  Despite recent developments, there are a number of conceptual issues on the
hadronic light-by-light (HLbL) contribution to the muon $(g-2)$ which remain
unresolved. One of the most controversial ones is the precise way in which
short-distance constraints get saturated by resonance exchange, particularly in
the so-called Melnikov-Vainshtein (MV) limit. In this paper we address this and
related issues from a novel perspective, employing a warped five-dimensional
model as a tool to generate a consistent realization of QCD in the large-$N_c$
limit. This approach differs from previous ones in that we can work at the
level of an effective action, which guarantees that unitarity is preserved and
the chiral anomaly is consistently implemented at the hadronic level. We use
the model to evaluate the inclusive contribution of Goldstone modes and
axial-vector mesons to the HLbL. We find that both anomaly matching and the MV
constraint cannot be fulfilled with a finite number of resonances (including
the pion) and instead require an infinite number of axial-vector states. Our
numbers for the HLbL point at a non-negligible role of axial-vector mesons,
which is closely linked to a correct implementation of QCD short-distance
constraints.
