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We consider a chiral gauge theory from which light composite Dirac neutrinosdynamically emerge, augmented by a QCD axion to solve the strong CP problem. Weshow that an interplay between the composite sector and the axion can alsosimultaneously lead to successful baryogenesis and generate a correct darkmatter abundance via "co-genesis" without contradicting present constraints. Wepredict $\Delta N_{\rm eff} \geq 0.14$ and $m_a$ in the range $O(1)\>{\rm \mueV}$--$O(10)\>{\rm meV}$, which can be firmly tested by upcoming CMBexperiments and axion searches.

Composite neutrinos and the QCD axion: baryogenesis, dark matter, small Dirac neutrino masses, and vanishing neutron EDM