I was thinking of C as an operation on a list of (generalized, not necessarily monotonically decreasing feerate) chunks - whether those are obtained by starting with the singletons from a linearization, or something else.
We want to prove that C(\epsilon_n + \delta_n) \geq \gamma_n I think. I suspect it’s possible to adapt your derivation above to work by placing C() invocations carefully, and using the rules around it.