Let G be a group with identity e, H an abelian group written additively, and \(\varphi :G\rightarrow G\) an endomorphism. We study solutions \(f:G\rightarrow H\) of the reflected Jensen equation, \( f\big (x\,\varphi (y)\big )+f\big (\varphi (y)^{-1}\,x\big )=2f(x), \qquad f(e)=0, \qquad (\forall x,y \in G). \) Writing \(u=\varphi (y)\in \textrm{Im}\,\varphi \) , this becomes \(f(xu)+f(u^{-1}x)=2f(x)\) for all \(x\in G\) and \(u\in \textrm{Im}\,\varphi \) . In contrast to the classical Jensen equation \(f(xy)+f(xy-1) = 2f(x)\) , the reflected endomorphism variant considered here exhibits a strong torsion-connement phenomenon. Our main result identifies a precise torsion obstruction in a purely elementary way. If \( K:=\langle \textrm{Im}\,\varphi \cap J(G)\rangle , \qquad H[4]:=\{h\in H:4h=0\}, \) then every solution satisfies the stronger right K-invariance modulo H[4]: \( f(x)-f(xk)\in H[4]\qquad (\forall x\in G,\ \forall k\in K). \) In particular, taking \(x=e\) yields the torsion confinement \(f(K)\subseteq H[4]\) . Consequently, if \(\varphi \) is surjective, \(G=\langle J(G)\rangle \) (e.g. Coxeter and dihedral groups), and H is 2-torsion-free, then \(H[4]=\{0\}\) and hence \(f\equiv 0\) on G. Finally, we give an explicit example on the dihedral group of order 8 with values in \(\mathbb {Z}_4\) showing that the exponent 4 is sharp: the confinement cannot, in general, be improved to H[2].
