Sharp Computability Bounds for Nonforking Sections

We study computability of sections of the restriction map \(S_1(N)\to S_1(M)\) for countable stable models \(M\preccurlyeq N\). Types are represented by characteristic functions on formulas with named parameters, and the two elementary diagrams are separately decidable. Classical stable definability gives a continuous section computable from the input type together with the fixed oracle \(0'\). This bound is sharp, already for the decidable, \(\omega\)-stable, \(\omega\)-categorical theory of infinitely many infinite equivalence classes. For a fixed decidable-range elementary inclusion in this theory, an oracle \(X\) computes a section exactly when it computes the set of classes of \(N\) that meet \(M\). Explicit inclusions realize every computably enumerable degree as this least auxiliary degree. Thus a computable section need not exist, although every individual type in the example is computable. In contrast, a decidable full diagram of the predicate expansion \((N,M)\) gives a computable section for every stable theory. The result addresses a Type-2 formulation of an AIM question whose printed statement leaves the effective presentation unspecified.