Thermal Alignment as a Pathway to Axion Dark Matter

Thermal alignment cannot be inferred from the axion mean alone because the dissipative bath that erases the initial displacement also prepares field and momentum fluctuations. We derive a phase space covariance bound that quantifies this memory and noise relation in the full inertial Langevin system. A renormalizable finite temperature gauge theory then links the temporary susceptibility, the Chern-Simons bath, bath termination, and the stable late potential through one scalar transition. Solving the coupled mean and covariance evolution demonstrates erasure of the incoming state, release of a causal infrared spectrum, and capture of the full phase space distribution as axion dark matter across the transition interval. Thermal alignment therefore determines a calculable late axion state rather than a homogeneous displacement alone.