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The relative thermodynamic stability of surface reconstructions including vacancies, adatoms and additional layers on both CaO and MnO$_{2}$ terminations is calculated to predict the surface phase diagram of ($\sqrt{2}\times\sqrt{2}$)$\it{R}$45$^{\circ}$ CaMnO$_{3}$ (001) using $\it{ab}$ $\it{initio}$ thermodynamics. Stoichiometric and nonstoichiometric reconstructions are considered. A set of boundary conditions driven by the binary and ternary sub-phases from CaMnO$_{3}$ defines its bulk stability region, enclosing the stable surface reconstructions that can be in equilibrium with the bulk. Most of the surfaces take the magnetic ordering of the bulk ground state, G-type antiferromagnetic (AFM); however, some of the MnO$_{2}$-terminated surfaces are more stable when the surface layer spins flip. At 573 K, the stoichiometric CaO- and MnO$_{2}$-terminated surfaces are predicted to be thermodynamically stable, as well as a CaO-terminated surface reconstruction where half the Ca are replaced by Mn. The MnO$_{2}$-terminated surface reconstructions dominate the phase diagrams at high temperatures, including phases with MnO and MnO$_{2}$ adatoms per surface unit cell., Comment: 21 pages, 4 figures