Continuity, Piecewise Corrections, and Functor Models in Function-Based Learning

Exact local correction of a learned function naturally introduces piecewise structure that conflicts with the global smoothness assumptions of classical analysis and optimization pipelines. Smoothing, quotient constructions, and discrete calculus each address aspects of this conflict, but none by itself serves heterogeneous downstream consumers with incompatible regularity demands. We argue that the conflict is architectural, not analytic: continuity is a requirement of particular consumers and should be enforced at designated interface functors rather than imposed as a global axiom on the internal mechanism. We make this precise in four steps. First, a descent condition under which internal corrections project lawfully to consumer views. Second, a fidelity theorem: for corrections of bounded scope there exists a smooth consumer adapter that is exact outside the correction region and within a stated tolerance on it. Third, an explicit, scope-parameterized construction of the adapter given a consumer’s contract, with the exactness–conditioning trade made quantitative. Fourth, an operational reading in which the smooth view is materialized lazily by a low-priority background service, with a concurrency analysis showing that domain separability reduces contention to a single, bounded race. We are explicit about what is not claimed: internal piecewise structure does not disappear, and the existence of a smooth adapter is a boundary property, not a smoothing of the mechanism.