)^3 \approx 1.03$, i.e., the present epoch is near the topological coincidence where the dilution factor is approximately unity. A rigorous evaluation of five candidate derivations for $N_\text{DM} = \pi$ (torus knot moduli, solid angle/writhe, Chern-Simons invariants, Jones polynomial, phase space volume) shows that all five fail --- the identification $N_\text{DM} = \pi$ is numerology, since topological multiplicities are integers while $\pi$ is transcendental. The physically motivated value is $N_\text{DM} = 3$ (from the trefoil's $C_3$ rotational symmetry, giving $3/8 = 0.375$, a 3.2% match). A new isotopy analysis (Section 3.5) proves that compact $H = 0$ knotted solitons are quantum-mechanically unstable: the FN path integral breaks ambient isotopy class because crossing-change configurations have finite action (codimension-1 in field space), unlike $H$-changing configurations which have infinite action. The trefoil quantum lifetime is $\tau \sim 10^{-22}$ s, ruling out Paper VI's compact knotted soliton picture. The vacuum condensate is the sole dark matter mechanism within the Hopf soliton framework. DOI: 10.5281/zenodo.19626079 Series: Paper LXVII in the Hopf Soliton Programme

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