Consolidation-Expansion Operator Mechanics:A Unified Framework for Adaptive Learning

Recursive reasoning systems alternate between acquiring new evidence and refining an accumulated understanding. Two design choices are typically left implicit: how to represent the evolving reasoning state, and when to stop iterating. This paper addresses both. We represent the reasoning state as an epistemic state graph encoding extracted claims, evidential relations, open questions, and confidence weights. We define the order-gap as the distance between the states reached by expand-then-consolidate versus consolidate-then-expand; a small order-gap suggests that the two orderings agree and further iteration is unlikely to help. Our main result gives a necessary and sufficient condition for the linearised order-gap to be non-degenerate near the fixed point, showing when the criterion is informative rather than algebraically vacuous. This is a local condition, not a global convergence guarantee. We apply the framework to recursive reasoning systems and sketch its application to agent loops, tree-of-thought reasoning, theorem proving, and continual learning.

Language agents depend on memory across interactions. However, the limited context windows of large language models (LLMs) and their inference costs constrain how much memory can be used at once. Existing systems mainly follow two strategies: memory retention and memory consolidation. Retention keeps raw records and preserves exact details, but relevant evidence may not fit under a tight budget; consolidation compresses and combines records, improving coverage per token but risking the loss of query-critical details. Neither strategy is universally preferable. This raises two central questions: when should consolidation replace retention, and which operator -- Merge, Abstract, or Rewrite -- should be selected? We formalize this decision by decomposing each operator's utility into a coverage effect on evidence omitted by retention and a signed replacement effect on raw evidence that already fits. Their balance explains why the preferred action changes with relative budget pressure. We implement this mechanism with Offline Abstraction-Safety (OAS), a lightweight learner that estimates action utilities from pre-generation features with held-out harm calibration. The public LongMemEval and LoCoMo benchmarks show the same budget-dependent pattern. On LongMemEval, consolidation improves absolute accuracy by up to 48% under tight budgets, whereas retention is preferable under loose budgets; LoCoMo replicates this crossover at a smaller budget, consistent with its shorter evidence. On both datasets, cross-note abstraction and merging generally outperform local rewriting when compression is necessary.

Offline reinforcement learning (RL) can fail spectacularly when bootstrapped temporal-difference (TD) updates amplify their own errors, driving the critic toward extreme and unusable Q-values. A key counterintuitive insight of this work is that collapse is not only a property of the backup rule or network architecture: optimizer dynamics themselves can directly trigger or suppress instability. From a control-theoretic viewpoint, we model offline TD learning as a feedback system and analyze Adam-based critic updates. This yields a necessary and sufficient condition for stability of the induced local update dynamics: within the regime we analyze, these dynamics are stable if and only if the spectral radius of the corresponding update operator is strictly below one. Further analysis suggests that standard Adam updates can inadvertently distort the parameter geometry, motivating explicit orthogonality constraints to prevent TD error amplification. To this end, we propose AdamO, an Adam-based optimizer with a decoupled orthogonality correction regulated by a strict task-alignment budget. We prove that this design theoretically guarantees worst-case task safety and preserves Adam's continuous-time dissipative dynamics. Empirically, AdamO is broadly compatible with diverse offline RL baselines, improving stability and returns across a broad suite of benchmarks.