= Energetic fission of an integer nematic disclination
In a strictly planar <nematic liquid crystal>, a charge-one <nematic disclination> is a winding-two map into the circle $\mathbb{RP}^1$. Its nonzero <fundamental group> class prevents complete unwinding while the surrounding loop stays fixed, but permits splitting into two charge-$1/2$ defects. For separation $r_0\ll d\ll L$, the two near fields contribute $\pi K\lambda_0^2(1/4+1/4)\log(d/r_0)$ while their common charge-one far field contributes $\pi K\lambda_0^2\log(L/d)$. Compared with the unsplit defect, the change is
$$
-\frac{\pi K\lambda_0^2}{2}\log(d/r_0)+\text{finite core-energy difference}.
$$
It is negative at sufficiently large scale separation. Thus integer-defect fission is energetically favorable despite conservation of total topological charge. This leading-log argument proves failure of global energetic stability, not absence of kinetic barriers or pinning. If <escape into the third dimension> is allowed, an integer winding can also become topologically trivial in $\mathbb{RP}^2$.
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