Given the known fact that realizations of modal logics T, K4, S4, and intuitionistic propositional logic IPC (via S4) each requires justification formula that is self-referential, it is reasonable to consider the fragment of each that can be realized non-self-referentially, called the non-self-referential realizable (NSR) fragment. Based on prehistoric graphs of G3-style sequent proofs of modal theorems, we are able to define the loop-free-provable (LFP) fragment of each logic that turns out to be a decidable subset of the NSR fragment. In this talk, we will explore properties of LFP fragments, their applications in the study of NSR fragments, and other related issues.