= Mode-number regularization of soliton masses
Compare the same total number of finite-box modes in the <soliton> and vacuum sectors before removing the <ultraviolet cutoff>. A localized <bound state> replaces a continuum mode, and its contribution must be retained. For the <Sine-Gordon kink fluctuation operator> and the odd phase branch $\delta(k)=2\arctan(1/k)$, the bound translation mode replaces the free $k=0$ oscillator. Thus the complete oscillator correction is $-m/2-(m/2)\int dk\,\delta(k)k/[2\pi\sqrt{k^2+1}]$ in this convention. The continuum integral alone misses the finite $-m/2$ contribution.
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