Separating Abelian and Homomorphic Entropy Cones

Chan and Yeung showed that finite groups suffice to determine which homogeneous linear information inequalities are universally valid. We compare two restricted group-characterizable entropy cones: the Abelian cone ${\tilde{\mathrm{\Gamma }}}_n^{\mathrm{A}\mathrm{b}\mathrm{l}}$ and the homomorphic cone ${\tilde{\mathrm{\Gamma }}}_n^{\mathrm{H}\mathrm{o}\mathrm{m}}$, the latter generated by coset systems of normal subgroups. We prove $${\tilde{\mathrm{\Gamma }}}_{16}^{\mathrm{A}\mathrm{b}\mathrm{l}}⊊{\tilde{\mathrm{\Gamma }}}_{16}^{\mathrm{H}\mathrm{o}\mathrm{m}},$$ and, if $n_{\mathrm{A}\mathrm{H}}$ is the least number of variables for which these cones differ, we show $6\le n_{\mathrm{A}\mathrm{H}}\le 16$. The separating functional is a class-restricted entropy inequality: it is valid on the Abelian cone but is not a universal information inequality. It is obtained by lifting the order dual of the P'alfy–Szab'o six-cross identity while quantifying errors at inexact subgroup joins. We then construct sixteen normal subgroups of a class-two $2$-group of order $2^{43}$ for which every join error vanishes while the endpoint containment fails by one bit. Since mixed-linear random variables are Abelian, the same example also separates the mixed-linear and homomorphic entropy cones.