Negative Molar Specific Heat of a Gas
I'm going to continue to resist calling C the heat capacity because, as used here, it is not path-independent.  Instead, I will take it to merely represent the derivative of the heat added per mole of ideal gas with respect to temperature for a polytropic process path. 

With this caveat, here is my interpretation of what is happening mathematically when C takes on negative values:  If the gas is expanding, a negative value of C means that we added heat to the gas during the expansion, but not enough to offset the expansion cooling that would occur in an adiabatic reversible expansion, so its temperature decreases while we add heat.  If the gas is being compressed, a negative value of C means that we removed heat from the gas during the compression, but not enough to offset the compression heating that would occur in an adiabatic reversible compression, so its temperature increases while we remove heat.  It's as simple as that.