Nonattainment under strict positivity for shifted Poisson fidelity (${\inf }_{u>0,\log u\in L^1}[\alpha \mathrm{TV}(u)+D_g(u)]=0\text{ is not attained}$)

On a bounded domain, suppose $0<g<1$, $\alpha >0$, and a bounded positivity-preserving operator satisfies $T1\ne 0$. Every strictly positive $u$ has $Tu\ne 0$: otherwise $T{\chi }_{\{u\ge 1/n\}}=0$ 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 $u=\epsilon$ have zero variation and energy at most $\epsilon \Vert T1{\Vert }_1\to 0$. Their excluded zero limit proves nonattainment. Coercivity and bounded-variation compactness cannot by themselves preserve a nonclosed strict-positivity constraint.