Daily Papers

Object detection methods trained on a fixed set of known classes struggle to detect objects of unknown classes in the open-world setting. Current fixes involve adding approximate supervision with pseudo-labels corresponding to candidate locations of objects, typically obtained in a class-agnostic manner. While previous approaches mainly rely on the appearance of objects, we find that geometric cues improve unknown recall. Although additional supervision from pseudo-labels helps to detect unknown objects, it also introduces confusion for known classes. We observed a notable decline in the model's performance for detecting known objects in the presence of noisy pseudo-labels. Drawing inspiration from studies on human cognition, we propose to group known classes into superclasses. By identifying similarities between classes within a superclass, we can identify unknown classes through an odd-one-out scoring mechanism. Our experiments on open-world detection benchmarks demonstrate significant improvements in unknown recall, consistently across all tasks. Crucially, we achieve this without compromising known performance, thanks to better partitioning of the feature space with superclasses.

Model overconfidence and poor calibration are common in machine learning and difficult to account for when applying standard empirical risk minimization. In this work, we propose a novel method to alleviate these problems that we call odd-k-out learning (OKO), which minimizes the cross-entropy error for sets rather than for single examples. This naturally allows the model to capture correlations across data examples and achieves both better accuracy and calibration, especially in limited training data and class-imbalanced regimes. Perhaps surprisingly, OKO often yields better calibration even when training with hard labels and dropping any additional calibration parameter tuning, such as temperature scaling. We demonstrate this in extensive experimental analyses and provide a mathematical theory to interpret our findings. We emphasize that OKO is a general framework that can be easily adapted to many settings and a trained model can be applied to single examples at inference time, without significant run-time overhead or architecture changes.

Distinct-n scoreLi2016 is a widely used automatic metric for evaluating diversity in language generation tasks. However, we observed that the original approach for calculating distinct scores has evident biases that tend to assign higher penalties to longer sequences. We refine the calculation of distinct scores by scaling the number of distinct tokens based on their expectations. We provide both empirical and theoretical evidence to show that our method effectively removes the biases existing in the original distinct score. Our experiments show that our proposed metric, Expectation-Adjusted Distinct (EAD), correlates better with human judgment in evaluating response diversity. To foster future research, we provide an example implementation at https://github.com/lsy641/Expectation-Adjusted-Distinct.

We investigate the mechanism design problem faced by a principal who hires multiple agents to gather and report costly information. Then, the principal exploits the information to make an informed decision. We model this problem as a game, where the principal announces a mechanism consisting in action recommendations and a payment function, a.k.a. scoring rule. Then, each agent chooses an effort level and receives partial information about an underlying state of nature based on the effort. Finally, the agents report the information (possibly non-truthfully), the principal takes a decision based on this information, and the agents are paid according to the scoring rule. While previous work focuses on single-agent problems, we consider multi-agents settings. This poses the challenge of coordinating the agents' efforts and aggregating correlated information. Indeed, we show that optimal mechanisms must correlate agents' efforts, which introduces externalities among the agents, and hence complex incentive compatibility constraints and equilibrium selection problems. First, we design a polynomial-time algorithm to find an optimal incentive compatible mechanism. Then, we study an online problem, where the principal repeatedly interacts with a group of unknown agents. We design a no-regret algorithm that provides mathcal{O}(T^{2/3}) regret with respect to an optimal mechanism, matching the state-of-the-art bound for single-agent settings.

We present a new methodology for detecting out-of-distribution (OOD) images by utilizing norms of the score estimates at multiple noise scales. A score is defined to be the gradient of the log density with respect to the input data. Our methodology is completely unsupervised and follows a straight forward training scheme. First, we train a deep network to estimate scores for levels of noise. Once trained, we calculate the noisy score estimates for N in-distribution samples and take the L2-norms across the input dimensions (resulting in an NxL matrix). Then we train an auxiliary model (such as a Gaussian Mixture Model) to learn the in-distribution spatial regions in this L-dimensional space. This auxiliary model can now be used to identify points that reside outside the learned space. Despite its simplicity, our experiments show that this methodology significantly outperforms the state-of-the-art in detecting out-of-distribution images. For example, our method can effectively separate CIFAR-10 (inlier) and SVHN (OOD) images, a setting which has been previously shown to be difficult for deep likelihood models.

Point-adjustment (PA), for years the default scoring protocol in time-series anomaly detection (TSAD), was shown by Kim et al. (2022) to award near-perfect F1 to random anomaly scores. The field adopted a suite of replacement metrics (PA%K, range-based precision/recall, affiliation precision/recall, and Volume-Under-the-Surface, VUS, ROC/PR). We ask, independently and adversarially, whether these resist no-skill detectors on real benchmarks, and find the answer turns entirely on one overlooked variable: N, the number of random attempts an adversary reports the best of. Under a single honest run (N=1), not one replacement metric is gameable on any of six benchmarks (UCR, SMD, SMAP, MSL, NAB, PSM): a random detector reaches 90% of the best real detector's score on at most 11% of series for affiliation-F1, 5% for the ROC family, and 2% for the PR-based metrics and PA%K. But under best-of-N reporting, the seed-shopping endemic to ML, the metrics split sharply. affiliation-F1 and every ROC-based metric inflate steeply, affiliation crossing gameable (25% of series) by N=3 and reaching 0.98 at the full pool (N=41), the ROC family crossing by N=9-11; the PR-based metrics and PA%K stay near-flat at every N, floored near the anomaly prevalence (the lone exception is NAB at large N). A paired test finds VUS-ROC inflated on 131 series where its sibling VUS-PR is not, and never the reverse. The ROC-vs-PR split follows from the order-statistic behaviour of AUC under extreme class imbalance (a random PR-AUC is floored at prevalence); affiliation inflates by a second route, its extreme single-run leniency (already fragile at N=1). We release a pip-installable stress-test harness, and recommend reporting single-run scores or disclosing N and preferring PR-based metrics, which resist best-of-N inflation on nearly every benchmark.

A benchmark score compares a system output against a reference, and methodological attention falls almost entirely on the first term. We measure the second. The retained annotation record of a six-language benchmark for personally identifiable information contains two independent annotator labellings, the aggregate shipped as gold, and a reviewer gold from independent expert re-annotation of a sample. Using it, we hold the scored output fixed and exchange the reference. The score moves by 4.95 F1 points for one output and 2.00 for the other (95% CIs [3.23, 6.27] and [0.26, 3.37]), and by 7.55 in the worst language. The comparison between two outputs moves as well: the paired interaction between reference and system is +2.95 points (CI [+1.97, +3.87]), survives correction for multiple testing, and changes one language's margin outright. The cause is an undocumented aggregation default that usually kept one annotator when adjudication did not fire, making the shipped reference a partial copy of an output being scored. We report the resulting reference-sensitivity band, show how to compute one from any retained annotation record, and argue that the quantity belongs beside the score.

Decision APIs that return a probability for each option are routinely used as if those probabilities were properties of the options themselves. That reading is valid only if the model satisfies independence of irrelevant alternatives (IIA): adding an option that is neither the right answer nor its strongest rival should leave the odds between those two unchanged. We test this in a pre-registered study of jev-1.13.0, a hosted decision model, using 299,463 calls on intent classification (BANKING77, CLINC150) and topic classification (AG News). Every added option moves the log-odds between the gold label and the strongest wrong label: the root-mean-square systematic shift is 0.20-0.52 nats, 1.2 to 3 times the run-to-run variation, even when the added option takes at most 0.1% of the probability mass. The shifts are directional. An option similar to the wrong answer draws probability from it (+0.10 nats toward the gold label). A paraphrase of the wrong answer produces an intermediate pattern: the gold label's probability is almost conserved, but the paraphrase receives only about a third of the wrong answer's share rather than the half that strict substitution implies. The distortion grows steadily with the logarithm of the number of options, reaching about 1 nat at 255 options, and off-domain padding pushes the odds against the gold label. In contrast, questions packed into one call do not interact (bias < 0.001). Scoring each option with an independent yes/no question removes the option-set dependence but picks the right answer 8.7 points less often. Jev sits between two local baselines, an option-independent scorer and a prompt readout. Its probabilities are best read as relative to the option list they were computed over.

We establish a parametric supercongruence related to Whipple's {}_5F_4 formula and Dwork's dash operation. As a typical consequence, we obtain the following result: for any prime pequiv3pmod4 and odd integer rgeq1, $ sum_{k=0}^{p^r-1}(8k+1)(frac14)_k^3(frac12)_k{(1)_k^3(frac34)_k}equiv 3p^r+27p^{3r}{4}H_{(p^r-3)/4}^{(2)}p^{r+3}, where (x)_n=x(x+1)\cdots(x+n-1) is the Pochhammer symbol and H_n^{(2)}=\sum_{k=1}^n1{k^2} is the n-th harmonic number of order 2$. This confirms a conjecture of Guo and Zhao [Forum Math. 38 (2026), 1099-1109]. Our proof rely on a new parametric WZ pair which allows us to transform the original sum to a computable form in the sense of congruence. Another essential ingredient of our proof involves the properties of Dwork's dash operation.

System One models answer typed questions in a single forward pass and promise the same answer for the same input. For choice questions, this promise fails: the answer depends on the order of the options, increasingly so as they grow in number, and averaging over their cyclic shifts reduces this dependence without removing it. Scoring each option alone, which the order cannot affect, shows that the order changes the answer mostly when the model is unsure among its top few options, even though the listwise answer looks confident. We therefore propose Runoff, which considers, then chooses: it scores each option alone to keep a few candidates, then compares them in listings where each candidate takes every position. It needs no training and can run in two forward passes, and its answer does not depend on the option order by construction. On two open models and five datasets, it changes no answer under shuffling, against 9.3% for cyclic averaging, and with 20 options it is about as accurate and three to four times faster. Wrapped around eight System One models, it removes the order dependence of the five benchmarks, while cyclic averaging changes up to 15.4%.

We study the Levine hat problem, a classic combinatorial puzzle introduced by Lionel Levine in 2010. This problem involves a game in which n geq 2 players, each seeing an infinite stack of hats on each of their teammates' heads but not on their own, must simultaneously guess the index of a black hat on their own stack. If one of the players fails to do so, the team loses collectively. The players must therefore come up with a good strategy before the game starts. While the optimal winning probability V_{n} remains unknown even for n=2, we make three key advances. First, we develop a novel geometric framework for representing strategies through measurable functions, providing a new expression of V_{n} and a unified treatment of the game for finite and for infinite stacks via integral formulations. Secondly, we construct a new strategy K_{5} that reaches the conjectured optimal probability of victory : 0.35. We also show that K_{5} is part of a larger class of strategies that allow us to improve current bounds and resolve conjectured inequalities. Finally, we introduce and entirely solve a continuous generalization of the problem, demonstrating that extending to uncountable hat stacks increases the optimal winning probability to exactly 1/2. This generalization naturally leads to a broader and smoother strategic framework, within which we also describe how to compute optimal responses to a range of strategies.

Post-hoc out-of-distribution (OOD) detection has garnered intensive attention in reliable machine learning. Many efforts have been dedicated to deriving score functions based on logits, distances, or rigorous data distribution assumptions to identify low-scoring OOD samples. Nevertheless, these estimate scores may fail to accurately reflect the true data density or impose impractical constraints. To provide a unified perspective on density-based score design, we propose a novel theoretical framework grounded in Bregman divergence, which extends distribution considerations to encompass an exponential family of distributions. Leveraging the conjugation constraint revealed in our theorem, we introduce a ConjNorm method, reframing density function design as a search for the optimal norm coefficient p against the given dataset. In light of the computational challenges of normalization, we devise an unbiased and analytically tractable estimator of the partition function using the Monte Carlo-based importance sampling technique. Extensive experiments across OOD detection benchmarks empirically demonstrate that our proposed ConjNorm has established a new state-of-the-art in a variety of OOD detection setups, outperforming the current best method by up to 13.25% and 28.19% (FPR95) on CIFAR-100 and ImageNet-1K, respectively.

In this paper we show that reporting a single performance score is insufficient to compare non-deterministic approaches. We demonstrate for common sequence tagging tasks that the seed value for the random number generator can result in statistically significant (p < 10^-4) differences for state-of-the-art systems. For two recent systems for NER, we observe an absolute difference of one percentage point F1-score depending on the selected seed value, making these systems perceived either as state-of-the-art or mediocre. Instead of publishing and reporting single performance scores, we propose to compare score distributions based on multiple executions. Based on the evaluation of 50.000 LSTM-networks for five sequence tagging tasks, we present network architectures that produce both superior performance as well as are more stable with respect to the remaining hyperparameters.

Closed-source frontier labs do not disclose parameter counts. Storing F facts requires at least F/(bits per parameter) weights, so factual recall lower-bounds parameter count--an intrinsic, serving-independent signal, though (as we show) a coarse one. We introduce Incompressible Knowledge Probes (IKPs), a benchmark of 1,400 factual questions spanning 7 tiers of obscurity, designed to isolate knowledge that cannot be derived by reasoning or compressed by architectural improvements. We score with no hallucination penalty (lambda = 0: IKP accuracy is simply the fraction of probed facts answered correctly), which removes both the penalty hyperparameter and the per-tier flooring choice; a full lambda x flooring ablation shows the calibration is robust across scoring choices while individual estimates are not, motivating the no-penalty default. We calibrate a log-linear mapping from IKP accuracy to parameter count on 93 open-weight models (135M-1,600B) spanning 19 vendors, achieving R^2 = 0.910; leave-one-out cross-validation confirms generalization (median fold error 1.48x, 72% within 2x, 86% within 3x). The instrument is deliberately coarse--its 90% prediction interval spans ~3x in either direction, wider than inference economics--so IKP recovers order-of-magnitude effective capacity and relative rankings, not precise parameter counts. For Mixture-of-Experts models, total parameters predict knowledge (R^2 = 0.67) better than active parameters (R^2 = 0.41). We evaluate 201 models from 27 vendors on a curated probe set (1,311 of 1,400 probes surviving name-collision and label-ambiguity filters) and report effective knowledge capacity for all major proprietary frontier models as prediction bands rather than point estimates; for heavily safety-tuned models these are lower bounds, since refusal policy can suppress tens of percentage points of otherwise-answerable capacity.

For a vertex v of a graph G, the triangle-degree td_G(v) is the number of triangles containing v. A graph is triangle-distinct, or K_3-irregular, if its vertex triangle-degrees are pairwise distinct. Chartrand, Erdős, and Oellermann asked whether a regular K_3-irregular graph exists. We prove that such graphs exist for every regularity rge 9. More precisely, for every integer kge 15 we construct a 2k-regular triangle-distinct graph on 4k+2 vertices; complementation gives a (2k+1)-regular example of the same order. The construction uses two antiregular threshold blocks joined by a zero-one matrix with prescribed margins, followed by matrix 2-switches that preserve those margins. After a uniform reference perturbation, exactly three triangle-degree collisions remain. Switches chosen according to parity remove two of them, and the last possible collision is controlled by a short quadratic discriminant argument. Odd kge 17 and even kge 62 are handled symbolically, while the remaining 24 values are settled by exact finite verification. Together with the known examples for 9le rle 29, this closes the positive existence problem for every rge 9.

In economy, markets are denoted as efficient when it is impossible to systematically generate profits which outperform the average. In the past years, the concept has been tested in other domains such as the growing sports betting market. Surprisingly, despite its large size and its level of maturity, sports betting shows traits of inefficiency. The anomalies indicate the existence of strategies which shift betting from a game of chance towards a game of skill. This article shows an example for an inefficiency detected in the German soccer betting TOTO 13er Wette, which is operated by state-run lottery agencies. Gamblers have to guess the outcome (win, draw, loss) of 13 soccer matches listed on a lottery tip. Applying stochastic methods, a recipe is presented to determine hit rates for single match outcomes. More important, the recipe provides the number of lottery tips required to achieve a specific number of strikes (number of correct match forecasts per lottery tip) for any given level of safety. An approximation is derived to cope with large numbers in hypergeometric distributions, valid under certain constraints. Overall, the strategy does lead to returns exceeding the aggregated lottery fees, resulting in moderate, but consistent profits. It is briefly discussed if lessions learned from soccer betting can be transferred back to financial markets, because gamblers and retail investors face similar challenges and opportunities.

We propose a new score-based model with one-step sampling. Previously, score-based models were burdened with heavy computations due to iterative sampling. For substituting the iterative process, we train a standalone generator to compress all the time steps with the gradient backpropagated from the score network. In order to produce meaningful gradients for the generator, the score network is trained to simultaneously match the real data distribution and mismatch the fake data distribution. This model has the following advantages: 1) For sampling, it generates a fake image with only one step forward. 2) For training, it only needs 10 diffusion steps.3) Compared with consistency model, it is free of the ill-posed problem caused by consistency loss. On the popular CIFAR-10 dataset, our model outperforms Consistency Model and Denoising Score Matching, which demonstrates the potential of the framework. We further provide more examples on the MINIST and LSUN datasets. The code is available on GitHub.

We introduce the first version of KWBench (Knowledge Work Bench), a benchmark for unprompted problem recognition in large language models: can an LLM identify a professional scenario before attempting to solve it. Existing frontier benchmarks have saturated, and most knowledge-work evaluations to date reduce to extraction or task completion against a specification. KWBench targets the step before that: recognizing the governing structure of the situation from raw inputs alone. The benchmark contains 223 tasks sourced from practitioners across acquisitions, contract negotiations, clinical pharmacy, organizational politics, fraud analysis, and incentive design. Each task encodes a formal game-theoretic pattern (principal-agent conflict, signaling, mechanism design failure, strategic omission, coalitional dynamics, strategic interdependence) and carries structured ground truth recording the expert reading of the situation and the anticipated failure modes. Models receive raw data and a task prompt with no indication of problem type. Scoring is a three-tier rubric gated by a mandatory conjunctive check. Mandatory criteria encode the predicted wrong paths. We evaluate 16 models. The best model passes on 27.9% of tasks. The top two models agree on only 31.7% of their passes. Among the top 8, 44 tasks are solved by exactly one model; routing across the top 8 covers 50.7% of the benchmark, nearly double the best single model. Conditional on passing, quality scores converge (approx 83% across models); unconditional scores do not. Same models articulate the relevant game-theoretic concept correctly when asked, then fail to apply it unprompted. We release KWBench to shift how frontier models are evaluated on knowledge work, scoring them on whether they recognize the right problem from the situation alone, not only on how well they execute once the problem has been framed for them.

We systematically study antithetic initial noise in diffusion models, discovering that pairing each noise sample with its negation consistently produces strong negative correlation. This universal phenomenon holds across datasets, model architectures, conditional and unconditional sampling, and even other generative models such as VAEs and Normalizing Flows. To explain it, we combine experiments and theory and propose a symmetry conjecture that the learned score function is approximately affine antisymmetric (odd symmetry up to a constant shift), supported by empirical evidence. This negative correlation leads to substantially more reliable uncertainty quantification with up to 90% narrower confidence intervals. We demonstrate these gains on tasks including estimating pixel-wise statistics and evaluating diffusion inverse solvers. We also provide extensions with randomized quasi-Monte Carlo noise designs for uncertainty quantification, and explore additional applications of the antithetic noise design to improve image editing and generation diversity. Our framework is training-free, model-agnostic, and adds no runtime overhead. Code is available at https://github.com/jjia131/Antithetic-Noise-in-Diffusion-Models-page.

Agentic on-policy distillation (OPD) trains a student agent on its own trajectories while a teacher scores every token. A natural explanation of where it goes wrong is compounding error: the student's mistakes carry its trajectories beyond the region where the teacher is reliable, so recent methods keep training inside that region, through episode-length curricula, teacher-guided turns, prefix replay, or teacher intervention where the two disagree. We show this explanation is incomplete. From the states where the student departs most from the teacher, the teacher's own takeovers still finish the task, yet the student trained on its token scores keeps running out of turns. We attribute this to the myopia of next-token scoring. The teacher rates each token without seeing where the episode ends, so its scores let through the safe, delaying moves that outcome-based RL would punish with zero reward, a penalty agentic OPD lacks. We call this difference between what the teacher does and what it scores the rollout-scorer gap. We propose Rollout-to-Scorer Distillation (R2S), which repairs the scorer before distillation by mining high-disagreement states from the untrained student, running teacher takeovers from them, keeping the states where enough takeovers pass a reward-free check such as consensus or an LLM judge, and fine-tuning the teacher to give its takeover-informed scores on every assistant token from the divergence turn onward, without the takeover in context. The repaired teacher is frozen as the sole scorer for standard OPD, requiring no task reward during repair or training. Across three environments, two teachers and four students, R2S outperforms three state-of-the-art agentic-OPD methods in all twelve teacher-student-environment cells, by 12.8 success points on average over the best of them in each cell. Project page: https://tree-shu-zhao.github.io/r2s/.

Selecting LLM-generated code candidates using LLM-generated tests is challenging because the tests themselves may be incorrect. Existing methods either treat all tests equally or rely on ad-hoc heuristics to filter unreliable tests. Yet determining test correctness requires knowing which codes are correct, creating a circular dependency. Our key insight is that we need not determine test correctness at all: test votes should rank, not merely count. What matters is not how many codes pass a test, but whether the test can distinguish correct from incorrect code. We break the circular dependency via leave-one-out evaluation: hold out one test, rank codes by their aggregate scores on all remaining tests, and measure whether the held-out test's pass/fail pattern agrees with this ranking. We formalize this agreement as the leave-one-out AUC~(LOO-AUC) and prove that the expected LOO-AUC is proportional to each test's ability to separate correct code from incorrect code. Building on this, we propose ACES~(AUC ConsistEncy Scoring) with two complementary variants: ACES-C provides closed-form weights that provably approximate the oracle in expectation under a mild assumption on average test quality; ACES-O drops this assumption and iteratively optimizes a differentiable LOO-AUC objective. Both operate solely on the binary pass matrix with negligible overhead, and achieve state-of-the-art Pass@k on multiple code generation benchmarks.

Can LLM agents make contributions to frontier computer science R&D? We introduce FrontierCS 2, a benchmark of 48 open-ended tasks. We take a paper and the code its authors released and delete the paper's contribution. The deleted code becomes the human reference that sets the bar. Earlier open-ended benchmarks give each agent a problem statement and grade one program after the run. FrontierCS 2 places the agent inside a research environment. The agent inherits a working repository and may submit a patch at any point. For each patch, a judge returns a development-suite score but keeps the evaluator, the held-out final suite and the reference private, and grades the final submission on that suite against the reference. The tasks span 13 domains, from machine-learning systems and high-performance computing to databases, security and robotics. Beyond these 48 tasks, FrontierCS 2 keeps a private set of more than 100 tasks, to be announced in the future. We evaluate 12 agents. The best one is Claude Opus 5.5, passing 58% of its runs. No run passes on 16 tasks. A higher score is not the same as a pass: most runs that beat the reference on the combined final score still fall short on some metric or workload. On a shared capability scale for agents and tasks, the median task is roughly as difficult as what the strongest agent can handle.

We consider a binary classification problem under group fairness constraints, which can be one of Demographic Parity (DP), Equalized Opportunity (EOp), or Equalized Odds (EO). We propose an explicit characterization of Bayes optimal classifier under the fairness constraints, which turns out to be a simple modification rule of the unconstrained classifier. Namely, we introduce a novel instance-level measure of bias, which we call bias score, and the modification rule is a simple linear rule on top of the finite amount of bias scores.Based on this characterization, we develop a post-hoc approach that allows us to adapt to fairness constraints while maintaining high accuracy. In the case of DP and EOp constraints, the modification rule is thresholding a single bias score, while in the case of EO constraints we are required to fit a linear modification rule with 2 parameters. The method can also be applied for composite group-fairness criteria, such as ones involving several sensitive attributes.

Outcome RL raises pass@1 but not pass@k at large k (Yue et al., 2025). But every pass@1 is scored under a token budget. Every recipe also trains under one. On physics and math problems at fixed token budgets, we compare the pass@1 of 20 released RL models (1.5B to 32B) with that of the thinking models they started from. We ask whether each pass@1 gain is changed behaviour or new knowledge. We divide each change in pass@1 into three parts. Finishing measures answering before the budget runs out. Reliability measures being right once finished. Reach measures problems RL solves that its base answered wrongly in all eight tries, minus problems the base solves that RL answered wrongly in all eight. We find four things. (1) Where the base often hits the token budget before answering, most of the change, gain or loss, is finishing (82% at 4k and 8k). Given more tokens, the base closes most of the finishing gap itself. (2) Where the base has enough tokens, the gain is reliability: the RL model is graded right more often, even on problems both finish. (3) By our definition, RL rarely solves what its base cannot. Reach adds at most 2.1 percentage points on any of our four physics and math sets. We checked 144 physics problems that only the RL model solved. Of these, 3.5% hold up when resampled or read. In a random sample, nine of ten new RL steps a human confirmed also appear in the base's samples. (4) In a small-scale RL run on an 8B model, the gain on new problems is finishing, with reliability flat. Supplying each problem's physical setup was not shown to improve on plain RL.

On-policy distillation (OPD) trains a student on its own responses with token-level feedback from a stronger teacher, yet the teacher can fail on questions the student already answers correctly, and how often each model succeeds varies across tasks. OPD ignores these outcomes and, on average, pushes down even the student's correct responses; gating feedback by student correctness fixes the direction but uses the teacher in the same way whether or not it succeeded. We introduce DuoOPD, in which the student's outcome sets the direction of feedback and the joint teacher-student outcome decides how the teacher supports it: when only the teacher succeeds, its verified answer becomes context for scoring the student's failed response, and when only the student succeeds, a weight shared within the task reinforces the whole response. A single rule covers all four outcome combinations without task-specific settings. Across Qwen3 and Llama, DuoOPD outperforms all five baselines in mean macro accuracy, improving over OPD by 2.58 and 5.98 percentage points, and it also leads on two further task mixtures spanning scientific calculation, instruction following, and code generation. Ablations show that outcome-based direction alone stays near the gated baseline, while the joint-outcome designs supply most of the gain.