= Opposite-flow fidelity identity for quadratic data
{title2=$\Delta F_++\Delta F_-=-\tfrac{\theta(1-\theta)}2\sum_\pm\|\delta_{\pm t}w\|_2^2+\theta\int\delta_tg\,\delta_tw+o(t)$}
Let $w\in BV\cap L^\infty$ and $g\in BV\cap L^2$ on a bounded domain, $F_g(v)=\tfrac12\|v-g\|_2^2$, and $w_{\pm t}=(1-\theta)w+\theta w\circ\Phi_{\pm t}$ for a smooth compactly supported <local flow>. The displayed formula is exact to an $o(t)$ remainder, and $g$ need not be bounded. Writing $J_t=\det D\Phi_t$, its cross-term identity is
$$
\int g(\delta_tw+\delta_{-t}w)=-\int\delta_tg\,\delta_tw+\int g(1-J_{-t})\delta_{-t}w.
$$
The remainder is $o(t)$ because $1-J_{-t}=O(t)$ and weighted $L^1$ convergence follows from
$$
\int|g||\delta_{-t}w|\le K\|\delta_{-t}w\|_1+2\|w\|_\infty\int_{|g|>K}|g|.
$$
First let $t\to0$, then $K\to\infty$. The mass change of $w^2$ under opposite flows is $O(t^2)\|w\|_2^2$. Combined with the <BV jump-product limit with one bounded factor>, this evaluates the jump contribution without a bounded fidelity derivative.
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