The Hassanzadeh-Nasrollah Nejad-Simis Conjecture on Euler Conductors

We prove the Hassanzadeh-Nasrollah Nejad-Simis conjecture: if $f\in (x_1,\dots ,x_n{)}^2⊊k[[x_1,\dots ,x_n]]$ has an isolated critical point and $\mathrm{char}k=0$, then its Euler conductor $J_f:f$ is not contained in the Tjurina ideal $(J_f,f)$. More generally, if $A=R\mathrm{/}I$ is a nonzero Noetherian $k$-algebra and $a=[f]\in A$ is nilpotent with $d_{A\mathrm{/}k}a=0$, then $I:f⊈(I,f)$. For a local ring $(R,\mathfrak{n})$ we obtain the stronger noncontainment $I:f⊈(I,f)+\mathfrak{n}(I:f)$. The proof reduces a hypothetical containment to a self-exact square-zero element and detects its differential by the trace of a regular representation over the dual numbers. Finally, an explicit five-variable isolated singularity satisfies $J_f:f\subseteq \overline{J_f}$, disproving the integral-closure strengthening proposed by Ma and Zuo.