Before revision:  We also prove that E(C') \leq E(C)  +3 if G does not contain faces of size five, which gives a new tool for proving results about cycle spectra and evidence that these face sizes obstruct long cycles . As a sample application, we test our motor on a conjecture on essentially 4-connected graphs.
After revision:  We also prove that E(C') \leq E(C) +3 if G does not contain faces of size five, which gives a new tool for proving results about cycle spectra and evidence that these face sizes obstruct long cycles in many graph classes. We test our motor on a conjecture on essentially 4-connected graphs.