VFScale: Intrinsic Reasoning through Verifier-Free Test-time Scalable Diffusion Model
Authors: Tao Zhang
Jia-Shu Pan
Ruiqi Feng
Tailin Wu
Venue: ICLR 2026 Poster (2026)
Keywords: Reasoning
Energy-based Diffusion model
Monte Carlo tree search
Test-time scaling
Primary area: generative models
TL;DR: We introduce the Verifier-free Test-time Scalable Diffusion Model (VFScale) to achieve scalable intrinsic reasoning, which equips number-of-sample test-time scaling with the intrinsic energy function of diffusion models as the verifier.

Abstract: Inspired by human SYSTEM 2 thinking, LLMs excel at complex reasoning tasks via extended Chain-of-Thought. However, similar test-time scaling for diffusion models to tackle complex reasoning remains largely unexplored. From existing work, two primary challenges emerge in this setting: (i) the dependence on an external verifier indicating a notable gap from intrinsic reasoning of human intelligence without any external feedback, and (ii) the lack of an efficient search algorithm. In this paper, we introduce the Verifier-free Test-time Scalable Diffusion Model (VFScale) to achieve scalable intrinsic reasoning, which equips number-of-sample test-time scaling with the intrinsic energy function of diffusion models as the verifier. Concretely, VFScale comprises two key innovations to address the aforementioned challenges. On the training side,  VFScale consists of a novel MRNCL loss and a KL regularization to improve the energy landscape, ensuring that the learned energy function itself serves as a reliable verifier. On the inference side, VFScale integrates the denoising process with a novel hybrid Monte Carlo Tree Search (hMCTS) to improve search efficiency. On challenging reasoning tasks of Maze and Sudoku, we demonstrate the effectiveness of VFScale's training objective and scalable inference method. In particular, trained with Maze sizes of up to 6×6, our VFScale solves 88\% of Maze problems with much larger sizes of 15×15, while standard diffusion model completely fails. The code can be found at https://github.com/AI4Science-WestlakeU/VFScale.

Published as a conference paper at ICLR 2026
VFSCALE: INTRINSIC REASONING THROUGH FRRIFIER-FREE TEST-TIME SCALABLE DIFFUSION MODEL
Tao Zhang Jia-Shu Pan
Ruiqi Feng"
Tailin Wu
1 1Zhejiang University
2Department of Artificial Intelligence, School of Engineering, Westlake University (zhangtao, pan iashu, Nutailing @west.lkee .dd..cn
ABSTRACT
Inspired by human SYSTEM thinking, LLMs excel at complex reasoning tasks via extended Chain-of- Thouugh.. However, similar test-time scaling for diffusion models to tackle complex reasoning remains largely unexplored. From existing work, two primary challenges emerge in this setting: (i) the dependence on external verifier indicating a notable gap from intrinsic reasoning of human in- telligence without any external feedback, and (ii) the lack of an efficient search algorithm. In this paper, we introduce the Verifier-free Test-time Scalable Diffusion Model (VFScale) achieve scalable intrinsic reasoning, which equips number- of-sample test-time scaling with the intrinsic energy function of diffusion models as the verifier. Concretely, VFScale comprises two key innovations to address the aforementioned challenges. On the training side, VFScale consists of novel MRNCL loss and KL regularization to improve the energy landscape, ensuring that the learned energy function itself serves reliable verifier. On the inference side, VFScale integrates the denoising process with a novel hybrid Monte Carlo Tree Search (hMCTS) to improve search efficiency. On challenging reasoning tasks of Maze and Sudoku, we demonstrate the effectiveness of VFScale's training objective and scalable inference method. In particular, trained with Maze sizes of up to 6 x 6, our VFScale solves 88% of Maze problems with much larger sizes of 15 x 15, while standard diffusion models completely fail. The code can be found at https://githu..com/
htt.e/uubb.oo/AAAAeieeee. AI4Science-SW U/sssaeeeeaaaaaaaaaaaaaa
INTRODUCTION
Human intelligence can allocate more computation to solve harder problems, known as SYSTEM 2 thinking (Kahneman| (011). Motivated by this, large language models (LLMs) based on autore- gressive models have exhibited promising performance in reasoning tasks through deliberate long Chain-of- -ooouuhht-hhhttt (Lightman al. 2023 DeepSeek-A1
al., (025). Alongside autoregressive
models, diffusion models have recently emerged as a compelling alternative. By casting reasoning as an optimization problem, these models iteratively refine candidate solutions toward higher-quality outputs. This approach has demonstrated strong potential, achieving notable results on tasks such as Sudoku solving, graph connectivity, and shortest-path computation (Du et al.) 2022 2024).
Although the iterative denoising procedure of diffusion models mirrors the repeated reasoning cycles required by complex reasoning tasks, their performance declines sharply once the problem difficulty exceeds the training distribution (Du et al.) 2022). Moreover, prior works show that simply increasing the number of sampling steps quickly leads to a performance plateau (Du et al.) 2024). Instead, in image generation, inference-time scaling to improve performance by increasing the number of samples has proven effective (Ma et al.] 2025). This success, however, heavily relies on a dense external verifier (Ouyang al. [2022] Lee et al.) 2024), which provides continuous guidance by scoring sample quality. Moreover, these external verifiers are often learned from extensive labeled
"Equal contribution.
1 "ooresponniig author.

---

Published as a conference paper at ICLR 2026
Training
Test
4x4
6x6
12x12
15x15
Figure 1: Visualizations of Maze training data and solutions generated by hMCTS denoising of our VFScale framework.
datasets that are particularly difficult obtain for reasoning tasks. In contrast, human intelligence can engage in deep reflective intrinsic reasoning without external feedback, revealing a clear gap with such methods (Lombrozo 2024; Chi et al., 1994 Huang al., 2024). These observations motivate our central question: Can we design Verifier-free Test-time Scalable Diffusion Model to achieve scalable intrinsic reasoning?
In pursuit of such test-time scalable intrinsic reasoning model, we leverage the fundamental mechanics of diffusion models. Diffusion models approximate the score function (Song al. 202 which is equivalent to the negative gradient of the energy function. Our key insight is that the energy function, as a measure of the learned probability, can serve as an effective verifier itself. Based on the above premise, in this work, we explore equipping number-of-sampe test-time scaling with diffusion models' intrinsic energy function.
However, this strategy presents two primary challenges: (1) Firstly, using number-of-sampl test-time scaling requires a verifier to evaluate and select samples. Prior work on test-time scaling has also established that verifier accuracy is crucial for performance (Setlur al., 2025 Ma et al. 2025). Consequently, using the energy function as an intrinsic verifier requires it to re reliably reflect the sample quality. Nevertheless, previous energy-band -based diffusion models have largely ignored this requirement. (2) Secondly, the efficiency of the search algorithm is critical when scaling by varying the number of samples (Snell al. 2025). However, the search efficiency of existing methods, such as best-of-N (BoN), warrants further enhancement.
To approach these challenges, we propose Verifier-Free Test-Time Scalable Diffusion Models (VFS- cale), a novel framework for solving complex reasoning problems. VFScale consists of innovations in both training objectives and inference scaling shown in Fig. to respectively address the two afore- mentioned challenges. (1) On the training side, VFScale incorporates novel Monotonic-Regee Regression Negative Contrastive Learning (MRNCL) objective and KL regularization to improve the energy landscape, especially to better align the energy function with sample quality with MRNCL loss. The MRNCL objective samples two negatives at different corruption levels for each positive sample, and requires their energy values to reflect their L2 distances to the positive. We refer to this alignment as performance-energ consistency P The KL regularization term backpropagates through the denoising process, smoothing the energy landscape. (2) On the inference side, we propose novel hybrid Monte Carlo Tree Search (hMCTS) that strategically balances exploration and exploitation during the denoising process. In the noisy initial stages, it employs Best-of-N (BoN) for broad, parallel exploration, preventing the premature elimination of promising solution pathways. As denoising progresses and the search space narrows, transitions MCTS for deep, fine -en--rranedd exploitation of the most viable candidates. This hybrid strategy enables VFScale to efficiently translate increased test-time computation into substantial gains in reasoning performance.
We demonstrate the effectiveness of our VFScale's training objective and scalable inference method on challenging reasoning tasks of Maze and Sudoku. In Sudoku tasks, our VFScale solves 43% problems when conditioned much less given digits oout-ff-distributio condition), while the standard diffusion model only solves 30% problems. Maze tasks, trained with Maze sizes of up
Here, positive samples refer to data points, while negative samples correspond to noise-perturbed versions of the data.
Detailed deseription for performance-energy consistency can he found in Appendix
2

---

Published as a conference paper at ICLR 2026
Previous
Ours Training
Training
Posttive
L MSE Contrast
LMSE + Contrast + MRNCL + LKL
Soreat WoOR
Inference
Inference
Narve inference
Basl-gi-N search
hMCTS Denolsing
[K]
MCTS Daneking
Exparsion
(+5
Bockpropagation
The
Exparitoo
Chocse
MCTS Denobingg
MCTS Simulation step
MCTS Sacipropagution
The dewest EXTERNAL roward FERO!!
Ealtal INTRRNSIC
MCTS Rollout
Figure 2: Overview of VFScale. This figure illustrates the key aspects of VFScale by contrasting its training and inference strategies with those of the previous method. (1) To qualify the intrinsic energy of diffusion models as a verifier, VFScale introduces LMRNCL. and LKL. to improve the energy landscape during training. (2) In order for higher search efficiency, VFScale proposes hybrid Monte Carlo Tree Search (hMCTS) that achieves a balance between best-of-N N and MCTS.
6 6, VFScale successfully solves 88% of substantially larger 15 15 instances (see Fig. D. whereas the standard diffusion model fails entirely.
In summary, our contributions are follows: (1) We introduce Verifier-free Test-time Scalable
Diffusion Models (VFScalc), novel framework that can scale up test-time computation for better rea- soning capability. (2) We introduce
Peootonic-Reeressien Negative Contrastive Learning (MRNCL) and KL. regularization into the training objective to improve the energy landscape. (3) We integrate
hybrid Monte Carlo Tree Search into the denoising process, enabling test-time scaling with better search efficiency. (4) We demonstrate the effectiveness of our method on challenging Sudoku and Maze datasets where the conditions are out-of-distribution or the problem sizes are much larger.
RELATED WORK
Diffusion for Reasoning:
The iterative refinement process inherent in diffusion models lends itself well to tasks requiring multistep reasoning, Du et al. (2022) firstly formulated multi-step step reasoning as an energy minimization problem. To approach harder problems with more complex energy landscapes. more compute budget is used for optimization steps. Du et al. (2024) formulated the energy-based model as diffusion model, and the performance on harder tasks was significantly improved. While effective, the performance gain by scaling the compute of these methods quickly saturates as the number of sampling steps increases. Building upon Du et al. (2024), VFScale instead explores verifier-free number-of-sample scaling and proposes corresponding training and inference strategies.
Test-time Scaling of Diffusion: Improving diffusion model samples by increasing compute resources increasingly important direction. Early approaches focused on increasing the number of sampling steps, leading to diminishing returns (Ho et al.) 2020; Karras
at inference time has emerged
al. 2022). More recent work (Ma et al. 2025 Yoon al. 2025) has explored alternative scaling
3

---

Published as a conference paper at ICLR 2026
dimensions, such as increasing the number of sampled candidates, though these methods typically rely on external verifiers to guide solution selection. VFScale advances beyond these methods by leveraging the intrinsic energy function to guide test-time search, capability further cultivated through customized training and inference strategies. These innovations make VFScale a versatile and scalable solution for real-world reasoning tasks.
PRELIMINARY
The Denoising Diffusion Probabilistic Model (DDPM) (Ho et al.) 2020) contains predefined forward process to corrupt data into Gaussian noises, and a learnable reverse process to generate new samples from them. The forward process follows Gaussian transition kernel (2+11 N(x, ya, (1 T), -
T, where the noise schedule {ar) and T is set to make ær follow approximately standard Gaussian distribution. The reverse process can be learned to predict the noise from the corrupted data by minimizing
MMSE = Exooee [|e- Eo (Vā + vI - āze, 1)
(I)
where the expectation is w.r.t. 2n ~ p(x), ~ NoO, 1), and f (1....T) oz = II =1 0, Eq.
is equivalent optimizing reweighted variational bound negative log- ligllii. Without Loss of generality, this work, we parameterize Eq as the negative gradient of the energy function V Egg (2t, t) as in Du al. (023).. To improve the energy landscape, Du al. (2024) introduced contrastive loss
Conntrstt E no 20 ,e,t
log - (c)..)) .) Eo(07
e-
where 2o p(æ), E NO,, H, Vāra0 + /1 āje. at Varay /1 -1-11-rp ā,e. Here To Puorruu ( (zp) are negative examples by corrupting the positive examples æn- This contrastive loss encourages the positive (ground-truth) examples to be local energy minima.
The reverse process starts with æt (O, I), and iteratively applies the learned denoising net cg. where Song et al. (2021a) introduces an adjustable noise scale d;:
Tt-1 01-1 vI= وgC9 (I,,t)
ā,- ofEa (æ,,t) + oyEi, Et (O. (3)
ac
Importantly, Song et al. 22021a) highlights that a diffusion model trained with {az )T= can be used sample with {ar,, P= where any increasing subsequence of 2.... of length S. This variable spacing significantly accelerates sampling and will serve as an efficient approach for MCTS simulation in Section
METHOD
4.1 PROBLEM SETUP
Let D (C, denote dataset for reasoning task, where each instance consists of an input E R and its corresponding solution x E R Our goal is to model the complex reasoning problem as an optimization task to optimize an objective [cc x) and capitalize the diffusion model's ability to iteratively refine the solution ac
In this work, we leverage an energy-basee -aasd diffusion model to learn an energy function Eg(c, æt, t) to model the optimization objective [cc æ). For the sake of brevity, we omit c in subsequent notations and denote it simply as (x t). During inference, we apply test-time scaling by varying the number of samples and use the learned energy function an intrinsic verifier, thereby eliminating the need for an external verifier. However, the reasoning performance gain by test-time scaling is highly dependent both the quality of the energy landscape and the inference-time search algorithm. To address these challenges, we propose VFScale as shown in Fig. 2 with our solutions detailed in Section 4.2 and Section 4.3
4

---

Published as a conference paper at ICLR 2026
4.2 TRAINING OF VFSCALE
Although the denoising loss LMSE (Eq, D and the contrastive loss LComras (Eq. 2 are effective, find that simply scaling up inference budget through BoN processes yields only a marginal gain (Sec. Through deeper analysis, we find that the core reason is the low quality of the energy landscape, especially the lack of performance- novel Monotonic-Regune
consistency, which we address by introducing a Negative Contrastive Learning (MRNCL) loss and incorporating KL regularization, which we detail as follows.
Memnotonic-Reresssin Negative Contrastive Learning. Specifically, while the contrastive loss drives positive examples to the local energy minimum, imposes no constraints on the relative energy ordering among negative samples. For example, it is likely that a negative example 2; that is further apart from the positive example St has lower energy (higher probability) than a negative example 2, that nearer. Thus, the inference process can be stuck around x; and can hardly move out. This will result in reduced performance- does not necessarily correspond to a more accurate predicted solution Ln. as will shown in Sec. 5.1
consistency, where lower energy at step t
This problem cannot be easily remedied by increasing inference budget through BoN during test; instead, we explore more fundamental way to solve this problem to shape the energy landscape by regularizing the energy among negative examples during training. The approach seeks to enforce con- sistency between the relative energy levels of negative samples and their corresponding performance quality, comprising the following key steps.
(1) Generate negative samples: Specifically, from positive example zo p(æ), we sample two negative examples Ti and Ty D and the latter has larger L,2 distance to zp- Specifically, the distances of Zp. Zy and zo to In are 0, 12,0 راح respectively.
zoll2- and 12,0 = ر| xo
(2) Obtain the energy for each sample: Then we obtain their corresponding noisy examples at step t via æ; = Vā + VI arC, a. = Vārzg + VI and a vo + vT a.c. Their energy values are E, (z,, E, = Eg(x, t), and E, Ex t), respectively.
(3) Apply Monotonic regression to get the slope and intercept: constrain the relationship between the energy of different samples and their distance to the ground truth using monotonic function. Linear regression is one specific and effective instance of this approact which will be used throughout the paper unless otherwise mentioned. Specifically, we fit linear function with the three points { E ), (12,0) ET) (2200 E }}. which is characterized by the slope k; and the intercept by-
The core idea is to directly
(4) Calculate the MRNCL loss: Then obtain the corresponding fitted points {(0, Ei), (2200 E ), 11,,5, E Er-)) from the linear function. We then compute the MRNCL loss as follows:
LMRNCL = 20 .5 .c max maxx0,77 k:)) + | Et + E
E,
|
E,
where 7 is hyperparameter and (0....; T). T.. The first term max(0, k2) encourages that any three samples It, क, 2,, have positive (and larger than 7) slope of energy vs. L2 distance. The latter three terms encourage the energy vs. L2 distance to be linear, making the energy landscape smoother.
KL regularization, In addition to LARNCL that improves performance-enery consistency, we further include Kl-regularizztton (Du al. 2021) to further improve the energy landscape:
LKL Et, (x)) Eaao-graddo (z)] + (w) (o) (x))
(5)
3 Detaill to generate negative samples are in Appendix A
'For its simplicity and effectiveness, we primarily use linear regression in this work and term the specitic loss LRNCL (eincar-eerrsssoon NCI.). We provide detailed ablation study in Appendix th.2 analyzes the impact other monotonic constraints (e.g., quadratic, rank-based) and further justifies the choice of linear function for achieving optimal test-time scalability.
Thhe details of linear regression algorithm can be found in Appendix
5

---

Published as a conference paper at ICLR 2026
where (z) is the probability distribution of H; at denoising step t. When optimizing w.r.t. CKL. it is essentially optimizing w.r.t the sampling (denoising) process by shaping the energy landscape to make it easier to sample. The first term in LKI. encourages the samples æ, to have low energy, and the second term is maximizing the entropy of the samples, encouraging the samples to be diverse. Both terms allow better test-time scaling. Unlike Du et al. (202T), we have this KL regularization on each denoising step i, and we employ a more accurate estimation of entropy (Lombardi & Pant (016)).
Together, the training objective of our VFScale is:
L LMSE. + LContsst + LMRNCI. + LKL.;
(6)
where the latter two terms improve the energy landscape and boost the test-time scalability.
4.3 INFERENCE OF VFSCALE
To fully exploit the potential of the diffusion model with a more efficient search algorithm, we propose a novel hybrid Monte Carlo Tree Search denoising (hMCTS denoising) method, which progressively applies best-of-N (BoN) and Monte Carlo Tree Search (MCTS) denoising in sequence, as detailed below, throughout the denoising process. As demonstrated in Algorithm [ we employ BoN for the diffusion process the early diffusion stages, which introduces T, initial noises to maintain consistent number of function evaluations (NFE) per example during the diffusion process Subsequently, hMCTS denoising utilizes the MCTS denoising to iteratively perform the denoising process until the termination state Zo is reached. This approach enables MCTS more accurately estimate node values when the noise is relatively small, thereby preventing the premature exclusion of potentially promising nodes. Next, will detail the MCTS denoise process.
We treat the current diffusion state Dt as the state, the noise to remove as the action, and model the denoising process of the diffusion model node in MCTS represents the state T; along with its current visit count N(xx)) and the state value Q(x:). A terminal node is defined as node whose denoising step is O. In this context, we use Vx, (x t) and Ex t) of the mergy-based diffusion as the policy network and value network, respectively. Each deepening of the search tree corresponds to a single denoising step. Similar to MCTS in AlphaGo (Silver et al., (016) and AlphaZero (Silver et al.) 2017), each MCTS rollout consists of four core steps: selection, expansion, simulation, and backpropagation, as illustrated in Fig. and Algorithm in Appendix A
Markov Decision Process (MDP). Therefore,
(1) Selection: Based on the Upper Confidence Bound (UCB) from AlphaGo (Silver et al., 2016). starting from the current root node state TG, select child node based on the following adjusted UCB of MCTS-enhance diffusion formula until a leaf node (xx, Q(xv)) ), N (wv)} is reached:
N
UCB(s,, a.) Qax,, a.) + c
n4
where Q(x,, az) represents the value of children node, action a includes predicted noise co and random Gaussian noise E, n4 represents the number of visits to node i, N₁ represents the number of visits to the parent node of i, and is the exploration hyperparameter.
(2) Expansion: Unless the state of the node reached a terminal state No. we expand the children of the selected node by choosing an action and creating new nodes based on the action. For the expansion step of MCTS denoising, we perform a denoising step and add different but limited Gaussian noise.
This results in K distinct branches
{ {2
(k) k 0, 1. K 1). where each child node (k) is
derived as the following equation adjusted from Eq. 3:
V1 OgtEg (Ztr, (')
(k) x Fe-1 -1
ār-1 -o ob (xx,t) t') (k)
(8)
Vār
with eø (xe,)') determined by œe and I', and (()) N(O, I) representing random Gaussian noise.
'In this paper, we primarily report K N₂ MCTS denoising, where ensuring K N., I. N, guarantees that the NFE (Number of Function Evaluations) for each case of BoN, MCTS denoising, and hMCTS denoising remains the same. Allocating an identical NFE is the basis for fair comparison the search strategies' effectiveness, as detailed in Appendix[F)
For a detailed explanation of the distinctions between hMCTS denoising, MCTS denoising, and BoN, as well the impact of MCTS denoising start step f,, please refer to Appendix D

---

Published as a conference paper at ICLR 2026
(3) Simulation: We randomly select a child node and perform random simulation of the MDP until a terminal state is reached. For the simulation of MCTS denoising, we use DDIM (Song et al. (021a) for fast sampling to obtain do (0 from the randomly chosen child node state x -/-1 and then use Eo (a
(k")
(axp- (=1)) )) as the reward for backpropagation instead of a reward from external verifier
(4) Backpropagation: Finally, we backpropagate the node values to the root node, updating the value of each node using the expected value along the path.
(k) After N, rollouts, we select x with the highest value ((a("1) )
N(a)" as the state æ for the next de- noising step. The next MCTS denoising process starts from this state and proceeds to the termination state Fg. Here, the depth of the search tree is decided by three elements: the number of rollout steps N,, the maximum number of branches K for each node, and the search policy.
In summary, our proposed paradigm, VFScale, enhances performance-enere consistency and ease of sampling through MRNCI. loss and KI. regularization, respectively (Sec. 4.2). while fully unlocking
the test-time scalability of diffusion models via hMCTS denoising (Sec. 4.3
EXPERIMENTS
In the experiments, we focus on investigating the challenges of replacing an external verifier with a learned energy function to enable test-time scaling of diffusion models via varying the number of samples. We then justify the effectiveness of our innovations in both training and inference. Specifically, we aim to address the following three key questions: (1) What factors hinder the success of test-time scaling of energy-basee diffusion models? (2) Does the training of VFScale exactly improve the energy landscape to unleash the test-time scalability of diffusion models? (3) Can the inference method of VFScale achieve better test-time scaling up?
To answer these questions, we conduct extensive experiments on two challenging reasoning tasks: Maze and Sudoku, evaluating them in distt--iistiiuution (OOD) settings where test instances are substantially more complex than the training data and cause naive methods to fail In Section 5.1 we find that the naïvely trained energy-base during test time and identify the key reason as the poor quality of the energy landscape, particularly its lack of performunce-enrry consistency. Addressing this In Section 5.2 the training methods of VFScale demonstrate significant impact on unbinding the scalability of diffusion models. As shown in Fig. [ with the model trained with VFScale training methods, hMCTS denoising can better release the scalability, enabling generalization to much more challenging tasks during inference (Sec. (.3)
diffusion model (Du et al.] 2024) falls short of scaling up
Maze Success Rate (Grid size 15)
Sudoku Success Rate (Harder Dataset)
oveevvl WATTS Perent Irant MCT therrd vitieete
0.8
0.4
vriese Zways
Success Rate
0.6
0.3
0.4-
0.2
0.1
10"
10a
10a
10a
10t
10a
Relative Compute Budget (N)
Relative Compute Budget (N)
Figure 3: Scalability of different approaches on Maze and Sudoku.
$ 'Although our method is hased an energy-based diffusion model, the proposed hMCTS is also applicable to conventional noise- predicting diffusion models, albeit requiring an external verifier. Additional results are provided in Appendix [E]
Moore experimental details can he found in Appendix Appendix C and Appendix respectively.
7

---

Published as a conference paper at ICLR 2026
5.1 TEST-TIME SCALING BOTTLENECK OF EIERGY-BASEL DIFFUSION MODELS
Test-time scaling up the diffusion model to solve harder reasoning problems is challenging. When tested on more challenging tasks, diffusion models trained with the original method fail to achieve substantial gains, even under larger inference budgets. Specifically, as shown in Table [ models trained on simpler tasks with original training methods exhibit significantly poorer performance on more complex tasks. Success rate plunges from 100% Maze size 6 x 6 to 6% on 15 x 15 and from 32% on Sudoku with 33 given entries to below 5% at 25 and 0% at 21 givens. Building upon this, we conduct best-of-N to examine whether test- time scaling can enhance the performance of the originally trained model. 3ss "Original, BoN" row reveals that enlarging the inference budget N offers marginal gains-Maze success rises by most 5%, while Sudoku remains below 30% even at N = 320. These indicate that naïve test-time scaling of diffusion models fails to deliver meaningful improvements. Next, we analyze the underlying causes from the perspectives of training and inference, respectively.
Table 1: Success rate across different grid sizes of Maze and various number of given entries for naïve inference (N~ 1) for comparison of the generalization ability of models obtained by different training methods. Let M denote the grid size of Maze and D denote the number of given digits in Sudoku. Original denotes the original training method in Du et al. (2024). VFScale tr. (ours) denotes our full training method, and this naming convention is used for following figures and tables. Bold font denotes the best model, and an underline denotes the second- model. The same markings are used in the tables below.
Maze suecess rate
Sudoku success rate
Methods
M 6
M 8
M
M
D 33
D 25
D
17
Original
1,0000
0.9062
0,5781
0.3750
0.0625
0.3203
0.1094
0.0234
0.0000
0.0000
VPScale wfo MRNCI.
1,0000
0,9922
0.7734
0.5625
0.2500
0.4219
0.1719
0.0469
0.0078
0.0000
VFSccll ir. wio KL
L0000
0.9844
0.6953
0.4375
0.2812
0.4219
0.2578
0.0391
0.0078
0.0000
VPScale tr. (ours)
L0000
1,0000
0.7750
0.5391
0.2812
0.1953
0.1016
0,0078
0.0000
0.0000
Table 2: Success rate of BoN for different training methods on Maze with grid size 15 and Sudoku harder dataset guided with ground truth. Here, L N.
Siddek
Methods
Nzl
Ne11
N=21
No41
N:81
Na166
Nal
N=11
N:21
No41
N:B1
N=161
Original, BuN (Eueggy)
0.0625
0,0469
0,0547
0,0781
0.1016
0.1094
0.0859
0.1641
0.2185
0.2344
02422
0.2656
0,2812
Origina, (6N (Gioind Truth)
0.0625
0. 1250
0,1094
0.1328
0.1719
0.1719
0.0039
0.1641
02188
0.2344
0.322
0.2066
0.2899
VIScale in. win MRNCI. BcN (Croand Trub)
0.2500
0,5078
0,5038
0.6622
0.7109
0.7422
0.0094
0.2878
0.2969
0.3438
0,3750
0.3828
0.4219
First, there is still considerable room for improvement on the training side to fully unlock the test-time scalability of diffusion model:")) 10) As shown in T even when ground truth is used to guide test-time scaling, the success rate on the Maze task increases by only 7% over the energy-guided scaling method at an inference budget of N = 161, remaining below 20%. In the Sudoku task, raising N to 321 yields only a 2% improvement compared with suergy-guided scaling, still below 30%. These findings indicate that employing the learned energy function as the verifier can lead to misestimation; the performance-energ) consistency must be further improved. Moreover, Table 4 reports that the originally trained model achieves only about 70% performance-enery consistency in quantitative evaluation, underscoring a significant mismatch between the mnergy-baser verifier and actual performance. In short, these results highlight the urgent need to refine training methods to improve the quality of the energy landscape.
Second, substantial advancements in test-time scaling methodologies are still required to fully harness the inherent scalability of diffusion models. As shown in Table first and second rows regardless of whether ground truth is as a verifier, the performance of BoN quickly plateaus. In the Maze task, increasing the inference budget N from to 161 raises the success rate from 6% to 17% a gain of only 10% and the rate of improvement slows markedly; a similar trend is observed in Sudoku, where success rate growth decelerates significantly and the maximum success
As shown in the Appendix D.I training an effective external verifier is non-trivial and dense reward signal is crucial, which underscores the value of using the learned energy as intrinsic verifier.
8

---

Published as a conference paper at ICLR 2026
rate remains below 30%. These findings suggest that there remains substantial room to improve the efficiency of test-time scaling methods.
5.2 THE EFFECT OF VFSCALE TRAINING METHODS
Table 3: Success rate on Maze with grid size 15 and Sudoku harder dataset for comparison of the model's ability to scale up under BoN with different training methods. Here, L = N.
Maze
Sudoku
Methods
N=1
N=11
N=21
N=41
N-51
N=161
N=1
N=11
N=21
X=41
N=81
N=161
N=321
Originad, RoN
0.0655
016409
0.0547
0.0781
0.1016
0.1094
0.0859
0.1641
0.2188
0.2344
0.2422
0.3655
0.2812
ViSsaale MRNCI, BoN
0.2500
0.4277
0.5100
0.5391
06016
0.6044
0.1122
0.3656
0.3125
0.3438
0.3750
0.3906
0.4141
VPScale KL. BuN
0.2812
0.4658
0.4922
0.5547
0.385)
0.6250
0.1562
0.3422
0.2578
0.2856
0.3656
0.2891
0.3047
VISScale (surs), BoN
0.2656
0,5839
0.6866
0.6875
0,682
0,7031
0.0708
0,2000
0,3281
0.348
0.3515
0.3501
0.196
Table 4: Performance-energy consistency of BoN on Maze with grid size 15 test the effect of MRNCL loss. Here, L = N. Details of consistency calculation can be found in Appendix
Performance-cnergg consistency
Methods
N=11
N=21
N=41
N=81
N=161
Original, BoN
0.7317
0,7333
0,7313
0.7283
0.7300
VFScale tr. w/o KL, BoN
0.8370
0.8445
0.8476
0.8375
0.8371
In this subsection, we evaluate the effectiveness of the MRNCL and KL losses in VFScale compared with the baseline model trained without these losses. First, as shown in Table "N=1" columns, even without test-time scaling, integrating either the MRNCL loss or the KL loss produces significant improvements: on the Maze task, the success rate rises from 6% to 28%, and on Sudoku it likewise increases from 9% to 16%. Second, Table also shows that adding the MRNCL loss alone produces steady improvement in performance- eneryy consistency of over 10%. Finally, when we apply BoN for test-time scaling of the diffusion model, Table 3 demonstrates that using only the MRNCL loss, only the KL loss, or both, boosts the Maze success rate by more than 60% and the Sudoku success rate by over 10%. Meanwhile, as illustrated in the Fig. after incorporating the MRNCL loss and the KL loss, the solutions produced by the model different denoising steps are all noticeably closer to the ground truth solutions. Together, these findings provide strong evidence that the MRNCL and KL losses significantly enhance the test-time scalability of diffusion models.
Figure 4: Comparison of the L2 distances between the solutions obtained by different training methods and the ground truth at various denoising steps.
Trade-off between Naïve Inference and Scalability. A critical observation from Table is that the full VFSeale model occasionally underperforms ablated variants (c.g., "w/o KL") in the naive N = setting. This is deliberate trade-off central to our contribution. The ablated models produce "sharper" energy landscapes that may yield a confident local minimum in a single step (hence higher N - scores) but are often riddled with poor local optima that trap search algorithms. In contrast, our joint objectives (CMRNCL + LKI. are explicitly designed to smooth the landscape, intentionally sacrificing N = sharpness ensure global navigability. The payoff for this sacrifice is superior scalability. For instance, in the Maze task (Table 3). the "w/o KL" model starts strong at 0.2812 but plateaus at 0.6250 (N = 161). Conversely, our full model starts lower (0.2656) but scales dramatically to 0.7031 (BoN) and further to 0.8828 with MMCTS (T) Thus, the lower = 1 performance is symptom of successful optimization for test-time search potential.

---

Published as a conference paper at ICLR 2026
5.3
TEST-TIME SCALABILITY OF VFSCALE
Table 5: Success rate of different approaches on Maze with grid size 15 and Sudoku harder dataset.
Here, N, N N, N.
Maze success
Sudake voccos
Methods
A-1
A-11
N-21
2141
.Y-11
N-161
X11
A1-11
21-21
2141
21-11
N-161
2V-331
Outgina:
0.1094
ViSccll
0.5859
0.701
00773
0.0899
00838
0.1094
0.125
0.1328
VPSialc (surd, BoN
0.2656
0.5859
0.6405
08875
0.653
0.3031
0.700
0.2000
0.3381
0.3438
0.336
0.3994
0.500
VTSccll tr. (ours) IMCTS denotling (0405)
0.3656
0.6816
0.7266
0.7909
03333
0,8338
0.3891
0.330
0.335
0,3072
0,406
0.4397
Using the diffusion model trained with VFScale training methods, we evaluate various inference approaches to validate the efficacy of our inference methods described in Section 4.3 As shown Table 5, in the Maze experiment, the MCTS denoising method slightly outperforms BoN, while hMCTS denoising yields a significantly higher success rate, with maximum improvement of approximately 18% than BoN. Moreover, as budget N increases, the performance gap between hMCTS denoising and BoN widens. In the Sudoku experiment, hMCTS denoising also consistently outperforms BoN, with maximum improvement 5%. As illustrated in the scaling curve in Fig. [ hMCTS denoising with the model trained with additional MRNCL loss and KL loss shows marked improvement compared to both other inference methods. At the same time, the rate of improvement for other inference methods clearly slows down compared to hMCTS denoising. These results provide strong evidence that our inference method can effectively scale during test time, offering clear advantage over BoN
CONCLUSION
In this work, we have introduced the Verifier-free Test-time Scalable Diffusion Model (VFScale), novel framework to achieve scalable intrinsic reasoning to tackle more complex reasoning tasks than in training. VFScale explores applying the varying
scaling method with Iearned energy verifier. Concretely, faced with low-quality energy landscape and the lack of efficient search algorithms, VFScale is composed of two innovations in training and inference to address these challenges. On the training side, VFScale introduces two auxiliary training losses, LMRNCL and LKL. to improve the energy landscape. This, in turn, ensures that the learned energy function is an effective verifier for test-time scaling. On the inference side, VFScale integrates hybrid Monte Carlo Tree Search (hMCTS) denoising to better leverage the model's test-time scalability. We have conducted extensive experiments on Maze and Sudoku validate the efficacy of VFScale and believe that VFScale offers robust solution for test-time scaling, unlocking the notable potential of diffusion models for complex reasoning. Limitations and future directions are discussed Appendix
ACKNOWLEDGMENTS
This work was supported by the Westlake University Center for High-performance Computing. The views and conclusions contained herein are those of the authors and should not be interpreted as representing the official policies or endorsements of the funding entities.
10

---

Published as a conference paper at ICLR 2026
REFERENCES
Michelene TH Chi, Nicholas De Leeuw, Mei-Hung Chiu, and Christian Va Eliciting self-explanations improves understanding. Cognitive science, 18(3)449-477, 1994.
DeepSeek-AI, Daya Guo, Dejian Yang. Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, Xiaokang Zhang. Xingkai Yu, Yu Wu, Z. F. Wu, Zhibin Gou, Zhihong Shao, Zhuoshu Li, Ziyi Gao, Aixin Liu, Bing Xue, Bingxuan Wang, Bochao Wu, Bei Feng. Chengda Lu, Chenggang Zhao, Chengqi Deng, Chenyu Zhang, Chong Ruan, Damai Dai, Deli Chen, Dongjie Ji, Erhang Li, Fangyun Lin, Fucong Dai, Fuli Luo, Guangbo Hao, Guanting Chen, Guowei Li. H. Zhang, Han Bao. Hanwei Xu, Haocheng Wang, Honghui Ding, Huajian Xin, Huazuo Gao, Hui Qu, Hui Li, Jianzhong Guo, Jiashi Li, Jiawei Wang. Jingchang Chen, Jingyang Yuan, Junjie Qiu, Junlong Li, J. L. Cai, Jiaqi Ni, Jian Liang, Jin Chen, Kai Dong, Kai Hu, Kaige Gao, Kang Guan, Kexin Huang, Kuai Yu, Lean Wang, Lecong Zhang, Liang Zhao, Litong Wang. Liyue Zhang, Lei Xu, Leyi Xia, Mingchuan Zhang. Minghua Zhang, Minghui Tang, Meng Li, Miaojun Wang, Mingming Li, Ning Tian, Panpan Huang, Peng Zhang, Qiancheng Wang. Qinyu Chen, Qiushi Du, Ruiqi Ge, Ruisong Zhang, Ruizhe Pan, Runji Wang, R. J. Chen, R. L.. Jin, Ruyi Chen. Shanghao Lu, Shangyan Zhou, Shanhuang Chen, Shengfeng Ye, Shiyu Wang, Shuiping Yu, Shunfeng Zhou, Shuting Pan, S. Li, Shuang Zhou, Shaoqing Wu, Shengfeng Ye, Tao Yun, Tian Pei, Tianyu Sun, T. Wang. Wangding Zeng, Wanjia Zhao, Wen Liu, Wenfeng Liang, Wenjun Gao, Wenqin Yu, Wentao Zhang, W. L. Xiao, Wei An, Xiaodong Liu, Xiaohan Wang, Xiaokang Chen, Xiaotao Nie, Xin Cheng, Xin Liu, Xin Xie. Xingchao Liu, Xinyu Yang, Xinyuan Li, Xuecheng Su, Xuheng Lin, X. Q. Li, Xiangyue Jin, Xiaojin Shen, Xiaosha Chen, Xiaowen Sun, Xiaoxiang Wang. Xinnan Song, Xinyi Zhou, Xianzu Wang. Xinxia Shan, Y. K. Li, Y. Q. Wang. Y. X. Wei, Yang Zhang, Yanhong Xu. Yao Li, Yao Zhao, Yaofeng Sun, Yaohui Wang, Yi Yu, Yichao Zhang, Yifan Shi, Yiliang Xiong, Ying He, Yishi Piao, Yisong Wang, Yixuan Tan, Yiyang Ma, Yiyuan Liu, Yongqiang Guo, Yuan Ou, Yuduan Wang, Yue Gong, Yuheng Zou, Yujia He, Yunfan Xiong, Yuxiang Luo, Yuxiang You, Yuxuan Liu, Yuyang Zhou, Y. X. Zhu, Yanhong Xu, Yanping Huang, Yaohui Li, Yi Zheng, Yuchen Zhu Yunxian Ma, Ying Tang, Yukun Zha, Yuting Yan, Z. Z. Ren, Zehui Ren, Zhangli Sha, Zhe Fu, Zhean Xu, Zhenda Xie, Zhengyan Zhang, Zhewen Hao, Zhicheng Ma. Zhigang Yan. Zhiyu Wu, Zihui Gu, Zijia Zhu, Zijun Liu, Zilin Li, Ziwei Xie, Ziyang Song, Zizheng Pan, Zhen Huang, Zhipeng Xu, Zhongyu Zhang, and Zhen Zhang. Deepseek-rl: Incentivizing reasoning capability in Ilms via reinforcement learning, 2025. 12948]
URL
Yilun Du, Shuang Li, Joshua Tenenbaum, and Igor Mordatch. Improved contrastive divergence training of energy-based models. In International Conference on Machine Leurning, pp. 2837- 2848. PMLR, 2021.
Yilun Du, Shuang Li, Joshua Tenenbaum, and Igor Mordatch. Learning iterative reasoning through energy minimization. In International Conference Machine Learning, pp. 5570-5582. PMLR, 2022.
Yilun Du, Conor Durkan, Robin Strudel, Joshua B Tenenbaum, Sander Dieleman, Rob Fergus, Jascha Sohl-Dickstein, Arnaud Doucet, and Will Sussman Grathwohl. Reduce, reuse, recycle: Composi- tional generation with energy-based diffusion models and meme. In International conference on machine learning. pp. 8489-8511 PMLR, 2023.
Yilun Du, Jiayuan Mao, and Joshua B Tenenbaum. Learning iterative reasoning through energy diffusion. In International Conference on Machine Learning, pp. 1p.111 11764-11776. PMLR, 2024.
Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840-68511 2020.
Jie Huang, Xinyun Chen, Swaroop Mishra, Huaixiu Steven Zheng, Adams Wei Yu, Xinying Song, and Denny Zhou. Large language models cannot self-correct reasoning yet. In The Twelfth International Conference on Learning Representations, 2024. URL utLpttpss: htattps://openrevieee net/ ftte//ormm222 id- 1KmD3EKBPO
Michael Igorevich Ivanitskiy, Rusheb Shah, Alex F Spies, Tilman Räuker, Dan Valentine, Can Rager, Lucia Quirke, Chris Mathwin, Guillaume Corlouer, Cecilia Diniz Behn, et al. A configurable library for generating and manipulating maze datasets. arXiv preprint arXiv:2309. 10498, 2023.
11

---

Published as a conference paper at ICLR 2026
Daniel Kahneman. Thinking, fast and slow. macmillan, 2011.
Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion- based generative models. Advances in neural information processing systems, 35:26565-26577, 2022.
David Lane, David Scott, Mikki Hebl, Rudy Guerra, Dan Osherson, and Heidi Zimmer. Introduction to statistics. Citeseer, 2003.
Harrison Lee, Samrat Phatale, Hassan Mansoor, Thomas Mesnard, Johan Ferret, Kellie Ren Lu, Colton Bishop, Ethan Hall, Victor Carbune, Abhinav Rastogi, et al. Rlaif vs. rlhf: Scaling reinforcement learning from human feedback with ai feedback. In International Conference Machine Learning, pp. 26874-26901 PMLR, 2024.
Hunter Lightman, Vineet Kosaraju, Yuri Burda, Harrison Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe. Let's verify step by step. In The Twelfth International Conference Learning Representations, 2023.
Damiano Lombardi and Sanjay Pant. Nonparametric k-nearest-neiihhor entropy estimator. Physical Review E, 93(1)0013300, 2016.
Tania Lombrozo. Learning by thinking natural and artificial minds. Trends Cognitive Sciences, 2024.
Nanye Ma, Shangyuan Tong, Haolin Jia, Hexiang Hu, Yu-Chuan Su, Mingda Zhang, Xuan Yang, Yandong Li, Tommi Jaakkola, Xuhui Jia, and Saining Xie. Scaling inference time compute for diffuuson models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2523-2534, June 2025.
Long Ouyang, Jeffrey Wu, Xu Jiang, Diogo Almeida, Carroll Wainwright, Pamela Mishkin, Chong Zhang, Sandhini Agarwal, Katarina Slama, Alex Ray, et al. Training language models to follow instructions with human feedback. Advances in neural information processing systems, 35:27730- 27744, 2022.
Rasmus Palm, Ulrich Paquet, and Ole Winther, Recurrent relational networks. Advances in neural information processing systems, 31, 2018.
Amrith Setlur, Nived Rajaraman, Sergey Levine, and Aviral Kumar. Scaling test-time compute without verification RL is suboptimal. In Forty-second International Conference Machine Learning, 2025, URL
David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. nature, 529(7587):484 489, 2016.
David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the game of go without human knowledge. nature, 550(7676) 359, 2017.
Charlie Victor Snell, Jachoon Lee, Kelvin Xu, and Aviral Kumar. Scaling LLM test-time compute optimally can be more effective than scaling parameters for reasoning. In The Thirteenth Inter- national Conference on Learning Representations, 2025.
Cozum?id=4PWWWW2E
Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. In Interna- tional Conference on Learning Representations, 2021a. URL Uttp net/
Yang Song, Jascha Sohl-Dickstein, Diederik Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. basred generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021b. URLhttpa /ttp://///p/eeeeee/eeee htwp://wwwwee.....eeeeeeeeo
idd=xTTG12RRH5
12

---

Published as a conference paper at ICLR 2026
Po-Wi Wei Wang, Priya Donti, Bryan Wilder, and Zico Kolter. Satnet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver. In International Conference on Machine Learning, pp. 6545-6554 PMLR, 2019.
Yangzhen Wu, Zhiqing Sun, Shanda Li Sean Welleck, and Yiming Yang. Scaling inference computation: Compute-optimal inference for problem-solving with language models. In The 4th Workshop on Mathematical Reasoning and Al at NeurlPS 2024. URL https wwwpreewwww..eeeooommmeeeee...
Jaesik Yoon, Hyeonseo Cho, Doojin Baek, Yoshua Bengio, and Sungjin Ahn. Monte carlo tree diffusion for system 2 planning. In Forly-second International Conference on Machine Learning, 2025. URL https: /ttp:ts:: htt.://wwww........c..ocooooeaaa
13

---

Published as a conference paper at ICLR 2026
Part I
Appendix
Table of Contents
A
Related algorithms and metric calculation
15
A.
Pereoryance-energy Consistency
15
A.2
Negative Sample Generation
15
A.3
Linear regresson algorithm
15
Inference algorithm
16
B
Details of experiments
16
Overview
16
B.2
Core Metrie Definition
17
Details Sudoku experiments
B. 4
Details of Maze experiments
17
B.5
Compute Resources
B.6
Data Representation and Processing for Discrete Tasks
18
Performance sensitivity hyperparameters
Additional results
20
The Challenge of External Verifiers and Value Intrinsic Energy
20
D.2
Ablation Study on Monotonic Regression Constraints for MRNCL
21
D.3
Analysis on switch-over step t, hMCTS
22
D.4
Impact of Negative Sample Generation Strategy
23
D.5
Stability Analysis and Multi- Results
23
D.6
From Local Constraints to Global Ranking Alignment
24
E
Extension of hMCTS Conventional Diffusion Models
24
F
Computational Cost and Fair Comparison
25
F.1 Training Cost
25
F2 Inference Cost and Fair Comparison
25
Broader Impacts
26
H The Use LLMS
26
Limitations and future work
26
Visualization results
27
J.1 Visualization of Maze experiments
27
1.2 Visualization Sudoku experiments
27
K Ethies Statement
27
Reproducibility statement
28
14

---

Published as a conference paper at ICLR 2026
A RELATED ALGORITHMS AND METRIC CALCULATION
A.1 PERFORMANCE-ENERGY CONSISTENCY
From a high-level perspective, higher refformance-energee consistency indicates that the learned energy function serves precise verification. In this paper, performance-energ consistency refers to the consistency between the results evaluated using an energy model and those evaluated using real-world metries for the same sample. Specifically, the consistency requires that good samples are assigned low energy, while poor samples are assigned high energy. Performance-eneree element pairs that maintain the same relative order in both permutations X and Y. where X and Y represent the index arrays obtained by sorting the original energy values E = (Ei, E2. ................................ EN) and performance metric values P (Pi, P2,... Px), respectively, in ascending order. In this paper, the energy values are calculated by energy model Eig (2o) for samples Fg- The performance metric values are calculated as the L2 distance between the generated samples zo and the ground truth under the given condition.
accurate intrinsic verifier; conversely, lower consistency denotes less
consistency measures the proportion of
Let X = X2.......... XN) and Y = (Y1, Y2. .2.....YN) YN) be the index arrays obtained by sorting the original energy values E = (E1, Ez.... .. EN) and performance metric values P = -P1. , ,N), respectively, in ascending order. Specifically, X, is the rank of the i-th sample in the sorted energy values E, and Y; is the rank of the i-th sample in the sorted performance metric values P.
Consistency Definition:
The consistency is defined the proportion of consistent pairs (i.j) where i < j and the relative order of and j in X the same as in Y. Specifically:
N
N
Consistency
1
Σ
I((X,
<Y)))Y))((X> MON,,XX,,,, X,K,> Y,)),
1=1+1
where:
N(N-1) 1) is the total number of pairs (i,j) with i < j.
I[[]] is the indicator function, which evaluates if the condition inside the brackets holds
(i.e., the relative order is consistent), and 0 otherwise.
A.2 NEGATIVE SAMPLE GENERATION
Negative samples are generated by introducing noise into the positive sample ag. the Maze and Sudoku experiments, permutation noise is applied to the channel dimension to induce significant changes in the solution. Other noise types can used, as this remains a hyperparameter choice. Specifically, we first randomly sample two scalars P1 and p2 from a uniform distribution in the interval [o, 1). i.e., P1; P2 ~ Uniform(0. 1) (P1 < P2). Then, for each channel position of the positive sample No. we swap the channel positions with probabilities P1 and P2, resulting in To and xo such that the L2 distance between 2n and Zg is smaller than the L2 distance between xo and zg- For other noise types, such as Gaussian noise, we normalize the L2 norm of the noise and apply noise at different scales to ensure that the 1.2 distance from To to zig is smaller than the 1.2 distance from Too to Fo-
A.3 LINEAR-REGRESSION ALGORITHM
Given three points (I1, (I2, and (æ3, Ua), we wish to fit line of the form
Lane et al. (2003)
y = kr + b
The mean of the zz-corddinates and the mean of the y-coordinates are:
F
(i) + x2 + xz),
y
= 3 (y1 + + + vs)
The slope k of the fest-fit line is given by the formula:
15

---

Published as a conference paper at ICLR 2026
Algorithm hMCTS denoising of VFScale
1: Input: EBM (.). Diffusion Steps T, MCTS denoising start step t,, Number of initial noise for Best-of-N L, Maximum MCTS branch count K, Maximum MCTS
rollout step Nr. A set of initial diffusion state { (k) Gaussian noise.
k = L..... L} sampled from
2: // BoN for T -
for T to 1 do
(6)
4: Use Eq. to get a, x;11 for L,... L
5: end for
(4) ( (K))
Ft, <- arg min
// MCTS denoising for to
7: for [= = l. to 1 do
8: Do MCTS Rollouts:
9:
for = to N, do
10:
Selection:
Use UCB from Eq. 7 to select the child node until a leaf node or a is reached and form path using the nodes
terminal node {æz, (,
accessed during the selection;
Expansion: Use Eq. 8 to generate child node state a. (k) for ay and initialize ((a;(-)) = 0, N (ax
11:
0 for l O.... K 1;
12:
Simulation: Randomly choose k" ~ {0, K 1} and do DDIM simulation from 2 (k*) to get (x({({))
13:
Backpropagation: Update the value and visit count of each node Tt, in the path
using (xx;,) + Q(xt,) Eg ( (z(xx=))) (x))))
14:
N(v,,) + ( +
15:
end for
16:
Et-1 mag.aax (2)
end for
18: return 20
3 (z+ Z)(Bs 77)
h:
Σ (x: - ))
This formula represents the least-squares solution for the slope. Once the slope determined, the intercept b can be calculated
The equation of the best-fit line is:
② = kr + b
A.4 INFERENCE ALGORITHM
The detailed algorithm of hMCTS VFScale inference method is in Algorithm I
B DETAILS OF EXPERIMENTS
B. OVERVIEW
In Maze experiments, the datasets are generated by Ivanitskiy et al. (2023), and the diffusion model mainly trained with Maze sizes of up 6 x 6 while tested on harder datasets with sizes significantly larger than 6 (Fig. M). For Sudoku experiments, the basic setting is adopted from Du et al. (024) where the diffusion model is trained on SAT-Net dataset with 31 to 42 given entries Wang al. 2019) and tested on the harder RRN dataset with 17 to 34 given entries Palm et al. (2018) with fewer given entries. All models and inference methods are evaluated with solving success rate. Here successful
16

---

Published as a conference paper at ICLR 2026
solving means the predicted solution exactly matches the ground-truth truth solution on all entries, very stringent metrid"
All training methods compared with the original training pipeline in Du ct al. 2024). LRNCL and KL represent the loss terms LRRCL and LKL. respectively in Eq. VFScalc tr. w/ LRNCL, VFScale w/o KI., and VFScale (ours) represent the three training methods of VFScale, the last being the full version. The MCTS denoising and hMCTS denoising are compared with BoN with the same computational budget. The experiment code be found the repo
B.2 CORE METRIC DEFINITION
For Maze, successful solving means finding a continuous path from starting location to target location without breaking or overlapping with the wall; for Sudoku, successful solving means filling all the missing numbers that exactly match the ground truth, both of which are very stringent metrics.
B.3 DETAILS SUDOKU EXPERIMENTS
For Sudoku experiment, the dataset, model architecture, and training configurations are adopted from Du al. (2024). We mainly use solving success rate to evaluate different models. The model backbone and training configurations can be found in Fig. 5 and Table respectively. All exploration hyperparameters are set as 100 for the Sudoku task.
3x3 Conv2D, 384
Resblook 384
Resblock 384
Resblock 384
Resblock
Resblock 384
Resblock 384
3x3 Conv2D,
Figure The model architecture for VFScale on Sudoku task. The energy value computed using the 1.2 norm of the final predicted output similar to Du al. (203)) while the output is directly used as noise prediction for the diffusion baseline.
Table 6: Details of training for Sudoku task.
Training configurations
Number of training steps
100000
Training batch size
64
Learning rate
0.0001
Diffusion steps
10
Inner loop optimization steps
20
Denoising loss type Optimizer
MSE
Adam
B.4 DETAILS OF MAZE EXPERIMENTS
The details of maze experiments and the model backbone are provided in Table 7 and Fig. 6 respectively. The key metric, the maze-solving success rate, is defined the proportion model- generated paths that have no breakpoints, do not overlap with walls, and begin and end at the start
For example, successful solving of 15 15 Maze needs to predict all 31 x 31 961 entries correctly (for each grid cell, predict path/not path/wall), where the path length from target the order of 10
17

---

Published as a conference paper at ICLR 2026
and target points, respectively. Maze datasets generated by Ivanitskiy al. (023) and detailed hyperparameter configurations are in Table 7 for Maze task.
All the exploration hyperparameters c are set as 100
3x3 Conv2D, 384
Resblock
Resblock 384
Reshlock 384
Resblock 384
Resblock 384
Resblock 384
3x3 Conv2D, 9
Figure 6: The model architecture for VFScale on Maze task. The energy value is computed using the L2 norm of the final predicted output similar to Du et al. (2023), while the output directly used as noise prediction for the diffusion baseline.
Table 7: Details of Maze dataset, training.
Dataset:
Size of training dataset with grid size 4 Size of training dataset with grid size
10219
9394
Size of training dataset with grid size 6
10295
Minimum length of solution path
5
Algorithm to generate the maze
DFS
Size of test dataset with grid size
837
Size of test dataset with grid size
888
Size of test dataset with grid size 10
948
Size of test dataset with grid size 12
960
Size of test dataset with grid size 15
975
Size of test dataset with grid size 20
978
Size of test dataset with grid size 30
994
Training configurations
Number of training steps
200000
Training batch size
64
Learning rate
0.0001
Diffusion steps Inner loop optimization steps Denoising loss type Optimizer
10
20
MSE + MAE Adam
B.5 COMPUTE RESOURCES
For training efficiency, we report the training time of our models along with the machine information (I GPU with 80 GB of VRAM, CPU: 16). On the Maze (100,000 steps) environment, naive training took hours, while incorporating LRNCL increased the training time hours. Adding KL regularization resulted in 3-hour training time, and combining LRNCL and KI. required 6 hours. For the Sudoku (200,000 steps) environment, naive training took hours. Training with +LRNCL took 5 hours, +KL took 6 hours, and +LRNCL+KL required 7 hours.
B.6 DATA REPRESENTATION AND PROCESSING FOR DISCRETE TASKS
key aspect our "continuous reasoning formulation" is its application to discrete tasks like Maze (and Sudoku) via dissrree-continuuusddiieeee pipeline. This approach, which is common
18

---

Published as a conference paper at ICLR 2026
for applying generative models discrete data (Du al. 2024), allows the model to leverage gradient-basee refinement in a continuous space. The process as follows:
Discrete Continuous Encoding:
The discrete x W Maze task is first encoded into continuous tensor. We treat each grid cell as having one of C states (e.g., "path," "wall," "empty") and use one-hot encoding for each cell. This transforms the discrete x W grid into continuous H x W x C' tensor, To. which serves as the model's ground truth.
Continuous Diffusion Process: Our energy function, Es, and all search algorithms (BoN/MMCTS) operate entirely this continuous x W x C' space. All diffusion, noise prediction, and energy calculations are performed on these cantiuuuusvvatue tensors.
To clarify overview figure, the intermediate states shown (e.g., mx.) are not discrete paths. They are renderings of the continuous probability distribution (e.g., in the "path" channel) of the predicted åo at intermediate step. This visualizes the model's 's evolving "belief" as it denoises from = T (pure noise) to = 0 (a confident solution).
Visualization (e.g., in Figure 2)
Decoding to Diserete: To obtain the final discrete solution for evaluation, apply standard argmax operation along the channel dimension (C) of the final output tensor 2o.
This "collapses" the probability distribution back to the single most likely discrete state for each grid cell. This final discrete grid is then evaluated for success.
C PERFORMANCE SENSITIVITY TO HYPERPARAMETERS
In this subsection, we analyze the impact of several hyperparameters on the experimental results. As shown in Table 9 the influence different noise scales on the performance of various methods is presented. The hMCTS denoising and BoN require a relatively larger noise scale to better expand the search space and improve final performance, while the diffusion model with naive inference performs best with smaller noise scale. As demonstrated in Table 3 and Fig. the effect of varying inner-loop optimization steps on the results is also analyzed. It can be observed that performance improves gradually with an increasing number of steps, and after 5 steps, the performance stabilizes and the improvement slows down. Therefore, we chose inner-loop optimization steps for the Maze experiments.
Success rate
0.20
0.55
6.38
2
10
Number of Optimization Step
Figure 7: Visualization of success rate across different number of inner-loop optimization steps Maze with grid size 15 x 15.
19

---

Published as a conference paper at ICLR 2026
Table 8: Success rate across the different number of inner-loop optimization step size 15.
Maze with grid
Number optimization step
Medhods
VFScale ir. (oun), Naive inference
0.0000
0.1562
0.2108
0.2734
0,2812
0.2734
0.2812
0.2969
0.2969
0.2969
Table 9: Success rate across different noise scales Maze with grid size 15.
Noise scale
Methods
VTScale tr. (ours), bMCTS denoising (energy)
0,3828
0.4375
0.5312
0.6094
0,6562
0.6953
0.7031
0.7344
0,7734
0.7969
VFScale inforence
0.3125
0.2656
0,2578
0.2344
0.2656
0.2578
0,2422
0,2500
0.2500
VTScale tr. (ours), BoN(energy)
0.3906
0.4453
0.5312
0.5703
0.5938
0.6328
0.6641
0.6719
0.6797
0.6562
D ADDITIONAL RESULTS
D.1 THE CHALLENGE EXTERNAL VERIFIERS AND THE VALUE OF INTRINSIC ENERGY
In the main text, we argue that leveraging the learned energy function as an intrinsic verifier is cornerstone of our verifier-free, test-time scaling framework. This subsection provides detailed comparative analysis to substantiate this claim, demonstrating two key points: (1) training high- performing external verifier for complex reasoning tasks is nontrivial, and (2) the dense, continuous reward signal provided by our intrinsic energy is crucial for effective scaling and performs on par with perfect oracle.
Intrinsic Verifier vs. Learned External Verifier: To assess the difficulty and effectiveness of using a separately trained model as a verifier, we constructed learnable external verifier. Specifically, trained classifier predict the element-wise accuracy of given Maze solution. While this classifier achieved a high correlation of 0.99 with the true ground truth scores, its performance as guide for test-time scaling was substantially inferior our intrinsic energy verifier.
As shown in Table 10 when using Best-of-N (BoN) sampling, the external verifier leads to performance drop of up to 30% compared using the intrinsic energy function. We attribute this significant gap to the external verifier's inability to distinguish between the subtle yet critical differences among high-quality candidate solutions generated during the scaling process. leading to suboptimal selections. This experiment highlights that even verifier with high statistical correlation may fail to provide the precise guidance needed for complex reasoning.
Table 10: Comparison of test-time scaling performance on the x Maze task using our intrinsic energy verifier versus trained external verifier. Both methods use the same set of generated samples under the Best-of-N (BoN) framework. Success rates are reported for different compute budgets (N).
Method
N=1
N=11
N=21
N=41
N=81
VFScale tr. (ours), BoN, Intrinsic Verifier
0.2656
0.5859
0.6406
0.6875
0,6562
VFScale tr. (ours), BoN, External Verifier
0.2656
0.3516
0.3594
0.3438
0.3203
The Importance of Dense Reward Signal:To further locate the advantage of our intrinsic verifier, we compared its performance against two perfect ground-truth (GT) oracles:
A sparse GT 0/1 verifier, which provides a binary success/failure signal only.
A continuous GT score verifier, which provides precise continuous quality score for any given sample (a perfect, dense reward).
The results in Table IT are highly informative. First, the sparse 0/1 verifier performs significantly worse (over 10% gap N 81) than our dense, energy-guided method. This starkly illustrates the
20

---

Published as a conference paper at ICLR 2026
advantage of continuous guidance; dense reward signal allows the search algorithm (hMCTS) to make more informed decisions at each step, rather than relying on a simple binary outcome.
Crucially, the performance of our intrinsic verifier is nearly identical to that of the perfect continuous GT score verifier. This result strongly validates the effectiveness of our training objectives (especially LRNCL) in shaping high-quality energy landscape that accurately reflects the true solution quality. In essence, our framework successfully learns an intrinsic verifier that functions as a near- perr-peffett, dense reward oracle, unlocking the full potential of test-time scaling without external supervision.
Table 11: Performance comparison on the 15 x 15 Maze task between our energy-guided intrinsic verifier and two perfect ground-truth (GT) oracles: a sparse binary verifier and continuous score verifier. All methods use the hMCTS search algorithm.
Method
N=1
N=11
N=21
N=41
N=81
VFScale (ours), hMCTS, Energy Guided (Intrinsic)
0.2656
0.6406
0.7266
0.7969
0.8203
VFScale (ours), hMCTS, GT 0/1 Guided (Sparse)
0.2656
0.5469
0.5938
0.6562
0.6719
VFScale (ours), hMCTS, GT Score Guided (Dense)
0.2656
0.6328
0.7266
0.7734
0.8359
D.2 ABLATION STUDY ON MONOTONIC REGRESSION CONSTRAINTS FOR MRNCL
To validate the effectiveness the linear constraint and analyze the impact of different energy-L.2 distance relationships within our Monotonic Regression Negative Contrastive Learning (MRNCL) framework, we conducted a comprehensive ablation study on the Maze task. We compare our proposed LRNCL (Linear) against several variants:
This method fits a quadratic function, encouraging the energy-L2 distance of samples to lie within its monotonically increasing region.
QRNCL (Quadratic Regression NCL):
RBNCL (Rank-based NCL): This uses the rank correlation between sample energies and their L2 distances as loss, without assuming specific functional form for the relationship.
FTNCL trasfortranntooomationn NCL): This requires the learned energy function ap- proximate pre-defined linear function with fixed, non-adaptive parameters.
Table 12: Success rates on the 15 15 Maze task for different MRNCL variants. While QRNCL achieves the best performance for single-shot generation (N=1), LRNCL demonstrates superior test-time scalability as the compute budget (N) increases, achieving the highest success rate at N=81.
Training Method
N=1
N=11
N=21
N=41
N=81
Original, BoN, Energy Guided
0,0625
0.0469
0.0547
0.0781
0.1016
+LRNCL (ours)
0.2812
0.4688
0.4922
0.5547
0,5859
+QRNCL
0.3516
0.3672
0.3906
0.3984
0.3828
+RBNCL
0.1875
0,4219
0,4766
0.5234
0,5781
*FTNCL
0.3125
0.4141
0.4844
0.4922
0.5156
The sealing results in Table 12 show that all MRNCL. variants significantly improve performance over the baseline. Notably, interesting trade-off emerges: QRNCL yiclds the largest performance gain without scaling (N=1), but offers the weakest benefit for test-time scalability, with the performance degrading at N=81. In contrast, RBNCL exhibits strong scalability, surpassed only by LRNCL, which provides the largest and most consistent scalability gain.
These findings collectively demonstrate that proposed LRNCL, chosen for its simplicity, is highly effective at calibrating the energy landscape to improve a model's test rest-time scalability, which the primary goal of this work.
21

---

Published as a conference paper at ICLR 2026
0.-
Success rate
Orgras, Enery denaicing,
MCTS start step ts
Figure 8: Visualization of Success rate across different MCTS start step t..
D.3 ANALYSIS ON SWITCH-OVER STEP to OF HMCTS
The parameter t, controls the proportion of the total inference budget allocated to MCTS denoising. When tg 9, means only MCTS denoising used, while f, means only BoN is used. For 0 < f. < 9 (maximum denosing step is 10.), hMCTS denoising is applied. As shown in Table [4] and Fig. there noticeable peak model performance as varies.
While manually tuned l, is effective, we also developed an adaptive mechanism to set the switch- over step on a per-sample basis, eliminating the need for hyperparameter tuning. This method achieves more dynamic global exploration and switching to MCTS for later-stage local exploitation based on the state of the search.
balance by using Best-of-N (BoN) for early-stage
Specifically, the mechanism utilizes the energy variance across the different search branches. The switch from BoN to MCTS occurs when this energy variance exceeds preset threshold, creating closed-loop system that dynamically balances the two strategies. As shown in Table [3] this adaptive approach shows slight advantage budgets (e.g., at N=11 and N=21), demonstrating its utility and robustness.
manually tuned f.s particularly with smaller inference
Table 13: Success rate the 15 x 15 Maze task comparing a manually tuned, fixed switch-over step (t,) with our proposed adaptive t, mechanism. The adaptive method shows notable advantage at smaller compute budgets.
Switch-Over Method
N=11
N=21
N=41
N=81
VFScale (ours), hMCTS, Tuned f.
0.6406
0.7266
0.7969
0.8203
V/FScale tr. (ours), hMCTS, Adaptive f,
0.7031
0.7734
0.7891
0.8203
Table 14: Success rate of hMCTS denoising on Maze with grid size 15 across different MCTS start steps.
Methods
5
Original, bMCTS denoising (energy)
0.0781
0.0703
0.0859
0,0781
0.1250
0.1484
0.1250
0.0781
0.0625
0.0703
VPScale iz. (ours), bMCTS denoising (energy)
0.6094
0.7969
0.6406
22

---

Published as a conference paper at ICLR 2026
Table 15: Success rate of BoN for different training methods Maze with grid size 15 and Sudoku harder dataset guided with ground truth accuracy. Untrained, BoN (gt) represents using ground truth to guide the BoN. Here, L N. Bold font denotes the best model.
Maze
Sudoku
Methods
Nv=21
NV-11
A-331
Unirained, (20)
Original,
0.1094
0.1719
0.2344
0.2969
DDPM, (g0
0L1587
0.2033
0.0009
0.0156
VFScalc tr. LRNCI. (8))
0.5078
0.5938
0.6562
0.7109
0.7422
0.2578
0.3438
0.3828
0.4219
Table 16: Success rate and element-wise accuracy of BoN for different training methods on Sudoku harder dataset guided with ground truth accuracy. Here, L = N. Bold font denotes the best model.
Mashasle
Nel
Nell
Ne21
Xelt
Ne81
Xe161
Ne331
Net
Ne1
Xe41
No81
Nm161
NNs311
Bock GT acconay gridit
0:0081
0.0200
B10000
0.000)
010200
0.0001
0.015
0.931
01009
01642
0.6641
Organnal Boll accuncy suided
0.1641
02666
0.3812
0.7731
LRNCL,
0.3438
D.4 IMPACT OF NEGATIVE SAMPLE GENERATION STRATEGY
To analyze the sensitivity of our framework to the negative sampling strategy, we compare the permutation- approach used in our main experiments against a more standard Gaussian noise perturbation method. In this alternative, negative samples are generated by adding scaled Gaussian noise to the positive samples.
The results presented in Table 17 lead to two key findings. First, the LRNCL framework robustly improves performance and unlocks test-time scalability regardless the specific negative sample generation strategy employed, demonstrating its general applicability. Both perturbation methods significantly outperform the original baseline.
Second, the choice of strategy is not without impact. While the Gaussian noise approach provides strong performance boost, the permutation based method yields superior test-time scalability, with its advantage growing as the compute budget (N) increases. This suggests that structured, problem- aware perturbations (like permutation noise for grid-based tasks) can create more informative negative samples for training the energy landscape compared unstructured noise.
Table 17: Success rate on the 15 x 15 Maze task comparing different negative sample generation strategies for LRNCL. While both strategies significantly outperform the baseline, the Permutation- based method shows better scalability.
Training Method
N=1
N=11
N=21
N=41
N=81
Original, BoN, Energy Guided
0.0625
0.0469
0.0547
0.0781
0.1016
ionuutatiionbbsede Perturb (ours)
0.2812
0.4688
0.4922
0.5547
0.5859
Gaussian Noise Perturb
0.3125
0.3750
0.3906
0.4062
0.4141
D.5 STABILITY ANALYSIS AND MULTI-SEED RESULTS
To validate the stability our framework, we computed the standard deviation over 10 runs with different random seeds for our key experiments the 15 x 15 Maze task. The results, presented in Table 18 summarize the performance (mean + std) for the primary methods as the compute budget (N) increases.
The data confirms that our VFScale framework not only superior in performance but also stable, as indicated by the tight standard deviations. Both training (VFScale tr.) and inference (hMCTS) methods show significant and consistent gains over the baseline, reinforcing the robustness of our approach.
23

---

Published as a conference paper at ICLR 2026
Table 18: Mean and standard deviation of success on the 15 15 Maze task 10 random seeds.
Methods
N=11
N=21
N=41
N=81
Original, BoN VFScale tr. (ours), BoN
0.0539 + 0.0055
0.0594 + 0.0087
0.0594 + 0.0063
0.0609 + 0.0058
0.6203 1 0.0230
0.6562 1 0.0148
0.6680 1 0.0176
0.6836 1 0.0165
VFScale tr. (ours), hMCTS (ours)
0.6727 1 0.0230
0.7336 1: 0.0180
0.7945 1 0.0086
0.8422 1 0.0147
D.6 FROM LOCAL CONSTRAINTS TO GLOBAL RANKING ALIGNMENT
A natural question arises regarding our training objective: since MRNCL enforces only local monotonicity (between positive sample and its perturbed negatives), how does this translate to globally aligned energy landscape? Our hypothesis is that by repeatedly enforcing this simple, local, linear consistency across billions of samples, batches, and diffusion timesteps during training, we implicitly shape globally structured and reliable energy landscape. We provide both direct and indirect evidence to support this.
Direct Evidence: Global Consistency Metries. First, our Performance- Energy Consistency (PEC) metric (defined in Appendix A serves as a measure of global rank-order consistency across a batch of samples. As shown in Table 4 our training improves this global consistency metric from 73% (Original) to 84% (VFScale).
To further rigorously quantify this, we conducted correlation a standard metric for global ranking alignment- between the energy values and the true solution quality across various denoising timesteps. As presented in Table 19 full method ("Learned energy") achieves a significantly higher and more positive correlation compared to both the external verifier and the baseline energy model without MRNCL. Notably, in the critical intermediate stages of denoising (t = 0 to 7), our method maintains a strong positive correlation (>> 0.4), indicating that the energy landscape reliably guides the search towards better solutions globally.
experiment measuring the Kendall-7 rank
Table 19: Kendall-7 rank correlation (mean and std) between the verifier score/energy and the ground-truth quality across denoising steps t. Our intrinsic energy verifier demonstrates superior global alignment compared an external verifier and the baseline (w/o MRNCL).
Denoising Step
Learned External Verifier
Intrinsic Energy (w/o MRNCL)
Intrinsic Energy (Ours)
0
-0.0978 (0.1750)
0.2300 (0.0854)
0.4760 (0.1492)
-0.0976 (0.1750)
0,2300 (0.0854)
0.4744 (0.1490)
-0.0980 (756)
0.2308 (0.0854)
0.4760 (0.1492)
-0.0910 (0. 1834)
0.2290 (0.0838)
0.4642 (0.1362)
-0.0824 (0. 1678)
0.2368 (0.0800)
0.4094 (0.1538)
-0.1292 (0. 1718)
0,2506 (0.0594)
0.4414 (0.1300)
-0.1318 (0. 1748)
0,2638 (0.0766)
0.3978 (0.2466)
-0.3618 (0. 1446)
0,2262 (0.1194)
0.4508 (0.2024)
-0.0316 (0.2854)
0.3214 (0.2138)
-0.1570 (0. 1852)
-0.2686 (0.0322)
0.3712 (0.0520)
-0.3994 (0.0368)
Indirect Evidence: Search Success. The most conclusive proof of global alignment the final outcome. The fact that framework successfully scales- improvinn the 15 x 15 Maze success rate from a baseline of 6.25% to 88.28% (Table 5 serves as strong empirical evidence that a global change the energy landscape has occurred. Such dramatic improvement in global search performance would be unattainable if the energy landscape remained locally fragmented and globally unaligned.
-ereees
E EXTENSION OF HMCTS TO CONVENTIONAL DIFFUSION MODELS
The inference method hMCTS of VFScale is not limited to energy mased diffusion models. With an external verifier, can be applied to conventional models that predict noise directly. We validated this in the Maze task by replacing the learned energy with MSE as the reward, confirming hMCTS's
24

---

Published as a conference paper at ICLR 2026
compatibilityy Its gains over best-of-N search from the Table 20 below further demonstrate broad applicability across diffusion models.
Table 20: Success rate on Maze (grid size 15) using a conventional hoise-predicting diffusion model (DDPM) guided by an external ground-truth verifier. We compare the performance hMCTS with BoN under different compute budgets (N). Here, the reward is defined as the negative MSE to the ground-truth solution. Results validate the compatibility of hMCTS with conventional diffusion and its improved efficiency over best-of-NV sampling.
Methods
N
N = 11
N = 21
N = 41
N = 81
N = 161
DDPM, hMCTS, Ground-truth guided
0.0312
0.1328
0.1406
0.1875
0.2188
0.2422
DDPM, BoN, sround-truth guided
0,0312
0,1094
0.1587
0.1746
0,2031
0.2422
COMPUTATIONAL COST AND FAIR COMPARISON
This section provides a transparent analysis of the computational costs associated with the VFScale framework, covering both training and inference. We also detail our methodology for ensuring fair comparison between different inference- time search strategies. All benchmarks were run on machine with one 80GB VRAM GPU and 16 CPU cores.
F.1 TRAINING COST
By design, VFScale tackles a more complex training objective than the baseline to shape a search- friendly energy landscape. This increased complexity results in a higher demand for training time and memory, as detailed Table and Table
While the LRNCL and KI. losses increase the training cost, this upfront investment is crucial trade- off. For the original method, performance plateaus once the loss converges, regardless of additional training. In contrast, our approach enables the base model's performance to continuously improve with larger training budget, which turn unlocks significantly enhanced test-time scalability.
Table 21:
Training time in hours for Sudoku and Maze task. The VFScale training objectives require more time but lead models with superior scalability.
Method
100,000 Steps (hours) 200,000 Steps (hours)
Original
2
4
+LRNCL
4
5
+KL
3
+LRNCL*KL
6
7
Table 22: Peak GPU memory usage (VRAM in GB) during training with a batch size of 64.
Method
Sudoku (GB)
Maze (GB)
Original
3.28
3.40
+LRNCL
4.56
5.86
+KL
4.44
21.21
+LRNCL+KL
5.57
23.05
F.2 INFERENCE COST AND FAIR COMPARISON
To ensure a fair and meaningful comparison between inference methods (BoN, MCTS, and our hMCTS), we allocate the same Number of Function Evaluations (NFE) to each method for given problem instance. Since the forward pass through the model the primary computational bottleneck, controlling for NFE allows to isolate the efficiency of the search strategy itself.
25

---

Published as a conference paper at ICLR 2026
Table 23 reports the wall-clock inference time. As expected, hMCTS exhibits modest overhead (up to 31% % longer than BoN at N=81). This known trade-off, as the sequential nature of tree expansion in MCTS-based methods is inherently less parallelizable than the embarrassingly parallel BoN approach. However, hMCTS is still significantly more efficient than a pure MCTS implementation.
Crucially, our central claim holds: for an identical computational budget (NFE), hMCTS delivers superior performance over both BoN and MCTS. The difference in wall-clock time stems from the search strategy's s implementation rather than from the core computational complexity. We view this as manageable engineering trade-off for the substantial performance gains achieved.
Table 23: Wall-clock inference time per sample (seconds) on the Maze task. For the same NFE, hMCTS provides superior results, justifying its modest time overhead compared to BoN.
Method
N=1
N=11
N=21
N=41
N=81
BoN
2.17
2.80
5.63
11.01
20.16
MMCTS (ours)
1.02
3.28
6,56
12.66
26.48
MCTS
1.44
7.59
14.03
28.56
57.66
G BROADER IMPACTS
This paper aims to advance machine learning, particularly diffusion- generative models. While improvements promise higher-quality Al-generated content, they also carry potential societal risks. Accordingly, we must remain vigilant to unintended negative outcomes and guard against any unethical illicit applications of this technology.
H THE USE OF LLMS
In the preparation of this manuscript, Large Language Models (LLMs) were used as assistive tool to improve the quality of the work. The specific uses include:
Writing and Language Refinement: LLMs were utilized to paraphrase sentences for clarity, correct grammatical errors, and refine the overall language to meet high academic standards.
Coding and Engineering Assistance: L.MMs were employed to generate boilerplate code, assist debugging, and accelerate the implementation of standard algorithms, which supported the engineering of our experimental framework.
The authors reviewed, edited, and validated all LLM-generated content (both text and code) to ensure its technical accuracy and originality. The authors take full responsibility for all content presented in this paper.
I LIMITATIONS AND FUTURE WORK
Our inference framework primarily relies on MCTS, which presents two key limitations: (1) limited compatibility with parallel computing, and (2) challenges in effectively evaluating node quality during the early stages of denoising. Future work could explore integrating alternative search strategies, such as those proposed by Wu al. (2024). Additionally, to enhance performance-energy consistency, we introduce linear-regression negative contrastive learning, which enforces linear relationship between energy and the distance to real samples. Further investigation is needed to assess the broader implications of this constraint and explore alternative regularization approaches. Lastly, while our current implementation utilizes Gaussian noise for branching, other diffusion- based branching mechanisms remain open for exploration.
26

---

Published as a conference paper at ICLR 2026
grid size 4
grid size 4
grid size 4
grid size 4
grid size
H
SD
grid size 5
grid size 5
grid size 5
grid size 5
grid size
grid
grid
grid
grid
grid
Figure 9; Visualization of training maze dataset.
J VISUALIZATION OF RESULTS
J.1 VISUALIZATION MAZE EXPERIMENTS
This section presents visualizations of the training in Fig. test Maze data in Fig. T0 and samples generated by different methods in Fig. II the visuals, black pixels denote walls, green represents the starting point, red represents the goal point, blue marks the solved path, and white represents feasible area. All visualizations are based on few representative samples. The results from training and test sets clearly show that the tasks in the test set are notably more challenging than those in the training set. Visual comparisons of samples generated by different methods reveal that the originally trained model, regardless of the inference strategy, performs consistently worse than VFScale.
J.2 VISUALIZATION SUDOKU EXPERIMENTS
This section presents visualizations of the training and test Sudoku data in Fig. 12 and representative samples generated by different methods Fig. I3 In the visuals, black numbers denote the condition. green numbers represent correct predictions, and red numbers represent wrong predictions. All visualizations are derived from a few representative samples. The comparison between the training and test sets clearly indicates that the tasks in the test are significantly more difficult than those in training set. When comparing the samples generated by different methods, evident that originally trained model, regardless of the inference strategy, consistently underperforms VFScale.
K
ETHICS STATEMENT
All authors have read and adhered to the ICLR Code of Ethics. This research focuses on the foun- dational capabilities of generative models for algorithmic reasoning tasks. The datasets used are synthetically generated and do not contain personally identifiable information sensitive data. Our work does not involve human subjects. While our primary contribution is algorithmic, we acknowl- edge that advancements in generative models can have broader societal impacts. We believe the potential for misuse of this specific technology low, but we encourage the responsible development and application of all AT systems.
27

---

Published as a conference paper at ICLR 2026
grid size
grid size 6
grid size 6
grid size 6
grid size
2
grid size
grid size 8
grid size 8
grid size 8
grid size
510
grid size 10
grid size 10
grid size 10
grid size 10
grid size 10
T
grid
grid size 12
grid size 12
grid size
grid size
grid
grid
grid size
Figure 10: Visualization of test maze dataset, where the blue paths are ground-truie solutions.
L REPRODUCIBILITY STATEMENT
To ensure the reproducibility of our research, we have made the complete source code, datasets, and pre-trained model checkpoints publicly available htttps:// 114science-Wesee wtw..taate///sssalllllll
com/
The repository contains the full implementation of our proposed VFScale framework, including all training objectives (e.g.. MRNCL) and the hMCTS inference algorithm. Furthermore, comprehensive details regarding our experimental setup, model architectures, hyperparameter settings, and computational resources are provided in the Appendix.
28

---

Published as a conference paper at ICLR 2026
(ours), hMCTS denolsing, Energy guided
VFScale (ours),
MCTS denoising,
Eneray guidad
VFScale (ours), Random search, Energy guided
Original, hMCTS denoising, Energy guided
Original, Random search, Energy guided
Figure 11: Visualization of samples generated by different training and inference methods.
Tat Brites
Figure 12: Visualization of training and test Sudoku dataset.
29

---

Published as a conference paper at ICLR 2026
VFScale (ours), hMCTS denoising, Energy guided
Test:
Entries
Test: 25 Entries
Test: 34 Entries
2
8 2
1
9
6
4
9
1
5
3
4
8
2
9
4
7
2
6
8
4
2
3
9
5
6
4
0
5
5
7
3
8
8
9
4
7
2
3
9
6
8
5
7
8
5
6
(ours),
denoising, Energy guided
5
VFScale (ours), BoN, Energy guided
Original, MMCTS denois ing, Energy guided
5
4
3
9
Original, BoN, Energy guided
2
2
9
8
2
3
9
6
9
2
9
6
7
R
9
3
5
7
4
3
9
6
8
2
7
5
7
8
2 9
5
Figure 13: Visualization of samples generated by different training and inference methods.
30


Reviews:
[{"review_id": "VYIY0jSYS5", "reviewer": "Reviewer_nnZk", "summary": "The paper introduces **VFScale**, a diffusion-based framework for intrinsic reasoning that scales test-time inference *without any external verifier*. The core idea is to use the model’s own **energy landscape** as a dense verifier, training it to correlate with solution quality. Two key mechanisms enable this:  \n1. **Monotonic-Regression Negative Contrastive Learning (MRNCL)**, enforcing *performance–energy consistency* (lower energy ↔ better performance), and  \n2. a **KL regularizer**, smoothing the diffusion trajectory.  \n\nAt inference, the paper proposes **hybrid MCTS (hMCTS)**: a transition from **Best-of-N** exploration in early noisy steps to **Monte Carlo Tree Search** exploitation as denoising progresses.  \n\nExperiments on **Maze** and **Sudoku** show substantial out-of-distribution scaling — e.g., a model trained on 6×6 mazes solves 15×15 mazes at ~88% success.", "questions": "1. Could you formally define **MRNCL**, provide its gradient, and state any conditions ensuring monotonic energy–performance ordering?  \n2. How exactly is the **BoN→MCTS switch** triggered — by diffusion timestep, entropy, or compute budget? Have you tested adaptive criteria?  \n3. Why does the **external verifier** underperform despite 0.99 correlation? Can you analyze the ranking errors?  \n4. Have you applied VFScale to **non-grid CSPs** (e.g., graph coloring or SAT)? If not, what prevents generalization?  \n5. How is the **performance–energy consistency (PEC)** metric computed (e.g., Kendall-τ)? Please include its formal definition.  \n6. Could you report **wall-clock cost and memory usage** for the largest setups to evaluate practical scalability?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- **Innovative framing:** Redefines verifier-free reasoning as intrinsic energy shaping.\n- **Clear empirical impact:** Demonstrates significant generalization to harder unseen settings.\n- **Methodological synergy:** MRNCL + KL training improves energy consistency; hMCTS improves search efficiency.\n- **Alignment with trustworthy AI:** Encourages interpretable internal uncertainty signals instead of opaque verifiers.\n- **Reproducibility:** Code, datasets, and checkpoints are to be released; strong commitment to open science.", "weaknesses": "1. **Limited experimental diversity.**  \n   Evaluation restricted to Maze and Sudoku; lacks heterogeneous reasoning domains (e.g., SAT solving, graph-based CSPs, symbolic reasoning). Limits generalization claims.\n\n2. **Insufficient theoretical analysis.**  \n   MRNCL is intuitively justified but mathematically underdeveloped. No formal calibration or ordering guarantees.\n\n3. **Heuristic inference strategy.**  \n   The hMCTS switching mechanism (BoN→MCTS) is empirical and lacks sensitivity or ablation analysis.\n\n4. **Statistical rigor.**  \n   No standard deviation, confidence intervals, or multi-seed statistics are reported.\n\n5. **Compute transparency.**  \n   Training with MRNCL+KL is more expensive; wall-clock and GPU-hour trade-offs are not quantified.\n\n6. **External verifier comparison.**  \n   The claim that intrinsic energy outperforms a 0.99-correlated verifier is interesting but needs deeper error analysis (e.g., top-k misrankings).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ryxR9rV92R", "reviewer": "Reviewer_bGxA", "summary": "This paper introduces VFScale, a framework for enhancing the test-time scalability of diffusion models for reasoning tasks, notably without requiring an external verifier. The core idea is to leverage the model's intrinsic energy function as a verifier. On the training side, the authors propose MRNCL loss and a KL regularization to improve the energy landscape. For inference, the authors adopt a hybrid MCTS for the denoising process to improve search efficiency. VFScale demonstrates good performance on challenging Maze and Sudoku tasks.", "questions": "Please refer to the weaknesses above. I am inclined to raise my score if my concerns are adequately addressed.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Strengths of this work are summarized below:\n- The paper tackles a highly relevant and challenging problem: enabling test-time scaling for diffusion models on reasoning tasks without relying on an external verifier. Overall, the research direction is interesting and valuable.\n- The test-time scaling capability of VFScale seems strong. For example, Figure 3 clearly illustrates that the proposed method scales significantly better with an increased computational budget (N) compared to all baselines. While other methods see diminishing returns and plateau quickly, the success rate of VFScale continues to climb under both MCTS and BoN.\n- This paper is well-written and clear.", "weaknesses": "Weaknesses of this work are summarized below:\n- My primary concern stems from the ablation study in Table 1. These results, which test the base model's generalization (N=1), contradict the paper's central claim about the utility of the proposed training losses. Specifically, on Sudoku: The full model, \"VFScale tr. (ours),\" performs significantly worse than the ablated models. For instance, at D=33, the full model achieves a 0.1953 success rate, whereas both \"w/o MRNCL\" and \"w/o KL\" achieve 0.4219. At D=29, the full model (0.0078) is drastically outperformed by \"w/o KL\" (0.2578). Similarly, on Maze for M=12, the \"w/o KL\" model achieves a perfect 1.0000 success rate, while the full model only reaches 0.5391. I would like to know the potential reasons behind the results, for example, providing an explanation for why their full model can be the worst performer in this naïve inference setting.\n\n- While the performance on Maze and Sudoku is good, the experiments are limited to grid-world toy problems. The paper's impact would be stronger if the authors demonstrated VFScale's applicability to more complex and practical reasoning or optimization domains where diffusion models are being applied.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RLt5iswhnO", "reviewer": "Reviewer_g4A4", "summary": "The paper proposes VFScale, a way to make diffusion models solve reasoning problems better without relying on an external verifier. It treats the model’s own energy as an intrinsic verifier and improves both training and inference: on the training side, a loss called MRNCL enforces that samples farther from the ground truth have monotonically higher energy and adds a KL term to smooth the landscape so lower energy better aligns with correctness; on the inference side, a hybrid search hMCTS uses Best-of-N early (broad exploration) and MCTS later (deep exploitation) to convert extra compute into higher success more efficiently.", "questions": "1. Have the authors tried using majority voting for the task? If the diffusion model can consistently output a path more often than the other, it can also be used as a verifier.\n2. How about pitting MRNCL against margin / listwise ranking losses—and showing how results shift when you swap the “near/far” distance metric for negatives?\n3. Any quick Kendall $\\tau$ / Spearman $\\rho$ between energy and correctness across timesteps to show the intrinsic-verifier story holds up?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Treats diffusion energy as an intrinsic verifier, removing dependence on external verifiers and focusing on verifier accuracy + search efficiency.\n- Training (MRNCL + KL) aligns low energy with correctness; inference (hMCTS) sensibly combines early Best-of-N with later MCTS to turn extra compute into better.", "weaknesses": "- Using diffusion energy as an intrinsic verifier and aligning it via MRNCL + KL is largely heuristic; the paper provides no formal results\n- MRNCL relies on how “near/far” negatives are defined and enforces only local monotonicity; the paper does not analyze robustness to the distance metric nor show that local constraints imply a global ranking alignment.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5vaQas7JYc", "reviewer": "Reviewer_FVwD", "summary": "This paper aims to address the lack of test-time scaling capabilities in diffusion models for complex reasoning tasks (e.g., maze solving and Sudoku). The core idea of this work is to use the diffusion model's own intrinsic energy function as a verifier to guide the search.\n\nTo achieve this, VFScale incorporates two key training innovations: Monotonic-Regression Negative Contrastive Learning (MRNCL) and KL regularization—to improve the energy landscape so that the model’s learned energy can act as an intrinsic verifier. At inference time, a hybrid Monte Carlo Tree Search (hMCTS) algorithm is proposed. This algorithm uses Best-of-N (BoN) for broad exploration during the early, noisy denoising stages and switches to MCTS for deep exploitation in the later, less-noisy stages.\n\nExperimental results show that VFScale (e.g., trained on 6x6 mazes) can achieve a high success rate on complex tasks far beyond the training distribution (e.g., 15x15 mazes), where baseline models fail completely.", "questions": "1. **Computation of $p_{\\theta, t}(x)$:**  \n   How is $p_{\\theta, t}(x)$ computed exactly? Please also clarify its relationship to the predicted energy function.\n\n2. **Representation of Discrete Data:**  \n   Please elaborate on how the Maze and Sudoku datasets are represented. Specifically, how are discrete states encoded into continuous vectors for diffusion modeling and energy computation?How are the intermediate Maze visualizations (e.g., Fig. 1) generated? Are they obtained by projecting continuous latent states, or by decoding discrete states from intermediate diffusion steps?\n\n4. **Energy–Distance Relationship:**  \n   What is the theoretical motivation for enforcing a *linear* relationship between the predicted energy and the \\( L_2 \\) distance? Would a monotonic but non-linear mapping (e.g., logistic or exponential) be equally valid or even more flexible?\n\n5. **MCTS vs. Best-of-N Sampling:**  \n   Since each MCTS expansion involves multiple Gaussian perturbations, what makes your MCTS procedure fundamentally different from a multi-step Best-of-N or beam search strategy?\n\n---\n\n#### Miscellaneous\n\n1. **Definition of “Guided with Ground Truth” (Table 2):**  \n   The term *“guided with ground truth”* in Table 2 is not clearly defined. Please clarify the exact experimental setup and what this guidance entails.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. **Interesting Problem Setup:**  \n   The paper tackles the idea of *intrinsic reasoning*—removing external verifiers for diffusion-based reasoning—which is conceptually appealing and timely. This aligns well with the emerging research trend on *test-time scaling* and reasoning within generative models.\n\n2. **Clear Positioning within Existing Literature:**  \n   The paper establishes clear connections to prior work, particularly Du et al. (2022, 2024) on energy-based diffusion, effectively situating the contribution within the broader context of diffusion-driven reasoning research.\n\n3. **Algorithmic Insight of hMCTS:**  \n   The hybrid design of hMCTS demonstrates sound algorithmic intuition. Employing Best-of-N (BoN) for parallel exploration during early noisy stages and transitioning to MCTS for fine-grained exploitation in later denoising stages is well-motivated and consistent with the diffusion dynamics.\n\n4. **Strong OOD Generalization Performance:**  \n   The experiments show notable improvements in out-of-distribution (OOD) generalization, suggesting that the proposed intrinsic reasoning framework may enhance robustness beyond the training distribution.", "weaknesses": "1. **Marginal Gains from MRNCL:**  \n   The improvement attributed to MRNCL appears small and inconsistent across tasks. The MRNCL objective, which enforces a linear energy–distance relationship, feels somewhat ad hoc. Imposing strict linearity between the $L_2$ distance and energy may oversimplify the energy landscape and lacks clear theoretical justification. Empirically, much of the observed gain seems to stem from the KL term rather than MRNCL itself (see Tables 1 and 3, particularly in the Sudoku experiments).\n\n2. **Questionable Inference Design in Continuous Space:**  \n   The proposed hMCTS in practice appears to reduce to a Best-of-N sampling strategy with branching Gaussian perturbations. It remains unclear what specific advantages MCTS provides in this continuous setting, as its benefits are typically most pronounced in discrete action spaces.\n\n3. **Unclear Experimental Setup and Visualization:**  \n   The implementation details for the Maze task are under-specified. While the paper implies continuous-valued diffusion states, the Maze itself is inherently discrete. It is unclear how intermediate diffusion states are visualized (e.g., Fig. 1) or discretized back into paths. This ambiguity raises questions about whether the claimed “continuous reasoning formulation” genuinely applies to these discrete tasks.\n\n4. **Limited Novelty of Components:**  \n   The major components—contrastive energy training, KL regularization, and MCTS-based inference—are largely adapted from existing literature (e.g., Du et al., 2024; Silver et al., 2016). The central contribution, MRNCL, introduces a monotonic regression term that offers modest empirical improvement, making the overall technical novelty somewhat incremental.\n\n5. **Limited Evaluation Scope:**  \n   The experiments are confined to synthetic reasoning tasks (Maze and Sudoku). While suitable for controlled evaluation, the absence of results on larger or real-world structured reasoning benchmarks limits the generality and impact of the work.\n\n---\n\n#### Miscellaneous Issue\n\n1. **Inconsistent Notation:**  \n   The definition of the energy function is inconsistent across sections. In Du et al. (2024) and line 198, it is defined with both the input and its corresponding solution, whereas in other parts of the paper, the energy function is presented as depending solely on the solution.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)