Generalized Residual Closure: General Learning Dynamics for Stability–Plasticity Compatibility

Learning algorithms typically specify the coordinates of adaptation before learning begins: parameters, latent states, policies, memories, or architectural variables are chosen first, and optimization determines how they should change. We introduce Generalized Residual Closure (GRC), a theory in which learning is instead the recursive least-lawful closure of future-relevant residuals. A learner's current representation opens into relations that constrain future continuation; disagreement with realized continuation produces a generalized residual that diagnoses insufficiency without prescribing its own write address. After fixing the learner–world boundary, we show that every persistent internal learning transition has only two primitive components: Transformation, which changes relations within the current representation, and Representation, which changes what must be represented as distinct. We define stability through reconstructible future-operational semantics rather than historical realization and derive a local necessary-and-sufficient condition, together with a least-cost construction, for nonzero semantic-safe learning; reconstructive preservation weakly enlarges the corresponding safe-plasticity operator. We further introduce dynamic sufficiency, future-closure viability, and Growth Learning, under which persistent learning must preserve future learnability and non-regressively improve the learner's closure-capability frontier. Optimization, deep/residual learning, and fixed-state reinforcement learning/control arise as restricted sectors of this dynamics. Under explicit operational assumptions, stable–plastic continual-growth learners admit a conditional GRC representation.