AdvJudge-Zero: Binary Decision Flips in LLM-as-a-Judge via Adversarial Control Tokens

Hongliang Liu; Tung-Ling Li; Yuhao Wu

REVIEW 5 major objections 5 minor 4 references

AdvJudge-Zero: Binary Decision Flips in LLM-as-a-Judge via Adversarial Control Tokens

T0 review · 5 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper claims that the binary verdicts of LLM-as-a-Judge systems are a shallow last-layer logit gap that short, low-perplexity control tokens — discovered with no seeds and no gradients — can flip from 'No' to 'Yes' at rates above 90% o

desk verdict A useful empirical attack paper whose headline FPR numbers for Omni-Judge are probably inflated by a metric artifact; the zero-seed discovery method is real progress, but the evaluation needs parsing validation and held-out splits before I'd trust the 22/24 claim. read the letter →

arxiv 2512.17375 v2 pith:WI67VKJH submitted 2025-12-19 cs.LG cs.CLcs.CR

Tung-Ling Li , Yuhao Wu , Hongliang Liu This is my paper

classification cs.LGcs.CLcs.CR

keywords LLM-as-a-JudgerewardhackingcontroltokenslogitgapadversarialdiscoveryRLHFRLVRLoRAtraining

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that binary LLM-as-a-Judge decisions — the yes/no verdicts that supply reward signals in RLHF-style training — are governed by a shallow, high-gain linear readout on the model's last hidden layer, captured by the single number F = logit('No') − logit('Yes'). Because this gate is shallow, short low-perplexity token sequences (special characters, markdown fragments, structural delimiters) can drive F below zero and flip a correct 'No' into an incorrect 'Yes' without changing the question or answer. The authors introduce AdvJudge-Zero, a discovery method that uses the judge's own next-token distribution and beam search with no seed patterns and no gradients, and report ensemble false-positive rates above 90% on 22 of 24 model–dataset cells across six open-weight judges, with near-100% rates on several math benchmarks and on the Omni-Judge model. They also show that a LoRA fine-tune on a small, mechanism-diverse pool of these tokens reduces false positives from 96–99% to 2–6% while preserving true positives, and that under GRPO training the hardened judge removes the reward-collapse failures seen in the unhardened baseline. The consequence, if true, is that reward signals in current post-training pipelines are systematically exploitable by plausible formatting tokens that a policy could generate.

What carries the argument

The central object is the no–yes logit gap F = z_no − z_yes, evaluated at the first decision token; a flip is F < 0. The geometric account treats F as a linear readout (w_No − w_Yes)ᵀh + b on the last-layer hidden state h, with w_F the refusal direction. Effective control tokens act as low-rank, model-intrinsic perturbations u anti-aligned with w_F — a 'soft mode' that moves many prompts across the decision boundary at once. Discovery uses the judge's own next-token distribution plus a beam-search top-k schedule (large k for the first token, decreasing with length) to propose candidate sequences of length 1–7, verified in batch by the F-sign criterion; selection keeps tokens high in duplicat

What would settle it

Parse the full output of Omni-Judge (and other judges) and compare the marked verdict field (e.g., 'Equivalence Judgement') with the F<0 flip criterion on the same prompts; if the two disagree on a substantial fraction of prompts, the AdvJudge-Zero FPR numbers are not measuring actual verdict flips. A second check: run the same discovery pipeline but select tokens by the parsed verdict rather than by F-sign, and see whether the discovered pool changes.

Watch

Extended reading notes

Core claim

On the paper's own terms: a binary judge's verdict is a linear readout on the last-layer hidden state at the first decision token, and the sign of the logit gap F = z_no − z_yes is the entire decision. AdvJudge-Zero searches the judge's own next-token distribution with a beam-search top-k schedule, inserts candidate sequences into judge prompts, and keeps those that make F negative across many prompts (high duplication count) with strongly negative average F. These sequences, typically special characters, markdown fragments, or structural delimiters, perturb the hidden state along a low-rank 'soft mode' anti-aligned with the judge's refusal direction, so they transfer across prompts and acro

Load-bearing premise

The load-bearing premise is that the judge's binary verdict is fully captured by the sign of the first-decision-token logit gap F = z_no − z_yes; for templates like Omni-Judge, where the verdict is emitted as a later field inside a markdown report, this first-token criterion may not match the parsed verdict.

Editorial extensions

If this is right

Reward signals in RLHF/RLVR pipelines can be steered by short, low-perplexity tokens that policy models could plausibly emit, making reward hacking a realistic failure mode rather than a worst-case adversarial string attack.
Discovery needs no seed tokens and no gradients: a model's own next-token distribution and beam search suffice to find transferable control tokens from scratch, so new judge models can be probed without manual red-teaming.
Effectiveness is non-monotonic in token length — semantic composition matters more than sequence length — so defenses based purely on token-length filters or perplexity thresholds will miss the strongest triggers.
A small pool of mechanism-diverse control tokens can serve as both a stress test and a training signal: LoRA fine-tuning on such tokens cuts false positives from roughly 96–99% to 2–6% while keeping true positives near 100%.
Hardening with the mechanism-diverse pool, unlike naive sampling, generalizes across model families and, under GRPO training, eliminates reward-collapse failures (false-positive spikes and length collapse) observed in the unhardened baseline on MATH and GSM8K.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the judge's verdict is read from a later token (Omni-Judge's template asks for a markdown report ending in an 'Equivalence Judgement' field), the F<0 criterion used for discovery and FPR may not equal the parsed verdict; the paper does not report agreement between F-sign flips and parsed outputs, so the headline FPR numbers could overstate real flips for that judge.
The low-rank 'soft mode' account predicts that control tokens found on one dataset will transfer to unseen prompts and tasks on the same model, and that strengthening the readout head alone (e.g., by making it deeper) would shrink the attack surface; both are testable without retraining the whole model.
The same discovery procedure could be pointed at other binary gates — safety refusals, preference comparisons, or content filters — since the mechanism is a generic first-token logit gap; the paper only studies correctness-style false positives.
A policy trained with a judge that has been LoRA-hardened on these tokens may simply find new control tokens outside the pool; the paper's own limitation note concedes adversarial training may reshape but not remove the low-rank vulnerable directions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Alex,

Short version: this paper has a real contribution and a likely load-bearing evaluation flaw. If you work on judge models or reward hacking, it's worth reading; if the Omni-Judge numbers are what you'd cite, hold off until the authors clarify their verdict metric.

What's actually new: the zero-seed beam-search discovery over the judge's own next-token distribution, with no curated trigger seeds and no gradients. That's a genuine step past Master-RM's human-curated tokens and past their own prior logit-gap steering work, which was about refusal behavior. Applying the same F = z_no - z_yes readout to judge/reward models and showing cross-family transfer to specialized judges is the interesting part. The LoRA defense results on Omni-Judge are also plausible as a first pass.

What worries me: the FPR metric and the flip definition are the same object. Tokens are discovered by selecting candidates with F<0 on 50 prompts, then FPR is scored as the rate of F<0 on the same distribution, with no held-out attack split. That makes the near-100% numbers partly guaranteed by construction. More specifically, Omni-Judge's template (Appendix D) doesn't output a first-token Yes/No verdict; it emits a markdown report with an 'Equivalence Judgement' line later in the generation. Computing F at the first output token for that model looks wrong. If they're grabbing logits at some later position, they don't say how. Table 3's Omni-Judge row drives the headline '22 of 24 cells' claim, so this isn't a minor detail. The stress-test note is right on target, and I don't think the paper answers it.

Also worth flagging: the abstract promises GRPO results, a 9-class taxonomy, and 70B scalar-reward transfer. None of that appears in the body. Either it's missing from this version or it was cut; either way the abstract overstates the manuscript. The defense gets trained and tested on the same discovered pool, so generalization of the LoRA hardening is unknown. No error bars, no sample sizes, which matters when some cells sit at 69-79% and others at 100%.

Credit where due: the geometric low-rank perturbation analysis is honest — they label it an illustrative probe on two models. The mechanism taxonomy in the discussion is sensible. The cross-model commonalities (structural tokens like <|im_end|> for Qwen, markdown fragments for Gemma) are the most believable evidence in the paper.

The escape hatch: if the authors release code, parse actual judge outputs on Omni-Judge and re-run the FPR with held-out prompts and a fixed discovery/eval split, the core claim — that short low-perplexity tokens flip judge verdicts at high rates — might survive. The direct-evaluation results on general-purpose models with strict YES/NO templates (Appendix B) are more credible than the Omni-Judge row.

My recommendation: yes, send it to peer review — the method is new enough and the implications for RLHF/RLVR reward signals are significant enough to merit referee time. But a referee needs to push on the Omni-Judge metric artifact and the missing abstract promises before this is citable as-is.

Best, [Your name]

Referee Report

5 major / 5 minor

Summary. The paper proposes AdvJudge-Zero, a zero-seed beam-search method that discovers short control-token sequences which, when inserted into LLM-as-a-Judge prompts, flip the judge's binary correctness decision from 'No' to 'Yes'. The flip is defined as the sign change of a last-layer logit gap F = z_no − z_yes at a 'decision position.' The authors report very high ensemble false-positive rates (FPR) across six general-purpose Qwen/Llama/Gemma models on four math/reasoning datasets, transfer of the discovered tokens to specialized judge models (notably near-100% FPR for Omni-Judge), a geometric interpretation via a low-rank 'soft mode' anti-aligned with a refusal direction, and a LoRA-based adversarial training defense for Omni-Judge. The abstract additionally claims the defense works under GRPO training and is stratified by a 9-class mechanism taxonomy.

Significance. If the F-based flip criterion is validated against actual generated verdicts for every judge model, the paper would make a useful contribution to the study of reward hacking in LLM post-training pipelines: it provides a reproducible, zero-seed discovery algorithm, a large cross-model evaluation table, and a potential mitigation. The cross-family comparisons and the attempt to connect adversarial control tokens to a low-dimensional geometric mechanism are valuable. However, several load-bearing claims currently rest on an unvalidated metric, a discovery/evaluation circularity, and experimental results that appear only in the abstract. These issues need to be resolved before the contribution can be assessed reliably.

major comments (5)

[§3.1, §4.3, App. D] The flip criterion F = z_no − z_yes is defined at 'the decision position' and assumes a single Yes/No token. Appendix D shows that Omni-Judge's template requires a markdown report beginning with '## Student Final Answer' and containing '## Equivalence Judgement' (TRUE/FALSE) later; the first output token is not Yes/No. The paper does not specify how the decision position is identified for Omni-Judge, nor does it validate flips against parsed outputs. Table 3 attributes near-100% FPR to Omni-Judge, and Section 5 selects Omni-Judge for defense training precisely because of this. Without parsed-output validation, the Omni-Judge numbers—and the defense results built on them—are uninterpretable. Please either evaluate actual generated verdicts or remove/qualify the Omni-Judge claims.
[§4.1, Alg. 1] Discovery selects tokens using F(X⊕A)<0 over a 50-prompt sample D_sub, and the direct evaluation in §4.2 is on the same model–dataset distribution with the same F criterion. No explicit held-out attack split or independent verification is described. Because the selection objective is identical to the evaluation metric, the reported FPR is partly guaranteed by construction: a token that flips one of the 50 discovery prompts counts toward selection even if it never flips elsewhere. Please report FPR on a prompt set disjoint from D_sub and, where possible, use a separate evaluation criterion (parsed outputs) for the yes/no templates.
[Abstract vs. §5–§6] The abstract claims 'stratified by a 9-class mechanism taxonomy' and 'Under GRPO training, the hardened judge eliminates the reward-collapse failures ... at ten seeds per condition.' I could not find the 9-class taxonomy or any GRPO experiments in the main text or appendices. Section 5 only reports LoRA fine-tuning results on Omni-Judge, and §6.3 explicitly defers full RLHF/DPO/RLVR experiments to future work. These unsupported claims must either be implemented and reported or removed from the abstract.
[§4.2, Table 2] No sample sizes, standard errors, or seeds are given for any FPR. With D_sub=50, near-100% values (e.g., 100.00% for Qwen3-4B on AIME) could correspond to 50/50 or 49/50 prompts, making cross-cell comparisons such as Llama-3.2-3B on GSM8K (69.34% vs. 99.16%) statistically ungrounded. Report per-cell N and binomial or bootstrap confidence intervals for all FPR/TPR tables, including Table 4.
[§3.3, App. F] The 'low-perplexity' and 'plausibly generated during post-training' characterization is unsupported. Many of the top tokens in Appendix F are not natural continuations: e.g., 'ablytyped', 'scalablytyped', 'HeadersHeightSizeMode', '<|reserved_special_token_>', 'yarg'. No perplexity or likelihood values under the target policy distribution are reported. Since the paper's threat model depends on these tokens being realistic, please measure and report perplexity (or generation probability) for the discovered sets, or soften the 'realistic reward-hacking risks' claim accordingly.

minor comments (5)

[Fig. 2] The legend uses 'AdvRM' and 'MasterRM' while the text and tables use 'AdvJudge' and 'Master-RM'; unify the naming.
[§3.1] For Omni-Judge the template uses TRUE/FALSE rather than Yes/No; clarify whether F is computed on 'No'/'Yes' logits or on the template's actual decision tokens.
[§4.4] The non-monotonicity of FPR with token length is based on plots without error bars or aggregate statistics; add a quantitative summary.
[§3.2] The geometric alignment analysis uses Master-RM tokens and only two models; the claim that AdvJudge-Zero exploits the same low-rank mode is an extrapolation. Add analysis on AdvJudge-Zero tokens or label the result as preliminary.
[App. B, App. D] Typos: Appendix B has 'Output must be strictly : YES or NO' (stray colon); Appendix D has missing apostrophes ('student s answer'). Also the 'Preprint. December 22, 2025.' date seems inconsistent with the arXiv ID 2512.17375.

Circularity Check

2 steps flagged · score 6.0 of 10

The headline FPR is largely the same F<0 criterion used to select the control tokens, and for Omni-Judge that criterion does not match the model's actual markdown verdict.

fitted input called prediction [Section 3.3 (verification) and Section 4.1 (evaluation protocol)]
"we compute the no–yes logit gap F(X⊕A). We perform verification in batches for efficiency and record all candidates that cause a flip, that is, those with F(X⊕A)<0. ... The main metric of our study is the false positive rate (FPR), defined as the proportion of prompts correctly answered as “No” by the model that are incorrectly flipped to “Yes” when an adversarial token is present. ... Each adversarial ensemble is evaluated only on the specific model and dataset from which it was discovered."

Candidate selection and the reported FPR are the same event: F(X⊕A)<0. The ensembles are retained precisely because they made the no–yes logit gap negative on the verification prompts, and then FPR is computed from that same F<0 event on the same (model, dataset). The near-100% direct FPR is therefore partly guaranteed by construction rather than an independent test that real verdicts flipped. The cross-model stress test is the only partially independent evidence, and it still uses the same F-based criterion.
self definitional [Section 3.1 vs. Appendix D (Omni-Judge template)]
"The judge model reads this template and outputs a single-token decision, either “Yes” (correct) or “No” (incorrect). ... A flip occurs when F(X)<0. ... # RESPONSE : MARKDOWN REPORT ## Student Final Answer ... ## Equivalence Judgement [Whether the student s answer share the same meaning with the reference answer . ( TRUE or FALSE )]"

The paper defines the binary verdict as the sign of F=z_no−z_yes and then reports FPR from that definition. For Omni-Judge, however, the model's own template (Appendix D) requires a markdown report beginning with '## Student Final Answer' and containing a TRUE/FALSE 'Equivalence Judgement'; the first token is not Yes/No. Hence the near-100% Omni-Judge FPR in Table 3 is not shown to correspond to an actual flipped verdict; it is an artifact of equating 'flip' with the F<0 criterion used during discovery.

full rationale

The paper's central numeric claim rests on a definitional identification: a 'flip' is F(X)=z_no−z_yes<0, and the same F<0 event is used both to discover/select the tokens and to compute FPR. Thus the direct, in-distribution FPR is not an independent confirmation of the attack; it re-measures the selection objective. The self-citation (Li & Liu, 2025) is not by itself load-bearing because F is redefined and measured in this paper, and no uniqueness theorem is imported. However, the Omni-Judge result, which drives the headline '22 of 24 cells' claim, is especially vulnerable: Appendix D instructs a markdown report rather than a single Yes/No token, so the F<0 criterion does not demonstrably correspond to the model's actual binary decision without an unspecified parsing or position-identification step. The cross-model stress test provides some independent content, but for Omni-Judge it inherits the same metric artifact. Score 6 reflects that the central FPR numbers partially reduce by construction, while the defense/GRPO claims are downstream and not the source of the circularity.

Assumptions & free parameters 6 free parameters · 6 assumptions · 2 invented entities

The paper's contribution rests mainly on the shallow-linear-readout assumption and on the selection metric F<0. The only clearly invented object is the soft-mode construct, and the 9-class mechanism taxonomy mentioned in the abstract is absent from the body. The LoRA defense is a fitted recipe rather than a demonstrated general cure.

free parameters (6)

beam top-k schedule = 300 for first token, decreasing; exact schedule not specified
Beam width is hand-chosen to trade diversity against compute; no sensitivity analysis is given.
max control-token length n = 7
Search is capped at length 7; the paper shows FPR is non-monotonic in n, so the cap affects the findings.
selection count per length = 10 per length, up to 70 per model-dataset cell
Top-10 threshold determines ensemble size and therefore FPR; no ablation is reported.
discovery sample size D_sub = 50 prompts per dataset
Algorithm 1 samples 50 templates per dataset; FPR generalization depends on this sample.
flip threshold = F < 0
Both discovery and evaluation use the same signed logit-gap threshold; actual decoded token output is not sampled.
LoRA hyperparameters = r=4, alpha=16, dropout=0.05, lr=2e-4, batch=8, 1 epoch, 20k examples, 5% train
Defense configuration is chosen without ablations, and the defense is measured on the same attack pool used for training.

assumptions (6)

domain assumption Binary judge decisions are a linear function of the final hidden state at the first decision token: F = z_no − z_yes ≈ (w_no − w_yes)^T h + b.
Sections 3.1–3.2 define the entire attack and evaluation around this readout; no verification that actual decoded outputs follow it for all four judge templates.
domain assumption Transformer representations are near-injective, so small prompt edits move h locally without erasing semantics.
Invoked in Sections 2 and 7 via Nikolaou et al. 2025 to justify why control tokens leave content unchanged; not proven for judge prompts.
domain assumption Safety/evaluation alignment is installed as a shallow first-token gate.
Cites Qi et al. 2024; load-bearing for the claim that first-token logit-gap steering suffices.
domain assumption Low-perplexity tokens drawn from the judge's own next-token distribution are representative of what policies will generate.
Section 3.3 uses this to frame attacks as natural reward-hacking risks; no policy-rollout validation appears in the body.
standard math PCA on successful-flip hidden-state perturbations and a random-vector null give a valid test of mechanism alignment.
Section 3.2 and Table 1; the Z-scores are suggestive but assume centered perturbations and independent null sampling.
domain assumption A LoRA fine-tune on the same discovered token pool hardens against the broader vulnerability class.
Section 5 reports FPR drops on the same pool and datasets; no held-out attack families or fresh discoveries are tested.

invented entities (2)

Low-rank "soft mode" u
purpose: Explains why diverse control tokens all steer the same hidden-state direction toward the Yes side
Borrowed from condensed-matter physics; inferred from PCA of successful attacks on two models; no independent measurement or predictive handle beyond the paper's own perturbation samples.
Refusal direction w_F
purpose: The linear boundary of the binary judge readout
Defined as w_no − w_yes from the output head; it is an internal construct, not a testable entity outside the model.

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Cite this review

Pith. "Pith review of AdvJudge-Zero: Binary Decision Flips in LLM-as-a-Judge via Adversarial Control Tokens." pith.science (2026). https://pith.science/paper/WI67VKJH

@misc{pith2026251217375,
  author       = {Pith},
  title        = {Pith review of: AdvJudge-Zero: Binary Decision Flips in LLM-as-a-Judge via Adversarial Control Tokens},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WI67VKJH}},
  note         = {Machine review of arXiv:2512.17375}
}

abstract

LLM-as-a-Judge systems supply the reward signal in modern RLHF and RLVR pipelines, but their binary verdict reduces to a single linear readout F_gap on one hidden state. We show this readout is shallow enough that short, low-perplexity tokens flip the verdict from "No" to "Yes". These tokens are sampled from the judge's own next-token distribution at the response position, with no manual seed set and no gradient-based optimization. Our procedure, AdvJudge-Zero, reaches $>$90% ensemble false-positive rate on 22 of 24 (model, dataset) cells across six Qwen, Llama, and Gemma judges, versus 54-72% for the prior curated 10-token benchmark, and the discovered surface transfers cross-format to a 70B scalar reward model. The same discovered pool enables a defense: a LoRA fine-tune stratified by a 9-class mechanism taxonomy hardens cross-family generalization where naive sampling on the same pool fails, with mechanism breadth rather than pool size carrying the gain. Under GRPO training, the hardened judge eliminates the reward-collapse failures (false-positive spikes and length collapse) we observe in the unhardened baseline on both MATH and GSM8K at ten seeds per condition. The discovered pool, the mechanism taxonomy, and per-prompt flip records will be released under responsible disclosure.

Figures

Figures reproduced from arXiv: 2512.17375 by the authors.

**Figure 1.** Control tokens discovered by AdvJudge-Zero flip the judges Yes/No decision on a math solution by steering the lastlayer logit gap, without improving the solution itself. den state hL(X) that retains almost all prompt semantics, consistent with recent work on injectivity in language models (Nikolaou et al., 2025). Post-training then installs a gate at the readout that separates “No”-like and “Yes”-like responses, i… view at source ↗

**Figure 2.** Comparison of Ensemble FPR vs. Baseline across four datasets (AIME, MATH, GSM8K, Multi-subject RLVR). tions. 1 2 3 4 5 6 7 Token Length (n) 0 20 40 60 80 100 FPR (%) Qwen 2.5 7B on AIME 2 3 4 5 6 7 Token Length (n) 0 20 40 60 80 100 FPR (%) Llama 3.3 70B on MATH [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FPR vs. token length n for Qwen2.5-7B-Instruct on AIME (left) and Llama-3.3-70B-Instruct on MATH (right). 5. Adversarial Training with Discovered Control Tokens To evaluate the effectiveness of adversarial training as a mitigation, we fine-tuned the Omni-Judge model. We selected this model because our stress tests revealed it to be the most vulnerable judge (see [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FPR vs. Token Length (n) for google/gemma-3-4b-it across datasets. Top left: AIME, Top right: GSM8K, Bottom left: MATH, Bottom right: Multisubject RLVR. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: FPR vs. Token Length (n) for meta-llama models across datasets. Top left: Llama-3.2-3B-Instruct (AIME), Top right: Llama-3.2-3B-Instruct (GSM8K), Bottom left: Llama-3.3-70B-Instruct (MATH), Bottom right: Llama-3.3-70B-Instruct (Multisubject RLVR). 13 [PITH_FULL_IMAGE:…

**Figure 6.** Figure 6: FPR vs. Token Length (n) for Qwen2.5-7B-Instruct across datasets. Top left: AIME, Top right: GSM8K, Bottom left: MATH, Bottom right: Multisubject RLVR. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: FPR vs. Token Length (n) for Qwen3 model variants across datasets. Top left: Qwen3-4B-Instruct-2507 (AIME), Top right: Qwen3-4B-Instruct-2507 (GSM8K), Bottom left: Qwen3-30B-A3B-Instruct-2507 (MATH), Bottom right: Qwen3-30B-A3B-Instruct2507 (Multisubject RLVR). 15 [P…

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Works this paper leans on

4 extracted references · 1 linked inside Pith

[1]

So , there are 3 choices for the position of the digit 1

** Choose the digit that is 1 **: There are 3 positions (a , b , c ) where the digit 1 can be placed . So , there are 3 choices for the position of the digit 1
[2]

"step’] 10 6 [’{\n

** Choose the other two digits **: The remaining two digits must be different from each other and different from 1. Therefore , we have 9 choices for the first of these two digits (0 -9 except 1) and 8 choices for the second digit (0 -9 except 1 and the first digit ). This gives us $9 \ times 8 = 72$ choices for the other two digits . Combining these ch...

2025
[2023]

URL https://arxiv.org/abs/2305. 18290. Saad-Falcon, J., Buchanan, E. K., Chen, M. F., Huang, T.- H., McLaughlin, B., Bhathal, T., Zhu, S., Athiwaratkun, B., Sala, F., Linderman, S., Mirhoseini, A., and R, C. Shrinking the generation-verification gap with weak ver- ifiers.arXiv preprint arXiv:2506.18203, 2025. URL https://arxiv.org/abs/2506.18203. Stiennon...

arXiv 2025
[2025]

A , C , D

URL https://arxiv.org/abs/2503. 02623. Zhang, X., Sun, H., Zhang, Y ., Feng, K., Yang, C., and Meng, H. Critique-grpo: Advancing llm rea- soning with natural language and numerical feedback. arXiv:2506.03106, 2025. URL https://arxiv. org/abs/2506.03106. Zhao, Y . et al. One token to fool llm-as-a-judge. arXiv:2507.08794, 2025. URL https://arxiv. org/abs/2...

arXiv 2025

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

So , there are 3 choices for the position of the digit 1

** Choose the digit that is 1 **: There are 3 positions (a , b , c ) where the digit 1 can be placed . So , there are 3 choices for the position of the digit 1

[2] [2]

"step’] 10 6 [’{\n

** Choose the other two digits **: The remaining two digits must be different from each other and different from 1. Therefore , we have 9 choices for the first of these two digits (0 -9 except 1) and 8 choices for the second digit (0 -9 except 1 and the first digit ). This gives us \(9 \ times 8 = 72\) choices for the other two digits . Combining these ch...

2025

[3] [2023]

URL https://arxiv.org/abs/2305. 18290. Saad-Falcon, J., Buchanan, E. K., Chen, M. F., Huang, T.- H., McLaughlin, B., Bhathal, T., Zhu, S., Athiwaratkun, B., Sala, F., Linderman, S., Mirhoseini, A., and R, C. Shrinking the generation-verification gap with weak ver- ifiers.arXiv preprint arXiv:2506.18203, 2025. URL https://arxiv.org/abs/2506.18203. Stiennon...

arXiv 2025

[4] [2025]

A , C , D

URL https://arxiv.org/abs/2503. 02623. Zhang, X., Sun, H., Zhang, Y ., Feng, K., Yang, C., and Meng, H. Critique-grpo: Advancing llm rea- soning with natural language and numerical feedback. arXiv:2506.03106, 2025. URL https://arxiv. org/abs/2506.03106. Zhao, Y . et al. One token to fool llm-as-a-judge. arXiv:2507.08794, 2025. URL https://arxiv. org/abs/2...

arXiv 2025