= Nonattainment under strict positivity for shifted Poisson fidelity
{title2=$\inf_{u>0,\,\log u\in L^1}[\alpha\operatorname{TV}(u)+D_g(u)]=0\text{ is not attained}$}
On a bounded domain, suppose $0<g<1$, $\alpha>0$, and a bounded <positivity-preserving operator> satisfies $T1\ne0$. Every strictly positive $u$ has $Tu\ne0$: otherwise $T\chi_{\{u\ge1/n\}}=0$ for all $n$, and <continuity> gives $T1=0$. Since $s-g\log(1+s)>0$ whenever $s>0$, every such $u$ has positive energy. The constants $u=\varepsilon$ have zero variation and energy at most $\varepsilon\|T1\|_1\to0$. Their excluded zero <limit> proves nonattainment. <Coercivity> and <bounded-variation compactness> cannot by themselves preserve a nonclosed strict-positivity constraint.
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