We discuss some relevant mathematical background first in Section S.2, those are directly used in the paper or proofs. We include some examples and properties related to OSBP and PPFs of OSBP in Section S.3. There are two lemmas and four theorems in the paper. We prove them here in Sections S.4, S.5, S.6 related to OSBP, PPF ofOSBP and SUMO respectively. We additionally provide one theorem (The...