Degenerate and connection-dependent cosmological sectors in f(Q,C) gravity

We investigate cosmological solutions in $f(Q,C)$ gravity formulated within symmetric teleparallel geometry, where gravitation is described by the nonmetricity scalar $Q$ and the boundary term $C$, related to the Ricci scalar of General Relativity through $\mathring{R}=Q+C$ in the absence of curvature and torsion. The symmetry requirements of FLRW spacetime admit three distinct realizations of the affine connection, leading in principle to three different cosmological sectors within the same theory. We show that the connection field equations play a crucial role in determining the cosmological dynamics. In their simplest realization, these equations impose a constraint that renders the theory dynamically equivalent to $f(\mathring{R})$ gravity, causing the cosmological background equations associated with the three connection realizations to coincide. This defines a degenerate cosmological sector of $f(Q,C)$ gravity, in which three a priori distinct geometric constructions converge to the same cosmological dynamics. We then consider the complementary class of genuinely nonequivalent $f(Q,C)$ models, for which the choice of connection becomes physically relevant. In this regime, the connection sector introduces additional dynamical degrees of freedom and gives rise to novel cosmological phenomenology absent in $f(\mathring{R})$ gravity.