The Six Primitives of Adaptation: Necessity, Independence, and Sequential Dependence

In this paper, I prove that sublinear regret across the environment class C requires six functional algorithmic properties, that these properties are mutually independent, and that they form a directed informational chain whose closing link makes the structure accumulative across cycles. All six properties are defined functionally—as conditions on the distributions an algorithm induces over actions and canonical summaries—so that the results are invariant under implementation and apply to any decision-making system that can be modelled within the class. Taken together, these results constitute the first unified functional theory of adaptive necessity in Class C. Class C is the union of all POMDPs satisfying at least one of six structural properties covering the fundamental qualitative dimensions of adaptive hardness: reward ambiguity (P1), absorbing traps (P2), local optima (P3), deterministic optimality (P4), constrained feasibility (P5), and nonstationarity (P6). Part I (Necessity). I define six algorithmic primitives X1–X6 as purely functional properties of algorithms. For each primitive I construct an explicit environment in C and prove an Ω (T) regret lower bound for any algorithm lacking that primitive. The six primitives are: Objective Tracking (X1), Cross-Context Safety Transfer (X2), Global Attractor Exploration (X3), Policy Simplification (X4), Feasibility Projection (X5), and Feedback Adaptation (X6). Part II (Independence). For every ordered pair (i, j) with i≠j I exhibit an explicit algorithm possessing Xj but lacking Xi, which suffers Ω (T) regret on E*. All thirty directed-pair independence results are verified on a single compound environment with full non-interference analysis. Part III (Sequential Dependence). I prove six Information Enhancement results establishing that the six primitives form a directed information chain: possessing Xi strictly increases the mutual information available to X₈+₁'s task (with the X4→X5 link behavioural rather than information-theoretic), and the closing link X6→X1 proves the chain accumulative via almost-sure martingale convergence of Bayesian posteriors across cycles. *All mathematical work is provided in full transparency and independent verification is highly encouraged. For any feedback or collaboration, please contact me via the email address listed on the paper. Updated: 07 April 2026 (V5. 0)