A counterexample to the E-derivation part of the LNED conjecture

For a field $K$ of characteristic zero and $c\in K\setminus\{0\}$, let $\delta_c f(x)=f(x)-f(x+c)$ and $J_c=(x(x-c/2))\subset K[x]$. We prove that $\delta_c$ is a locally nilpotent $K$-$\mathcal{E}$-derivation such that the image of $J_c$ under $\delta_c$ contains $x^{2m}$ and excludes $x^{2m+1}$ for every $m\ge1$, refuting the $\mathcal{E}$-derivation part of Zhao's locally nilpotent derivation and $\mathcal{E}$-derivation (LNED) conjecture already in one variable. The proof specializes Zhao's Bernoulli-polynomial membership criterion for images of quadratic ideals at $x=c/2$. For any $v$ in an index set $T$, the shift $x_v\mapsto x_v+1$ in $K[x_i:i\in T]$, fixing the other variables, gives the same failure of the Mathieu–Zhao condition for the corresponding quadratic ideal. Checked in Lean from the definitions; not yet reviewed by experts. Produced by AI models under the author's direction.