  A number of proposals with differing predictions (e.g. Borel group
cohomology, oriented cobordism, group supercohomology, spin cobordism, etc.)
have been made for the classification of symmetry protected topological (SPT)
phases. Here we treat various proposals on an equal footing and present
rigorous, general results that are independent of which proposal is correct. We
do so by formulating a minimalist Generalized Cohomology Hypothesis, which is
satisfied by existing proposals and captures essential aspects of SPT
classification. From this Hypothesis alone, formulas relating classifications
in different dimensions and/or protected by different symmetry groups are
derived. Our formalism is expected to work for fermionic as well as bosonic
phases, Floquet as well as stationary phases, and spatial as well as on-site
symmetries. As an application, we predict that the complete classification of
3-dimensional bosonic SPT phases with space group symmetry $G$ is $H^4_{\rm
Borel}\left(G;U(1)\right) \oplus H^1_{\rm group}\left(G;\mathbb Z\right)$,
where the $H^1$ term classifies phases beyond the Borel group cohomology
proposal.
