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: $SU(2)\to U(1)\to\mathbb{Z}_2$. The firstbreaking produces 't Hooft-Polyakov monopoles; the second, via a complex symmetrictraceless 5-plet Higgs, confines magnetic flux into $\mathbb{Z}_2$ 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 ($M \sim 2\pi v_\Phi/g$),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, obtaining$\Omega h^2 \simeq 0.12\,(M/10^{12}\,\text{GeV})^2$, naturally matchingthe observed density for $v_\Phi \sim 10^{12}$ GeV. The string network predicts astochastic gravitational wave background with $G\mu \sim 10^{-6}$--$10^{-12}$ andpeak frequency $f_* \sim 10^{-7}(v_\Sigma/10^{10}\,\text{GeV})$ Hz, within LISA reach.Kinetic mixing gives $\epsilon \sim 10^{-5}$--$10^{-2}$, potentially testable atfuture colliders. Entanglement monogamy follows from charge conservation.Condensed-matter analogues in chiral magnets and liquid crystals provide table-top tests.