Steering Recurrent Reasoners at Inference Time with Readout Feedback

# Steering Recurrent Reasoners at Inference Time with Readout Feedback
Source: [https://arxiv.org/html/2608.24136](https://arxiv.org/html/2608.24136)
Masanori KoyamaAffiliation:Graduate School of Engineering, The University of TokyoSeongcheol JeongAffiliation:Graduate School of Engineering, The University of TokyoFumiya UchiyamaAffiliation:Graduate School of Engineering, The University of TokyoKenji KuboAffiliation:Graduate School of Engineering, The University of TokyoKohei HayashiAffiliation:Graduate School of Engineering, The University of TokyoMasahiro SuzukiAffiliation:Graduate School of Engineering, The University of TokyoYutaka MatsuoAffiliation:Graduate School of Engineering, The University of Tokyo

###### Abstract

Recurrent models, which repeatedly update latent states with shared computation blocks, have emerged as powerful architectures for solving complex reasoning tasks\. Existing inference\-time methods scale computation by running more steps or sampling more trajectories, but ignore information revealed within each trajectory\. Here we show that recurrent models can be improved at inference time by using their own readout probabilities to steer latent dynamics without retraining\. We introduce Readout Feedback \(RoFB\), a test\-time intervention that converts intermediate predictions into token\-wise pairwise coupling forces injected into the latent dynamics\. Across three recurrent models \(AKOrN, ItrSA\+\+, TRM\) on Sudoku and Maze, RoFB yields clear gains in four of six model\-task pairs, achieving performance unattainable by merely running more steps or selecting from multiple trajectories, at comparable or lower computational cost\. These results suggest that closed\-loop steering of latent dynamics can serve as a complementary inference\-time control mechanism for recurrent reasoning models\.

## 1Introduction

Human experts often solve difficult problems by thinking longer: mathematicians refine intermediate structures, and chess players search and revise candidate lines before committing to a move\. Modern reasoning systems increasingly exploit an analogous principle —additional computation at inference time can improve performance\. In large language models, this idea appears in chain\-of\-thought prompting\([Wei et al\., 2022](https://arxiv.org/html/2608.24136#bib.bib1)\), self\-consistency\([Wang et al\., 2022](https://arxiv.org/html/2608.24136#bib.bib2)\), and search\-based reasoning\([Yao et al\., 2023](https://arxiv.org/html/2608.24136#bib.bib3)\), where inference\-time computation is used to generate, evaluate, or aggregate candidate reasoning paths\. These developments have motivated inference\-time computation as an additional scaling axis beyond model size and training data\.

A complementary line of work studies recurrent reasoning models, which solve problems by repeatedly updating latent states with shared computation blocks\. Unlike feedforward models with a fixed computational depth, recurrent reasoners expose inference\-time computation explicitly through the number of recurrent updates\. This principle has appeared both in algorithmic and language\-model settings: recurrent networks can extrapolate learned algorithms by iterating their computation\([Bansal et al\., 2022](https://arxiv.org/html/2608.24136#bib.bib4)\), while looped or recurrent\-depth Transformers use repeated latent computation to increase effective depth and improve reasoning performance\([Yang et al\., 2023](https://arxiv.org/html/2608.24136#bib.bib5);[Saunshi et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib6);[Geiping et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib7)\)\. In parallel, compact recurrent reasoners designed for structured reasoning, such as HRM\([Wang et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib8)\), TRM\([Jolicoeur\-Martineau, 2025](https://arxiv.org/html/2608.24136#bib.bib9)\), URM\([Gao et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib10)\), AKOrN\([Miyato et al\., 2024](https://arxiv.org/html/2608.24136#bib.bib11)\), and related iterative transformer and self\-attention models\([Kubo et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib12)\)have shown strong performance on puzzle\-style reasoning benchmarks including Sudoku, Maze, and ARC\-AGI 1 & 2\([Chollet, 2019](https://arxiv.org/html/2608.24136#bib.bib13);[Chollet et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib14)\)\.

Despite this progress, current inference\-time improvements for recurrent reasoners mostly exploit their iterative structure in passive ways: running the dynamics for more steps\([Wang et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib8);[Geiping et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib7)\), or sampling multiple trajectories and selecting one according to a criterion, such as one with the lowest potential defined on the dynamics\([Miyato et al\., 2024](https://arxiv.org/html/2608.24136#bib.bib11)\), or the most confident one\([Kubo et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib12)\)\. Although being effective, these strategies come with inherent limitations: longer trajectories may saturate or remain trapped in unsuccessful dynamical regimes, while multi\-trajectory voting increases compute roughly linearly with the number of candidates\. This raises a natural question: can we improve a frozen recurrent reasoner by actively steering each latent trajectory during inference, rather than merely running longer or sampling more trajectories?

In this work, we propose Readout Feedback \(RoFB\), a closed\-loop inference\-time intervention for recurrent reasoning models \(Fig\.[1](https://arxiv.org/html/2608.24136#S1.F1)\)\. RoFB is motivated by a simple observation about successful recurrent inference: as a trajectory approaches a correct solution, tokens develop a class\-dependent cluster structure in latent space\. Tokens predicted to belong to the same class become aligned, while tokens assigned to different classes become separated\. RoFB uses the model’s own readout probabilities to turn this observation into a feedback signal\. At each selected inference step, it injects a coupling term into the latent dynamics that encourages token clustering based on the distance between tokens’ readout probabilities, while leaving all model weights unchanged\.

We evaluate RoFB on Sudoku and Maze using three complementary recurrent reasoners with distinct architectural designs: AKOrN, a single\-level oscillator\-inspired model\([Miyato et al\., 2024](https://arxiv.org/html/2608.24136#bib.bib11)\); ItrSA\+\+\([Kubo et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib12)\), a strong iterative self\-attention model for these benchmarks; and TRM\([Jolicoeur\-Martineau, 2025](https://arxiv.org/html/2608.24136#bib.bib9)\), a compact hierarchical recursive model\. Across these models and tasks, RoFB improves inference performance without retraining, and its gains are complementary to confidence\-based voting when voting is applicable\. We further analyze performance as a function of inference steps, trajectory count, and normalized inference compute, showing that RoFB improves not only final accuracy but also the compute\-accuracy trade\-off of recurrent reasoning\. These results suggest that frozen recurrent reasoners contain underutilized inference\-time capability that can be unlocked by closed\-loop control of their latent dynamics\.

![Refer to caption](https://arxiv.org/html/2608.24136v1/fig_1_ver5.png)Figure 1:Conventional inference vs\. our method\.\(a\) Conventional inference without feedback for a recurrent model\. \(b\) Inference with RoFB\. RoFB injects a feedback coupling into the latent dynamics based on the distance between readout probabilities, leaving all model weights unchanged\.
## 2Preliminaries

### 2\.1Recurrent Models for Reasoning

We first give the general setups for the recurrent models considered in this paper \(Fig\.[1](https://arxiv.org/html/2608.24136#S1.F1)\(a\)\)\. In a nutshell, these models consist of three main components: \(I\) aninput embedding mapφ\\varphi, \(II\) arecurrent moduleRR, and \(III\) areadout headψ\\psi\.

More specifically, the input embeddingφ\\varphimaps the raw input tokens𝐱raw=\{𝐱raw\(i\)\}i=1N∈\{1,…,V\}N\\mathbf\{x\}\_\{\\text\{raw\}\}=\\\{\\mathbf\{x\}^\{\(i\)\}\_\{\\text\{raw\}\}\\\}\_\{i=1\}^\{N\}\\in\\\{1,\\ldots,V\\\}^\{N\}to a fixed embedding𝐱=\{𝐱\(i\)=φ\(𝐱raw\(i\)\)\}i=1N∈ℝN×D\\mathbf\{x\}=\\\{\\mathbf\{x\}^\{\(i\)\}=\\varphi\(\\mathbf\{x\}^\{\(i\)\}\_\{\\text\{raw\}\}\)\\\}\_\{i=1\}^\{N\}\\in\\mathbb\{R\}^\{N\\times D\}, whereVVis the vocabulary size,NNis the number of tokens, andDDis the embedding dimension\. The recurrent moduleRRiteratively applies an update to the latent state𝐳t=\{𝐳t\(i\)\}i=1N∈\(ℝD\)N\\mathbf\{z\}\_\{t\}=\\\{\\mathbf\{z\}^\{\(i\)\}\_\{t\}\\\}\_\{i=1\}^\{N\}\\in\(\\mathbb\{R\}^\{D\}\)^\{N\}overTTsteps:

𝐳t\+1\(i\)=R\(i\)\(𝐳t;𝐱\),t=0,1,…,T−1\.\\mathbf\{z\}^\{\(i\)\}\_\{t\+1\}=R^\{\(i\)\}\(\\mathbf\{z\}\_\{t\};\\mathbf\{x\}\),\\quad t=0,1,\\ldots,T\{\-\}1\.\(1\)This stepwise update can be a discretization of a continuous\-time update:𝐳˙t\(i\)=f\(i\)​\(𝐳t,𝐱\),t∈\[0,T\]\.\\dot\{\\mathbf\{z\}\}^\{\(i\)\}\_\{t\}=f^\{\(i\)\}\(\\mathbf\{z\}\_\{t\};\\mathbf\{x\}\),~~t\\in\[0,T\]\.Each latent token𝐳t\(i\)\\mathbf\{z\}^\{\(i\)\}\_\{t\}is typically confined to a compact manifoldℳ⊂ℝD\\mathcal\{M\}\\subset\\mathbb\{R\}^\{D\}\(e\.g\., theDD\-dimensional hypersphere\)\. The initial value𝐳0∈ℳ\\mathbf\{z\}\_\{0\}\\in\\mathcal\{M\}is either randomly sampled from a certain distribution, or learned as a fixed parameter\. AfterTTsteps, the readout headψ:ℳ→ℝC\\psi:\\mathcal\{M\}\\to\\mathbb\{R\}^\{C\}, usually a multi\-layer perceptron \(MLP\) of a few layers, produces a per\-token probability vector𝐩\(i\)=softmax⁡\(ψ⁡\(𝐳T\(i\)\)\)\\mathbf\{p\}^\{\(i\)\}=\\operatorname\{softmax\}\(\\psi\(\\mathbf\{z\}^\{\(i\)\}\_\{T\}\)\), whereCCis the number of output classes\.

As specific models, we selected three representative recurrent models for reasoning tasks in the present study: AKOrN\([Miyato et al\., 2024](https://arxiv.org/html/2608.24136#bib.bib11)\), ItrSA\+\+\([Kubo et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib12)\), and TRM\([Jolicoeur\-Martineau, 2025](https://arxiv.org/html/2608.24136#bib.bib9)\)\. We made this choice based on their strong performance in reasoning tasks and their diverse architectural designs including hierarchical structures and latent state geometry\. The key architectural differences are summarized in Table[1](https://arxiv.org/html/2608.24136#S2.T1)\. In what follows, we describe the specific architectures of the three recurrent models in detail\.

Table 1:Architectural differences of the models used in this work\.The table summarizes the key architectural features of the three recurrent models \(AKOrN, ItrSA\+\+, and TRM\) analyzed in this paper, including their latent state geometry, coupling mechanism, and readout strategy\.
### 2\.2AKOrN

Artificial Kuramoto Oscillatory Neurons \(AKOrN\)\([Miyato et al\., 2024](https://arxiv.org/html/2608.24136#bib.bib11)\)is a recurrent model with strong performance in reasoning tasks, such as Sudoku, Maze, and object recognition\. In AKOrN, each latent token𝐳t\(i\)\\mathbf\{z\}\_\{t\}^\{\(i\)\}resides on the manifold\(𝕊m−1\)D/m\(\\mathbb\{S\}^\{m\-1\}\)^\{D/m\}, wheremmis the oscillator dimension, and follows a Kuramoto oscillator\([Kuramoto, 1984](https://arxiv.org/html/2608.24136#bib.bib15)\)\-like dynamics;

Δ​𝐳t\(i\)=Proj𝐳t\(i\)⁡\(𝐱\(i\)\+∑j𝐌t\(i​j\)​\(𝐳t\)​𝐳t\(j\)\),𝐳t\+1\(i\)=Π⁡\[𝐳t\(i\)\+γ​Δ​𝐳t\(i\)\]\.\\Delta\\mathbf\{z\}\_\{t\}^\{\(i\)\}=\\operatorname\{Proj\}\_\{\\mathbf\{z\}\_\{t\}^\{\(i\)\}\}\\Big\(\\mathbf\{x\}^\{\(i\)\}\+\\sum\_\{j\}\\mathbf\{M\}^\{\(ij\)\}\_\{t\}\(\\mathbf\{z\}\_\{t\}\)\\mathbf\{z\}\_\{t\}^\{\(j\)\}\\Big\),\\quad\\mathbf\{z\}\_\{t\+1\}^\{\(i\)\}=\\Pi\\left\[\\mathbf\{z\}\_\{t\}^\{\(i\)\}\+\\gamma\\Delta\\mathbf\{z\}\_\{t\}^\{\(i\)\}\\right\]\.\(2\)Here,𝐌t\(i​j\)\\mathbf\{M\}^\{\(ij\)\}\_\{t\}is a learnable coupling matrix,Proj𝐳⁡\(𝐟\)=𝐟−⟨𝐟,𝐳⟩​𝐳\\operatorname\{Proj\}\_\{\\mathbf\{z\}\}\(\\mathbf\{f\}\)=\\mathbf\{f\}\-\\langle\\mathbf\{f\},\\mathbf\{z\}\\rangle\\mathbf\{z\}projects𝐟\\mathbf\{f\}onto the tangent space of\(𝕊m−1\)D/m\(\\mathbb\{S\}^\{m\-1\}\)^\{D/m\}at𝐳\\mathbf\{z\},Π\(⋅\)=⋅/∥⋅∥\\Pi\(\\cdot\)=\\cdot/\\\|\\cdot\\\|is the token\-wise normalization operator that projects each token back to the product sphere,𝐳t=\{𝐳t\(i\)\}i=1N\\mathbf\{z\}\_\{t\}=\\\{\\mathbf\{z\}^\{\(i\)\}\_\{t\}\\\}\_\{i=1\}^\{N\}is the collection of all latent states, andγ\\gammais a learnable step size\. The coupling matrix𝐌t\(i​j\)\\mathbf\{M\}\_\{t\}^\{\(ij\)\}is implemented as the self\-attention output of the current latent state, allowing learnable all\-to\-all coupling among tokens\. We deliberately drop the natural\-frequency term𝛀i\\boldsymbol\{\\Omega\}\_\{i\}present in the original formulation, for simplicity\.

In this work, after a brief exploration of several design choices, we implement AKOrN in a slightly modified form using a normalization\-free transformer:

Δ​𝐳t\(i\)=Proj𝐳t\(i\)⁡\(mlp⁡\[\(𝐳t\+𝐱\+SelfAttn⁡\(𝐳t\+𝐱\)\)\(i\)\]\),\\Delta\\mathbf\{z\}^\{\(i\)\}\_\{t\}=\\operatorname\{Proj\}\_\{\\mathbf\{z\}^\{\(i\)\}\_\{t\}\}\\left\(\\mathrm\{mlp\}\\left\[\\bigl\(\\mathbf\{z\}\_\{t\}\+\\mathbf\{x\}\+\\mathrm\{SelfAttn\}\(\\mathbf\{z\}\_\{t\}\+\\mathbf\{x\}\)\\bigr\)^\{\(i\)\}\\right\]\\right\),\(3\)wheremlp\\mathrm\{mlp\}is a two\-layer feed\-forward network with a GELU activation, andSelfAttn\\mathrm\{SelfAttn\}is a multi\-head self\-attention block with a positional encoding\.

### 2\.3ItrSA\+\+

ItrSA\+\+ is another recurrent model proposed in\([Kubo et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib12)\), which shows even stronger reasoning capabilities in Sudoku and Maze\. Unlike AKOrN, ItrSA\+\+ is designed to have a two\-level hierarchy, where the low\-level process𝐳\\mathbf\{z\}is designed to capture fast\-changing finer details, while the high\-level process𝐲\\mathbf\{y\}is expected to capture more slower\-changing global structure\. In the low\-level process𝐳\\mathbf\{z\}, the dynamics is updated as follows:

𝐳t\+1\(i\)=\{RN⁡\(𝐳t\(i\)\+SelfAttn⁡\(𝐳t\)\)ift≢0\(modL\),RN⁡\(𝐳t\(i\)\+CrossAttn⁡\(RN⁡\(𝐲t\),RN⁡\(𝐱\)\)\)ift≡0\(modL\),\\mathbf\{z\}\_\{t\+1\}^\{\(i\)\}=\\begin\{cases\}\\operatorname\{RN\}\\left\(\\mathbf\{z\}\_\{t\}^\{\(i\)\}\+\\operatorname\{SelfAttn\}\(\\mathbf\{z\}\_\{t\}\)\\right\)&\\text\{if $t\\not\\equiv 0\\pmod\{L\}$,\}\\\\ \\operatorname\{RN\}\\left\(\\mathbf\{z\}\_\{t\}^\{\(i\)\}\+\\operatorname\{CrossAttn\}\\left\(\\operatorname\{RN\}\(\\mathbf\{y\}\_\{t\}\),\\operatorname\{RN\}\(\\mathbf\{x\}\)\\right\)\\right\)&\\text\{if $t\\equiv 0\\pmod\{L\}$,\}\\end\{cases\}\(4\)whereRN\\operatorname\{RN\}denotes the RMS normalization\. The high\-level process𝐲\\mathbf\{y\}is updated everyLLsteps:

𝐲t\+1\(i\)=\{RN⁡\(𝐳t\(i\)\+SwiGLU⁡\(𝐳t\(i\)\)\)ift≡0\(modL\),𝐲t\(i\)otherwise\.\\mathbf\{y\}\_\{t\+1\}^\{\(i\)\}=\\begin\{cases\}\\operatorname\{RN\}\\left\(\\mathbf\{z\}\_\{t\}^\{\(i\)\}\+\\operatorname\{SwiGLU\}\(\\mathbf\{z\}\_\{t\}^\{\(i\)\}\)\\right\)&\\text\{if $t\\equiv 0\\pmod\{L\}$,\}\\\\ \\mathbf\{y\}\_\{t\}^\{\(i\)\}&\\text\{otherwise\.\}\\end\{cases\}\(5\)The latent state space of ItrSA\+\+ can be identified with𝕊D−1​\(D\)\\mathbb\{S\}^\{D\-1\}\(\\sqrt\{D\}\)thanks to the RMS normalization\.

### 2\.4Tiny Recursive Model \(TRM\)

TRM\([Jolicoeur\-Martineau, 2025](https://arxiv.org/html/2608.24136#bib.bib9)\)is another recurrent model with strong performance in reasoning benchmark tasks, such as Sudoku, Maze, and ARC\-AGI\([Chollet, 2019](https://arxiv.org/html/2608.24136#bib.bib13);[Chollet et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib14)\)\. Proposed as a simplified version of the groundbreaking Hierarchical Reasoning Model \(HRM\)\([Wang et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib8)\), TRM shows even more powerful reasoning capabilities while reducing model size and computational complexity\. TRM has a two\-level hidden\-state hierarchy, which is designed to capture both global structure and local details in the reasoning process\. The low\-level process𝐳\\mathbf\{z\}, designed to capture local details, is updated as follows:

𝐳t\+1\(i\)=\[TFn⁡\(𝐳t\+𝐲t\+𝐱\)\]\(i\)\.\\mathbf\{z\}\_\{t\+1\}^\{\(i\)\}=\\left\[\\operatorname\{TF\}\_\{n\}\\\!\\bigl\(\\mathbf\{z\}\_\{t\}\+\\mathbf\{y\}\_\{t\}\+\\mathbf\{x\}\\bigr\)\\right\]^\{\(i\)\}\.\(6\)Here,TFn\\operatorname\{TF\}\_\{n\}is annn\-layer transformer block\. The high\-level process𝐲\\mathbf\{y\}, on the other hand, is updated only everyLLsteps to capture the global structure of the problem:

𝐲t\+1\(i\)=\{TFn⁡\(𝐳t\(i\)\+𝐲t\(i\)\)ift≡0\(modL\),𝐲t\(i\)otherwise\.\\mathbf\{y\}\_\{t\+1\}^\{\(i\)\}=\\begin\{cases\}\\operatorname\{TF\}\_\{n\}\\\!\\bigl\(\\mathbf\{z\}\_\{t\}^\{\(i\)\}\+\\mathbf\{y\}\_\{t\}^\{\(i\)\}\\bigr\)&\\text\{if $t\\equiv 0\\pmod\{L\}$,\}\\\\ \\mathbf\{y\}\_\{t\}^\{\(i\)\}&\\text\{otherwise\.\}\\end\{cases\}\(7\)Each𝐳t\(i\)\\mathbf\{z\}^\{\(i\)\}\_\{t\}lies on𝕊D−1​\(D\)\\mathbb\{S\}^\{D\-1\}\(\\sqrt\{D\}\)thanks to an RMS normalization\([Zhang and Sennrich, 2019](https://arxiv.org/html/2608.24136#bib.bib16)\)at the end of each update\. The readout is applied to the high\-level state𝐲t\\mathbf\{y\}\_\{t\}after a certain number of recursions, producing the final output of the model\.

## 3Readout Feedback \(RoFB\)

### 3\.1Inter\-token Clusterization

Figure 2:\(Top\) Correct cell count and total cell entropy \(i\.e\.,∑i​jH⁡\(pi​j\)\\sum\_\{ij\}H\(\\textbf\{p\}^\{ij\}\)\) in two inference trajectories of the same Sudoku puzzle\.The inference steps unroll untilt=256t=256\.\(Bottom\) Cosine similarity matrices at each timestep\.The token indices are rearranged so that blank tokens classified into1,2,⋯,91,2,\\cdots,9are followed by input tokens classified into1,2,⋯,91,2,\\cdots,9in this order\.Before explaining our method RoFB, we first inspect how tokens in the latent space behave during the inference dynamics, taking inference trajectories of AKOrN on a Sudoku task as examples\.

We observe that the inference dynamics typically has three qualitatively different phases: \(i\) a “wandering” phase where the prediction stays only partially correct and unconfident \(i\.e\., large total cell entropy,t=0t=0tot≈120t\\approx 120in Fig\.[2](https://arxiv.org/html/2608.24136#S3.F2)\), \(ii\) an “Aha” phase where the entropy suddenly plummets and the prediction becomes very certain \(t≈120t\\approx 120\), and \(iii\) a solution phase where the prediction is correct and confident \(t≈120t\\approx 120tot=256t=256\)\.

We observe that this sudden plummet in entropy corresponds to the point where the prediction becomes correct\. That is, if the model is run forT=256T=256inference steps, then a trajectory that can solve a puzzle has its “Aha” phase beforet=256t=256, while a trajectory that cannot solve the puzzle does not exhibit such a phase withint=256t=256\. The “Aha” timing depends on the initial value even when the input puzzle is identical\. This can be explained by the dynamics first being trapped at a local minima, after which it escapes the local minima and converges to a global minimum\([Li et al\., 2023](https://arxiv.org/html/2608.24136#bib.bib17)\)\.

Although the above picture provides an overview of the inference behaviors, what has been overlooked is the token\-wise behavior during these phases, specifically,how tokens are positioned relative to each other in the latent space\. The lower half of Fig\.[2](https://arxiv.org/html/2608.24136#S3.F2)demonstrates the token pair\-wise cosine similarity along the inference dynamics\. Note that the token indices are rearranged so that blank tokens classified into1,2,⋯,91,2,\\cdots,9are followed by input tokens classified into1,2,⋯,91,2,\\cdots,9in this order\. We can see that, initially during the wandering phase, almost every blank token pair has a large similarity value≈1\.0\\approx 1\.0, indicating that the tokens are aligned regardless of the classes to which they truly belong\. But once the puzzle has been solved, the tokens make cluster representation; tokens of the same class are well aligned, having large cosine similarity around1\.01\.0, while those representing different classes have lower with cosine similarity \(Fig\.[2](https://arxiv.org/html/2608.24136#S3.F2), Bottom, Solved sample,t=200t=200\)\. On the other hand, in an unsolved sample, the tokens do not make such a cluster representation, and the cosine similarity values stay larger regardless of the targets and are less structured \(Fig\.[2](https://arxiv.org/html/2608.24136#S3.F2), Bottom, Unsolved sample,t=200t=200\)\.

From these observations, we hypothesize that for the recurrent models to solve a problem, the tokens need to be clustered according to the classes they represent\. That is, the tokens representing the same class need to be aligned, while those representing different classes need to be separated\. Based on this hypothesis, we propose RoFB, which is described in the following section\.

### 3\.2RoFB

RoFB realizes the above idea by inserting a feedback coupling term that depends on the distance between readout probability vectors\. If two tokens have similar readout probability vectors, we encourage the alignment of the two\. Conversely, if two tokens have different readout probability vectors, we discourage the alignment of the two\.

We designed RoFB so that the dynamics retains the original forward information along with the feedback of the readout probability vectors\. We implement this by a simple addition, that is,

𝐳t\+1\(i\)=Πℳ⁡\(R\(i\)​\(𝐳t,𝐱\)\+λ⋅g⁡\(𝐳t\)⋅Proj𝐳t\(i\)​\(∑j≠iJ~t\(i​j\)​\(𝐩t\(i\),𝐩t\(j\)\)​𝐳t\(j\)\)\)\.\\mathbf\{z\}^\{\(i\)\}\_\{t\+1\}=\\operatorname\{\\Pi\}\_\{\\mathcal\{M\}\}\\left\(R^\{\(i\)\}\(\\mathbf\{z\}\_\{t\},\\mathbf\{x\}\)\+\\hbox\{\\pagecolor\[HTML\]\{F0F8FF\}$\\displaystyle\\lambda\\cdot g\(\\mathbf\{z\}\_\{t\}\)\\cdot\\text\{Proj\}\_\{\\mathbf\{z\}^\{\(i\)\}\_\{t\}\}\\left\(\\sum\_\{j\\neq i\}\\tilde\{J\}^\{\(ij\)\}\_\{t\}\\Bigl\(\\mathbf\{p\}^\{\(i\)\}\_\{t\},\\mathbf\{p\}^\{\(j\)\}\_\{t\}\\Bigr\)\\,\\mathbf\{z\}^\{\(j\)\}\_\{t\}\\right\)$\}\\right\)\.\(8\)Here,J~t\(i​j\)\\tilde\{J\}^\{\(ij\)\}\_\{t\}is the normalized coupling term that reflects the distance between the readout probability vectors𝐩t\(i\)\\mathbf\{p\}^\{\(i\)\}\_\{t\}and𝐩t\(j\)\\mathbf\{p\}^\{\(j\)\}\_\{t\},gga gate function, andλ\\lambdaa scaling hyperparameter\. TheProj\\operatorname\{Proj\}operator projects the feedback term to the tangent space of the manifoldℳ\\mathcal\{M\}at𝐳t\(i\)\\mathbf\{z\}\_\{t\}^\{\(i\)\}, which ensures that the dynamics stays on the manifold, andΠℳ\\operatorname\{\\Pi\}\_\{\\mathcal\{M\}\}is a projection to the manifoldℳ\\mathcal\{M\}\. We provide more detailed explanation on the coupling termJ~t\(i​j\)\\tilde\{J\}^\{\(ij\)\}\_\{t\}and the gate functionggin the following\.

#### CouplingJ~\\tilde\{J\}\.

The coupling termJ~t\(i​j\)\\tilde\{J\}^\{\(ij\)\}\_\{t\}is a scalar defined as

J~t\(i​j\)=Jt\(i​j\)/ν\(i\),Jt\(i​j\)​\(𝐩t\(i\),𝐩t\(j\)\)=h⁡\(d⁡\(𝐩t\(i\),𝐩t\(j\)\)\),\\tilde\{J\}^\{\(ij\)\}\_\{t\}=J^\{\(ij\)\}\_\{t\}/\\nu^\{\(i\)\},\\quad J^\{\(ij\)\}\_\{t\}\\Bigl\(\\mathbf\{p\}^\{\(i\)\}\_\{t\},\\mathbf\{p\}^\{\(j\)\}\_\{t\}\\Bigr\)=h\\left\(d\\Bigl\(\\mathbf\{p\}^\{\(i\)\}\_\{t\},\\mathbf\{p\}^\{\(j\)\}\_\{t\}\\Bigr\)\\right\),\(9\)wherehhis a non\-increasing function,ν\(i\)\\nu^\{\(i\)\}is a certain normalization term \(e\.g\.,ν\(i\)=∑k≠i\|Jt\(i​k\)\|\\nu^\{\(i\)\}=\\sum\_\{k\\neq i\}\|J^\{\(ik\)\}\_\{t\}\|: signed sum\),dda bounded distance function such asd⁡\(x,y\)=1−x⊤​y∈\[0,1\]d\(x,y\)=1\-x^\{\\\!\\top\}y\\in\[0,\\,1\],𝐩t\(i\)\\mathbf\{p\}^\{\(i\)\}\_\{t\}the readout probability vector of theii\-th token,𝐩t\(i\)=softmax⁡\(ψ⁡\(𝐳t\(i\)\)\)∈ΔC−1:=\{𝐩∈\[0,1\]C∣p1\+⋯\+pC=1\}\\mathbf\{p\}^\{\(i\)\}\_\{t\}=\\operatorname\{softmax\}\\\!\\left\(\\psi\\Bigl\(\\mathbf\{z\}^\{\(i\)\}\_\{t\}\\Bigr\)\\right\)\\in\\Delta^\{C\-1\}:=\\\{\\mathbf\{p\}\\in\[0,1\]^\{C\}\\mid p\_\{1\}\+\\cdots\+p\_\{C\}=1\\\}, We typically sethhash⁡\(0\)⩾0⩾h⁡\(1\)h\(0\)\\geqslant 0\\geqslant h\(1\), so that the coupling is attractive for tokens with similar readout distributions \(d→0d\\to 0\) and repulsive for dissimilar ones \(d→1d\\to 1\)\.

Although Eq\. \([9](https://arxiv.org/html/2608.24136#S3.E9)\) permits an attractive\-repulsive coupling, all experiments in this paper use the repulsive instantiationh⁡\(d\)=−dh\(d\)=\-d\. Therefore, the implemented RoFB should be interpreted as discouraging collapse among tokens with dissimilar readout distributions, rather than explicitly attracting tokens with similar readouts\.

For tasks with pre\-given clue tokens \(Sudoku, Maze\), RoFB is applied only to blank\-cell tokens: the update in Eq\. \([8](https://arxiv.org/html/2608.24136#S3.E8)\) is restricted toi∈ℬi\\in\\mathcal\{B\}and the inner sum to∑j∈ℬ,j≠i\\sum\_\{j\\in\\mathcal\{B\},\\,j\\neq i\}, whereℬ⊂\{1,…,N\}\\mathcal\{B\}\\subset\\\{1,\\dots,N\\\}is the set of blank\-cell indices\. We retain the unrestricted form in Eq\. \([8](https://arxiv.org/html/2608.24136#S3.E8)\) for notational simplicity\.

#### Gategg\.

The gate functiong⁡\(𝐳t\)g\(\\mathbf\{z\}\_\{t\}\)is designed to turn on the feedback term only when the model is considered to be in the wandering phase\. One way to implement this is, after running the original inference dynamics for a certain number of steps, using a confidence check function that turns on when the entropy is higher than a certain threshold\. That is, we can set

g⁡\(𝐳t\):=1\[t⩾tmin\]⏟Warmup​σ⁡\(maxi⁡Ht\(i\)−α​Hunifτ\)⏟Confidence check,g\(\\mathbf\{z\}\_\{t\}\):=\\underbrace\{1\[t\\geqslant t\_\{\\text\{min\}\}\]\}\_\{\\text\{Warmup\}\}~\\underbrace\{\\sigma\\Big\(\\frac\{\\max\_\{i\}H^\{\(i\)\}\_\{t\}\-\\alpha H\_\{\\text\{unif\}\}\}\{\\tau\}\\Big\)\}\_\{\\text\{Confidence check\}\},\(10\)where1​\[⋅\]1\[\\cdot\]is the indicator function,σ\\sigmais the sigmoid function with temperatureτ\\tau,Ht\(i\)H^\{\(i\)\}\_\{t\}is the entropy of the readout probability vector of tokeniiat timett,HunifH\_\{\\text\{unif\}\}is the entropy of the uniform distribution, andα\\alphais a preset threshold\.

#### Hyperparameter Setups\.

RoFB typically requires only a small number of hyperparameters\. For example, in the above gate function design, we have four hyperparameters:λ\\lambda,tmint\_\{\\text\{min\}\},α\\alpha, andτ\\tau\. The selection of these hyperparameters can be done by grid search or by using automated hyperparameter tuning tools such as Optuna\([Akiba et al\., 2019](https://arxiv.org/html/2608.24136#bib.bib18)\)\.

## 4Related Work

#### Iterative Reasoning Models\.

Dating back to early works such as Universal Transformer\([Dehghani et al\., 2018](https://arxiv.org/html/2608.24136#bib.bib19)\)and Deep Equilibrium Models\([Bai et al\., 2019](https://arxiv.org/html/2608.24136#bib.bib20)\), the concept of using shared computation blocks multiple times has demonstrated the enormous potential in solving complex tasks such as learning various algorithms\([Yang et al\., 2023](https://arxiv.org/html/2608.24136#bib.bib5)\)and end\-to\-end algorithm synthesis with recurrent networks\([Bansal et al\., 2022](https://arxiv.org/html/2608.24136#bib.bib4)\)\. Such recurrent architectures have been also applied to language models and have been shown to be effective in logical reasoning such as math problems\([Geiping et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib7);[Bae et al\., 2024](https://arxiv.org/html/2608.24136#bib.bib21);[Zhu et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib22)\)\. More recently, a series of compact recurrent/looped models explicitly targeting reasoning benchmarks, AKOrN\([Miyato et al\., 2024](https://arxiv.org/html/2608.24136#bib.bib11)\), HRM\([Wang et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib8)\), TRM\([Jolicoeur\-Martineau, 2025](https://arxiv.org/html/2608.24136#bib.bib9)\), URM\([Gao et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib10)\)and ItrSA\+\+\([Kubo et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib12)\)have been proposed, showing that recursive latent updates themselves provide a strong inductive bias for combinatorial reasoning tasks like Sudoku, Maze, and ARC\-AGI\.

#### Internal Dynamics of Recurrent Reasoners\.

While the internal mechanisms of recurrent models in LLMs are gradually being analyzed\([Geiping et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib7);[Lu et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib23);[Blayney et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib24)\), the internal mechanisms of distilled compact models \(HRM, TRM, etc\.\) are less well understood\.[Ren and Liu \(2026\)](https://arxiv.org/html/2608.24136#bib.bib25)propose a learning rule that scales resolution in time from the dynamic characteristics of HRM, but RoFB is designed to leverage these characteristics to steer inference, which has been an open question\. Theoretical analyses of the internal dynamics of recurrent Transformers include[Geshkovski et al\. \(2023\)](https://arxiv.org/html/2608.24136#bib.bib26);[Geshkovski et al\. \(2024\)](https://arxiv.org/html/2608.24136#bib.bib27), which show the cluster\-forming dynamics of recurrent self attention dynamics\.

#### Test\-time Intervention\.

Inference\-time improvement has been studied broadly in large language models through prompting, sampling, and compute\-allocation techniques\([Wei et al\., 2022](https://arxiv.org/html/2608.24136#bib.bib1);[Wang et al\., 2022](https://arxiv.org/html/2608.24136#bib.bib2);[Yao et al\., 2023](https://arxiv.org/html/2608.24136#bib.bib3);[Zhou et al\., 2022](https://arxiv.org/html/2608.24136#bib.bib28);[Zhai et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib29)\), all of which compare or aggregate candidate outputs rather than intervening in the latent computation\. The nearest neighbor specialised for recurrent reasoners is C\-voting\([Kubo et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib12)\), which selects among multiple latent trajectories using a confidence signal; RoFB differs in two respects—it operates on a single trajectory rather than a set, and uses the readout signal to*deform*the latent dynamics during inference rather than to*score*completed candidates\.

#### Output\-space Refinement\.

A broad line of work refines predictions or representations by exploiting relationships in output space, such as confidence, neighborhood structure, or similarity between predicted probability vectors\. In structured prediction, dense CRFs and CRF\-RNN refine unary neural predictions through pairwise inference, encouraging mutually compatible labels for related variables\([Krähenbühl and Koltun, 2012](https://arxiv.org/html/2608.24136#bib.bib30);[Zheng et al\., 2015](https://arxiv.org/html/2608.24136#bib.bib31)\)\. In source\-free adaptation and clustering, methods such as NRC\([Yang et al\., 2021](https://arxiv.org/html/2608.24136#bib.bib32)\), AaD\([Yang et al\., 2022](https://arxiv.org/html/2608.24136#bib.bib33)\), and related batch\-wise similarity objectives\([Pathak and Balasubramanian, 2026](https://arxiv.org/html/2608.24136#bib.bib34)\)use prediction consistency, attraction, dispersion, or anti\-collapse terms to improve target\-domain predictions without source labels\. Deep clustering methods similarly refine soft assignments by sharpening confident predictions or enforcing consistency among neighboring samples\([Xie et al\., 2015](https://arxiv.org/html/2608.24136#bib.bib35);[Ji et al\., 2018](https://arxiv.org/html/2608.24136#bib.bib36);[Van Gansbeke et al\., 2020](https://arxiv.org/html/2608.24136#bib.bib37)\)\. RoFB shares the principle that output distributions contain useful relational structure, but rather than optimizing a training or adaptation loss, RoFB uses the intermediate readout probabilities of a frozen recurrent reasoning model at inference time\.

## 5Experiments

We evaluate RoFB on two benchmarks, Sudoku and Maze, with the three models described earlier\.

Sudokuis a logic\-based combinatorial number\-placement puzzle\. The objective is to fill a9×99\\times 9grid with digits so that each column, each row, and each of the nine3×33\\times 3subgrids contains all of the digits from 1 to 9\. Following the prior works\([Wang et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib8);[Kubo et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib12)\), we usedSudoku Extremedataset, and applied the same preprocessing and augmentation techniques\. We tested with 20,000 puzzle boards randomly sampled from all test boards\.

Mazeis a pathfinding problem where the goal is to find a shortest path from a starting point to a target point in a30×3030\\times 30grid environment with obstacles\. Following the prior works\([Wang et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib8);[Kubo et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib12)\), we usedMaze Harddataset, and applied the same preprocessing techniques\. We tested with all 1,000 puzzle boards\.

#### Training\.

For each task, we trained the models on each task using truncated backpropagation through time\([Aicher et al\., 2019](https://arxiv.org/html/2608.24136#bib.bib38)\)following the prior works\([Wang et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib8);[Kubo et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib12)\); see Appendix[A](https://arxiv.org/html/2608.24136#A1)\.

#### Evaluation\.

For AKOrN and ItrSA\+\+, we evaluated the performance of a model based on the readout afterTeval∈\{64,128,256\}T\_\{\\text\{eval\}\}\\in\\\{64,128,256\\\}recurrence steps\. We also evaluated model performances using confidence\-based voting\([Kubo et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib12)\)withKvote∈\{1,2,⋯,64\}K\_\{\\text\{vote\}\}\\in\\\{1,2,\\cdots,64\\\}, where trajectories are sampled with random initial values and the one minimizing the sum of entropy over all cells is selected as the final output\.

For TRM, we evaluated the performance based on the readout afterTeval=16T\_\{\\text\{eval\}\}=16recurrence steps following the original evaluation protocol of this model\([Wang et al\., 2025](https://arxiv.org/html/2608.24136#bib.bib8)\)\. We did not apply voting for TRM, as TRM learns a fixed initial value of the latent dynamics, rather than sampling it from a distribution, and thus the voting procedure is not applicable\.

For Sudoku, we report board\-level exact accuracy, where a prediction is correct only if all cells match the ground\-truth board\. For Maze, we report two metrics: shortest\-path accuracy and valid\-path accuracy\. The task is to find a shortest path from the start to the goal, and the training label is given as one such shortest path\. Since the labeled path is not necessarily the only path to the goal, shortest\-path accuracy measures whether the model finds an optimal path, while valid\-path accuracy measures whether the model reaches the goal with any valid path, regardless of optimality\.

Hyperparameters are tuned on a validation set; see Appendix[B](https://arxiv.org/html/2608.24136#A2)\.

#### Additional Computational Cost of RoFB\.

Since RoFB increases the computational cost per inference step, we evaluate performance based on FLOPs at eachKvoteK\_\{\\text\{vote\}\}andTevalT\_\{\\text\{eval\}\}value for a fair comparison\. The increase in compute per trajectory and per step due to RoFB was kept small, within \+20% \(see Appendix Table[5](https://arxiv.org/html/2608.24136#A3.T5)\)\. We also provide results based on wall time, including hyperparameter search, in the Appendix[B](https://arxiv.org/html/2608.24136#A2),[C](https://arxiv.org/html/2608.24136#A3)\. Since hyperparameter search is performed only once, the computational efficiency of RoFB improves asKvoteK\_\{\\text\{vote\}\}andTevalT\_\{\\text\{eval\}\}increase\.

### 5\.1Performance

Figure 3:Compute\-accuracy comparison of baseline vs\. RoFB on \(model, task\) pairs\.*Color*encodesKvoteK\_\{\\text\{vote\}\}count \(Baseline blue / RoFB red, shade darkens withKvoteK\_\{\\text\{vote\}\}from1/1to64/64\)\.*Shape*encodes the inference time stepsTT\(▲,■\\blacktriangle,\\blacksquare,∙\\bulletforT=64,128,256T=64,128,256;Nblock∈\{8,12,16\}N\_\{\\text\{block\}\}\\in\\\{8,12,16\\\}for TRM\)\. TRM on Maze panels use a compressed y\-axis due to baseline saturation\.Fig\.[3](https://arxiv.org/html/2608.24136#S5.F3)shows the compute\-accuracy comparison of the baseline and RoFB across all model\-task pairs\. Blue and red colors represent the baseline and RoFB models, respectively, with darker shades representing largerKvoteK\_\{\\text\{vote\}\}values\. The shape of the marker represents the inference time steps\. Since TRM does not use voting, only one shade is shown for TRM\.

For every model\-task pair except TRM on Maze, we observe that the performance generally improves with increasingKvoteK\_\{\\text\{vote\}\}andTevalT\_\{\\text\{eval\}\}values in both baseline and RoFB models\. The performance of TRM on Maze both at baseline and with RoFB shows a plateau, only a0\.1%0\.1\\%decrease in performance with increasing inference time steps with RoFB\.

#### Performance Gain of RoFB\.

We observe a clear performance boost with RoFB in four of the six model\-task pairs \(AKOrN on both tasks, ItrSA\+\+ on Maze, and TRM on Sudoku\) at the same or lower computational cost, while the performance was largely unchanged in the remaining two pairs \(ItrSA\+\+ on Sudoku, TRM on Maze\) \(Fig\.[3](https://arxiv.org/html/2608.24136#S5.F3)\)\. Crucially, in the four positive cells, RoFB reaches operating points that are unattainable by either prolongingTevalT\_\{\\text\{eval\}\}or increasingKvoteK\_\{\\text\{vote\}\}in the baseline\. For instance, for ItrSA\+\+ on Maze, RoFB at\(Kvote=1,Teval=128\)\(K\_\{\\text\{vote\}\}\{=\}1,T\_\{\\text\{eval\}\}\{=\}128\)achieves83\.6%83\.6\\%shortest\-path accuracy, exceeding the baseline at\(Kvote=64,Teval=256\)\(K\_\{\\text\{vote\}\}\{=\}64,T\_\{\\text\{eval\}\}\{=\}256\),81\.7%81\.7\\%, at roughly100×100\\timesless inference compute\. For AKOrN on Sudoku, RoFB at\(Kvote=4,Teval=256\)\(K\_\{\\text\{vote\}\}\{=\}4,T\_\{\\text\{eval\}\}\{=\}256\)reaches91\.5%91\.5\\%board accuracy,6\.46\.4% above the baseline at the same operating point and not attained even by the baseline atKvote=64K\_\{\\text\{vote\}\}\{=\}64\. For TRM on Sudoku, where confidence\-based voting is unavailable, RoFB lifts board accuracy from68\.4%68\.4\\%to74\.2%74\.2\\%\(\+5\.8\+5\.8%\) atNblock=16N\_\{\\text\{block\}\}\{=\}16, demonstrating that the gain does not depend on the availability of a voting mechanism\.

Although the performance boost with RoFB was small in ItrSA\+\+ on Sudoku at the standard horizon, extending the inference steps further improved the RoFB performance, with about11pp gain atTeval=1024T\_\{\\text\{eval\}\}\{=\}1024\(Appendix[D](https://arxiv.org/html/2608.24136#A4)\)\.

Table 2:Performance deterioration with sign\-flipped coupling\.Evaluated atKvote=4K\_\{\\text\{vote\}\}=4andTeval=256T\_\{\\text\{eval\}\}=256for AKOrN and ItrSA\+\+, and atNblock=16N\_\{\\text\{block\}\}=16for TRM\. Maze is evaluated with shortest\-path accuracy\.

### 5\.2Mechanistic Analysis of Coupling

#### Sign\-flipped RoFB\.

To better understand the mechanism of how RoFB works, we performed an intervention study by flipping the sign of the feedback signal\. We did this to see whether the direction of the feedback signal drives the trajectory towards the correct attractor or cluster, which we suppose to be a key mechanism of RoFB\. If this is the case, flipping the sign of the feedback should deteriorate the performance, as it would push the trajectory away from the correct cluster\.

We find that the four model\-task pairs with a performance boost from RoFB exhibit a large performance deterioration when the sign of the feedback is flipped, while the other two pairs show little to no change\. This indicates that pushing the latent state along the clustering direction, which guides the trajectory toward the correct cluster when that cluster is not yet well formed, is crucial for RoFB’s effectiveness\.

## 6Limitations and Discussion

Our evidence is confined to two puzzle benchmarks \(Sudoku Extreme,Maze Hard\) and three small\-scale recurrent reasoning architectures, and does not cover language models, to which a direct transfer is non\-trivial\. If RoFB is applied directly to LLMs, the large vocabulary size may dilute the inter\-token coupling on which RoFB relies\. Whether RoFB or such a variant provides additional gains to LLMs is left to future work\.

Two of the six model\-task pairs in our suite show no meaningful improvement under RoFB: ItrSA\+\+ on Sudoku and TRM on Maze \(Section[5](https://arxiv.org/html/2608.24136#S5)\)\. Notably, neither RoFB nor its sign\-flipped counterpart shifts the accuracy of these two pairs by more than11% \(Table[2](https://arxiv.org/html/2608.24136#S5.T2)\), in stark contrast to the four positive pairs, where sign\-flipping degrades accuracy by99to4040% — suggesting that in these two pairs the coupling, in either sign, finds no direction in the latent dynamics on which to act\. We currently lack an*a priori*diagnostic that would predict from the model and task alone whether RoFB will help; designing such a diagnostic is an important direction for future work\.

Of the two null model\-task pairs, TRM on Maze deserves a separate caveat: the baseline already attains 92\.0% shortest\-path and 98\.8% valid\-path accuracy, leaving little room for the validation\-based hyperparameter search to discriminate RoFB configurations from the baseline\. The flat objective surface during hyperparameter selection may itself contribute to the null result, independently of whether the underlying dynamics admits a useful coupling direction\.

## 7Conclusions

We proposed RoFB to enhance the inference performance of recurrent reasoning models\. RoFB utilizes the readout probabilities during inference to provide feedback to the latent states\. Our results showed that RoFB improves the inference performance in four model×\\timestask pairs out of six examined without requiring any retraining and with lower computational cost\. Our results provide evidence that readout\-based steering can complement existing inference\-time strategies such as longer rollouts and multi\-trajectory voting\. While the present evidence is limited to puzzle\-style benchmarks and compact recurrent reasoners, it suggests a promising direction for using intermediate predictions to control frozen recurrent dynamics at test time\.

## Acknowledgments

Authors SK, MK, SJ, FU, KK, and KH are supported by computational resources of the TSUBAME4\.0 supercomputer provided by Institute of Science Tokyo through the HPCI System Research Project \(Project ID: hp260232\)\.

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## Appendix AExperimental setup

#### Tasks and datasets\.

We evaluate three iterative reasoners — AKOrN\[[Miyato et al\., 2024](https://arxiv.org/html/2608.24136#bib.bib11)\], ItrSA\+\+\[[Kubo et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib12)\], and TRM\[[Jolicoeur\-Martineau, 2025](https://arxiv.org/html/2608.24136#bib.bib9)\]— on two combinatorial reasoning tasks:*Sudoku Extreme*\(9×\\times9 grid, 1,000 base puzzles augmented to∼1,001,000\{\\sim\}1\{,\}001\{,\}000training samples; 422,786 test puzzles\) and*Maze Hard*\(30×\\times30 grid, 1,000 train / 1,000 test\)\. Both datasets are shared across all three models — same training split, same test split, same vocabulary — so that any across\-model gap reflects architecture and training choices, not data\.

#### Training protocol\.

All three models are trained from scratch with AdamW\[[Loshchilov and Hutter, 2017](https://arxiv.org/html/2608.24136#bib.bib39)\], truncated backpropagation through time \(TBPTT\)\[[Aicher et al\., 2019](https://arxiv.org/html/2608.24136#bib.bib38),[Kubo et al\., 2026](https://arxiv.org/html/2608.24136#bib.bib12)\], where the backpropagation gradients are computed only on the last fixed number of steps\. The parameters were renewed with an exponential moving average \(EMA\)\. We follow each model’s published recipe for the model\-specific hyperparameters\. Full hyperparameters are in Appendix[E](https://arxiv.org/html/2608.24136#A5)\(Tables[6](https://arxiv.org/html/2608.24136#A5.T6)–[8](https://arxiv.org/html/2608.24136#A5.T8)\)\.

## Appendix BHyperparameter Search for RoFB

For each model\-task pair, we perform a single hyperparameter sweep over a scaling parameterλ\\lambdaand three RoFB gate parametersα,tmin,τ\\alpha,t\_\{\\min\},\\taudefined in Eqs\. \([8](https://arxiv.org/html/2608.24136#S3.E8)\) and \([10](https://arxiv.org/html/2608.24136#S3.E10)\)\. Beyond these four parameters, we fixed the coupling function toh⁡\(d\)=−dh\(d\)=\-d, representingrepulsivefeedback, which is held constant across the tasks and models\.

We used Optuna\[[Akiba et al\., 2019](https://arxiv.org/html/2608.24136#bib.bib18)\]with the TPE sampler and additionally enable the Successive Halving Pruner\[[Li et al\., 2018](https://arxiv.org/html/2608.24136#bib.bib40)\]\(an ASHA\-style pruner with reduction factorη=3\\eta=3andminresource=1\\min\_\{\\text\{resource\}\}=1\) for all model\-task pairs, runningntrial=30n\_\{\\text\{trial\}\}=30trials per pair\. The pruner reports intermediate validation accuracy after each ofnchunksn\_\{\\text\{chunks\}\}disjoint chunks of the validation set:nchunks=4n\_\{\\text\{chunks\}\}=4for Sudoku tasks \(Nval=500N\_\{\\text\{val\}\}=500, i\.e\.125125boards per chunk\) andnchunks=2n\_\{\\text\{chunks\}\}=2for Maze tasks \(Nval=100N\_\{\\text\{val\}\}=100, i\.e\.5050boards per chunk\)\. Each trial evaluates a single\(λ,α,tmin,τ\)\(\\lambda,\\alpha,t\_\{\\min\},\\tau\)tuple by running inference with RoFB on the full validation set at a single operating point \(Kvote=1K\_\{\\text\{vote\}\}=1andTevalT\_\{\\text\{eval\}\}matching the canonical training\-time inference budget per model:256256Kuramoto steps for AKOrN,256256self\-attention steps \(=64=64outer blocks\) for ItrSA\+\+,1616outer steps for TRM\), and computes the aggregated accuracy over all validation boards \(board accuracy for Sudoku; shortest path accuracy for Maze\)\. The validation set𝒱aug\\mathcal\{V\}\_\{\\text\{aug\}\}consists of the firstNvalN\_\{\\text\{val\}\}boards of the training set, with one task\-specific symmetry\-group augmentation applied per board\. The best tuple is selected by the aggregated validation accuracy and is then*reused at every\(K,T\)\(K,T\)operating point*in Fig\.[3](https://arxiv.org/html/2608.24136#S5.F3), treating the sweep as a one\-time per\-cell compute cost that we account for separately in Section[C](https://arxiv.org/html/2608.24136#A3)\.

Table 3:Search space and Optuna configuration of the four\-axis RoFB hyperparameter sweep\.λ\\lambdaandτ\\tauare sampled log\-uniformly;α\\alphauniformly;tmint\_\{\\min\}as integers in steps of 8\. The Successive Halving Pruner usesη=3\\eta=3,minresource=1\\min\_\{\\text\{resource\}\}=1for both tasks\.Table 4:Best hyperparameter values selected by the sweep and the achieved aggregated validation accuracy\.tmint\_\{\\min\}is reported in the sweep\-internal time unit \(Kuramoto steps for AKOrN; self\-attention steps for ItrSA\+\+; L\-level steps for TRM, where one TRM outer step = 18 L\-level steps on Sudoku and 12 on Maze\)\.
## Appendix CComputational cost of RoFB

Here we explain the additional compute introduced by RoFB at test time, relative to a baseline forward pass\. The two sources of overhead are \(i\) the one\-time hyperparameter sweep and the \(ii\) the per\-step RoFB block inserted into every test forward pass\. The former is a fixed cost across all\(K,T\)\(K,T\)operating points, while the latter scales with the baseline compute and shifts the RoFB curve to the right by a constant factor on a log\-compute axis\.

We measure the test\-time cost of one\(Kvote,T\)\(K\_\{\\text\{vote\}\},T\)operating point as the total number of forward steps,\(boards\)×\(trajectories per board\)×\(steps per trajectory\)\(\\text\{boards\}\)\\times\(\\text\{trajectories per board\}\)\\times\(\\text\{steps per trajectory\}\), normalized so that the baseline operating point\(Kvote=1,T=Tmax\)\(K\_\{\\text\{vote\}\}\{=\}1,T\{=\}T\_\{\\max\}\)without RoFB equals11\. Under this normalization, the cost decomposes into a per\-inference forward term and a one\-time\-per\-cell hyperparameter\-sweep term:

𝒞~base​\(K,T\)\\displaystyle\\widetilde\{\\mathcal\{C\}\}^\{\\text\{base\}\}\(K,T\)=K⋅TTmax,\\displaystyle=K\\cdot\\frac\{T\}\{T\_\{\\max\}\},\(11\)𝒞~RoFB\-fwd​\(K,T\)\\displaystyle\\widetilde\{\\mathcal\{C\}\}^\{\\text\{RoFB\-fwd\}\}\(K,T\)=K⋅TTmax⋅rtot​\(T\),\\displaystyle=K\\cdot\\frac\{T\}\{T\_\{\\max\}\}\\cdot r\_\{\\text\{tot\}\}\(T\),\(12\)𝒞~RoFB\-tot​\(K,T\)\\displaystyle\\widetilde\{\\mathcal\{C\}\}^\{\\text\{RoFB\-tot\}\}\(K,T\)=𝒞~RoFB\-fwd​\(K,T\)\+𝒞~sweep,\\displaystyle=\\widetilde\{\\mathcal\{C\}\}^\{\\text\{RoFB\-fwd\}\}\(K,T\)\+\\widetilde\{\\mathcal\{C\}\}\_\{\\text\{sweep\}\},\(13\)whereKKis the vote count,TTis the inference horizon \(in each model’s canonical step unit, see Table[5](https://arxiv.org/html/2608.24136#A3.T5)\),TmaxT\_\{\\max\}is the training\-time canonical horizon,rtot​\(T\)≥1r\_\{\\text\{tot\}\}\(T\)\\geq 1is the per\-step overhead of an RoFB\-augmented forward pass over the baseline \(defined in[Per\-step RoFB block cost\.](https://arxiv.org/html/2608.24136#A3.SS0.SSS0.Px2)\), and𝒞~sweep\\widetilde\{\\mathcal\{C\}\}\_\{\\text\{sweep\}\}is the hyperparameter\-sweep cost amortized over a single test evaluation \(defined in[Hyperparameter\-sweep cost\.](https://arxiv.org/html/2608.24136#A3.SS0.SSS0.Px1)\)\.

The main result figure \(Fig\.[3](https://arxiv.org/html/2608.24136#S5.F3)\) plots the per\-inference forward cost only — baseline against𝒞~RoFB\-fwd\\widetilde\{\\mathcal\{C\}\}^\{\\text\{RoFB\-fwd\}\}— because the sweep cost is incurred once per cell and is small relative to the K\-curve test evaluation suite reported in the figure \(<2%<2\\%for the four AKOrN/ItrSA\+\+ cells; see[Hyperparameter\-sweep cost\.](https://arxiv.org/html/2608.24136#A3.SS0.SSS0.Px1)\)\. For completeness, the appendix figure Fig\.[4](https://arxiv.org/html/2608.24136#A3.F4)re\-plots the same operating points with𝒞~RoFB\-tot\\widetilde\{\\mathcal\{C\}\}^\{\\text\{RoFB\-tot\}\}on thexx\-axis, confirming that the Pareto improvement is preserved when the sweep cost is included\. We additionally report the same comparison on a wall\-clock axis in Fig\.[6](https://arxiv.org/html/2608.24136#A3.F6)\.

#### Hyperparameter\-sweep cost\.

We pay the sweep cost once per cell, regardless of\(K,T\)\(K,T\)\. With the lighter sweep protocol of Appendix[B](https://arxiv.org/html/2608.24136#A2)\(ntrial=30n\_\{\\text\{trial\}\}\{=\}30Optuna TPE trials with a SuccessiveHalving pruner, each evaluated onNvalN\_\{\\text\{val\}\}validation puzzles forTsweepT\_\{\\text\{sweep\}\}inference steps withKvote=1K\_\{\\text\{vote\}\}\{=\}1\), the FLOP cost of one sweep, expressed in baseline forward steps, is𝒞sweepboard⋅step=ntrial⋅Tsweep⋅Nval⋅rtot​\(Tsweep\)\\mathcal\{C\}\_\{\\text\{sweep\}\}^\{\\text\{board\}\\cdot\\text\{step\}\}=n\_\{\\text\{trial\}\}\\cdot T\_\{\\text\{sweep\}\}\\cdot N\_\{\\text\{val\}\}\\cdot r\_\{\\text\{tot\}\}\(T\_\{\\text\{sweep\}\}\), which we divide by the baseline costTmax⋅NtestT\_\{\\max\}\\cdot N\_\{\\text\{test\}\}at\(Kvote=1,T=Tmax\)\(K\_\{\\text\{vote\}\}\{=\}1,T\{=\}T\_\{\\max\}\)to obtain the dimensionless𝒞~sweep\\widetilde\{\\mathcal\{C\}\}\_\{\\text\{sweep\}\}shown in Eq\. \([13](https://arxiv.org/html/2608.24136#A3.E13)\)\. The numerical values in Table[5](https://arxiv.org/html/2608.24136#A3.T5)additionally include a factorηASHA=0\.5\\eta\_\{\\text\{ASHA\}\}\{=\}0\.5to reflect that the SuccessiveHalving pruner removes, on average, half of the trials before they reachTsweepT\_\{\\text\{sweep\}\}depth\.

Across the four AKOrN/ItrSA\+\+ cells,𝒞~sweep\\widetilde\{\\mathcal\{C\}\}\_\{\\text\{sweep\}\}ranges from0\.380\.38to1\.651\.65\(in units of one baseline test evaluation atKvote=1K\_\{\\text\{vote\}\}\{=\}1,T=TmaxT\{=\}T\_\{\\max\}\), which is<1\.7%<1\.7\\%of the K\-curve test evaluation suite \(∑K∈\{1,2,4,8,16,32,64\}K=127\\sum\_\{K\\in\\\{1,2,4,8,16,32,64\\\}\}K=127baseline test evaluations\) — the reason the main figure omits this term\. For the two TRM cells the same ratio is larger \(TRM cells useKvote=1K\_\{\\text\{vote\}\}\{=\}1only, so∑K=1\\sum K=1, inflating the ratio to6\.8×6\.8\\timeson Sudoku and28\.7×28\.7\\timeson Maze\), but the absolute sweep cost is in fact the smallest of any cell \(2\.182\.18M and0\.300\.30M board⋅\\cdotstep respectively, see Table[5](https://arxiv.org/html/2608.24136#A3.T5)\); the inflated ratio reflects only the cheap denominator \(TRM usesbf16forward and a single trajectory\)\.

#### Per\-step RoFB block cost\.

At each inference step where the gateg⁡\(𝐳t\)g\(\\mathbf\{z\}\_\{t\}\)is non\-zero \(i\.e\.,t≥tmint\\geq t\_\{\\min\}, controlled by the warm\-up parameter selected by the sweep\), the RoFB block adds one\(B,N,N\)×\(B,N,D\)\(B,N,N\)\\times\(B,N,D\)batched matrix multiplication on top of the baseline forward step — the dominant additional FLOP \(≈N2⋅D\\approx N^\{2\}\\cdot DMACs per board, whereNNis the number of cells andDDthe latent dimension; the readout, similarity, and normalization steps account for less than15%15\\%\)\. Writing the active fraction asactive​\(T\)=max⁡\(0,\(T−tmin\)/T\)\\text\{active\}\(T\)=\\max\\bigl\(0,\\,\(T\-t\_\{\\min\}\)/T\\bigr\)and the per\-step overhead ratio asrstepr\_\{\\text\{step\}\}, the per\-step total isrtot​\(T\)=1\+rstep⋅active​\(T\)r\_\{\\text\{tot\}\}\(T\)=1\+r\_\{\\text\{step\}\}\\cdot\\text\{active\}\(T\)\. Becauserstepr\_\{\\text\{step\}\}scales asN2N^\{2\}per cell and the baseline forward step scales sub\-quadratically inNN,rstepr\_\{\\text\{step\}\}ranges from1\.2%1\.2\\%on TRM×\\timesSudoku \(N=81N\{=\}81\) to19\.5%19\.5\\%on AKOrN×\\timesMaze \(N=900N\{=\}900\), as shown in Table[5](https://arxiv.org/html/2608.24136#A3.T5)\.

#### Per\-model\-task\-pair numerical values\.

Table[5](https://arxiv.org/html/2608.24136#A3.T5)lists the model\-task\-pair\-specific values needed to evaluate Eq\. \([13](https://arxiv.org/html/2608.24136#A3.E13)\)\. At theKvote=1K\_\{\\text\{vote\}\}\{=\}1,T=TmaxT\{=\}T\_\{\\max\}operating point, the per\-inference forward overhead𝒞~RoFB\-fwd/𝒞~base=rtot​\(Tmax\)\\widetilde\{\\mathcal\{C\}\}^\{\\text\{RoFB\-fwd\}\}/\\widetilde\{\\mathcal\{C\}\}^\{\\text\{base\}\}=r\_\{\\text\{tot\}\}\(T\_\{\\max\}\)ranges from1\.0081\.008on TRM×\\timesSudoku to1\.0981\.098on AKOrN×\\timesMaze\. At the largest operating point reported in the main figure \(K=64K\{=\}64for AKOrN/ItrSA\+\+,K=1K\{=\}1for TRM\), the same ratio is essentially unchanged becausertotr\_\{\\text\{tot\}\}does not depend onKK\.

Table 5:Per\-cell parameters for the RoFB compute model of Eq\. \([13](https://arxiv.org/html/2608.24136#A3.E13)\)\.TmaxT\_\{\\max\}is each model’s canonical inference horizon \(Kuramoto step for AKOrN, self\-attention step for ItrSA\+\+, outer block for TRM\);TsweepT\_\{\\text\{sweep\}\}is the same horizon expressed in the unit used internally for thetmint\_\{\\min\}search \(TRM: one outer block=Hcycles⋅Lcycles=H\_\{\\mathrm\{cycles\}\}\\cdot L\_\{\\mathrm\{cycles\}\}LL\-level steps=18=18on Sudoku and1212on Maze, hence16×18=28816\\times 18=288and16×12=19216\\times 12=192\)\.rstepr\_\{\\text\{step\}\}is the FLOP fraction of the batched matrix multiplication \(torch\.bmm\) of shape\(B,N,N\)×\(B,N,D\)\(B,N,N\)\\times\(B,N,D\)relative to one baseline forward step\.rtot\(Tmax\)r\_\{\\text\{tot\}\}^\{\(T\_\{\\max\}\)\}isrtot​\(Tmax\)r\_\{\\text\{tot\}\}\(T\_\{\\max\}\)evaluated at the cell’stmint\_\{\\min\}\.𝒞~sweep\\widetilde\{\\mathcal\{C\}\}\_\{\\text\{sweep\}\}is the amortized sweep cost of Eq\. \([13](https://arxiv.org/html/2608.24136#A3.E13)\), computed withntrial=30n\_\{\\text\{trial\}\}\{=\}30,ηASHA=0\.5\\eta\_\{\\text\{ASHA\}\}\{=\}0\.5, and the\(Nval,Ntest\)\(N\_\{\\text\{val\}\},N\_\{\\text\{test\}\}\)shown\. The𝒞~sweep\\widetilde\{\\mathcal\{C\}\}\_\{\\text\{sweep\}\}column is plotted only on the appendix figures \(Figs\.[4](https://arxiv.org/html/2608.24136#A3.F4),[6](https://arxiv.org/html/2608.24136#A3.F6)\); the main figure \(Fig\.[3](https://arxiv.org/html/2608.24136#S5.F3)\) shows only the per\-inference forward overhead\.
#### Including the sweep cost \(appendix figure\)\.

Fig\.[4](https://arxiv.org/html/2608.24136#A3.F4)re\-plots the operating points of Fig\.[3](https://arxiv.org/html/2608.24136#S5.F3)after adding𝒞~sweep\\widetilde\{\\mathcal\{C\}\}\_\{\\text\{sweep\}\}to the RoFBxx\-coordinate \(Eq\. \([13](https://arxiv.org/html/2608.24136#A3.E13)\), third term\)\. The RoFB curve shifts to the right by a constant offset per cell \(the values listed in the last column of Table[5](https://arxiv.org/html/2608.24136#A3.T5)\) and the Pareto improvement of RoFB over baseline is preserved on all six cells\.

#### Wall\-clock axis \(appendix figure\)\.

Figs\.[5](https://arxiv.org/html/2608.24136#A3.F5)–[6](https://arxiv.org/html/2608.24136#A3.F6)report the same operating points on a wall\-clockxx\-axis instead of FLOPs\. Wall and FLOPs are not interchangeable across cells: AKOrN and ItrSA\+\+ run their forward in fp32, while TRM uses bf16, yielding a per\-step wall ratio of approximately one order of magnitude between the two dtype families\. Within each cell, however, baseline and RoFB share the same forward dtype \(Tables[6](https://arxiv.org/html/2608.24136#A5.T6)–[8](https://arxiv.org/html/2608.24136#A5.T8)\), so the within\-cell baseline\-vs\-RoFB comparison shown in each panel is dtype\-matched and therefore fair\.

Figure 4:Same operating points as Fig\.[3](https://arxiv.org/html/2608.24136#S5.F3), with the per\-cell hyperparameter\-sweep cost \(𝒞~sweep\\widetilde\{\\mathcal\{C\}\}\_\{\\text\{sweep\}\}in Table[5](https://arxiv.org/html/2608.24136#A3.T5)\) added to the RoFBxx\-coordinate\. Sweep cost shifts the RoFB scatter rigidly to the right by a constant per cell; the Pareto improvement is preserved across all six cells\.Figure 5:Same operating points as Fig\.[3](https://arxiv.org/html/2608.24136#S5.F3), with thexx\-axis replaced by measured wall\-clock time on a single NVIDIA GH200 \(forward only, no sweep cost\)\. Within\-cell baseline\-vs\-RoFB comparisons are dtype\-matched \(fp32 for AKOrN/ItrSA\+\+; bf16 for TRM\); cross\-cell wall comparisons should be read together with this dtype split\.Figure 6:Same as Fig\.[5](https://arxiv.org/html/2608.24136#A3.F5)but with the hyperparameter\-sweep wall added to the RoFBxx\-coordinate \(RoFB only\)\.

## Appendix DItrSA\+\+ on Sudoku with Longer Inference Steps

Figure 7:ItrSA\+\+ onSudoku Extreme, with inference horizons extended toT=1024T=1024steps\. The same RoFB hyperparameters selected atT=256T=256are reused at all horizons\.*Color*encodesKvoteK\_\{\\text\{vote\}\}count \(Baseline blue / RoFB red, shade darkens withKvoteK\_\{\\text\{vote\}\}from 1 to 64\)\.*Shape*encodes the inference time stepsTT\(▲,■\\blacktriangle,\\blacksquare,∙\\bullet,⧫\\blacklozenge,★\\bigstarforT=64,128,256,512,1024T=64,128,256,512,1024\)\.
## Appendix EDetailed Training Setups

Table 6:AKOrN: full architecture and training hyperparameters\.
Table 7:ItrSA\+\+: full architecture and training hyperparameters\. We use the affine\-free variant \(𝚛𝚖𝚜𝚗𝚘𝚛𝚖​\_​𝚊𝚏𝚏𝚒𝚗𝚎=False\\mathtt\{rmsnorm\\\_affine\}\{=\}\\mathrm\{False\}\) as the canonical baseline\. Inference horizons follow the conventionT=R⋅nblockT=R\\cdot n\_\{\\mathrm\{block\}\}withR=num​\_​rep​\_​attn=4R\{=\}\\mathrm\{num\\\_rep\\\_attn\}\{=\}4\.Table 8:TRM: full architecture and training hyperparameters\.§The Maze launcherscripts/run\_miyabi\.shcontains a double assignmentpuzzle\_emb\_lr=1e\-4 puzzle\_emb\_lr=1e\-3; the Hydra later\-wins rule yields1×10−31\{\\times\}10^\{\-3\}\.

## Appendix FCompute Resources

All training and evaluation runs were performed on NVIDIA GH200 GPUs\. ItrSA\+\+ cells and TRM×\\timesSudoku were trained on 8\-GPU nodes; TRM×\\timesMaze on a 4\-GPU node; AKOrN cells on a single GPU\. The total compute used for training across all six cells is approximately 100 GPU\-days\. The hyperparameter sweeps for RoFB were run on the same hardware \(∼\\sim10 GPU\-days total, amortized over the K\-curve test evaluation suite as discussed in Sec\.[C](https://arxiv.org/html/2608.24136#A3.SS0.SSS0.Px1)\)\.