State Representation and Termination for Recursive Reasoning Systems

# State Representation and Termination for Recursive Reasoning Systems
Source: [https://arxiv.org/html/2605.06690](https://arxiv.org/html/2605.06690)
Debashis Guha Amritendu Mukherjee Sanjay Kukreja Tarun KumarS P Jain School of Global Management;debashis\.guha@spjain\.orgIndian Statistical Institute;amritendum@alum\.iisc\.ac\.inS P Jain School of Global Management;sanjay\.ds18dba008@spjain\.orgeClerx Services Ltd\.;Tarun\.Kumar06@eclerx\.com

###### Abstract

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\.

Keywords\.recursive reasoning; knowledge graph; state representation; iterative refinement; termination criterion; operator non\-commutativity; convergence detection; long\-context reasoning\.

## 1\. Introduction

### 1\.1\. Recursive Reasoning as Adaptive Inference

A growing class of machine learning systems operates by iterative refinement rather than single\-pass inference\. At each step such a system acquires new information \(a retrieved document, an observed action outcome, a sampled hypothesis, or a newly generated chain\-of\-thought step\) and integrates it into an accumulated internal state before deciding whether to iterate further or commit to an output\. This pattern appears in iterative retrieval\-augmented generation\(Trivedi et al\.,[2023](https://arxiv.org/html/2605.06690#bib.bib10)\), agent action\-observation loops\(Yao et al\.,[2023a](https://arxiv.org/html/2605.06690#bib.bib12)\), recursive language models\(Zhang et al\.,[2025](https://arxiv.org/html/2605.06690#bib.bib14)\), tree\-of\-thought and graph\-of\-thought reasoning\(Yao et al\.,[2023b](https://arxiv.org/html/2605.06690#bib.bib13); Besta et al\.,[2023](https://arxiv.org/html/2605.06690#bib.bib4)\), and continual learning\(Kirkpatrick et al\.,[2017](https://arxiv.org/html/2605.06690#bib.bib2)\)\. We call this class*recursive reasoning systems*\.

Despite their diversity, recursive reasoning systems share a common structure\. There is an internal statests\_\{t\}representing what the system holds at steptt\. There is an*expansion*step that brings in new external evidence\. There is a*consolidation*step that refines the state using only what is already present\. And there is a termination decision: iterate further, or commit\.

### 1\.2\. Two Missing Components

In current systems, both the state and the termination criterion are handled implicitly or imposed from outside, and their absence leads to characteristic failure modes across four well\-studied architectures as discussed below

Chain\-of\-thought and tree\-of\-thought reasoning\.Chain\-of\-thought prompting\(Wei et al\.,[2022](https://arxiv.org/html/2605.06690#bib.bib11)\)and its tree\-structured extension\(Yao et al\.,[2023b](https://arxiv.org/html/2605.06690#bib.bib13)\)represent the reasoning state as a text trace\. There is no explicit record of which claims are supported, which are in conflict, and which remain unresolved\. Contradictions accumulate without detection; reasoning drifts across steps; and there is no mechanism to revisit or revise earlier conclusions in light of later evidence\. Termination is fixed by depth or iteration count, with no notion of reasoning being complete\.

Retrieval\-augmented generation and multi\-hop QA\.Iterative retrieval systems such as IRCoT\(Trivedi et al\.,[2023](https://arxiv.org/html/2605.06690#bib.bib10)\)alternate between retrieving evidence and generating reasoning steps\. Retrieved facts are not tracked relationally across iterations, provenance is not preserved in a structured way, and the system has no representation of what it still needs to find\. This leads to redundant retrieval, missing critical evidence, and inconsistent synthesis across documents\. Termination is a fixed hop count or token budget, with no signal whether further retrieval would change the answer or not\.

Agent action\-observation loops\.In ReAct\-style agents\(Yao et al\.,[2023a](https://arxiv.org/html/2605.06690#bib.bib12)\), the state is a sequence of actions and observations with no canonical representation of goals, subgoals, or resolved versus unresolved tasks\. This produces looping behaviour, repeated tool calls, and failure to recognise completion\. Termination is an arbitrary maximum step count or manual stop condition\.

Iterative self\-refinement\.Self\-Refine\(Madaan et al\.,[2023](https://arxiv.org/html/2605.06690#bib.bib9)\)and related approaches iterate a generate\-critique\-refine loop without an explicit representation of what has been fixed, what remains uncertain, or what is stable\. Without such a state, the system has no signal for when further refinement is beneficial\. Empirically, intrinsic self\-correction without external feedback frequently fails to improve and can degrade performance\(Huang et al\.,[2023](https://arxiv.org/html/2605.06690#bib.bib6)\): the system oscillates between answers, over\-refines, or hallucinates improvements\. Termination is a fixed iteration count or a subjective heuristic\.

Across all four cases, state is implicit in transient text or hidden activations rather than structured, persistent, and inspectable; and termination is imposed by fixed limits rather than derived from the system’s own trajectory\.

### 1\.3\. The Proposal

We propose treating state representation and termination as explicit design choices, and give concrete definitions to both\. For state, we introduce the*epistemic state graph*\([Section˜2](https://arxiv.org/html/2605.06690#S2)\): a structured graph whose nodes encode claims, partial answers, and open questions, and whose edges encode evidential support, logical dependency, and inconsistency, all weighted by confidence\. This representation is structured, persistent, and inspectable across iterations\. For termination, we introduce the*order\-gap*\([Section˜5](https://arxiv.org/html/2605.06690#S5)\): a criterion derived from the system’s own operator structure that measures whether the system’s state is still sensitive to the ordering of expansion and consolidation\. When the order\-gap is small, both orderings produce nearly the same result, suggesting that further iteration is unlikely to matter\. When it is large, the ordering still matters and the system has not yet settled\.

### 1\.4\. Contributions

1. 1\.State representation\([Section˜2](https://arxiv.org/html/2605.06690#S2)\)\. The epistemic state graph, with a smooth Euclidean embedding that makes the expansion and consolidation operators amenable to analysis\.
2. 2\.Operator framework\([Section˜3](https://arxiv.org/html/2605.06690#S3)\)\. Formal definitions of the expansion operatorPeP\_\{e\}and consolidation operatorQQon the epistemic state, with their dynamics and key properties\.
3. 3\.Order\-gap termination\([Section˜5](https://arxiv.org/html/2605.06690#S5)\)\. The order\-gap termination criterion, a windowed stopping rule, and a non\-degeneracy theorem \([Proposition˜5\.2](https://arxiv.org/html/2605.06690#S5.Thmtheorem2)\) characterising when the criterion is informative\.
4. 4\.Stopping algorithm\([Section˜6](https://arxiv.org/html/2605.06690#S6)\)\. Pseudocode for recursive reasoning with order\-gap termination\.
5. 5\.Other recursive reasoning systems\([Section˜8](https://arxiv.org/html/2605.06690#S8)\)\. How the framework applies to agent loops, tree\-of\-thought reasoning, theorem proving, and continual learning\.
6. 6\.Illustration\([Section˜9](https://arxiv.org/html/2605.06690#S9)\)\. A closed\-form 2\-dimensional example verifying the dynamics and order\-gap formula\.

## 2\. State Representation

### 2\.1\. Requirements for a Reasoning State

A recursive reasoning system iterates by processing new evidence and refining its accumulated understanding\. The state must support two operations: expansion, which adds information from a new piece of evidence; and consolidation, which refines what is already present\. For both operations to be principled, the state must make their inputs and outputs explicit\. Expansion needs to know what is already established \(to avoid redundancy\) and what is still open \(to direct retrieval\)\. Consolidation needs to know which claims conflict \(to resolve them\) and which are mutually supporting \(to strengthen them\)\. The following six objects are the minimum needed to support both operations correctly:

- •Claims: extracted facts, with their provenance and confidence\.
- •Evidential relations: which claims support which others, and how strongly\.
- •Conflicts: which claims are inconsistent with each other\.
- •Partial conclusions: the system’s current best answer to the question or sub\-questions, together with the claims that support it\.
- •Open questions: dependencies that have been identified but not yet resolved: things the system knows it needs but has not yet found\.
- •Confidence: a weight on each claim and each relation, reflecting how strongly the evidence supports it\.

Without these, the system cannot direct later extraction based on what earlier extraction found, cannot detect when new evidence contradicts what it has already consolidated, and cannot identify what it still needs\. A state that holds all of this explicitly is the foundation for principled expansion and consolidation\.

### 2\.2\. The Epistemic State Graph

###### Definition 2\.1\(Epistemic State Graph\)\.

An*epistemic state graph*is a tupleS=\(V,E,ℓ,c,w\)S=\(V,E,\\ell,c,w\)where:

- •V=VC∪VA∪VOQV=V\_\{C\}\\cup V\_\{A\}\\cup V\_\{\\mathrm\{OQ\}\}is a finite vertex set partitioned intoClaimnodes \(VCV\_\{C\}\),PartialAnswernodes \(VAV\_\{A\}\), andOpenQuestionnodes \(VOQV\_\{\\mathrm\{OQ\}\}\)\.
- •E⊆V×V×𝒯E\\subseteq V\\times V\\times\\mathcal\{T\}is a set of typed directed edges, with𝒯=\{Supports,Requires,Contradicts\}\\mathcal\{T\}=\\\{\\textsc\{Supports\},\\textsc\{Requires\},\\textsc\{Contradicts\}\\\}\.
- •ℓ:V→ℝk\\ell:V\\to\\mathbb\{R\}^\{k\}assigns akk\-dimensional attribute vector to each node, encoding the claim text embedding and source identifier\.
- •c:V→\(0,1\]c:V\\to\(0,1\]assigns a confidence weight to each node\.
- •w:E→\(0,1\]w:E\\to\(0,1\]assigns a confidence weight to each edge\.

Node types\.Claim nodes record extracted facts\. PartialAnswer nodes hold the system’s current best answer to the question or a sub\-question, together with its supporting claims\. OpenQuestion nodes mark dependencies the system has identified but not yet resolved\.

Edge types\.Supportsedges run from claims to claims or to partial answers, encoding evidential backing\.Requiresedges run from partial answers or claims to open questions, encoding logical dependency on unresolved evidence\.Contradictsedges connect claims that are mutually inconsistent given current evidence\.

Consistency\.The graph is*consistent*at thresholdδ∈\(0,1\)\\delta\\in\(0,1\)if no two nodesu,vu,vconnected by aContradictsedge satisfyingc​\(u\)\>δc\(u\)\>\\deltaandc​\(v\)\>δc\(v\)\>\\deltaat the same time\. Consolidation drives the graph toward consistency\.

### 2\.3\. Smooth Euclidean Embedding

The epistemic graph is a combinatorial object; raw graph operations such as adding nodes, merging nodes, and retyping edges are discrete and not differentiable\. The operator analysis in[Section˜5](https://arxiv.org/html/2605.06690#S5)requires smooth maps\. We therefore embed the graph in a fixed\-dimension Euclidean space using differentiable relaxations\.

###### Definition 2\.2\(Graph Embedding\)\.

Fix a maximum node countNmaxN\_\{\\max\}and attribute dimensionkk\. The*graph embedding*φ:𝒢→ℝd\\varphi:\\mathcal\{G\}\\to\\mathbb\{R\}^\{d\}encodes the graph state by:

1. 1\.Encoding each node by its attribute vectorℓ​\(v\)∈ℝk\\ell\(v\)\\in\\mathbb\{R\}^\{k\}together with its confidence weightc​\(v\)∈\(0,1\]c\(v\)\\in\(0,1\], padded toNmaxN\_\{\\max\}slots in a fixed canonical order by node type, giving a block of dimension\(k\+1\)​Nmax\(k\+1\)N\_\{\\max\}\.
2. 2\.Appending a flattened adjacency\-weight tensor indexed by ordered node pairs and edge type, of dimensionNmax2​\|𝒯\|N\_\{\\max\}^\{2\}\|\\mathcal\{T\}\|\.

The total dimension isd=\(k\+1\)​Nmax\+Nmax2​\|𝒯\|d=\(k\+1\)N\_\{\\max\}\+N\_\{\\max\}^\{2\}\|\\mathcal\{T\}\|\.

The embeddingφ\\varphiconvertsSSinto a vectorθ=φ​\(S\)∈ℝd\\theta=\\varphi\(S\)\\in\\mathbb\{R\}^\{d\}, with graph operations implemented as differentiable coordinate updates so thatPeP\_\{e\}andQQareC1C^\{1\}on a neighbourhood of the states of interest\. We work inℝd\\mathbb\{R\}^\{d\}with the Euclidean norm throughout\.

## 3\. Expansion and Consolidation Operators

### 3\.1\. State Space and Operators

Let\(𝒮,∥⋅∥\)\(\\mathcal\{S\},\\left\\lVert\\cdot\\right\\rVert\)be a normed complete \(Banach\) space\. The embedded stateθt=φ​\(St\)∈ℝd⊂𝒮\\theta\_\{t\}=\\varphi\(S\_\{t\}\)\\in\\mathbb\{R\}^\{d\}\\subset\\mathcal\{S\}represents the system’s accumulated understanding at steptt\.

The*expansion operator*Pe:𝒮→𝒮P\_\{e\}:\\mathcal\{S\}\\to\\mathcal\{S\}updates the state by incorporating a new piece of evidencee∈ℰe\\in\\mathcal\{E\}, drawn from a distributionP\(⋅∣θt\)P\(\\cdot\\mid\\theta\_\{t\}\)that may depend on the current state\. In the epistemic graph setting,PeP\_\{e\}adds Claim nodes for newly extracted facts, links them to existing nodes via typed edges, and promotes any OpenQuestion node thateedirectly resolves to a PartialAnswer node\. Expansion is the only means by which new external information enters the state\.

The*consolidation operator*Q:𝒮→𝒮Q:\\mathcal\{S\}\\to\\mathcal\{S\}refines the state using only what is already present, without acquiring new evidence\. In the epistemic graph,QQresolvesContradictspairs by removing or downweighting the lower\-confidence endpoint, merges redundant Claim nodes, and shifts weight toward the current best PartialAnswer\.

The expansion\-consolidation decomposition and its formal properties are developed in full generality inGuha \([2026](https://arxiv.org/html/2605.06690#bib.bib5)\); this paper applies the decomposition to recursive reasoning systems and uses it to derive the order\-gap termination criterion\.

For the operator analysis, we impose the following assumption\. The concrete graph operations described above do not automatically satisfy it under all embeddings; it is a condition on the implemented consolidation map, which must be verified for a given implementation\.

Assumption \(Contractivity\)\.QQis a*contraction*on\(𝒮,∥⋅∥\)\(\\mathcal\{S\},\\left\\lVert\\cdot\\right\\rVert\): there existsρ∈\[0,1\)\\rho\\in\[0,1\)such that‖Q​\(θ1\)−Q​\(θ2\)‖≤ρ​‖θ1−θ2‖\\left\\lVert Q\(\\theta\_\{1\}\)\-Q\(\\theta\_\{2\}\)\\right\\rVert\\leq\\rho\\left\\lVert\\theta\_\{1\}\-\\theta\_\{2\}\\right\\rVertfor allθ1,θ2∈𝒮\\theta\_\{1\},\\theta\_\{2\}\\in\\mathcal\{S\}\.

Since𝒮\\mathcal\{S\}is a normed complete space andQQmaps𝒮\\mathcal\{S\}into itself, the Banach fixed\-point theorem then guarantees a unique fixed pointθ⋆∈𝒮\\theta^\{\\star\}\\in\\mathcal\{S\}satisfyingQ​\(θ⋆\)=θ⋆Q\(\\theta^\{\\star\}\)=\\theta^\{\\star\}\. This is a fixed point of the consolidation mapQQalone, not necessarily of the full stochastic dynamicsθt\+1=Q​\(Pet​\(θt\)\)\\theta\_\{t\+1\}=Q\(P\_\{e\_\{t\}\}\(\\theta\_\{t\}\)\), which depends on the expansion operator and the evidence distribution as well\.

### 3\.2\. Dynamics

The update rule at each recursive reasoning step is:

θt\+1=Q\(Pet\(θt\)\),et∼P\(⋅∣θt\)\.\\theta\_\{t\+1\}=Q\(P\_\{e\_\{t\}\}\(\\theta\_\{t\}\)\),\\qquad e\_\{t\}\\sim P\(\\cdot\\mid\\theta\_\{t\}\)\.\(1\)The system expands to incorporate new evidence, then consolidates what it holds\. This is the structure underlying the Recursive Language Model’s read\-then\-aggregate loop\(Zhang et al\.,[2025](https://arxiv.org/html/2605.06690#bib.bib14)\)and the Tiny Recursive Model’szz\-then\-yyiteration\(Jolicoeur\-Martineau,[2025](https://arxiv.org/html/2605.06690#bib.bib7)\); the two systems arrive at the same expand\-then\-consolidate pattern from opposite ends of the model\-scale spectrum\.

### 3\.3\. Non\-Commutativity

A key structural property of the operatorsPeP\_\{e\}andQQis that they do not, in general, commute\. Consolidating before expansion commits the state to its current best answer before new evidence arrives; expanding before consolidation allows new evidence to influence the state before commitment\. These two orderings produce different results, and the magnitude of the difference reflects how much the system’s current understanding would change if it paused to look at one more piece of evidence before consolidating\. This difference is the quantity we use as a termination criterion\.

## 4\. The Termination Problem

### 4\.1\. Why Termination is Hard Without State

Without an explicit state, the termination decision has nothing to act on\. A system whose state is an unstructured text trace cannot inspect whether it has found evidence for all its open questions, whether conflicts have been resolved, or whether further evidence would change the answer\. It must rely on external proxies: a step count, a token budget, or a self\-reported confidence level\.

Each proxy fails predictably\. Step counts are right for neither easy instances \(which converge early\) nor hard ones \(which may need more depth\)\. Token budgets conflate the cost of computation with the value of additional reasoning\. Self\-reported confidence is systematically miscalibrated on precisely the instances that are hardest\(Kadavath et al\.,[2022](https://arxiv.org/html/2605.06690#bib.bib1)\)\.

### 4\.2\. What a Principled Termination Criterion Needs

A principled termination criterion should have three properties\. First, it should be*endogenous*: derived from the system’s own state trajectory, not imposed from outside\. Second, it should be*sensitive*: it should be large when further iteration would materially change the answer and small when the system has settled\. Third, it should be*computable*: calculable from quantities the system already has access to, without requiring knowledge of the ground truth or the full document\.

The termination criterion we propose in[Section˜5](https://arxiv.org/html/2605.06690#S5)is designed to satisfy all three\. It is derived from the operator structure of the system itself, it reflects whether the ordering of operations still matters, and it is computable at each step from the current state and the current evidence\.

## 5\. Order\-Gap Termination

### 5\.1\. The Order\-Gap Criterion

###### Definition 5\.1\(Order\-Gap\(Guha,[2026](https://arxiv.org/html/2605.06690#bib.bib5)\)\)\.

The*order\-gap*at stateθ∈𝒮\\theta\\in\\mathcal\{S\}given evidenceeeis

Ω​\(θ;e\)=‖Q​\(Pe​\(θ\)\)−Pe​\(Q​\(θ\)\)‖\.\\Omega\(\\theta;\\,e\)\\;=\\;\\left\\lVert Q\(P\_\{e\}\(\\theta\)\)\-P\_\{e\}\(Q\(\\theta\)\)\\right\\rVert\.\(2\)

The definition holds the realised evidence itemeefixed across both orderings\. The first orderingQ​\(Pe​\(θ\)\)Q\(P\_\{e\}\(\\theta\)\)is what the system actually computes \(expand then consolidate\)\. The secondPe​\(Q​\(θ\)\)P\_\{e\}\(Q\(\\theta\)\)is the hypothetical in which consolidation happened first\. Since the evidence distribution may depend on the state,Pe​\(Q​\(θ\)\)P\_\{e\}\(Q\(\\theta\)\)uses the same realizedeerather than resampling fromP\(⋅∣Q\(θ\)\)P\(\\cdot\\mid Q\(\\theta\)\); this makes the comparison well\-defined and computable from quantities already available at each step\.

WhenΩ​\(θ;e\)\\Omega\(\\theta;e\)is large, the two orderings of expansion and consolidation produce substantially different states: the system’s answer is sensitive to whether it consolidates before or after reading the next piece of evidence, indicating it has not settled\. WhenΩ​\(θ;e\)\\Omega\(\\theta;e\)is small, both orderings produce nearly the same result: the ordering no longer matters, and further expansion is less likely to change the outcome materially\.

### 5\.2\. Windowed Stopping Rule

This motivates the following termination criterion: halt iteration when the windowed empirical order\-gap

Ω^t,w=1w​∑τ=t−w\+1tΩ​\(θτ;eτ\)\\widehat\{\\Omega\}\_\{t,w\}\\;=\\;\\frac\{1\}\{w\}\\sum\_\{\\tau=t\-w\+1\}^\{t\}\\Omega\(\\theta\_\{\\tau\};\\,e\_\{\\tau\}\)\(3\)falls below a thresholdε\>0\\varepsilon\>0\.

Scope\.A small order\-gap is a useful termination criterion, but it does not by itself guarantee that the state has converged to the true answer\.[Proposition˜5\.2](https://arxiv.org/html/2605.06690#S5.Thmtheorem2)below establishes when the linearised order\-gap near the fixed point is non\-degenerate, so that a small order\-gap is not an algebraic artefact\. Connecting this to a global convergence guarantee \(via Lipschitz constants, noise bounds, and finite\-sample concentration ofΩ^t,w\\widehat\{\\Omega\}\_\{t,w\}\) would require additional assumptions and is beyond the present scope\.

### 5\.3\. Linearised Commutator and Gramian

To characterise non\-degeneracy formally, we linearise the operators at the consolidation fixed pointθ⋆\\theta^\{\\star\}\. We impose one additional assumption:

Assumption \(Redundancy at the fixed point\)\.Pe​\(θ⋆\)=θ⋆P\_\{e\}\(\\theta^\{\\star\}\)=\\theta^\{\\star\}for alle∈ℰe\\in\\mathcal\{E\}\.

This says that at the fully consolidated fixed point, processing additional evidence leaves the state unchanged, since all relevant information is already incorporated\. Under this assumption, the chain rule applied toFe​\(θ\)=Q​\(Pe​\(θ\)\)−Pe​\(Q​\(θ\)\)F\_\{e\}\(\\theta\)=Q\(P\_\{e\}\(\\theta\)\)\-P\_\{e\}\(Q\(\\theta\)\)atθ⋆\\theta^\{\\star\}givesD​Fe​\(θ⋆\)=D​Q​\(Pe​\(θ⋆\)\)​D​Pe​\(θ⋆\)−D​Pe​\(θ⋆\)​D​Q​\(θ⋆\)=D​Q​\(θ⋆\)​D​Pe​\(θ⋆\)−D​Pe​\(θ⋆\)​D​Q​\(θ⋆\)DF\_\{e\}\(\\theta^\{\\star\}\)=DQ\(P\_\{e\}\(\\theta^\{\\star\}\)\)DP\_\{e\}\(\\theta^\{\\star\}\)\-DP\_\{e\}\(\\theta^\{\\star\}\)DQ\(\\theta^\{\\star\}\)=DQ\(\\theta^\{\\star\}\)DP\_\{e\}\(\\theta^\{\\star\}\)\-DP\_\{e\}\(\\theta^\{\\star\}\)DQ\(\\theta^\{\\star\}\), where the last equality usesPe​\(θ⋆\)=θ⋆P\_\{e\}\(\\theta^\{\\star\}\)=\\theta^\{\\star\}\. We define the*linearised commutator*for evidenceeeas the following quantity:

Σe:=D​Q​\(θ⋆\)​D​Pe​\(θ⋆\)−D​Pe​\(θ⋆\)​D​Q​\(θ⋆\)∈ℝd×d\.\\Sigma\_\{e\}\\;:=\\;DQ\(\\theta^\{\\star\}\)\\,DP\_\{e\}\(\\theta^\{\\star\}\)\-DP\_\{e\}\(\\theta^\{\\star\}\)\\,DQ\(\\theta^\{\\star\}\)\\;\\in\\;\\mathbb\{R\}^\{d\\times d\}\.\(4\)The*commutator Gramian*is

Gθ⋆:=𝔼e∼P\(⋅∣θ⋆\)​\[Σe⊤​Σe\]∈ℝd×d,G\_\{\\theta^\{\\star\}\}\\;:=\\;\\mathbb\{E\}\_\{e\\sim P\(\\cdot\\mid\\theta^\{\\star\}\)\}\\bigl\[\\Sigma\_\{e\}^\{\\top\}\\Sigma\_\{e\}\\bigr\]\\;\\in\\;\\mathbb\{R\}^\{d\\times d\},\(5\)where the expectation is taken under the evidence distribution evaluated at the fixed pointθ⋆\\theta^\{\\star\}\. For anyv∈ℝdv\\in\\mathbb\{R\}^\{d\}, the quadratic form satisfies:

v⊤​Gθ⋆​v=𝔼e∼P\(⋅∣θ⋆\)​\[‖Σe​v‖2\]\.v^\{\\top\}G\_\{\\theta^\{\\star\}\}v\\;=\\;\\mathbb\{E\}\_\{e\\sim P\(\\cdot\\mid\\theta^\{\\star\}\)\}\\bigl\[\\left\\lVert\\Sigma\_\{e\}v\\right\\rVert^\{2\}\\bigr\]\.\(6\)For finite or countable evidence distributions this expands as∑e∈supp⁡\(P\)P​\(e∣θ⋆\)​‖Σe​v‖2\\sum\_\{e\\in\\operatorname\{supp\}\(P\)\}P\(e\\mid\\theta^\{\\star\}\)\\,\\left\\lVert\\Sigma\_\{e\}v\\right\\rVert^\{2\}; for general evidence spaces the sum is replaced by an integral and the support condition by aP\(⋅∣θ⋆\)P\(\\cdot\\mid\\theta^\{\\star\}\)\-almost\-sure condition\. The proposition below is stated for the finite/countable case; the general case follows by replacing sums with integrals throughout\.

### 5\.4\. Non\-Degeneracy of the Commutator Gramian

###### Proposition 5\.2\(Local Non\-Degeneracy of the Commutator Gramian\)\.

LetW⊆ℝdW\\subseteq\\mathbb\{R\}^\{d\}be a subspace\. For finite or countable evidence distributions, the commutator Gramian satisfiesGθ⋆≻0G\_\{\\theta^\{\\star\}\}\\succ 0onWW\(i\.e\.,v⊤​Gθ⋆​v\>0v^\{\\top\}G\_\{\\theta^\{\\star\}\}v\>0for all nonzerov∈Wv\\in W\) if and only if

⋂e∈supp\(P\(⋅∣θ⋆\)\)ker⁡\(Σe\)∩W=\{0\}\.\\bigcap\_\{e\\in\\operatorname\{supp\}\(P\(\\cdot\\mid\\theta^\{\\star\}\)\)\}\\ker\(\\Sigma\_\{e\}\)\\;\\cap\\;W\\;=\\;\\\{0\\\}\.\(7\)For general evidence spaces, the equivalent condition is: for every nonzerov∈Wv\\in W,Σe​v≠0\\Sigma\_\{e\}v\\neq 0on a set of positiveP\(⋅∣θ⋆\)P\(\\cdot\\mid\\theta^\{\\star\}\)\-measure\.

###### Proof\.

For finite or countable distributions,v⊤​Gθ⋆​v=∑e∈supp\(P\(⋅∣θ⋆\)\)P​\(e∣θ⋆\)​‖Σe​v‖2v^\{\\top\}G\_\{\\theta^\{\\star\}\}v=\\sum\_\{e\\in\\operatorname\{supp\}\(P\(\\cdot\\mid\\theta^\{\\star\}\)\)\}P\(e\\mid\\theta^\{\\star\}\)\\,\\left\\lVert\\Sigma\_\{e\}v\\right\\rVert^\{2\}\. SinceP​\(e∣θ⋆\)\>0P\(e\\mid\\theta^\{\\star\}\)\>0for alle∈supp\(P\(⋅∣θ⋆\)\)e\\in\\operatorname\{supp\}\(P\(\\cdot\\mid\\theta^\{\\star\}\)\), each summand is non\-negative\. Hencev⊤​Gθ⋆​v=0v^\{\\top\}G\_\{\\theta^\{\\star\}\}v=0if and only if‖Σe​v‖2=0\\left\\lVert\\Sigma\_\{e\}v\\right\\rVert^\{2\}=0for everyeein the support, i\.e\.,v∈ker⁡\(Σe\)v\\in\\ker\(\\Sigma\_\{e\}\)for every suchee\. Thereforev⊤​Gθ⋆​v=0⇔v∈⋂e∈supp\(P\(⋅∣θ⋆\)\)ker⁡\(Σe\)v^\{\\top\}G\_\{\\theta^\{\\star\}\}v=0\\iff v\\in\\bigcap\_\{e\\in\\operatorname\{supp\}\(P\(\\cdot\\mid\\theta^\{\\star\}\)\)\}\\ker\(\\Sigma\_\{e\}\)\. Restricting toWW:Gθ⋆≻0G\_\{\\theta^\{\\star\}\}\\succ 0onWWholds iff no nonzerov∈Wv\\in Whasv⊤​Gθ⋆​v=0v^\{\\top\}G\_\{\\theta^\{\\star\}\}v=0, which is exactly condition \([7](https://arxiv.org/html/2605.06690#S5.E7)\)\. For the general\-distribution case,v⊤​Gθ⋆​v=∫‖Σe​v‖2​𝑑P​\(e∣θ⋆\)v^\{\\top\}G\_\{\\theta^\{\\star\}\}v=\\int\\left\\lVert\\Sigma\_\{e\}v\\right\\rVert^\{2\}\\,dP\(e\\mid\\theta^\{\\star\}\)\. Since the integrand is non\-negative, the integral is zero if and only ifΣe​v=0\\Sigma\_\{e\}v=0holdsP\(⋅∣θ⋆\)P\(\\cdot\\mid\\theta^\{\\star\}\)\-almost surely\. HenceGθ⋆≻0G\_\{\\theta^\{\\star\}\}\\succ 0onWWholds if and only if every nonzerov∈Wv\\in WsatisfiesΣe​v≠0\\Sigma\_\{e\}v\\neq 0on a set of positiveP\(⋅∣θ⋆\)P\(\\cdot\\mid\\theta^\{\\star\}\)\-measure\. ∎

### 5\.5\. Per\-Question Coverage Check

## 6\. Algorithm

[Algorithm˜1](https://arxiv.org/html/2605.06690#alg1)formalises the recursive reasoning procedure with order\-gap termination\. Each iteration computes bothQ​\(Pet​\(θt\)\)Q\(P\_\{e\_\{t\}\}\(\\theta\_\{t\}\)\)\(the actual state update\) andPet​\(Q​\(θt\)\)P\_\{e\_\{t\}\}\(Q\(\\theta\_\{t\}\)\)\(the alternative ordering, computed only to evaluate the order\-gap\)\. This roughly doubles the per\-iteration consolidation cost, which is typically small relative to the cost of expansion \(document retrieval, model forward passes\)\.

Algorithm 1Recursive Reasoning with Order\-Gap Termination1:Initial state

θ0\\theta\_\{0\}; operators

PeP\_\{e\},

QQ; evidence distribution

P\(⋅∣θ\)P\(\\cdot\\mid\\theta\); threshold

ε\>0\\varepsilon\>0; window width

ww; budget

TmaxT\_\{\\max\}
2:Final state

θT\\theta\_\{T\}; stopping step

TT
3:Initialise buffer

B←\(\)B\\leftarrow\(\)
4:for

t=0,1,…,Tmax−1t=0,1,\\ldots,T\_\{\\max\}\-1do

5:Sample

et∼P\(⋅∣θt\)e\_\{t\}\\sim P\(\\cdot\\mid\\theta\_\{t\}\)
6:Compute

Ωt←‖Q​\(Pet​\(θt\)\)−Pet​\(Q​\(θt\)\)‖\\Omega\_\{t\}\\leftarrow\\left\\lVert Q\(P\_\{e\_\{t\}\}\(\\theta\_\{t\}\)\)\-P\_\{e\_\{t\}\}\(Q\(\\theta\_\{t\}\)\)\\right\\rVert
7:Update

θt\+1←Q​\(Pet​\(θt\)\)\\theta\_\{t\+1\}\\leftarrow Q\(P\_\{e\_\{t\}\}\(\\theta\_\{t\}\)\)
8:Append

Ωt\\Omega\_\{t\}to

BB
9:if

\|B\|≥w\|B\|\\geq wand

1w​∑i=\|B\|−w\+1\|B\|Bi≤ε\\frac\{1\}\{w\}\\sum\_\{i=\|B\|\-w\+1\}^\{\|B\|\}B\_\{i\}\\leq\\varepsilonthen

10:return

\(θt\+1,t\+1\)\(\\theta\_\{t\+1\},\\;t\+1\)
11:endif

12:endfor

13:return

\(θTmax,Tmax\)\(\\theta\_\{T\_\{\\max\}\},\\;T\_\{\\max\}\)

## 7\. Application to Recursive Language\-Model Reasoning

We now apply the framework to recursive language\-model reasoning, the setting in which the state and termination problems are sharpest and the engineering infrastructure most developed\.

### 7\.1\. The Long\-Context Problem and Recursive Approaches

Single\-pass transformer models read all tokens simultaneously\. As document length grows, performance on tasks requiring integration of information across the document degrades, not because the model lacks capacity in principle, but because attention, spread across tens of thousands of tokens, diffuses\(Liu et al\.,[2024](https://arxiv.org/html/2605.06690#bib.bib8)\)\. The Recursive Language Model \(RLM\), developed byZhang et al\. \([2025](https://arxiv.org/html/2605.06690#bib.bib14)\)addresses this by treating the document as an external environment to be explored programmatically; the model reads fragments, extracts structured information, and spawns recursive calls for sub\-questions, folding their results into a growing aggregate state\. On the OOLONG long\-context benchmark, the RLM maintained strong performance at one million tokens where single\-pass models degraded substantially\.

The Tiny Recursive Model \(TRM\) ofJolicoeur\-Martineau \([2025](https://arxiv.org/html/2605.06690#bib.bib7)\), developed independently for abstract visual reasoning, alternates azz\-update \(refining the latent state given fixed input\) with ayy\-update \(committing to an output\)\. At seven million parameters it achieved 45% on ARC\-AGI\-1, surpassing models orders of magnitude larger\.

Both systems follow the dynamics of \([1](https://arxiv.org/html/2605.06690#S3.E1)\) but leave its two components implicit\. The RLM’s aggregate state is an unstructured REPL buffer; the TRM’s is a latent vector\. Neither derives its termination criterion from the operator structure\.

### 7\.2\. Operator Identification

In the language\-model setting, the epistemic graph of[Definition˜2\.1](https://arxiv.org/html/2605.06690#S2.Thmtheorem1)is the stateSt=θtS\_\{t\}=\\theta\_\{t\}\. The operators are:

Expansion\.Pet​\(S\)P\_\{e\_\{t\}\}\(S\)incorporates document chunkete\_\{t\}: it adds Claim nodes for newly extracted facts, links them to existing nodes via typed edges, and promotes any OpenQuestion node thatete\_\{t\}resolves to a PartialAnswer node\.

Consolidation\.Q​\(S\)Q\(S\)refines the accumulated state: it resolvesContradictspairs by removing or downweighting the lower\-confidence endpoint, merges redundant Claim nodes, and shifts weight toward the current best PartialAnswer\.

### 7\.3\. Control Applications

Three control applications follow directly from the order\-gap termination criterion\.

Stopping\.Halt chunk processing whenΩ^t,w≤ε\\widehat\{\\Omega\}\_\{t,w\}\\leq\\varepsilon\. Assuming that reprocessing a chunk already fully represented leaves the state unchanged, this follows[Algorithm˜1](https://arxiv.org/html/2605.06690#alg1)with formal motivation from[Proposition˜5\.2](https://arxiv.org/html/2605.06690#S5.Thmtheorem2)when the coverage check of[Remark˜5\.4](https://arxiv.org/html/2605.06690#S5.Thmtheorem4)holds\.

Consolidation scheduling\.TriggerQQwhen accumulated order\-gap since the last consolidation exceeds a cost threshold, rather than on a fixed schedule\.

Adaptive extraction\.Weight chunk selection toward evidence that addresses OpenQuestion nodes failing the coverage check of[Remark˜5\.4](https://arxiv.org/html/2605.06690#S5.Thmtheorem4), directing retrieval toward what is genuinely missing\.

## 8\. Other Recursive Reasoning Systems

The expansion\-consolidation decomposition and the order\-gap termination criterion apply beyond recursive language models\. We sketch five additional cases to establish the generality of the framework\.

Agent action\-observation loops\.In systems such as ReAct\-style agents\(Yao et al\.,[2023a](https://arxiv.org/html/2605.06690#bib.bib12)\), the state is the agent’s current understanding of its goals, sub\-goals, and environmental observations\. Expansion is taking an action and observing the outcome; consolidation is updating the internal goal structure in light of the observation\. The order\-gap measures whether the agent’s goal understanding would shift if it took one more action before updating its goal model\. The failure modes of repeated tool calls and looping correspond precisely to a large order\-gap that the system cannot detect because it has no state to measure it against\.

Tree\-of\-thought and graph\-of\-thought reasoning\.These systems\(Yao et al\.,[2023b](https://arxiv.org/html/2605.06690#bib.bib13); Besta et al\.,[2023](https://arxiv.org/html/2605.06690#bib.bib4)\)generate multiple reasoning paths and aggregate them\. Expansion is generating a new reasoning branch; consolidation is aggregating evidence across branches\. The order\-gap measures whether a new branch would change the aggregated conclusion before the current branches are consolidated\. The absence of an explicit state means that contradictions across branches accumulate without detection, in analogy with theContradictsedge problem in the epistemic graph\.

Iterative self\-refinement\.Self\-refinement systems\(Madaan et al\.,[2023](https://arxiv.org/html/2605.06690#bib.bib9)\)iterate a generate\-critique\-refine loop\. Expansion is generating a new candidate output or critique; consolidation is integrating the critique into a revised answer\. The order\-gap measures whether the system’s current answer would change if it consolidated its existing critique before generating a new one\. Without an explicit state tracking what has been fixed and what remains uncertain, the system cannot detect when refinement has converged, which is precisely the condition under which further iteration degrades rather than improves the answer\(Huang et al\.,[2023](https://arxiv.org/html/2605.06690#bib.bib6)\)\.

Theorem proving\.In neural theorem provers\(Lample et al\.,[2022](https://arxiv.org/html/2605.06690#bib.bib3)\), the state is the current proof context: established lemmas, open subgoals, and failed attempts\. Expansion is applying a tactic or lemma; consolidation is simplifying the proof context and closing sub\-goals\. The order\-gap measures whether the proof context is sensitive to the order in which tactics are applied, a useful indicator of whether the current strategy is well\-founded\.

Continual learning\.A continual learning system\(Kirkpatrick et al\.,[2017](https://arxiv.org/html/2605.06690#bib.bib2)\)must expand to incorporate new task evidence while consolidating to protect previously learned knowledge\. The order\-gap measures the conflict between new task evidence and existing knowledge: large values indicate that the system is in a regime of high task conflict where plasticity and retention trade against each other, and adaptive regularisation is warranted\.

In all these cases the two components \(an explicit structured state and the order\-gap as a termination criterion\) apply naturally within the general framework of[Sections˜3](https://arxiv.org/html/2605.06690#S3)and[5](https://arxiv.org/html/2605.06690#S5)\.

## 9\. Closed\-Form Illustration

We trace the order\-gap on a synthetic example with analytically specified operators to verify that the dynamics \([1](https://arxiv.org/html/2605.06690#S3.E1)\) and the order\-gap formula \([2](https://arxiv.org/html/2605.06690#S5.E2)\) are internally consistent and to illustrate the criterion’s qualitative behaviour\. This is expository, not empirical\.

### 9\.1\. Operators and Closed\-Form Order\-Gap

We use a 2\-dimensional stateθ=\(c,u\)∈\[0,1\]2\\theta=\(c,u\)\\in\[0,1\]^\{2\}, whereccis the system’s confidence in its current answer anduuis its residual uncertainty\. The operators are:

Pe​\(c,u\)\\displaystyle P\_\{e\}\(c,\\,u\)=\(c\+αe​\(1−c\),u\),\\displaystyle=\\bigl\(c\+\\alpha\_\{e\}\(1\-c\),\\;\\;u\\bigr\),\(8\)Q​\(c,u\)\\displaystyle Q\(c,\\,u\)=\(c,ρ​\(1−c\)​u\),\\displaystyle=\\bigl\(c,\\;\\;\\rho\(1\-c\)\\,u\\bigr\),\(9\)whereαe∈\(0,1\)\\alpha\_\{e\}\\in\(0,1\)is the relevance weight of evidenceeeandρ∈\(0,1\)\\rho\\in\(0,1\)is the consolidation rate\.PeP\_\{e\}raises confidence proportionally to relevance;QQreduces uncertainty proportionally to remaining headroom\(1−c\)\(1\-c\), so consolidation has diminishing effect as confidence approaches one\.

Both operators extend toC∞C^\{\\infty\}maps on an open neighbourhood of\[0,1\]2\[0,1\]^\{2\}\. ForQQalone, every point\(c,0\)\(c,0\)is a fixed point\. Under repeated relevant expansion withαe\>0\\alpha\_\{e\}\>0, the coupled dynamicsθt\+1=Q​\(Pet​\(θt\)\)\\theta\_\{t\+1\}=Q\(P\_\{e\_\{t\}\}\(\\theta\_\{t\}\)\)drivec→1c\\to 1andu→0u\\to 0, so\(1,0\)\(1,0\)is the limiting terminal state of the full process\. This example does not satisfy the global contraction assumption on the full\(c,u\)\(c,u\)state, sinceQQleavesccunchanged, but it illustrates the order\-gap calculation and decay behaviour in a transparent setting\.

Closed\-form order\-gap\.Direct computation gives:

Q​\(Pe​\(c,u\)\)\\displaystyle Q\(P\_\{e\}\(c,u\)\)=\(c\+αe​\(1−c\),ρ​\(1−c\)​\(1−αe\)​u\),\\displaystyle=\\bigl\(c\+\\alpha\_\{e\}\(1\-c\),\\;\\;\\rho\\,\(1\-c\)\(1\-\\alpha\_\{e\}\)\\,u\\bigr\),\(10\)Pe​\(Q​\(c,u\)\)\\displaystyle P\_\{e\}\(Q\(c,u\)\)=\(c\+αe​\(1−c\),ρ​\(1−c\)​u\)\.\\displaystyle=\\bigl\(c\+\\alpha\_\{e\}\(1\-c\),\\;\\;\\rho\\,\(1\-c\)\\,u\\bigr\)\.\(11\)The two orderings agree on the first coordinate and differ on the second:

Ω​\(\(c,u\);e\)=ρ​αe​\(1−c\)​u\.\\Omega\\bigl\(\(c,u\);\\,e\\bigr\)\\;=\\;\\rho\\,\\alpha\_\{e\}\\,\(1\-c\)\\,u\.\(12\)The signal vanishes asc→1c\\to 1\(high answer confidence\), asu→0u\\to 0\(uncertainty exhausted\), or asαe→0\\alpha\_\{e\}\\to 0\(irrelevant evidence\)\. It is large when confidence is low, uncertainty is high, and the evidence is relevant, the regime where further expansion is most likely to matter\.

### 9\.2\. Trajectory

We setρ=0\.9\\rho=0\.9, initial stateθ0=\(0\.20,0\.80\)\\theta\_\{0\}=\(0\.20,\\;0\.80\), and present five pieces of evidence with relevance weightsα1=0\.05\\alpha\_\{1\}=0\.05,α2=0\.35\\alpha\_\{2\}=0\.35,α3=0\.40\\alpha\_\{3\}=0\.40,α4=0\.05\\alpha\_\{4\}=0\.05,α5=0\.05\\alpha\_\{5\}=0\.05\. Evidence itemse2e\_\{2\}ande3e\_\{3\}are the pertinent ones;e1e\_\{1\},e4e\_\{4\},e5e\_\{5\}are tangential\. Statesθt\+1=Q​\(Pet​\(θt\)\)\\theta\_\{t\+1\}=Q\(P\_\{e\_\{t\}\}\(\\theta\_\{t\}\)\)are computed from \([8](https://arxiv.org/html/2605.06690#S9.E8)\)–\([9](https://arxiv.org/html/2605.06690#S9.E9)\); order\-gap valuesΩt=Ω​\(θt−1;et\)\\Omega\_\{t\}=\\Omega\(\\theta\_\{t\-1\};\\,e\_\{t\}\)are computed from \([12](https://arxiv.org/html/2605.06690#S9.E12)\)\. Values are computed from the closed\-form equations and rounded to three decimals\.

Table 1:Order\-gap trajectory\. The signal rises when pertinent evidence \(e2e\_\{2\},e3e\_\{3\}\) arrives and falls sharply as confidence rises and uncertainty falls\. Tangential evidence \(e4e\_\{4\},e5e\_\{5\}\) produces a near\-zero signal\. Expository, not empirical\.Sample computation \(step 1\)\.θ0=\(0\.20,0\.80\)\\theta\_\{0\}=\(0\.20,0\.80\),α1=0\.05\\alpha\_\{1\}=0\.05:Pe1​\(θ0\)=\(0\.240,0\.800\)P\_\{e\_\{1\}\}\(\\theta\_\{0\}\)=\(0\.240,0\.800\);θ1=Q​\(0\.240,0\.800\)=\(0\.240,0\.9×0\.760×0\.800\)=\(0\.240,0\.547\)\\theta\_\{1\}=Q\(0\.240,0\.800\)=\(0\.240,\\;0\.9\\times 0\.760\\times 0\.800\)=\(0\.240,\\;0\.547\);Ω1=0\.9×0\.05×0\.800×0\.800=0\.029\\Omega\_\{1\}=0\.9\\times 0\.05\\times 0\.800\\times 0\.800=0\.029\.

Stopping behaviour\.With thresholdε=0\.025\\varepsilon=0\.025,[Algorithm˜1](https://arxiv.org/html/2605.06690#alg1)halts at step 4 \(Ω^4,2=0\.022<0\.025\\widehat\{\\Omega\}\_\{4,2\}=0\.022<0\.025\) at which pointc4=0\.718c\_\{4\}=0\.718\. A fixed budget ofTmax=5T\_\{\\max\}=5continues throughe5e\_\{5\}, reachingc5=0\.732c\_\{5\}=0\.732, a change of0\.0140\.014relative to the step\-4 state\. The coverage check of[Remark˜5\.4](https://arxiv.org/html/2605.06690#S5.Thmtheorem4)correctly identifies the failure mode: ife2e\_\{2\}ande3e\_\{3\}had zero sampling probability, the order\-gap would stay near zero throughout, not because the state had converged, but because the pertinent evidence was never reached\.

## 10\. Conclusion

Recursive reasoning systems require two components that are consistently left implicit in current practice \- \(i\) an explicit, structured representation of the evolving reasoning state, and \(ii\) a termination criterion derived from the system’s own dynamics rather than imposed by an external budget\. We have proposed concrete definitions for both\. The epistemic state graph, with its typed nodes, edges and smooth Euclidean embedding, provides a state that is structured, persistent, and amenable to operator analysis\. The order\-gap, measuring the non\-commutativity of expansion and consolidation, provides a termination criterion that is derived from the system’s own dynamics, sensitive to whether further iteration matters, and computable at each step\. The non\-degeneracy theorem characterises exactly when the linearised criterion is non\-degenerate, meaning that a small order\-gap is not explained merely by algebraic cancellation in the linearised commutator\. We are explicit that this is a local condition\. The framework applies naturally to recursive language\-model reasoning, agent loops, tree\-of\-thought reasoning, theorem proving, and continual learning\. The closed\-form illustration confirms the qualitative behaviour of the criterion; large when pertinent evidence still changes the state, small once the system has settled\. The near\-term extensions are: a global convergence guarantee connecting the non\-degeneracy condition to answer error via Lipschitz and concentration arguments, empirical evaluation of[Algorithm˜1](https://arxiv.org/html/2605.06690#alg1)against fixed\-budget and confidence\-threshold baselines on long\-context QA benchmarks and further applications to continual learning and theorem proving\.

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