= Solution
The total charge of the widely separated $+1,-1$ pair is zero, so a uniform planar <nematic director> is compatible with the far boundary if it imposes no additional winding. Each integer defect is susceptible to <energetic fission of an integer nematic disclination>: the positive defect can split into two $+1/2$ defects and the negative defect into two $-1/2$ defects. Like-sign elementary defects repel, whereas opposite signs attract through their elastic distortion. This commonly produces subsequent encounters and annihilations of opposite half-integer defects.
The attraction can also be seen before fission. In the fixed-amplitude outer theory, a neutral pair of charges $q,-q$ separated by $R$ has
$$
F_{\rm pair}=2\pi K\lambda_0^2q^2\log(R/r_0)+\text{core terms},\qquad -\frac{dF_{\rm pair}}{dR}=-\frac{2\pi K\lambda_0^2q^2}{R}.
$$
Its energy therefore falls as the separation decreases. In <nematic defect-antidefect annihilation>, opposite defects meet, the amplitude can vanish in their merging cores, and the local winding disappears; the remaining smooth director distortion relaxes towards the uniform energy minimum.
\b[Expected route: integer-defect fission and attraction of opposite elementary defects, followed by annihilation and relaxation of the remaining texture.]
Direct approach and annihilation of the unsplit integer pair is also topologically allowed. Whether fission happens first, the paths of the four elementary defects, and the annihilation time require a specified dynamical law and core mobilities; the <free energy> alone does not fix them. Pinning, imposed boundary textures, or sustained thermal production of defect pairs can prevent a strictly defect-free final state. The stated final state presumes dissipative relaxation with no such obstruction.
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