Which came first, the lung or the breath?

TLDR: Air ventilatory mechanisms in the actinopterygian lineage are similar among different groups, including Polypteriformes, but are distinct from those of lungfish and tetrapods, however, there is extreme variation within this apparent dichotomy.

Abstract: Lungs are the characteristic air-filled organs (AO) of the Polypteriformes, lungfish and tetrapods, whereas the swimbladder is ancestral in all other bony fish. Lungs are paired ventral derivatives of the pharynx posterior to the gills. Their respiratory blood supply is the sixth branchial artery and the venous outflow enters the heart separately from systemic and portal blood at the sinus venosus (Polypteriformes) or the atrium (lungfish), or is delivered to a separate left atrium (tetrapods). The swimbladder, on the other hand, is unpaired, and arises dorsally from the posterior pharynx. It is employed in breathing in Ginglymodi (gars), Halecomorphi (bowfin) and in basal teleosts. In most cases, its respiratory blood supply is homologous to that of the lung, but the vein drains to the cardinal veins. Separate intercardiac channels for oxygenated and deoxygenated blood are lacking. The question of the homology of lungs and swimbladders and of breathing mechanisms remains open. On the whole, air ventilatory mechanisms in the actinopterygian lineage are similar among different groups, including Polypteriformes, but are distinct from those of lungfish and tetrapods. However, there is extreme variation within this apparent dichotomy. Furthermore, the possible separate origin of air breathing in actinopterygian and 'sarcopterygian' lines is in conflict with the postulated much more ancient origin of vertebrate air-breathing organs. New studies on the isolated brainstem preparation of the gar (Lepisosteus osseus) show a pattern of efferent activity associated with a glottal opening that is remarkably similar to that seen in the in-vitro brainstem preparation of frogs and tadpoles. Given the complete lack of evidence for AO in chondrichthyans, and the isolated position of placoderms for which buoyancy organs of uncertain homology have been demonstrated, it is likely that homologous pharyngeal AO arose in the ancestors of early bony fish, and was pre-dated by behavioral mechanisms for surface (water) breathing. The primitive AO may have been the posterior gill pouches or even the modified gills themselves, served by the sixth branchial artery. Further development of the dorsal part may have led to the respiratory swimbladder, whereas the paired ventral parts evolved into lungs.