What question did this study set out to answer?

The central aim is to propose a mathematical framework for understanding dark matter as stable, uncharged topological electromagnetic structures.

February 19, 2026Open Access

Dark Matter as Topological Electromagnetic Structures: Mathematical Framework for H = 0 Stable Configurations and SO (4, 2) Dark Representations

Key Points

The central aim is to propose a mathematical framework for understanding dark matter as stable, uncharged topological electromagnetic structures.
Extended the toroidal electron model using knot theory and SO(4,2) representation theory.
Classified candidate configurations and proved conservation of Hopf charge.
Validated mass spectrum against Battye-Sutcliffe numerical soliton calculations.
Conducted numerical simulations of Boltzmann freeze-out and gamma-ray flux predictions.
Proved topology's role in dark matter stability and charge conservation.
Estimated the mass of the trefoil structure as 2.0+0.7−0.8 MeV.
Predicted null results from nuclear recoil experiments and specific gamma-ray emissions.
Outlined multiple discrete dark matter species with specified mass ratios and consistency with cosmological observations.

Abstract

Dark Matter as Topological Electromagnetic Structures Mathematical Framework for H = 0 Stable Configurations and SO(4,2) Dark Representations We extend the toroidal electron model—where electric charge emerges as the Hopf linking number (H = ±1) of electromagnetic field configurations—to propose a framework for dark matter as stable, uncharged (H = 0) topological EM structures. Working within the Faddeev-Niemi nonlinear sigma model, we classify candidate configurations using knot theory (trefoil, figure-8, Whitehead link) and conformal group SO(4,2) representation theory. We prove that the Hopf charge is exactly conserved (topological stability), that the transverse magnetic dipole moment vanishes by CN symmetry for N ≥ 3, and that the comoving free-streaming length satisfies λfs < 10−2 Mpc (cold dark matter behavior). A ropelength-based mass spectrum, validated against Battye-Sutcliffe numerical soliton calculations, yields a best estimate of mtrefoil = 2.0+0.7−0.8 MeV. The structural relationship between the Faddeev-Niemi and Skyrme models provides a unified topological classification of leptons (H ≠ 0), baryons (B ≠ 0 Skyrmions), and dark matter (H = 0 knotted solitons). Key predictions: Null results in nuclear recoil experiments Monoenergetic MeV gamma-ray lines from annihilation into photon pairs Multiple discrete dark matter species with specific mass ratios Consistency with BBN, Bullet Cluster, and structure formation constraints Numerical simulations of the Boltzmann freeze-out, gamma-ray flux predictions, and sensitivity comparisons with current (INTEGRAL/SPI, COMPTEL) and projected (COSI, AMEGO-X) instruments are presented. The upcoming COSI mission (2027) will provide a decisive test. Contents: 95 pages, 19 figures (9 SVG diagrams + 10 simulation plots), 3 formal theorems with proofs, 65 references, Appendix on dark energy.

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