Resonance-enhanced super-superexchange yields giant chiral magnon splitting in rutile altermagnets

Altermagnets host momentum-selective spin splitting and chiral-split magnonic excitations despite vanishing net magnetization, enabling spin transport without ferromagnetism. In rutile structures, establishing altermagnetism spectroscopically has been challenging, motivating the search for a rutile platform with a resolvable exchange-driven chiral magnon splitting. Here we combine hybrid-functional first-principles calculations with linear spin-wave theory to show that rutile CuF$_2$ exhibits a meV-scale splitting between magnon modes of opposite chirality along momentum directions dictated by its $d$-wave altermagnetic symmetry. The splitting originates from an anomalously strong long-range super-superexchange channel Cu--F$\cdots$F--Cu, which enhances the symmetry-allowed difference between seventh-neighbour exchanges, $J_{7b} - J_{7a}$, controlling the chiral-mode splitting. We identify an orbital-resonance mechanism: energetic alignment between Cu $3d_{z^2}$ and F $2p_z$ states strengthens virtual hopping along the Cu--F$\cdots$F--Cu path and amplifies the anisotropic long-range exchange. Rutile CuF$_2$ therefore provides an ideal platform to validate rutile altermagnetism and suggests an orbital-energy description for engineering large chiral magnon splittings in insulating altermagnets.