RTM of Every Formal Methods A Partition Framework for Knowledge in Formal Reasoning

We model any formal method as a Reasoning Turing Machine (RTM) that evolves a base of established knowledge K0 by iteratively applying formal procedures fi . For a fixed ambient theory T and verifier family V , every RTM induces a canonical trichotomy of statements into: already known K, future–recognizable F (verifiable by certificates though not yet proved), and independent I. We prove K, F, I form a partition and argue (under standard assumptions) I ̸= ∅ (Gödel) and, for constructive growth regimes, F ̸= ∅. A uniform ability to decide RTM non-termination would decide co-Halting and collapse the trichotomy, which is impossible. This separates constructive growth from inherent incompleteness.