Free phases of Majorana fermions: Tenfold ways compared

We provide a mathematically rigorous classification of symmetry-protected topological (SPT) phases of neutral free fermions. Our approach utilizes Karoubi triples with negative squares, thought of as polarizations. We prove that neutral free fermion SPT phases protected by a symmetry algebra $A$ are classified by the real $K$-theory group $K_2(A^{op})$, and demonstrate how our classification reproduces known results in the presence of charge. Our formalism also allows for symmetries described by groups, potentially with time-reversal, using the formalism of fermionic groups and their fermionic group $C^*$-algebras. Our classification extends to positive spatial dimensions and includes weak phases using the crystalline equivalence principle. Our approach clarifies and unifies various existing tenfold way classifications by establishing their equivalence through Morita equivalences of symmetry algebras. We expect our classification to be the natural domain for the free-to-interacting map proposed by Freed and Hopkins.