This paper tackles two design questions for recursive reasoning systems (systems that alternate between gathering evidence and refining understanding): how to represent the evolving state and when to stop iterating. The authors propose an epistemic state graph encoding claims, evidence, open questions, and confidence weights, and introduce the "order-gap"—the difference between expand-then-consolidate versus consolidate-then-expand. A small order-gap suggests the two orders agree and further iteration won't help. They provide a necessary and sufficient condition for the linearized order-gap to be informative (non-degenerate) near a fixed point, and sketch applications to agent loops, tree-of-thought, theorem proving, and continual learning.
Main takeaways:
- Recursive reasoning systems need a state representation (epistemic state graph) and a stopping criterion (order-gap).
- The order-gap measures whether expand-then-consolidate ≈ consolidate-then-expand; small gap means stop iterating.
- A formal condition tells you when the order-gap is informative versus algebraically vacuous near the fixed point (local, not global).
- Framework applies to agent loops, tree-of-thought reasoning, theorem proving, and continual learning.
- Addresses the "when to stop" problem in iterative reasoning without relying on heuristics or fixed iteration counts.