Counterexamples to additivity of minimum output $p$-Rényi entropy of quantum channels for $p>3/4$ and $0\leq p<1/4$

Additivity of minimum output entropies is a central problem in quantum information theory. Nonadditivity is known for every Rényi order $p>1$, at the von Neumann point $p=1$, and near $p=0$, while most of the interval $03/4$ or $0\leq p<1/4$, there exist finite-dimensional projection-induced quantum channels such that additivity of the minimum output $p$-Rényi entropy fails. The proof combines two correlated random-projection constructions: a product-conjugate Bell-state witness for $p>3/4$, and a transpose-complement rank-defect witness for $p<1/4$. Thus the unresolved part of $0