Flavor-dependent neutrino transport is described by a well-known kinetic equation for occupation-number matrices in flavor space. However, in the context of fast flavor conversion, we identify an unforeseen predicament: the pivotal self-induced exponential growth of small inhomogeneities strongly violates conservation of neutrino-neutrino refractive energy. We prove that it is traded with the huge reservoir of neutrino kinetic energy through gradients of neutrino flavor coherence (the off-diagonal piece of the flavor density matrix) and derive the missing gradient terms. The usual equations remain sufficient to describe flavor evolution, at the cost of renouncing energy conservation, which cannot play any role in explaining the numerically observed final state.