[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
Manually writing hardware description language (HDL) code (e.g., Verilog) is an unavoidable part of the current hardware design process. This step is often boring and cumbersome, consuming a significant amount of engineers' time. As LLMs have demonstrated excellent performance in natural language processing and code generation, researchers try to explore the use of LLMs to generate HDL code as an aid in hardware design [2, 1, 21, 10, 11].

Recently, large language models (LLMs) have demonstrated excellent performance in understanding human instructions and generating code, which has inspired researchers to explore the feasibility of generating RTL code with LLMs. However, the existing approaches to fine-tune LLMs on RTL codes typically are conducted on fixed datasets, which do not fully stimulate the capability of LLMs and require large amounts of reference data. To mitigate these issues, we introduce a simple yet effective iterative training paradigm named ITERTL. During each iteration, samples are drawn from the model trained in the previous cycle. Then these new samples are employed for training in this loop. Through this iterative approach, the distribution mismatch between the model and the training samples is reduced. Additionally, the model is thus enabled to explore a broader generative space and receive more comprehensive feedback. Theoretical analyses are conducted to investigate the mechanism of the effectiveness. Experimental results show the model trained through our proposed approach can compete with and even outperform the state-of-the-art (SOTA) open-source model with nearly 37% reference samples, achieving remarkable 42.9% and 62.2% pass@1 rate on two VerilogEval evaluation datasets respectively. While using the same amount of reference samples, our method can achieve a relative improvement of 16.9% and 12.5% in pass@1 compared to the non-iterative method. This study facilitates the application of LLMs for generating RTL code in practical scenarios with limited data.

(2) There is a mismatch between the distributions of training samples and the LLM under training, which can lead to estimation errors [8, 13] in optimization process. Specifically, if the training samples are not directly generated by the LLM intended for training, their distributions are obviously misaligned. However, even if the training samples are generated by the LLM intended for training at the beginning, their distributions will still mismatch because the distribution of the LLM will shift during the training process.

In order to make LLMs more professional in RTL code generation, a common method is to use the corresponding database to fine-tune LLMs. However, most existing methods conform to the conventional paradigm of deep learning, which involves initially gathering data and then training the model. This approach may lead to two negative effects:

Peiyang Wu, Nan Guo, Xiao Xiao, Wenming Li, Xiaochun Ye, and Dongrui Fan. 2018. ITERTL: An Iterative Framework for Fine-tuning LLMs for RTL Code Generation. In Proceedings of Make sure to enter the correct conference title from your rights confirmation email (Conference acronym 'XX). ACM, New York, NY, USA, 7 pages. https://doi.org/XXXXXXX.XXXXXXX

Due to these two negative effects, the RTL code generation capability of existing fine-tuned models are limited. Additionally, a large number of reference samples are required for fine-tuning, which can be costly to obtain. In order to foster exploration and mitigate the distribution deviation between training samples and LLMs that is

(1) Since the model is trained on a limited amount of collected data, the room available for the model to explore is thus constrained, resulting in a narrow coverage of feedback signals.

RTL Code Generation, Large Language Model, Iteration, Reward Function

Institute of Computing Technology, Chinese Academy of Sciences
Beijing, China
guonan@ict.ac.cn

Institute of Computing Technology, Chinese Academy of Sciences
Beijing, China
yexiaochun@ict.ac.cn

Institute of Computing Technology, Chinese Academy of Sciences
Beijing, China
xiaoxiao@ict.ac.cn

Peiyang Wu
Institute of Computing Technology, Chinese Academy of Sciences
Beijing, China
wupeiyang@ict.ac.cn

Institute of Computing Technology, Chinese Academy of Sciences
Beijing, China
liwenming@ict.ac.cn

Dongrui Fan
Institute of Computing Technology, Chinese Academy of Sciences
Beijing, China
fandongrui@ict.ac.cn

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org.

Conference acronym 'XX, June 03–05, 2018, Woodstock, NY                                                                        Trovato et al.

being trained, an intuitive idea is to introduce reinforcement learning (RL) methods to allow LLMs to refine their policies through the interaction with the environment, like reinforcement learning from human feedback (RLHF)[17, 20] for common LLMs or CodeRL[7], RLTF[9], etc. for code LLMs. However, it is well-known that the process of RL is often complex, resource-intensive, and unstable. Typically, RL algorithms need to maintain several models at the same time (such as policy model, value model and reference model in PPO algorithm[19]) and conduct extensive hyper-parameters tuning, which will incur significant computational overhead and pose substantial obstacles for users.

To tackle the aforementioned issues, we introduce a simple yet effective iterative training scheme to fine-tune LLMs for Verilog code generation. Unlike previous methods learning from the same training set, our approach updates the training samples through sampling iteratively during the training process, thus enhancing the exploration range of LLMs and enriching feedback. Since the training samples are iteratively sampled from the updated model, the mismatch between distributions of the training samples and LLMs, as well as the resulting estimation error, are mitigated accordingly. As the first method which iteratively updates training samples in the RTL code generation field, our method enables the model to achieve state-of-the-art (SOTA) performance even with a smaller number of reference samples. Moreover, our approach is much easier to implement than complicated reinforcement learning. Only minimal hyper-parameter tuning is required, and there is no need to handle multiple models during the training process.

Our contributions can be outlined as follows:

(i) We develop an iterative supervised fine-tuning scheme for training LLMs to generate Verilog code. By expanding the exploration scope of models and reduces estimation errors caused by distribution mismatches, this solution can effectively boost the Verilog code generation capability and substantially decreases the number of externally-sourced reference samples needed for training.

(ii) Through a reward-maximization viewpoint, we theoretically analyze the limitations of previous methods and reveal the superiority of our approach.

(iii) With only about 37% reference samples, we outperform the state-of-the-art (SOTA) open-source LLM, attaining 42.9% and 62.2% pass@1 rate on VerilogEval-human and VerilogEval-machine benchmarks respectively. Utilizing the same quantity of reference samples, our method demonstrates a 16.9% and 12.5% relative enhancement in pass@1 scores compared to the non-iterative approach. Relative to commercial closed-source models, we surpass GPT-3.5 and approach the level of GPT-4 on the benchmarks.

## 2 BACKGROUND

### 2.1 Large Language Models for RTL Code Generation

Previous works[1, 2] have attempted to directly prompt LLMs to generate RTL code, achieving notable results. Chip-chat[1] design a 8-bit accumulator-based microprocessor with the assistance of commercial LLMs. In practical applications, it is challenging to directly generate usable code through human instructions. As a result, recent works[22, 4] explore optimizing the RTL codes generated by LLMs with feedback from Verilog tools. AutoChip[22] utilize error reports from compilers and simulators to help LLMs to rectify faulty code. [4] develops a Monte Carlo tree-search (MCTS) algorithm to enhance LLMs to generate correct and PPA-efficient code with the feedback from compilers and synthesis tools.

Instead of using off-the-shelf LLMs directly, some researchers[10] opt to train LLMs to make it specialized in hardware design. VeriGen[21] leverage corpora from GitHub and Verilog textbooks to fine-tune open source LLMs, defeating the state-of-the-art commercial Codex[3] LLM on 17 Verilog problems. ChipNeMo[11] customize LLaMa2[23] for applications in chip design such as chatbot, generating EDA tool script, and bug summarization. RTLCoder[12] develops a automated flow to generate instruction-code pairs for supervised fine-tuning (SFT) and propose a new SFT method leveraging code quality assessment.

In order to better measure the effect of LLMs on Verilog code generation, researchers have proposed corresponding evaluation benchmarks. VerilogEval[10] has released an open-source evaluation dataset, including 156 questions along with their corresponding golden solutions. RTLLM[15] present an open-source benchmark evaluating the quality of the generated code from three progressive perspectives: syntax, functionality, and design quality.

### 2.2 Fine-tuning LLMs with RTL data

To tailor LLMs for RTL code generation, researchers often need to fine-tune LLMs with domain-specific data. VeriGen[21] fine-tune LLMs by predicting next token on corpora from open-source code and textbooks, which can be regarded as a continual pre-training. To facilitate the model's ability to follow instructions, VerilogEval[10] applies SFT on LLMs using synthetic instruction-code pairs. Furthermore, RTLCoder[12] introduces a new fine-tuning algorithm which harnesses code quality evaluation upon candidates sampled from pretrained LLMs. However, during the training process, the aforementioned methods are constrained by static and unchanging training samples. According to previous research[24, 14] in natural language, updating train samples using in-training LLMs can significantly bootstrap the model performance. However, related research in the task of Verilog code generation remains scarce. Our work can be seen as a pioneering attempt.

## 3 APPROACH

In this section, we firstly detail the workflow of our proposed approach. Then analyze the deficiencies of related methods and reveal the improvements of our approach from a theoretical viewpoint.

### 3.1 Framework

As an iterative training scheme, our approach boost the generation capability by alternately sampling and training. During a single loop, we basically follows RTLCoder[12] for ease of implementation. The key distinction lies in our core idea of alternating iterations, which can significantly enhances the performance.

We briefly introduce the procedure in a single loop at first, and then explain the iteration process. As shown in Figure 1, in the t-th round of training, for each input instruction s, there are K corresponding output responses ak^t, 1 ≤ k ≤ K. Among these, the first K − 1 responses are sampled from the model πt acquired from previous training iteration. While the last response aK^t, which

Ideally, the true reward for the generated Verilog code should incorporate the evaluation of its functional correctness, as well as PPA metrics. However, it’s intractable to implement because verifying the functional correctness requires a comprehensive corresponding testbench, which is nearly impossible for researchers to develop given the large amount of samples for fine-tuning. Additionally, assessing PPA relies on logic synthesis, which also consumes a significant amount of time. Therefore, it is necessary to approximate the reward function to make it more manageable.

serve as the reference data, can be obtained from another teacher LLM or human, denoted as π_teacher. Assume the distribution of π_teacher remains unchanged, there is little impact from repeated sampling. Therefore, we omit the superscript t from a^t_K. Each response a^t_K is assigned a quality score z^t_k utilizing Pyverilog and Rough-L following [12].

After the initial round of training, we obtained the converged model π_{t+1}. Even though the capability of model π_{t+1} has improved comparing to π_t, there remains room for further enhancement. So we propose to resample new responses a^{t+1}_k using the new model π_{t+1} and assess new quality scores z^{t+1}_k. Resampling can bring

new feedback as the model's distribution has shifted after previous round of training. We compute the loss function L^{t+1} and update parameters with new data. This process is repeated until the training concludes.

where 𝔼 denotes the expected value, s represents the instruction (prompt), following the distribution ρ(.). a represents the response (code) generated by the LLM π which is being optimized. r represents the reward for the response, in other words, the quality of the generated code.

Let's first consider a naive approach that directly fine-tunes LLMs using reference data drawn from π_teacher, which can be considered a form of knowledge distillation. The optimization objective can be derived as follows:

The first equality holds because P_teacher is independent of π. While the approximate equality in the third line is based on the law of large numbers. Comparing equations 5 and 6, it is clear that this approach leads to systematic errors in two aspects^1: (1) replacing the distribution of the model being trained π(. | s) with that of π_teacher(. | s), and (2) the use of the negative Kullback-Leibler divergence (KL divergence) as the surrogate reward function between the probability distribution of the reference data and π.

Next, let's consider a non-iterative degenerate version of our approach, where a constant distribution π_c is used to generate training samples. This version is quite similar with RTLCoder [12]. We can also transform the optimization objective into a reward maximization problem. To simplify the derivation, here we only present the ranking loss component from Equation 2. And the

To control the scale relationship between the two loss functions we set λ equal to sg(L_ce), where sg(.) represents a stop-gradient operation.

Inspired by the previous work [8], we regard fine-tuning LLMs as a reward maximization problem, which can be formulated as follows:

Combining the quality score z and log probability p, the ranking loss [12, 24, 14] can be computed:

p_k = (Σ_j log P_π(a^t_{k,j} | s, a^t_{k,<j})) / ||a^t_k||       (1)

min_π Σ_j -log P_π(a_j | s, a_{<j}) = min_π Σ_j log (P_teacher(a_j | s, a_{<j}) / log P_π(a_j | s, a_{<j}))
= min_π Σ_j KL(P_teacher,j || P_π,j)
≈ max_π 𝔼_{s~ρ(.)} 𝔼_{a~π_teacher(.|s)} [Σ_j -KL(P_teacher,j || P_π,j)]     (6)

L^t_ranking = Σ_{z^t_k < z^t_r - β} max(p_k - p_r + α, 0)         (2)

Conditional log probability (length-normalized) is calculated leveraging the model π that is being trained:

max_π 𝔼_{s~ρ(.)} 𝔼_{a~π(.|s)} [r(s,a)],                        (5)

Unlike RTLCoder [12], we discard the use of softmax normalization to p based on subsequent experimental results.

Another loss component is the common cross entropy loss relative to the reference sample a_K:

The overall loss function is a weighted sum of both

L_ce = -Σ_i log P_π(a_{K,j} | s, a_{K,<j})                     (3)

The loss function comprises two components: the ranking loss [12, 24, 14] and the Maximum Likelihood Estimation (MLE) loss.

L^t = L_ce + λ L^t_ranking                                    (4)

ITERTL: An Iterative Framework for Fine-tuning LLMs for RTL Code Generation

^1If s is synthesized by another model rather than obtained from real samples, the estimation is also biased. We omit this aspect in this work since it's not the focus of our research.

term associated with the reference data in the ranking loss are also omitted, which does not interfere with our conclusions.

$$
\min_{\pi}(L_{\text{ranking}}) = \min_{\pi}\left[ \sum_{z_k < z_{\tau} - \beta} \max(p_k - p_{\tau} + \alpha, 0) \right]
$$

$$
\simeq \min_{\pi} [\max(p_k - p_{\tau} + \alpha, 0) I(z_k < z_{\tau} - \beta)]
$$

$$
\simeq \max_{\pi} \mathbb{E}_{s \sim \rho(\cdot)} \mathbb{E}_{a_k \sim \pi_c (\cdot | s)} \left\{
\mathbb{E}_{a_{\tau} \sim \pi_c (\cdot | s)} \left[ - \max(p_k - p_{\tau} + \alpha, 0) I(z_k < z_{\tau} - \beta) \right] \right\}
$$

$$
= \max_{\pi} \mathbb{E}_{s \sim \rho(\cdot)} \mathbb{E}_{a \sim \pi_c (\cdot | s)} [r_{\pi}(s, a)]
$$

Here, we denote the surrogate reward function as $r_{\pi}$ to show its dependency on $\pi$. Comparing equations 6 and 7, we find that this approach offers two improvements over the previous naive approach. (1) $\pi_c$ can be adjusted to an appropriate distribution to reduce the mismatch with $\pi$, for instance, selecting the initial distribution $\pi_0$. (2) The surrogate reward function incorporates the evaluation of code quality, which is more reasonable than a simple KL divergence.

Based on the analysis of the above two cases, we can formally reveal the superiority of our method. Using similar notations, our optimization scheme can be formulated as:

$$
\pi_{t+1} = \arg\max_{\pi} \mathbb{E}_{s \sim \rho(\cdot)} \mathbb{E}_{a \sim \pi_t (\cdot | s)} [r_{\pi}(s, a)]
$$

After t-th iteration, $\pi$ is updated, progressively deviating from the previous distribution $\pi_{old}$ ($\pi_0, ..., \pi_{t-1}$). By replacing $\pi_{old}$ with $\pi_t$ for sampling, our method effectively mitigates the distribution mismatch. Furthermore, due to the law of large numbers, increasing the number of sampling promotes better estimation expected values.

we can also analyze the potential bottleneck of our approach from Equation 8. Since the surrogate reward function $r_{\pi}$ evaluates code quality merely from a syntax perspective, ignoring functional correctness and PPA, there is a mismatch between $r_{\pi}$ and the true reward function $r$. As a result, although $r_{\pi}$ may increase with each iteration, the potential improvement for model is still constrained, which is verified by our experiments in Section 4. We anticipate that a better-designed surrogate reward function could further enhance the effectiveness of our approach in the future.

From another perspective, our approach can also be regarded as a degenerate variant of the generalized Expectation-Maximization (EM) algorithm. Consider the following optimization problem:

$$
\max_{\pi} \mathbb{E}_{s \sim \rho(\cdot)} \mathbb{E}_{a \sim \pi (\cdot | s)} [r_{\pi}(s, a)]
$$

Since $\pi$ appears in both the subscript of the expected symbol and the reward function, an EM-style optimization procedure would be like:

$$
\pi_{t+1} = \arg\max_{\pi} \mathbb{E}_{s \sim \rho(\cdot)} \mathbb{E}_{a \sim \pi_t (\cdot | s)} [r_{\pi}(s, a)]
$$

$$
\pi_{t+2} = \arg\max_{\pi} \mathbb{E}_{s \sim \rho(\cdot)} \mathbb{E}_{a \sim \pi (\cdot | s)} [r_{\pi_{t+1}}(s, a)]
$$

Equation 10 corresponds to our optimization approach, while optimizing Equation 11 typically requires reinforcement learning methods like policy gradients, which may increase the complexity and instability of the whole scheme. So we simply set $\pi_{t+2}$ equal to $\pi_{t+1}$ to avoid these issues. We leave the exploration of more precise optimization methods to future researches.

## 4 EXPERIMENTAL EVALUATION

### 4.1 Experimental Setup

**Benchmark:** We choose a comprehensive evaluation dataset, named VerilogEval [10], as the benchmark to measure the performance. This dataset consists of diverse Verilog tasks ranging from simple to complex, such as combinational circuits, finite state machines, code debugging, constructing testbenches, and so on. Two sets of design instructions are provided: the first one is generated by LLM, named **VerilogEval-machine**, containing 143 samples; the other one is manually written, named **VerilogEval-human**, comprising 156 samples. Functional correctness evaluation is conducted via ICARUS Verilog simulator by comparing the outputs of the generated design with that of the golden solution.

**Metric:** As with many code generation researches, pass@k metric [6] is employed to measure the performance, by regard a problem as solved if at least one of the $k$ generated code samples passes the unit tests. Specifically, for each problem, the model generated $n \geq k$ candidates, where $c \leq n$ samples pass the unit tests. The pass@k metric is estimated unbiasedly use the following expression [3]:

$$
\text{pass@}k = \mathbb{E}_{\text{Problems}} \left[1 - \frac{\binom{n-c}{k}}{\binom{n}{k}}\right]
$$

To avoid the impact of randomness, we mainly focus on the pass@1 value under greedy decoding. For a comprehensive evaluation, we also measure the pass@k = {1, 5, 10} metrics setting n = 10 under Top-p decoding.

**Decoding Strategy:** During the training stage, in order to enhance the diversity and promote the exploration, we use Top-p decoding strategy with $top_p = 0.95$ and $temperature = 0.5$. In the testing stage, to mitigate random errors, the model is prompted to generate responses with $temperature = \{0 (\text{greedy decoding}), 0.2, 0.5, 0.8\}$ and $top_p = 0.95$. For each test metric (pass@1, pass@5, and pass@10), the best result is chosen.

**Training Details:** DeepSeek-Coder-Instruct-6.7B [5] is chosen as the pre-trained model. We use the open-source instruction-code pairs from [12] as reference data. This dataset contains nearly 27k samples. We randomly sample only 10,000 entries to train the final version of our model. The learning rate is set to $10^{-5}$. AdamW optimizer is employed with $\beta_1 = 0.9$ and $\beta_2 = 0.999$. The bp16 Mixed-precision Training is adopted to avoid overflow. During each iteration, the model is trained for 3 epochs. The total number of iterations is set to 7. We conduct all experiments on 4 NVIDIA A800 GPUs. As for parameters mentioned in Section 3.1, we set the number of candidates $K = 4$. The hyper-parameters $\alpha$ and $\beta$ in Equation 2 are set to 0.3 and 0.2, respectively.

### 4.2 Comparison to State-of-the-Art Methods

As Table 1 shows, our model reach the state-of-the-art level among open-source models. On Pass@1 metric, our model trained with only 10k reference samples surpasses RTLCoder with 27k reference.

Table 1: Performance on VerilogEval Benchmark[10]. The results of GPT-3.5, GPT-4, VerilogEval, Codegen and VeriGen come from [10]. The results of RTLCoder come from [12]. The best results are marked in bold. If the best result comes from a closed-source model (like GPT-4), then the best result from the open-source model is also highlighted in bold. Additionally, the second best result is underlined.

samples by 1% and 1.3% on VerilogEval-machine and VerilogEval-human respectively. Leveraging an equal amount of reference samples, our model significantly outperforms RTLCoder-DeepSeek-10k on all metrics, with a relative improvement of 12.5% and 16.9% in pass@1 especially. Considering the size of the parameters, with only 6.7 billion parameters, our model significantly outperforms VerilogEval[10] with 16 billion parameters and a similar volume of reference samples (8502), further proving the efficiency and superiority of our method. Additionally, we incorporated a general-purpose code model, Codegen[16], into our comparison. Its subpar performance on the VerilogEval benchmark underscores the importance of tailoring LLMs for RTL code.

Relative to closed-source models, our model comprehensively surpasses GPT-3.5 on the benchmark. Against GPT-4, our model performs better on VerilogEval-machine but lags on VerilogEval-human. Considering the vast differences in training costs and the lightweight advantage of our model, these performance deficits are acceptable.

To delve deeper into the impact of iteration, we plot how pass@1 on VerilogEval-human varies with the number of iterations in Figure 2. It can be clearly seen that, with 5 iterations and only 10k reference data, our model beats the baseline model with with 27k reference data. From the first to the fifth iteration, the pass@1 rate exhibits a clear upward trend, indicating the efficacy of iterative training. From the fifth to the seventh iteration, the pass@1 rate gradually decreases, which can be explained by the mismatch between the surrogate reward function and the true reward function mentioned in Section 3.2. In fact, similar discoveries were found on LLMs trained using reward models on natural languages. Our work reveals that the surrogate reward function based on code quality evaluation and reward model on natural languages have similar properties, providing inspiration for subsequent improvement work.

Figure 3 depicts different loss functions curves across each iteration. Even though the loss function converges within each round, the loss value can still decrease by leveraging newly sampled data points from updated model, validating the effectiveness of the proposed iterative training approach. Another observed phenomenon is that as the number of iterations increases, the marginal decrease in the loss function gradually diminishes, which indicates that the surrogate reward function is nearing its optimization ceiling.

The baseline method[12] employ a softmax normalization for conditional log probability after Equation 1:

$$ p'_k = \frac{e^{p_k}}{\sum_{i=1}^K e^{p_i}} \qquad (13) $$

We conduct an ablation study to investigate its impact in iteration training. All other settings remain unchanged except for softmax normalization. From Figure 4, it can be observed that the pass@1 of the approach with softmax shows smoother variations across iterations, which may be caused by vanishing gradients brought by

Reviewing previous theoretical and experimental analyses, we believe that exploring two directions could further enhance the capability of LLMs. The first is to find a more suitable surrogate reward mechanism to provide feedback. Current methods simply rely on evaluating similarity or syntax checking, which is too superficial to reflect the true quality of RTL code. A more comprehensive reward mechanism is likely to further unleash the potential of training methods.

Another direction is to develop more advanced optimization algorithms. In this work, to enhance simplicity and usability, we avoid softmax saturation. We choose to discard softmax to maximize performance. While the softmax normalization can also be considered when a more stable and smooth training process is required.

using complex reinforcement learning methods, which might better optimize models with careful tuning of hyper-parameters. Drawing inspiration from Direct Preference Optimization (DPO) [18] in natural language generation, we look forward to seeing both simpler and more advanced optimization solutions in the RTL code generation domain in the future.

Conference acronym 'XX, June 03–05, 2018, Woodstock, NY

summarize with human feedback. Advances in Neural Information Processing Systems, 33, 3008–3021.

Received 20 February 2007; revised 12 March 2009; accepted 5 June 2009


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
We begin by recalling some notations and notions. Let κ be a number field and κ̄ an algebraic closure of κ. Let O_κ denote the ring of integers of κ. Let M_κ be the canonical set of distinct inequivalent valuations of κ satisfying the product formula

\[
\prod_{v \in M_\kappa} \|x\|_v = \prod_{v \in M_\kappa} |x|_v^{n_v} = 1, \ x \in \kappa^* = \kappa - \{0\}.
\]

Here \(\| \cdot \|_v\) is normalized by its absolute value \(| \cdot |_v\) as follows. A non-Archimedean place \(v \in M_\kappa\) corresponds to a nonzero prime ideal \(\mathfrak{p} = \mathfrak{p}_v \subseteq O_\kappa\), and we set

\[
\|x\|_v = |x|_v^{n_v} = \mathcal{N}(\mathfrak{p})^{-v_\mathfrak{p}(x)},
\]

where \(v_\mathfrak{p} = \operatorname{ord}_v\) is the valuation defined by \(\mathfrak{p}\), \(\mathcal{N}(\mathfrak{p})\) is the absolute norm of \(\mathfrak{p}\), i.e., the number of residue classes mod \(\mathfrak{p}\), and the multiplicity \(n_v = [\kappa_v : \mathbb{Q}_p]\) is the local degree of \(v\) if \(v|p\) for some \(p \in M_\mathbb{Q}\) such that \(\sum_{v|p} n_v = [\kappa : \mathbb{Q}]\). If \(v\) is Archimedean, then \(v\) corresponds to a real embedding \(\sigma : \kappa \to \mathbb{R}\) or a complex conjugate pair of complex embeddings \(\sigma, \bar{\sigma} : \kappa \to \mathbb{C}\), and we set \(\|x\|_v = |x|_v = |\sigma(x)|\) or \(\|x\|_v = |x|_v^2 = |\sigma(x)|^2\). Let \(M_\kappa^0\) be the set of non-Archimedean places in \(M_\kappa\) and \(M_\kappa^\infty = M_\kappa - M_\kappa^0\) be the set of all Archimedean places. Let \(S\) be a finite subset of \(M_\kappa\) containing the subset of all Archimedean valuations in \(M_\kappa\).

If \( v \) is non-Archimedean, set
\[
\chi_v(x, a) = \begin{cases}
\frac{|x - a|_v}{|x|_v \vee |a|_v} & : x, a \in \kappa \\
\frac{1}{|x|_v} & : a = \infty,
\end{cases}
\]
where, by definition,
\[
r^\vee = \max\{1, r\} \quad (r \in \mathbb{R}).
\]

If \( v \) is Archimedean, set
\[
\chi_v(x, a) = \begin{cases}
\frac{|x - a|_v}{(1 + |x|_v^2)^{1/2}(1 + |a|_v^2)^{1/2}} & : x, a \in \kappa \\
\frac{1}{\sqrt{1 + |x|_v^2}} & : a = \infty.
\end{cases}
\]

The proximity function for \( a \) is defined by
\[
m(x, a) = \frac{1}{[\kappa : \mathbb{Q}]} \sum_{v \in S} \log \frac{1}{\chi_v(x, a)^{n_v}}
\]
and similarly the valence function for \( a \)
\[
N(x, a) = \frac{1}{[\kappa : \mathbb{Q}]} \sum_{v \in M_\kappa - S} \log \frac{1}{\chi_v(x, a)^{n_v}}.
\]

Fix \( a, x \in \kappa \). Obviously, there exists a constant \( C \) depending only on \( |a|_v \), such that
\[
\max \left\{ 1, \frac{1}{|x - a|_v} \right\} \leq \frac{1}{\chi_v(x, a)} \leq C \max \left\{ 1, \frac{1}{|x - a|_v} \right\}.
\]

Thus we have
\[
m(x, a) = \frac{1}{[\kappa : \mathbb{Q}]} \sum_{v \in S} \log^+ \frac{1}{\| x - a \|_v} + O(1)
\]
and similarly,
\[
N(x, a) = \frac{1}{[\kappa : \mathbb{Q}]} \sum_{v \in M_\kappa - S} \log^+ \frac{1}{\| x - a \|_v} + O(1),
\]
where by definition,
\[
\log^+ r = \log r^\vee = \max\{0, \log r\} \quad (r \in \mathbb{R}_+).
\]

The relative multiplicative heights of an element \( x \) of the number field \( \kappa \) is defined by
\[
H_\kappa(x) = \left( \prod_{v \in M_\kappa^\infty} \left( \sqrt{1 + |x|_v^2} \right)^{n_v} \right) \left( \prod_{v \in M_\kappa - M_\kappa^\infty} \max\{1, |x|_v^{n_v}\} \right),
\]
\[
H_{*, \kappa}(x) = \prod_{v \in M_\kappa} \max\{1, |x|_v^{n_v}\},
\]
and the absolute (logarithmic) height \( h(x) \) by
\[
h(x) = \frac{1}{[\kappa : \mathbb{Q}]} \log H_\kappa(x).
\]

S. Lang [4] observed that there is no reason not to let \( x \) approach infinity; for example, Roth's Theorem can be changed to the following form (see [3], Theorem 6.14):

Theorem 1.1. Let κ be a number field and S ⊂ M_κ a finite subset of absolute values on κ. Assume that each absolute value in S has been extended in some way to κ̄. Let a₁, ..., a_q be distinct elements in κ̄ and ε a positive constant. Then there are only finitely many x ∈ κ such that

$$
\prod_{v \in S} \left( \min\left\{1, \frac{1}{\|x\|_v}\right\} \prod_{j=1}^q \min\{1, \|x - a_j\|_v\} \right) < \frac{1}{H_{*,κ}(x)^{2+\epsilon}}.
$$
                                                                    (3)

Next, assume M_κ^∞ ⊆ S. Without loss of generality, we may assume a_j ∈ κ for j ≥ 1. The inequality (3) can be rewritten into the following form:

$$
(q - 1) h(x) \leq N(x, \infty) + \sum_{j=1}^q N(x, a_j) + \epsilon h(x) + O(1),
$$
                                                                    (4)

where

$$
N(x, a_j) = \frac{1}{[\kappa : \mathbb{Q}]} \sum_{v \in S^c(x-a_j)} \operatorname{ord}_v(x - a_j) \log N({\mathfrak{p}}_v) + O(1)
$$
                                                                    (5)

for each j = 1, ..., q, and

$$
N(x, \infty) = \frac{1}{[\kappa : \mathbb{Q}]} \sum_{v \in S^c(x^{-1})} \operatorname{ord}_v(x^{-1}) \log N({\mathfrak{p}}_v) + O(1),
$$
                                                                    (6)

where

$$
S^c(y) = \{ v \in M_\kappa - S \mid \operatorname{ord}_v(y) > 0 \}.
$$

Define

$$
\overline{N}(x, a_j) = \frac{1}{[\kappa : \mathbb{Q}]} \sum_{v \in S^c(x-a_j)} \log N({\mathfrak{p}}_v)
$$
                                                                    (7)

and

$$
\overline{N}(x, \infty) = \frac{1}{[\kappa : \mathbb{Q}]} \sum_{v \in S^c(x^{-1})} \log N({\mathfrak{p}}_v).
$$
                                                                    (8)

It is a simple fact (see [3], p. 358) that the following Conjecture 1.2, which strengthens the inequality (4), implies the following Conjecture 1.3, the abc-conjecture (see [7], [9]).

Conjecture 1.2. Let a₁, ..., a_q be distinct elements in κ̄ and ε a positive constant. All but finitely many x ∈ κ satisfy the inequality

$$
(q - 1) h(x) \leq \overline{N}(x, \infty) + \sum_{j=1}^q \overline{N}(x, a_j) + \epsilon h(x) + O(1).
$$
                                                                    (9)

Conjecture 1.3. Given ε > 0, there exists a number C(ε) having the following property. For any nonzero relatively prime integers a, b, c such that a + b = c, we have

$$
\max \{ |a|, |b|, |c| \} \leq C(\epsilon) r(abc)^{1+\epsilon},
$$
                                                                    (10)

where r(abc) is the radical of abc defined by r(abc) = ∏_{p | abc} p, i.e. the product of distinct primes dividing abc.

In this paper, we will prove Conjecture 1.2 for algebraic integers, that is, we obtain the following result:

Theorem 1.4. Let \( a_1, \ldots, a_q \) be distinct elements in \( \overline{\kappa} \). All but finitely many algebraic integers \( x \in \kappa \) satisfy the inequality

\[
(q-1) h(x) \leq \sum_{j=1}^{q} N(x, a_j) + O(1).
\]
(11)

Theorem 1.5. Let \( a_1, a_2 \) be distinct elements in \( \overline{\kappa} \). All but finitely many simple numbers \( x \in \kappa \) satisfy the inequality

\[
h(x) \leq N(x, \infty) + N(x, a_1) + N(x, a_2) + O(1).
\]
(12)

In Theorem 1.5, a number \( x \in \kappa \) is called simple if negative powers of ideals occurred in \( x \) are all \(-1\). Theorem 1.5 immediately yields a special case of the \( abc \)-conjecture as follows:

Theorem 1.6. For any nonzero relatively prime integers \( a, b, c \) such that \( a + b = c \) and that one of \( a, b, c \) only has prime factors of power 1, we have

\[
\max\{ |a|, |b|, |c| \} \leq C r(abc),
\]
(13)

where \( C \) is an absolute constant.

Some results that would follow from the \( abc \)-conjecture can be found in [1], [8], pp. 185-188, [14]; see also [2], [5], [6], [15]. In [10], C. L. Stewart and R. Tijdeman proved that

\[
\max\{ |a|, |b|, |c| \} < \exp\left\{ C r(abc)^{15} \right\},
\]

where \( C \) is an absolute constant. In [11], C. L. Stewart and K. Yu obtained that

\[
\max\{ |a|, |b|, |c| \} < \exp\left\{ C(\varepsilon) r(abc)^{2/3+\varepsilon} \right\}.
\]

In [12], C. L. Stewart and K. Yu further proved that

\[
\max\{ |a|, |b|, |c| \} < \exp\left\{ C(\varepsilon) r(abc)^{1/3+\varepsilon} \right\}.
\]

## 2 Proofs of Theorem 1.4 and Theorem 1.5

We consider the following rational function

\[
Q(X) = \frac{1}{(X-a_1)\cdots(X-a_q)} = \sum_{j=1}^{q} \frac{A_j}{X-a_j},
\]

where

\[
A_j = \frac{1}{(a_j-a_1)\cdots(a_j-a_{j-1})\cdot(a_j-a_{j+1})\cdots(a_j-a_q)}.
\]

Theorem 1.4 and Theorem 1.5 will follow from the following result:

**Theorem 2.1.** Let \( a_1, \ldots, a_q \) be distinct elements in \( \kappa \) and \( \varepsilon \) a positive constant. There exist an extension field \( K \) of \( \kappa \) and an element \( x' \in K \) such that all but finitely many \( x \in \kappa \) satisfy the inequality

\[
(q-1)h(x) \leq \overline{N}(x, \infty) + \sum_{j=1}^q \overline{N}(x, a_j) + 2m\left\lceil x' Q(x), \infty \right\rceil + O(1). \tag{14}
\]

Proof. Set

\[
\rho_v = \min_{1 \leq j \leq q} |A_j|_v, \quad \sigma_v = \max_{1 \leq j \leq q} |A_j|_v, \quad \delta_v = \min_{1 \leq i < j \leq q} |a_i - a_j|_v,
\]
\[
E_{vj} = \left\{ x \in \kappa \,\middle|\, |x - a_j|_v < \frac{\delta_v}{2\varkappa_v} \right\},
\]

where

\[
\varkappa_v = \frac{1}{2} + (q-1)\frac{\sigma_v}{\rho_v}.
\]

When \( i \neq j \), \( x \in E_{vj} \), we have

\[
|x - a_i|_v \geq |a_i - a_j|_v - |x - a_j|_v \geq \delta_v \left(1 - \frac{1}{2\varkappa_v}\right) \geq \frac{\delta_v}{2\varkappa_v}.
\]

Since

\[
Q(x) = \frac{A_j}{x - a_j} \left\{1 + \sum_{i \neq j} \frac{A_i}{A_j} \cdot \frac{x - a_j}{x - a_i}\right\},
\]

we deduce that

\[
|Q(x)|_v > \frac{|A_j|_v}{|x - a_j|_v} \left\{1 - (q-1) \frac{\frac{\delta_v}{2q}}{\delta_v \left(1 - \frac{1}{2q}\right)}\right\} \geq \frac{\rho_v}{2|x - a_j|_v}
\]

and thus that

\[
\log^+ \|Q(x)\|_v > \log^+ \frac{1}{\|x - a_j\|_v} - n_v \log \frac{2}{\rho_v}
\]
\[
\geq \sum_{i=1}^q \log^+ \frac{1}{\|x - a_i\|_v} - q n_v \log^+ \frac{2\varkappa_v}{\delta_v} - n_v \log \frac{2}{\rho_v}.
\]

Obviously, this inequality is also true if \( x \not\in \cup_i E_{vi} \). Thus we obtain

\[
m(Q(x), \infty) \geq \sum_{j=1}^q m(x, a_j) - C_S,
\]

where

\[
C_S = \frac{1}{[\kappa : \mathbb{Q}]} \sum_{v \in S} \left( q n_v \log^+ \frac{2\varkappa_v}{\delta_v} + n_v \log \frac{2}{\rho_v} \right).
\]

On the other hand, for some \( x' \in K^* \), where \( K \) is an extension field of \( \kappa \) (see below for its construction), we have

\[
m(Q(x), \infty) = \frac{1}{[K : \mathbb{Q}]} \sum_{w \in S_K} \log^+ \|Q(x)\|_w + O(1)
\]
\[
= \frac{1}{[K : \mathbb{Q}]} \sum_{v \in S} \sum_{w|v} \log^+ \|Q(x)\|_v^{[K_w\!:\!\kappa_v]} + O(1)
\]
\[
= \frac{1}{[\kappa : \mathbb{Q}]} \sum_{v \in S} \log^+ \|Q(x)\|_v + O(1),
\]

where \( S_K \) is the set of \( w \in M_K \) such that \( w|v \) for some \( v \in S \) so that

\[
m(Q(x), \infty) \leq m(x' Q(x), \infty) + m(x', 0)
\]
\[
\leq m(x' Q(x), \infty) + h(x') - N(x', 0) + O(1).
\]

Hence for each \( j \in \{1, \ldots, q\} \), we have

\[
h(x') = m(x', \infty) + N(x', \infty)
\]
\[
\leq m(x - a_j, \infty) + N(x', \infty) + m\left(\frac{x'}{x - a_j}, \infty\right) + O(1)
\]
\[
\leq h(x) + N(x', \infty) - N(x, \infty) + m\left(\frac{x'}{x - a_j}, \infty\right) + O(1).
\]

Therefore

\[
m(x, \infty) + \sum_{j=1}^{q} m(x, a_j) \leq 2h(x) - N_{x'}(x) + S_{j,x'}(x) + O(1), \tag{15}
\]

where

\[
N_{x'}(x) = 2N(x, \infty) - N(x', \infty) + N(x', 0),
\]

\[
S_{j,x'}(x) = m\left(\frac{x'}{x - a_j}, \infty\right) + m(x' Q(x), \infty),
\]

and hence

\[
(q-1)h(x) \leq N(x, \infty) + \sum_{j=1}^{q} N(x, a_j) - N_{x'}(x) + S_{x'}(x) + O(1), \tag{16}
\]

where

\[
S_{x'}(x) = \frac{1}{q} \sum_{j=1}^{q} S_{j,x'}(x) = \frac{1}{q} \sum_{j=1}^{q} m\left(\frac{x'}{x - a_j}, \infty\right) + m(x' Q(x), \infty).
\]

We claim that there exists a constant \( c_v \) satisfying

\[
c_v \prod_{j=1}^{q} \left| \frac{x'}{x - a_j} \right|_{v+} \leq \left| \frac{x'}{(x - a_1) \cdots (x - a_q)} \right|_{v+}^q, \tag{17}
\]

where \( x_+ = x^\vee = \max\{1, x\} \). When \( |x - a_i|_v \leq \delta_v/2 \) for some \( i \), then for \( j \neq i \)

\[
|x - a_j|_v \geq |a_j - a_i|_v - |x - a_i|_v \geq \delta_v/2,
\]
and
\[
|x - a_j|_v \leq |x - a_i|_v + |a_i - a_j|_v \leq \delta_v/2 + \lambda_v,
\]
where
\[
\lambda_v = \max_{i,j} |a_i - a_j|_v.
\]

Hence we have either

\[
\prod_{j=1}^q \left| \frac{x'}{x - a_j} \right|_{v+} \leq \left| \frac{x'}{x - a_i} \right|_{v+}
\]
when \( |x'|_v \leq \delta_v/2 \), or
\[
\prod_{j=1}^q \left| \frac{x'}{x - a_j} \right|_{v+} \leq \left( \frac{2|x'|_v}{\delta_v} \right)^{q-1} \left| \frac{x'}{x - a_i} \right|_v \leq \left| \frac{x'}{x - a_i} \right|_v^q
\]

if \( |x'|_v > \delta_v/2 \).
Note that

\[
\left| \frac{x'}{(x - a_1)\cdots(x - a_q)} \right|_v \geq \left( \frac{1}{\delta_v/2 + \lambda_v} \right)^{q-1} \left| \frac{x'}{x - a_i} \right|_v.
\]

Thus when \( |x'|_v \leq \delta_v/2 \), there exists a constant \( c_v \) such that

\[
c_v \prod_{j=1}^q \left| \frac{x'}{x - a_j} \right|_{v+} \leq \left| \frac{x'}{(x - a_1)\cdots(x - a_q)} \right|_{v+},
\]
so that (17) follows. If \( |x'|_v > \delta_v/2 \), then

\[
\left| \frac{x'}{(x - a_1)\cdots(x - a_q)} \right|_v^q \geq \left( \frac{1}{\delta_v/2+\lambda_v} \right)^{(q-1)q} \prod_{j=1}^q \left| \frac{x'}{x-a_j} \right|_{v+}
\]
which also yields the estimate (17).

If \( |x - a_i|_v > \delta_v/2 \) for all \( i \), then

\[
\prod_{j=1}^q \left| \frac{x'}{x - a_j} \right|_{v+} \leq \left( \frac{2}{\delta_v} \right)_+^q \leq \left( \frac{2}{\delta_v} \right)_+^q \left| \frac{x'}{(x - a_1)\cdots(x - a_q)} \right|_{v+},
\]

and hence (17) follows again.

Hence, we obtain the estimate

\[
S_{x'}(x) \leq 2m (x'Q(x), \infty) + O(1), \tag{18}
\]

which implies by (16) that

\[
(q - 1)h(x) \leq \overline{N}(x, \infty) + \sum_{i = 1}^q \overline{N}(x, a_j) + 2m (x'Q(x), \infty) + O(1). \tag{19}
\]

It is easy to show that if (11) is true for all algebraic integers \( a_j \), then it is true for all algebraic numbers \( a_j \) so that we may assume that all \( a_j \) are algebraic integers. We may construct the above \( x' \) so that it is in \( \kappa \) but it would be hard to compute the term \( m(x'Q(x), \infty) \) (see Remark below). We turn to use the following result in [3], Theorem 2.32 (see also [13]): For a number field \( \kappa \), there is a number field \( K \supset \kappa \) such that for each ideal \( a \) in the ring of integers \( \mathcal{O}_\kappa \) of \( \kappa \), it holds that

(I) \( \mathcal{O}_K a \) is a principal ideal;

(II) \( (\mathcal{O}_K a) \cap \mathcal{O}_\kappa = a \).

Using this result we obtain an extension field \( K \supset \kappa \) such that for each ideal \( a \) in the ring of integers \( \mathcal{O}_\kappa \) of \( \kappa \), (I) and (II) hold. Thus there exist \( x_0, x_\infty \in \mathcal{O}_K \) such that

\( (x_0) = \mathcal{O}_K a_0, \, (x_\infty) = \mathcal{O}_K a_\infty, \)

and

\( (\mathcal{O}_K a_0) \cap \mathcal{O}_\kappa = a_0, \, (\mathcal{O}_K a_\infty) \cap \mathcal{O}_\kappa = a_\infty. \)

We then take \( x' = x_0 / x_\infty \in K \). This completes the proof. \(\Box\)

Remark. As noted in the above proof, \( x' \) can be constructed so that it belongs to \( \kappa \) (it would however be hard to compute the term \( m(x' Q(x), \infty) \)). To see this, write

\[
(x) = P_1^{t_1} \cdots P_l^{t_l} h^{-1}, \quad h = Q_1^{u_1} \cdots Q_h^{u_h} Q_{h+1}^{u_{h+1}} \cdots Q_{h+g}^{u_{h+g}},
\]

where \( t_i, u_j \) are positive integers, and

\( Q_i \in M_\kappa - S \) (i = 1, ..., h); \( Q_{h+j} \in S \) (j = 1, ..., g).

Similarly, we can write

\[
(x - a_j) = h^{-1}
p_{m_{j-1}+1}^{r_{m_{j-1}+1}} \cdots p_{m_j}^{r_{m_j}}
q_{n_{j-1}+1}^{s_{n_{j-1}+1}} \cdots q_{n_j}^{s_{n_j}},
\]

where \( r_i, s_j \) are positive integers, \( m_0 = n_0 = 0 \), and

\( p_i \in M_\kappa - S \) (i = 1, ..., \( m_q \)); \( q_j \in S \) (j = 1, ..., \( n_q \)).

We first assume that \( p_1, ..., p_{m_q} \) are distinct. Write \( a_0 = \prod_{i=1}^{m_q} p_i^{r_i - 1} \), and further define ideals \( d_i \) by

\[
p_i^{r_i} d_i = a_0 p_1 \cdots p_{m_q}, \quad i = 1, 2, ..., m_q,
\]

so that \( d_i \) is relatively prime to \( p_i \). Since these \( d_i \) in their totality are relatively prime, there are elements \( \delta_i \in d_i \) satisfying

\[
\delta_1 + \delta_2 + \cdots + \delta_{m_q} = 1.
\]

Since \( d_i | \delta_i \), hence \( p_j | \delta_i \) (\( j \neq i \)). Consequently, \( p_i \nmid \delta_i \) since \( p_i \nmid (1) \). We now determine elements \( \alpha_i \) such that

\[
p_i^{r_i - 1} \mid \alpha_i, \quad p_i^{r_i} \nmid \alpha_i, \quad i = 1, ..., m_q,
\]

which is obviously always possible since for this to happen αᵢ needs only to be an element from pᵣⁱ⁻¹ᵢ which does not occur in pʳⁱᵢ. Then the element

x₀ = α₁δ₁ + α₂δ₂ + ⋯ + α_{m_q}δ_{m_q}

has the property a₀ | x₀. For each of the prime ideals pᵢ occurs in m_q − 1 summands at least to the power pʳⁱᵢ; however, it occurs precisely to the power pʳⁱ⁻¹ᵢ in the i-th summand; consequently x₀ is divisible by precisely the (rᵢ − 1)-th power of pᵢ, but no higher power. If p₁, ..., p_{m_q} are not distinct, say p₁ = p₂, but p₂, ..., p_{m_q} are distinct, replace δ₂ by

p₁^{r₁} p₂^{r₂} δ₂ = a₀ p₁ ⋯ p_{m_q}

and determine the element α₂ such that

p₁^{r₁+r₂-2} | α₂,  p₁^{r₁+r₂-1} ∤ α₂.

Then the element x₀ is replaced by

x₀ = α₂δ₂ + α₃δ₃ + ⋯ + α_{m_q}δ_{m_q}.

Similarly, if we define

a_∞ = Q₁^{u₁+1} ⋯ Q_h^{u_h+1},

then we can find an element x_∞ such that when Q₁, ..., Q_h are distinct, each of the prime ideals Qᵢ occurs in h − 1 summands at least to the power Qᵢ^{uᵢ+2}; however, it occurs precisely to the power Qᵢ^{uᵢ+1} in the i-th summand; consequently x_∞ is divisible by precisely the (uᵢ + 1)-th power of Qᵢ, but no higher power. We then take x' ∈ κ_* satisfying x' = x₀ / x_∞.

Proof of Theorem 1.4: Note that the arguments in the above proof are true over K. Hence if x is an algebraic integer, then y = 1 / x'Q(x) is an algebraic integer, so that

m ( x' Q(x), ∞ ) = m(y, 0) = h(y) - N(y,0) + O(1)
= m(y,∞) - N(y,0) + O(1).

By using Dirichlet's unit theorem (see Theorem 2.36 and Lemma 4.3 in [3]), there exists a constant c(κ) such that |y|_v ≤ c(κ)|y|_w for any Archimedean v, w. Now we choose v ∈ M_κ^∞ such that

1 ≤ |y|_v = max_{w ∈ M_κ^∞} |y|_w.

Then we have

m(y, 0) = ∑_{w ∈ S} log^+ (1 / ‖y‖_w) + O(1) = O(1).

Thus we have

(q - 1) h(x) ≤ N(x, ∞) + ∑_{j=1}^q N(x, a_j) + O(1), j = 1, ..., q.  (20)

□

Proof of Theorem 1.5: Since x is simple, y = 1 / (x' Q(x)) is an algebraic integer, where q = 2.
The proof can be completed in the same way as for Theorem 1.4.
□

Since one of \( a, b, c \) only has prime factors of power 1, say, \( c \) does, taking an \( abc \)-point \( y \in \mathbb{P}^2(\kappa) \) with a reduced representation \( (a, b, c) \in \mathcal{O}_\kappa^3 \) and applying (12) to \( x = a/c \), we obtain

\( h\left(\frac{a}{c}\right) \leq \overline{N}\left(\frac{a}{c}, 0\right) + \overline{N}\left(\frac{a}{c}, -1\right) + \overline{N}\left(\frac{a}{c}, \infty\right) + O(1), \)      (21)

Since \( a + b + c = 0 \), and the elements \( a, b, c \) are relatively prime, we obtain

\[
\begin{aligned}
\overline{N}\left(\frac{a}{c}, 0\right) &= \frac{1}{[\kappa : \mathbb{Q}]} \sum_{v \in S^c(a)} \log \mathcal{N}(p_v), \\
\overline{N}\left(\frac{a}{c}, -1\right) &= \frac{1}{[\kappa : \mathbb{Q}]} \sum_{v \in S^c(b)} \log \mathcal{N}(p_v), \\
\overline{N}\left(\frac{a}{c}, \infty\right) &= \frac{1}{[\kappa : \mathbb{Q}]} \sum_{v \in S^c(c)} \log \mathcal{N}(p_v).
\end{aligned}
\]

Thus (21) becomes

\( h\left(\frac{a}{c}\right) \leq \overline{N}(y, E) + O(1), \)      (22)

where the definition of \( E \) is referred to Section 5.6.1 in [3]. Similarly, we can obtain

\( h\left(\frac{b}{c}\right) \leq \overline{N}(y, E) + O(1), \)      (23)

It is easy to show that

\( h(y) = \max \left\{ h\left(\frac{a}{c}\right), h\left(\frac{b}{c}\right) \right\} + O(1). \)

Combining (22) and (23), we obtain

\( h(y) \leq \overline{N}(y, E) + O(1), \)      (24)

and (13) thus holds. □

[5] Lang, S., Old and new conjectured Diophantine inequalities, Bull. Amer. Math. Soc. 23 (1990), 37-75.

[6] Lang, S., Algebra, revised third edition, Springer-Verlag, New York, 2002.

[7] Masser, D. W., Open problems, Proc. Symp. Analytic Number Th., W. W. L. Chen (ed), London: Imperial College, 1985.

[8] Nathanson, M. B., Elementary methods in number theory, Springer-Verlag, New York, 2000.

[9] Oesterlé, J., Nouvelles approches du "théorème" de Fermat (New approaches to Fermat's last theorem), Semin. Bourbaki, 40ème Année, Vol. 1987/88, Exp. No.694, Astérisque 161/162 (1988), 165-186.

[10] Stewart, C. L. and Tijdeman, R., On the Oesterlé-Masser conjecture, Monatsh. Math. 102 (1986), 251-257.

[11] Stewart, C. L. and Yu, K., On the abc conjecture, Math. Ann. 291(2) (1991), 225-230.

[12] Stewart, C. L. and Yu, K. On the abc conjecture. II, Duke Math. J. 108 (1) (2001), 169-181.

[13] Stewart, I. and Tall, D., Algebraic number theory and Fermat's last theorem, A K Peters, Ltd., Natick, Massachusetts, 2002.

[14] van Frankenhuysen, M., The abc conjecture implies Vojta's height inequality for curves, J. Number Theory 95 (2002), no. 2, 289–302.

[15] Vojta, P., Diophantine approximation and value distribution theory, Lecture Notes in Math. 1239, Springer, 1987.

Department of Mathematics, Shandong University, Jinan 250100, Shandong, P. R. China
e-mail: pchu@sdu.edu.cn

Department of Mathematics, Florida International University, Miami, FL 33199 USA
E-mail: libaoqin@fiu.edu


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
Density matrices are the most general descriptions of quantum states, covering both pure and mixed states. Positive semidefiniteness is a physical requirement of density matrices, imposing nonnegative probabilities of measuring physical values. Separately, nonlocality is a property shared by some bipartite quantum systems, indicating a correlation of the component parts that cannot be described by local classical variables. In this work, we show that breaking the positive-semidefinite requirement and allowing states with a negative minimal eigenvalue arbitrarily close to zero, allows for the construction of states that are nonlocal under one component labelling but local when the labelling is interchanged. This is an observer-dependent nonlocality, showing the connection between nonlocal objectivism and negative quasi-probabilities.

A 2-qubit density matrix ρ is a 4 × 4 matrix describing a (generally mixed) quantum system composed of two qubits. They can be written as a linear combination of 2-qubit spin operators including the identity:

\[
\rho = \frac{1}{4} \left( I_4 + \sum_{i,j} c_{ij} \sigma_i \otimes \sigma_j \right)
\tag{1}
\]

†Laboratoire Interdisciplinaire Carnot de Bourgogne, ICB/UTBM,
UMR 6303 CNRS, Université de Technologie de Belfort-Montbéliard,
90010 Belfort Cedex, France

July 31, 2024

Colm Kelleher†

where \( c_{ij} \in \mathbb{R} \), \( \sigma_i \) refer to 1-qubit Pauli spin operators hereafter labelled \( X, Y, Z, I \), and the tensor products of two identities \( I_4 \) is singled out with coefficient \( \frac{1}{4} \). Density states describing physical systems obey the following two conditions:

\[
\text{Tr}(\rho) = 1, \quad \lambda(\rho) \geq 0 \tag{2}
\]

where one also has \( \text{Tr}(\rho^2) \leq 1 \), with the equality saturated for pure states. The second condition above is the statement that \( \rho \) is positive semidefinite, in that all of its eigenvalues \( \lambda(\rho) \) are nonnegative. As 2-qubit operators are hermitian, \( \rho \) is hermitian by construction.

Nonlocality is a quantum-specific property that some bipartite systems have, indicating correlations that cannot be described by classical "hidden" variables. It was first described by Bell in 1964 [1], with an experimentally testable inequality discovered in 1966 [2]. Then, in 1995 the Horodecki brothers gave an equivalent mathematical statement [3] for nonlocality for 2-qubit density matrices. The statement is as follows:

**Theorem 1.1** (Horodecki). A 2-qubit density matrix \( \rho \) describes a nonlocal state iff the matrix \( \beta \) of coefficients of maximally mixed operators obeys the following:

\[
\mathcal{M} > 1 \tag{3}
\]

where \( \mathcal{M} = u + \bar{u} \), the two largest eigenvalues of \( \beta^T \beta \).

The matrix \( \beta \) of maximally mixed coefficients is the matrix whose entries are \( c_{ij} \) from (1) where neither \( \sigma_i \) nor \( \sigma_j \) are the identity operator.

In this work, we examine nonlocal conditions on a certain subset of states. We introduce the following definition:

**Definition 1.2** (Qubit-swapping). For a given density matrix \( \rho = \frac{1}{4}\left(I + \sum_{ij} c_{ij}\sigma_i \otimes \sigma_j\right) \), we define the *qubit-swapped* state \( \rho' \) by interchanging 1-qubit Pauli matrices without changing the associated coefficients:

\[
\rho' = \frac{1}{4}\left(I + \sum_{ij} c_{ij}\sigma_j \otimes \sigma_i\right) \tag{4}
\]

Definition 1.2 amounts to relabelling the qubits \( A \otimes B \mapsto B \otimes A \). One would expect that a simple relabelling would not affect the physical properties of a system described by \( \rho \). However one can define states where nonlocality is dependent on the operator-order labelling:

Definition 1.3. A swap-intolerant nonlocal (SIN) state is a state ρ that is nonlocal, but whose qubit-swapped state ρ' is local.

Now we can move on to the main theorem of this article:

Theorem 1.4. For a given λ₀ < 0 arbitrarily close to zero, one can construct a non-positive-semidefinite density state ρ whose smallest eigenvalue is λ₀, such that it is swap-intolerant nonlocal.

Theorem 1.4 shows that even arbitrary breaking of positive-semidefiniteness allows for unphysical interpretations of nonlocality. It will be proven by construction later in this work. First we must build up a particular type of state called perp states.

Perp states are a subset of hyperplane states introduced in [4]. Hyperplane states are formed by allowing nonzero coefficients c_ij ≠ 0 in (1) only for 2-qubit operators existing in a hyperplane of W(3,2), the symplectic polar space of 2-qubit operators. This space contains all 2-qubit operators up to a phase factor, and commutation relations between subsets of operators of size 3. It can be represented visually by a finite geometry known as the doily, see Fig. 1, and [5, 6, 7, 8] for further reading on its relationship with quantum information.

A hyperplane H of W(3,2) is a subset of points of W(3,2) such that any line of W(3,2) is either fully contained in H or intersects H at exactly one point. A perp-set P(p) is a hyperplane constructed by taking one point p of W(3,2), with associated operator O_p, and all points that commute with O_p:
P(p) := {O_i ∈ W(3,2), [O_i, O_p] = 0}

Perp-sets are hyperplanes of W(3,2) (see Fig. 1, right). Another type of hyperplane are grids (or hyperbolic quadrics in the mathematical literature), formed by 9 points lying on 6 intersecting lines with each point lying on two lines. An example of a grid is the set of maximally mixed operators G₂, i.e. those without identities in their qubit components. The grid G₂ contains all negative lines of the doily, see Fig. 2.

We define a perp state ρ as a state of the form (1) whose only nonzero coefficients c_ij are those associated to the operators in P(p), the perp set of the point p. For example, taking p = ZZ as in Fig. 1, one has that the only nontrivial operators furnishing ρ_ZZ are {ZZ, XX, YY, XY, YX, IZ, ZI}.

Finally one defines *semi-trivial* perp states ρ as those whose defining point p contains one identity operator, i.e. \( p \notin G_2 \). It is these states that we will show can become swap-intolerant nonlocal under arbitrary positive-definite breaking.

It was found in [4] that the eigenvalues of perp states are given by the following:

\[
\lambda(\rho) =
\begin{cases}
1 + c_p \pm \sqrt{\sum_{\text{pos}} (c_{i1} + c_{i2})^2 + \sum_{\text{neg}} (c_{i1} - c_{i2})^2} \\
1 - c_p \pm \sqrt{\sum_{\text{pos}} (c_{i1} - c_{i2})^2 + \sum_{\text{neg}} (c_{i1} + c_{i2})^2}
\end{cases}
\tag{6}
\]

where \( c_p \) is the coefficient associated to the point \( p \), \( c_{i1}, c_{i2} \) are colinear coefficients sharing the \( i \)th context with \( p \), and the sums are taken over positive and negative lines as defined by the operator products.

We now split the coefficients into two types: ones associated to the fully nontrivial points in \( G_2 \), which we label \( \beta_{ij} \), and the rest, which we label \( \tau_{ij} \). It is easy to see that for a point \( p \notin G_2 \), its incident lines contain one operator within \( G_2 \) and one outside it. Also, all of these contexts are positive, as the only negative lines are fully contained in \( G_2 \). Thus for a semi-trivial perp state we can label the eigenvalues as such:

\[
\lambda(\rho) =
\begin{cases}
1 + \tau_p \pm \sqrt{ \sum_{i=1}^3 (\beta_i + \tau_i)^2 } \\
1 - \tau_p \pm \sqrt{ \sum_{i=1}^3 (\beta_i - \tau_i)^2 }
\end{cases}
\tag{7}
\]

We let the \( \tau_i, i = 1,2,3 \) coefficients be coordinates of a point \( C \) in 3-dimensional parameter space \( S \). We let the coordinates \( \beta_i, i = 1,2,3 \) define a point \( E \) which we call the state point. The distance \( r_\pm := 1 \pm \tau_p \) define radii of spheres centred at the point \( C \), which we call \( (C, r_\pm) \). Then positive definiteness becomes a condition on the coordinates of the points:

\textbf{Proposition 2.1.} \textit{The semi-trivial perp state \( \rho \) is positive-semidefinite iff the state point \( E \) and its reflected point \( -E \) are contained on the surface or in the interior of the spheres \( (C, r_+) \), \( (C, r_-) \) respectively.}

This is proven in [4], but is evident upon inspection of the eigenvalues. What's more, the eigenvalues are dependent upon the radii \( r_\pm \) and the distances \( |\pm E, C| \), so under translations and rotations of the parameter space \( S \), any eigenvalue-dependent property such as positive-definiteness is unchanged. Therefore for any perp state \( \rho \) we are free to form a state \( R(\rho) \) by rotating the parameters in \( S \) by some rotation \( R \), then \( \rho \) and \( R(\rho) \) will have the same positive-definite property.

Now we introduce the nonlocality condition (3) into the parameter space \( S \).

Let the matrix \( \beta \) be the matrix of coefficients \( \beta_{ij} \) of fully nontrivial operators for a given state \( \rho \). Then for \( \rho \) a semi-trivial perp state, it is easy to see that \( \beta \) will only

have one column or row vector as nonzero entries. This is because for a given \( p \notin G_2 \), points in \( G_2 \) commuting with it must share the same nontrivial 1-qubit operator as the one in \( p \). Whether it is a row or column vector depends on whether the identity is in the right-hand or left-hand of \( p \):

\[
\beta^T \beta = \begin{cases}
\begin{pmatrix}
\beta_{ix}^2 \\
\beta_{iy}^2 \\
\beta_{iz}^2
\end{pmatrix}, & p = i \otimes I \\[2ex]
\begin{pmatrix}
\sum_j \beta_{ji}^2 \\
0 \\
0
\end{pmatrix}, & p = I \otimes i
\end{cases}
\tag{8}
\]

where the index \( i \) matches the chosen nontrivial operator in \( p \). This asymmetry in \( \beta^T \beta \) furnishes a swap-intolerant nonlocality as \( \mathcal{M} \) is given by the sum of the largest two eigenvalues of the above matrix. Let \( (\beta_1, \beta_2, \beta_3) \) be the vector entries of \( \beta \) for a given semi-trivial perp state \( \rho \), and WLOG let \( \beta_3 \) be the smallest of the three. Then the following conditions give a SIN state:

\[
\begin{aligned}
\beta_1^2 + \beta_2^2 + \beta_3^2 &> 1 \\
\beta_1^2 + \beta_2^2 &\leq 1 \\
\beta_3^2 &\leq \min(\beta_1^2, \beta_2^2)
\end{aligned}
\tag{9}
\]

These conditions give \( \mathcal{M} > 1 \) for \( p = I \otimes i \) (left-identity) but not \( p = i \otimes I \) (right-identity). The first describes the region outside the unit sphere in \( S \), the second describes the interior of the unit cylinder along the \( \beta_3 \) axis, and the third gives the exterior of the cone along the \( \beta_3 \) axis with endpoint at the origin. The intersection of these regions is nonempty in \( S \). We need to place our state point \( E \) in this intersection and find suitable coordinates \( \tau_i \) and \( \tau_p \) to describe our state.

First we prove the following about semi-trivial perp states.

**Proposition 3.1.** Semi-trivial perp states \( \rho \) cannot be both positive-semidefinite and nonlocal.

*Proof.* Let \( (\beta_1, \beta_2, \beta_3) \) be the coordinates of \( E \) in \( S \). Nonlocality in both left- or right-identity cases implies \( |E|^2 = \sum_{i=1}^3 \beta_i^2 > 1 \). Now for a given set of values for \( \tau_p,\, (\tau_i) = C \) one places \( C \) in the axis connecting \( E, -E \) (see Fig. 3). This provides the maximum distance \( E \) that leaves \( \rho \) positive semidefinite. But here the distance \( 2|E| = 1 + \tau_p + 1 - \tau_p = 2 \), in contradiction to the condition that \( |E| > 1 \) as required by nonlocality. □

We now prove Theorem 1.4. For a given λ₀ < 0, we let |E| = 1 + ½|λ₀| - outside the unit sphere in S thus giving a nonlocal state. By 3.1 this state is non-positive-semidefinite. Set the remaining parameters such that C is on the axis joining E and −E, and the point −E lies on the sphere (C, r₋). This then puts E a distance of |λ₀| from the sphere (C, r₊), making the smallest eigenvalue of ρ equal to λ₀. Now perform a rotation R in S such that R(E) is lying in the region described in (9), making the state swap-intolerant nonlocal.

We have shown here a connection between positive semidefiniteness of density matrices and label-dependence of nonlocality. If one were to relax the positive-semidefiniteness criterion even slightly then one can construct a 2-qubit state where the nonlocality depends on the order of the qubit labels. Furthermore, the boundary for nonlocality in general in this case (including swap-intolerant nonlocality) is the same as the boundary for positive semidefiniteness, identifying a boundary for physicality with one separating classical from quantum mechanics. It is the boundary that defines

|E| = 1, with |E| the point given by coordinates (\beta_1, \beta_2, \beta_3), the nonzero vector in \beta. We have used the symplectic polar space W(3,2) and perp sets of this space to construct these SIN states in the most general fashion.

This work is supported by the Graduate school EIPHI (contract ANR-17-EURE-0002) through the project TACTICQ. The author is grateful to Frédéric Holweck and Metod Saniga for feedback and corrections.

[8] Axel Muller et al. "Multi-qubit doilies: enumeration for all ranks and classification for ranks four and five". In: Journal of Computational Science 64 (Oct. 2022), p. 101853. arXiv:2206.03599 [quant-ph]. ISSN: 1877-7503. DOI: 10.1016/j.jocs.2022.101853. URL: http://arxiv.org/abs/2206.03599 (visited on 07/13/2023).


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
Reconstructing textureless areas in MVS poses challenges due to the absence of reliable pixel correspondences within fixed patch. Although certain methods employ patch deformation to expand the receptive field, their patches mistakenly skip depth edges to calculate areas with depth discontinuity, thereby causing ambiguity. Consequently, we introduce Multi-granularity Segmentation Prior Multi-View Stereo (MSP-MVS). Specifically, we first propose multi-granularity segmentation prior by integrating multi-granularity depth edges to restrict patch deformation within homogeneous areas. Moreover, we present anchor equidistribution that bring deformed patches with more uniformly distributed anchors to ensure an adequate coverage of their own homogeneous areas. Furthermore, we introduce iterative local search optimization to represent larger patch with sparse representative candidates, significantly boosting the expressive capacity for each patch. The state-of-the-art results on ETH3D and Tanks & Temples benchmarks demonstrate the effectiveness and robust generalization ability of our proposed method.

Multi-view Stereo (MVS) aims to rebuild dense 3D scenes or objects from a series of calibrated RGB images. It has a broad range of applications in fields such as virtual reality, 3D printing, and autonomous driving, leading to numerous efficient MVS frameworks (Kuhn et al. 2017; Shen 2013; Zheng et al. 2014; Furukawa and Ponce 2010). Nonetheless, despite the rapidly improving accuracy of the latest methods (Yao et al. 2018; Luo et al. 2019), these techniques frequently face challenges when dealing with textureless areas.

Recently, PatchMatch Stereo based-methods (Heise et al. 2013; Kuhn, Lin, and Erdler 2019) have accomplished remarkable successes in reconstructing dense 3D models from large-scale imagery. These methods typically follow an organized pipeline that includes initialization, propagation and refinement, positioning them uniquely suited for reconstructing scenes with unstructured viewpoints. However, due to the unreliable photometric consistency, most methods (Schönberger et al. 2016; Xu et al. 2020; Yuan et al. 2024a) still exhibit unsatisfactory performance when dealing with large textureless areas.

Some traditional methods (Xu and Tao 2019, 2020; Xu et al. 2022) leverage multi-scale geometric consistency and triangulated planar prior to address this challenge. Through obtaining planar hypotheses at coarser scales and propagating reliable ones into finer scales, these methods manage to effectively reconstruct partial textureless areas. Yet, the integration of larger textureless areas remains problematic due to their limited patch sizes. Partial approaches (Zhang et al. 2015; Romanoni and Matteucci 2019; Kuhn, Lin, and Erdler 2019) attempt to utilize superpixels for 3D planar surface fitting to alleviate the problem. However, constraints such as a fixed threshold and strict assumption of data distribution result in poor planarization for certain superpixels. APD-MVS (Wang et al. 2023) utilizes an adaptive deformable patch with cascade architecture. Nevertheless, excessive PatchMatch iterations often make the process time-consuming, leading it impractical for application on large-scale datasets.

Some learning-based methods, such as (Yang et al. 2020; Luo et al. 2019) leverage a coarse-to-fine cascading architecture to address the issue of textureless areas. By refining and upsampling depth maps at low resolution, these methods can effectively expand the receptive field while decreasing the cost volumes. However, these methods typically require substantial memory and time consumption. To circumvent this, several methods like (Wang et al. 2021) attempt to employ lightweight gated recurrent units (GRU) to encode the depth probability distribution for each pixel. However, the adoption of GRU makes capacity for spatial recognition becomes constrained, thereby exhibit limitations in reconstructing textureless areas. Other methods, like (Sormann et al. 2023; Ma et al. 2021) introduce an epipolar module to assemble high-resolution images into limited-size cost volumes. However, they often exhibit limited generalization, inhibiting their application in reconstructing scenarios with diverse characteristics.

Large man-made scenarios are frequently characterized by textureless areas across varying scales, which can typically be predicted through their surrounding reliable pixels. Therefore, an inspiring idea is to identify textureless areas across different scales and extract their reliable pixels for

plane fitting, thereby achieving both reconstruction of textureless areas and refinement of detailed regions.

Therefore, we propose TSAR-MVS, a generalized post-processing MVS method that sequentially applies confidence-based outliers filtering, superpixel-based local refinement, and feature-based area generation to planarize textureless areas. Its distinctive design enables it to be applied after most methods and enhance their reconstruction results. Although ACMP (Xu and Tao 2020) constructs triangulation planar priors, it not only heavily relies on the precision of each reliable pixel but also struggles from incomplete image coverage. Differently, our method can exploit more robust and comprehensive plane prior insight, thus can recover precise and complete planes for reconstruction. Moreover, unlike TAPA-MVS (Romanoni and Matteucci 2019) which are limited to planarizing localized areas, our textureless-aware segmentation can extract textureless areas across broader scales, thereby not limited by superpixels size.

Specifically, we first introduce a joint hypothesis filter consisting of a confidence estimator and a disparity discontinuity detector to provisionally remove outliers. Inspired by (Kuhn et al. 2020), we first combine three confidence features extracted from the matching cost to filter wrong estimates. Furthermore, we maintain areas with continuous disparity, underpinning the assumption brought by (Hirschmuller 2008). Through integrating both confidence estimator and disparity discontinuities into an aggregation score, we can treat small disparity discontinuities as outliers, while preserving reliable object edges which are continuous in disparity.

Subsequently, to propagate the confident pixels with correct depth, we propose an iterative correlation refinement procedure. In each iteration, we first cluster pixels with similar perceptual features into same superpixels, then we apply RANSAC to planarize each superpixel, resulting in pixels within a superpixel sharing a consistent depth. Following this, a pixel-wise weighted median filtering (WMF) is used to propagate the correct depth to the surrounding areas. This iterative process is conducted over several rounds to ensure the propagation of reliable depth. Finally, another WMF is adopted to interpolate the depths and normals of the remaining pixels.

Additionally, we propose a textureless-aware segmentation method combining edge detection and line detection. Typically, in man-made scenarios (Schops et al. 2017; Strecha et al. 2008; Knapitsch et al. 2017), pixels within the same textureless area exhibit a high degree of similarity. Thus, we attempt to separate textureless areas and independently estimate the depth of their homogeneous surfaces. Specifically, we first apply the Roberts edge detection to extract high-frequency information, followed by the Hough line detection that enhances region discrimination by concatenating separated edges into lines. By aggregating residual pixels into miscellaneous areas, we can distinguish and further planarize them, thereby improving reconstructing result.

Experiments on extensive datasets, including the ETH3D high-resolution dataset, Tanks & Temples dataset and Strecha dataset demonstrate that our method significantly outperforms other MVS methods. Furthermore, we demonstrate that our method can be incorporated as a component into other reconstruction pipelines to improve performance.

In summary, our contributions are as follows:
1) We propose a novel joint hypothesis filter by adaptly merging a confidence estimator and disparity discontinuities to yield an aggregation score that effectively eliminates outliers.
2) We present an iterative correlation refinement procedure that iteratively employs RANSAC and WMF to refine pixel estimates based on the surrounding confidence neighbors.
3) We introduce a textureless-aware segmentation technique that merges edge and line detection, thereby achieving the discrimination and planarization of textureless areas.
4) Through extensive experiments, we verified that our method achieves state-of-the-art results and possesses strong generalization capability.

The fundamental concept of PatchMatch (Barnes et al. 2009) employs random initialization, propagation and refinement to match approximate patch pairs between images. PMS (Bleyer, Rhemann, and Rother 2011) pioneered to introduce this in stereo vision, facilitating the emergence of numerous further algorithms. For acceleration, Gipuma (Galliani, Lasinger, and Schindler 2015) proposes checkerboard diffusion propagation to achieve algorithm deployment on GPUs. To reconstruct textureless areas, ACMM (Xu and Tao 2019) presents both multi-scale architecture and multi-view consistency for guidance, improved by ACMMP (Xu et al. 2022) which incorporates a probabilistic graphical model as plane priors for reconstruction. Different, TAPA-MVS (Romanoni and Matteucci 2019), PCF-MVS (Kuhn, Lin, and Erdler 2019) and TSAR-MVS (Yuan et al. 2024b) attempt to employ superpixel planarization to reconstruct textureless areas, while their performance are severely affected by limited superpixel size. Moreover, MAR-MVS (Xu et al. 2020) utilizes differential geometry and epipolar constraints to mitigates matching ambiguity. HPM-MVS (Ren et al. 2023) further introduces non-local extensible sampling patterns to avoid locally optimal solutions within textureless areas.

Concerning patch deformation, APD-MVS (Wang et al. 2023) separated patches of unreliable pixels into several outward-spreading anchors with high reliability. Differently, SD-MVS (Yuan et al. 2024a) adopts the SAM (Kirillov et al.

2023) to extract depth edges for patch deformation, while its single-granularity lacks accuracy for patch deformation.

MVSNet (Yao et al. 2018) is the first to utilizes differentiable 3D cost volumes to construct deep neural network, thereby becoming the foundation for abundant subsequent methods. To reduce memory, Cas-MVSNet (Yang et al. 2020) adopted a coarse-to-fine strategy to retain depths across multi-scale. IterMVS-LS (Wang et al. 2022a) leveraged a lightweight probability estimator to encodes depth distribution during regularization. For feature extraction, AAR-MVSNet (Wei et al. 2021) employed deformable convolutions to achieve adaptive feature aggregation. Differently, MVSTER (Wang et al. 2022b) leveraged the transformer structure to introduce the multi-head attention mechanism. Concerning geometrical topology, Geo-MVSNet (Zhang et al. 2023) proposed a two-branch geometry fusion network to enhance geometry perception. Regarding surface topology, RA-MVSNet (Zhang, Zhu, and Lin 2023) associated hypothesis planes with surface patches to enhance perception field. However, most learning-based methods suffer from either unaffordable memory usage or limited generalization capabilities, leaving room for further improvement.

Due to the absence of depth edge as constraints, APD-MVS erroneously skips depth edges to select anchor within heterogeneous areas, thereby causing potential inaccuracy. While by introducing semantic awareness and granularity richness, Semantic-SAM can achieve panoramic segmentation at any granularity level, which contains potential depth edges. Therefore, we adopt the Semantic-SAM to extract multi-granularity depth edges for patch deformation.

Since APD-MVS lacks depth edge guidance, its anchors typically skip depth edges to be selected within green heterogeneous areas, causing potential inaccuracy. Therefore, we attempt to extract depth edges as guidance for patch deformation. However, without accurate depth maps, it is rather difficult to extract depth edges. Although depth edges and image edges have a certain degree of co-occurrence, they are easily confused by factors such as illumination, shadow, occlusion, etc. Differently, by introducing semantic awareness and granularity richness, Semantic-SAM can achieve panoramic segmentation at any granularity level, which may contain potential depth edges. Therefore, we attempt to adopt Semantic-SAM to extract multi-granularity depth edges for patch deformation.

Although edge-confined patch deformation effectively improves the reliability of deformed patches compared to APD-MVS, it may cause the loss of anchors within sectors fully enclosed by depth edges. Addressing this, we propose adaptive equidistribution, which uniformly divides the homogeneous area of each unreliable pixel into eight sectors based on the number of its surrounding reliable pixels to recover previously-discarded anchors.

We evaluate our work on ETH3D benchmark (Schops et al. 2017) and Tanks & Temples benchmark (TNT) (Knapitsch et al. 2017). We compare our work against learning-based methods including PatchMatchNet (Wang et al. 2021), IterMVS-LS (Wang et al. 2022a), MVSTER (Wang et al. 2022b), EPP-MVSNet (Ma et al. 2021), EP-Net (Su and Tao 2023) and traditional methods including PCF-MVS (Kuhn, Lin, and Erdler 2019), ACMM (Xu and Tao 2019), ACMMP (Xu et al. 2022), SD-MVS (Yuan et al. 2024a), APD-MVS (Wang et al. 2023) and HPM-MVS (Ren et al. 2023).

Qualitative results on ETH3D are illustrated. It is evident that our method delivers the most complete and realistic reconstructed point clouds, especially in large textureless areas like the floors and walls, without introducing detail distortions. More qualitative results on the ETH3D and the TNT dataset can be referred in supplementary material.

Quantitative results on the ETH3D and the TNT dataset are respectively presented. Note that the first group is learning-based methods and the second is traditional methods. Meanwhile, we mark the best results in bold and the second-best results in red. Our method achieves the highest F1 score and completeness on the ETH3D dataset and the TNT Intermediate dataset, validating its state-of-the-art performance and strong generalization capability. Meanwhile, our method achieves the second best results in the TNT advanced datasets, falling short by less than 0.3% in F1 score compared to EPNet (Su and Tao 2023). Furthermore, we achieve comparative runtime compared to SOTA methods like APD-MVS and HPM-MVS. A detailed memory and runtime comparison is shown in supplementary material.

Multi-granularity Segmentation Prior  Regarding segmentation prior, we separately remove SAM prior (w/o. SAM), multi-granularity SAM prior (w/o. Her), aggregation (w/o. Agr.), and edge correction (w/o. CRF.). Compared to w/o. SAM, w/o. Her. which only introduce single-granularity SAM prior led to notable improvements, validating the effectiveness of depth edges for patch deformation. Additionally, Both w/o. Her. and w/o. Agr. produce similar better results than w/o. CRF., yet are inferior to MSP-MVS, indicating that both multi-granularity SAM prior and aggregation are equally crucial than edge correction.

Anchor Equidistribution  Concerning anchor selection, we respectively remove sector averaging (w/o. Avg.), anchor clustering (w/o. Cls.) and both (w/o. Equ.). w/o. Equ. has neither sector averaging nor anchor clustering achieves the worst results. w/o. Cls. slightly outperformed w/o. Avg., yet fell short to MSP-MVS, implying that sector averaging contribute more than anchor clustering.

Iterative Local Search Optimization  For candidate optimization, we respectively eliminated the variance component in ILS (w/o. Var.), the cost component in ILS (w/o. Cst.), and both (w/o. ILS). Due to the absence of ILS, w/o. ILS achieves the poorest outcomes. w/o. Var. yields better results than w/o. Cst., suggesting that cost plays a more critical role than variance during candidate optimization.

In this paper, we introduce MSP-MVS, a novel MVS method that leverages multi-granularity segmentation prior to constraint patch deformation within homogeneous areas. Specifically, we first leverage Semantic-SAM to extract multi-granularity depth edges and further aggregate and correct them for accurate patch deformation. Subsequently, we introduce anchor equidistribution to offer patch more uniformly distributed anchors. Moreover, we present ILS optimization to represent patches with sparse representative candidates. The state-of-the-art results on the ETH3D and Tanks & Temples benchmarks validate the effectiveness of our proposed method. In the future, we attempt to leverage Semantic-SAM to extract various 3D instances from point clouds, thereby facilitating object-level scene editing and improving scene’s interactivity.

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