Dark Matter as Half-Knots from a Two-Stage SU(2) Symmetry Breaking: Confined Half-Monopoles on Z_2 Strings

We propose that, dark matter consists of confined half-monopoles (``beads'') arisingfrom a two-stage SU(2) symmetry breaking: SU(2)→ U(1)₂. The firstbreaking produces 't Hooft-Polyakov monopoles; the second, via a complex symmetrictraceless 5-plet Higgs, confines magnetic flux into Z₂ strings carryinghalf the Dirac quantum. A monopole splits into two half-monopoles connected by astring (a ``rope''). When two ropes cross and reconnect, an unpaired half-monopolebead remains at the junction. This bead is massive (M ~ 2π v_Φ/g),spin-$1/2$, absolutely stable, and if the gauge sector is dark, interacts onlygravitationally. In this paper, We derive the relic abundance from first principles, obtainingΩ h² 0.12\,(M/10¹²\,GeV)², naturally matchingthe observed density for v_Φ ~ 10¹² GeV. The string network predicts astochastic gravitational wave background with Gμ ~ 10⁻⁶--10⁻¹² andpeak frequency f_* ~ 10⁻⁷(v_Σ/10¹⁰\,GeV) Hz, within LISA reach.Kinetic mixing gives ε ~ 10⁻⁵--10⁻², potentially testable atfuture colliders. Entanglement monogamy follows from charge conservation.Condensed-matter analogues in chiral magnets and liquid crystals provide table-top tests.