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

arXiv: https://arxiv.org/abs/2605.09968 · HTML: https://mlsanity.com/paper/2605.09968/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 44 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Prioritized Experience Replay](https://mlsanity.com/paper/1511.05952/) (2015)
- [PackNet: Adding Multiple Tasks to a Single Network by Iterative Pruning](https://mlsanity.com/paper/1711.05769/) (2017)
- [Universal Transformers](https://mlsanity.com/paper/1807.03819/) (2018)
- [Cyclical Learning Rates for Training Neural Networks](https://mlsanity.com/paper/1506.01186/) (2015)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Large Batch Training of Convolutional Networks](https://mlsanity.com/paper/1708.03888/) (2017)
- [Learning to Optimize Via Posterior Sampling](https://mlsanity.com/paper/1301.2609/) (2013)
- [A Finite Time Analysis of Temporal Difference Learning With Linear Function Approximation](https://mlsanity.com/paper/1806.02450/) (2018)
- [PonderNet: Learning to Ponder](https://mlsanity.com/paper/2107.05407/) (2021)
- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Recursive Language Models](https://mlsanity.com/paper/2512.24601/) (2025)

## Abstract (arXiv)

Every adaptive learning system must alternate between two operations: consolidating what it already knows and expanding into new evidence. We propose Consolidation-Expansion Operator Mechanics (OpMech), a framework that makes this structure precise. The central object is the order-gap Ogap(θ; e), the degree to which a consolidation operator~Q and an expansion operator~Pe fail to commute at a given knowledge state. Because the order-gap is computable from the system's own trajectory, it serves as a real-time control signal: large values indicate that the system is still sensitive to the ordering of consolidation and expansion; once the order-gap falls and stays small, further processing is unlikely to change the outcome. Three results give the signal precise meaning: the order-gap decays along convergent trajectories; a persistently large order-gap implies the system is far from its settled state; and an order-gap-based stopping rule terminates with provable guarantees in both noiseless and bounded-noise settings. The framework applies across five domains: bandits, reinforcement learning, stochastic optimization, continual learning, and recursive language models. We give conditions under which the order-gap reliably tracks convergence in three representative cases. We develop the recursive language model application in detail, showing how OpMech replaces heuristic stopping rules and fixed recursion budgets with principled, evidence-driven alternatives.