22.6 Completion and coalgebraic stabilization

Repeated diagnosis and repair produce an unfolding

whose behavior can be studied coalgebraically. Existing final-coalgebra theorems give existence under class-based or accessible set semantics [ Aczel and Mendler , 1989 , Barr , 1993 ] . Contractive metric realizations can give convergence under stronger analytic assumptions.

The central comparison problem is to relate a freely generated LINCS completion to the final semantics of the INC unfolding:

When is this comparison fully faithful? When is it essentially surjective? Can its hypotheses be checked from a learning sketch rather than supplied externally?

One route is through accessibility. If admissible learning sketches and their models form an accessible category, if \(T_{\operatorname {INC}}\) preserves the relevant filtered colimits, and if presentable objects generate the required semantics, then free-completion and final-behavior views may meet in one construction [ Adámek and Rosický , 1994 , Makkai and Paré , 1989 ] .

The analytic frontier is equally important. A contraction factor \(\rho {\lt}1\) should ideally be derived from the sketch, tangent structure, observation profile, and admission map. Stochastic variants must accommodate martingale noise, biased tangent estimates, changing covers, and data-dependent contraction. Only then can stabilization become a verifiable certificate rather than an assumed property of the optimizer.