3.6 Coinductive realization

A predictive theory need not terminate in a finite syntax. In the coalgebraic case, finite observations are unfoldings of an indefinitely continuing behavior, and behavioral equivalence is the natural success criterion. Universal imitation games use precisely this final-coalgebra perspective [ Mahadevan , 2024 ] .

ORACLE generalizes the move: the realized target may be a universal coalgebra, a category presented by generators and relations, a homotopy-coherent category, or an indexed family of local categories. Coinduction is therefore not replaced. It becomes one realization doctrine inside categorical identification.