Scalar total variation splitting under clipping ($\mathrm{TV}(u)=\mathrm{TV}(T_{M}u)+\mathrm{TV}(u-T_{M}u)$)

For a scalar BV space function, clipping to $[-M,M]$ partitions its coarea formula for BV functions into middle and tail levels. The clipped function has the original superlevel sets at heights in $(-M,M)$. The positive tail levels of $u-T_{M}u$ correspond to original heights above $M$, and its negative tail levels to heights below $-M$. Thus their total variation seminorms add exactly. This is stronger than the triangle inequality and is specific to scalar monotone clipping. It supports residual-preserving clipping of an ROF minimizer.