Signed layer-cake identity for quadratic fidelity ($u^2/2-fu={\int }_{\mathbb{R}}(t-f)({\chi }_{\{u>t\}}-{\chi }_{\{0>t\}})dt$)

For positive $u$, integrate $t-f$ from zero to $u$; for negative $u$, reverse the integral from $u$ to zero. This proves the identity without a positivity restriction on the data. For $u,f\in L^2$, the absolute integral is bounded by $u^2/2+|fu|$, so Fubini's theorem applies. The zero-function baseline removes the divergence from the negative-height tail.