Recursive Self-Identification at the Σ1/Π1 Boundary

We analyze recursively enumerable arithmetical predicates intended to represent epistemic knowledge. We isolate three structural conditions — recursive presentability, factivity, and fragment-level truth alignment —whose conjunction is called Recursive Self-Identification (RSI). The central result is a definability-theoretic classification: for every syntactically decidable fragment Γ, RSI(Γ) holds if and only if the set of true Γ-sentences in the standard model is recursively enumerable. In particular, RSI(Σ1) is consistent, whereas RSI(Π𝑛) for 𝑛 ⩾ 1 and RSI(Σ𝑛) for 𝑛 ⩾ 2 are inconsistent. The analysis identifies two distinct obstruction mechanisms with structurally different proofs: a diagonal self-reference obstruction for fragments containing Π1, formally equivalent to Gödel’s first incompleteness theorem, and a recursion-theoretic complexity obstruction used here for the higher existential fragments. Among the standard existential and universal levels, Σ1 is the unique one at which recursive self-identification is possible.