This paper presents a formal protocol for transitioning from discourse based on non-formal conceptual operators (Φ-operators) to rigorous mathematical reasoning. We define the class Φ of operators characterized by the absence of a complete formal system axiomatizing their derivability predicate. The central theorem demonstrates that for any Φ-operator O, the set O (K) ∖K generated from a consistent knowledge base K contains only (1) statements lacking determinate truth value, (2) logical tautologies, or (3) empirical hypotheses. We provide constructive isomorphisms mapping five canonical Φ-operators (labeled OH, OD, OG, ODe, OW) to formal systems, showing these mappings are either exact or involve bounded approximation errors. The protocol's soundness is established via complexity bounds derived from the Universal Parameterization Framework (UPF). A verifiable algorithmic criterion completes the transition. This work provides formal justification for focusing exclusively on mathematical systems for generating non-trivial necessary knowledge, while itself operating as a meta-mathematical analysis rather than philosophical discourse.
Daniil Osipenkov (Fri,) studied this question.