Dual-Resolution Recursive Energy: Certified Contract–Expand Inference for Sequential Decision Making

Fast habits and slow deliberation are usually modeled as separate systems. We propose a different view: System 1 and System 2 are two resolutions of the same recursive energy. A candidate decision is represented by a tree-structured energy whose root score determines the preferred action. The tree can be evaluated coarsely, by replacing subtrees with learned macro-energies, or finely, by expanding selected subtrees into their lower-level reasoning steps. In this view, a habit is a certified compressed evaluation, while deliberation is selective expansion of the same decision structure. The model switches resolution through learned certificates rather than a separate heuristic gate. If the contracted energy is sufficient to certify the current action choice, the model acts immediately; otherwise, it expands only the uncertain parts of the tree. Training follows a wake–sleep procedure: wake fits expert actions and records useful expansions, while sleep distills these expanded evaluations into macro-energies and calibrates contraction certificates. At test time, the same certificates guide online contract–expand decision making under a compute budget. This yields an imitation-based model of adaptive deliberation in which reasoning effort is allocated only when needed, while expansion traces expose over-contraction failures and learned certificates guide additional recursive expansion when compressed evaluations become unreliable.