Papers

Coding agents now sustain hours-long, tool-driven loops, yet their ability to carry long-horizon analog and mixed-signal circuits to electrical specification remains unmeasured. We introduce Analog Design Bench, a long-horizon agentic benchmark of 50 transistor-level design tasks contributed by 17 chip designers. Agents work with an open-source simulator, while an isolated verifier evaluates the submitted circuit using specification-based electrical tests. We evaluate 15 agent configurations across 2,250 two-hour attempts and observe full-specification pass rates from 8.0% to 78.0%. Coding-benchmark performance correlates with analog results but leaves much of the performance spread unexplained. Our failure analysis shows that most unsuccessful submissions have no recorded legality rejection but fail electrical acceptance, identifying electrical closure as the dominant endpoint challenge. We test time, reasoning effort, agent harness, and supplied design knowledge as interventions. Longer budgets and higher reasoning effort improve performance, while general skill documents provide little benefit and sometimes reduce performance. Supplying a task-matched reference topology, an idealized form of circuit-IP retrieval, raises DeepSeek V4 Pro by 18.7 percentage points and mainly accelerates GPT-5.6 Sol.

Masked diffusion language models (MDMs) admit flexible generation orders, making the unmasking strategy an inference decision. Existing methods vary in how they prioritize positions, control parallelism, restrict selection regions, revise predictions, or plan future denoising, yet it remains unclear when these choices should change during generation. We study this question through strategy reversals, where an alternative action becomes preferable to a fixed choice. We organize MDM inference into five axes–score, cardinality, region, commitment, and planning–and define adaptation opportunity as the one-step utility advantage of the best candidate action over a validation-selected fixed action. This view shows that adaptation value depends on both the frequency and magnitude of such reversals. Across three MDMs and ten tasks, adaptation opportunities are highly heterogeneous, with some regimes exhibiting concentrated and predictable one-step gains. This motivates selective adaptation: lightweight detectors calibrated on validation prompts identify high-opportunity states, capturing, for example, 56.9 percent of the candidate-set oracle opportunity by adapting only the top 10 percent of states on LLaDA-8B constrained JSON filling. Our transition-level results suggest that state adaptation is most useful when applied selectively rather than uniformly.

Sign data collection is costly, and teleoperation scales poorly, motivating reuse of large video corpora. Humanoid signing requires converting video-derived human motion into robot trajectories while preserving linguistic motion cues. Errors from fitting, human-motion repair, retargeting, robot geometry repair, and control are hard to separate from the final trajectory alone. We introduce HumanoidCSL-20K, a dataset and benchmark of 20,648 sentence-level Chinese Sign Language sequences, each with four aligned versions: the source, the repaired human motion, the direct robot reference, and the geometry-repaired robot reference. Observation-supported local human-motion repair, full-robot geometry repair, and cross-representation provenance make each transformation traceable. Paired evaluations measure human-motion continuity and content preservation, robot-reference feasibility, and physical execution. A sign-specific kinematic-reference protocol scores handshape, location, palm orientation, and inter-hand relation over the full planned motion. Full-corpus results show fewer abnormal arm / hand steps and less inter-hand and hand-body penetration after repair. Control experiments separate reference learnability from curriculum effects, while component scores expose remaining execution errors.

Multi-class anomaly classification in industrial vision remains challenging due to noisy/incomplete anomaly representations and the unknown number of anomaly classes. To overcome this, we propose MACO, a novel multi-class anomaly classification framework that learns comprehensive representations and dynamically estimates class number without prior knowledge. First, a soft-focus attention uses anomaly maps to concentrate on relevant abnormal regions, while suppressing background noise. Second, auxiliary classification ([A-CLS]) tokens complement the [CLS] token. They collectively attend to diverse anomaly sub-regions, yielding more holistic and discriminative features. These [A-CLS] tokens are also effective across more tasks and domains. To infer the class number, we propose Correlation-based Number Estimation strategy. It computes the average correlation among labeled classes and transfers its separability cue to the unlabeled set. Experiments on MVTec AD and MTD datasets demonstrate our superiority. Under known class number, MACO improves ARI by 6.5% and $\textbf{16.3%}$ on both datasets, respectively. In the more challenging unknown number scenario, it achieves an $\textbf{11.2%}$ NMI gain on MTD and outperforms existing number estimation strategies by $\textbf{24.1%}$ UPS on MVTec AD. Code will be released at https://github.com/HUST-SLOW/MACO.

Set-valued classifiers, whether derived from precise probabilities and an adapted cost function, from convex sets with a robust inference mechanism, or from conformal methods, are routine options to obtain more robust, trustworthy predictions. However, there is a lack of operational tools to measure how robust or imprecise a given user is ready to be when receiving predictions, that is how much precision he/she is ready to let go in exchange of more accuracy. This is why we propose, in this paper, practical and operational elicitation procedures to measure the user proneness to set-valued predictions. The effectiveness of the iterative elicitation procedure in converging to the target parameter value is demonstrated on both tabular and image datasets drawn from standard machine learning benchmarks. The results show that the procedure also presents the user with a small number of instances, highlighting the practicality of the approach for real-world applications aimed at identifying the decision maker's optimal behavior when faced with imprecision.

Fidelity-based quantum retrieval ranks candidates by the fidelity between query and archive states. Applying a shared input-independent unitary after fixed state encoding leaves that fidelity unchanged, so training the circuit cannot alter the ranking. Quantum parameter-efficient input-dependent retrieval adaptation (QuPID) repairs this by making the circuit input-dependent through data re-uploading and by comparing measurement readouts, vectors of local Pauli expectations, rather than states. The result is a small readout for adapting frozen image features to a local archive with limited data: training simulates the circuit classically, and inference runs on a GPU with fixed learned parameters. We characterize the class as a structured factorization of input-modulated quadratic feature maps, bound the frequency support of its re-uploading channel, and give a parameter-count generalization bound that motivates its small budget. Under a shared frozen backbone and a label-free protocol, QuPID's 60 parameters give higher precision-at-5 (P@5) on ChestX-ray14 and MURA than frozen medical encoders, and than adapters and low-rank adaptation (LoRA) with up to 5.25 million trainable parameters. On ChestX-ray14, the P@5 gain over the frozen encoder is +0.116, the lead over retuned adapters is widest at 512 adaptation examples (+0.040), and the full-budget margin over an equally compact classical rotation-plane head is +0.023 with a 95% interval excluding zero. Medical imaging is the primary testbed; the pattern recurs on two non-medical benchmarks, in report generation, and under simulated gate noise and finite-shot readout.

Learning population dynamics from temporally sparse, unpaired distribution snapshots is a fundamental challenge in developmental biology. Recent approaches based on neural differential equations and flow matching can interpolate between observed population snapshots, but may struggle to extrapolate beyond the training horizon and often lack an explicit mechanism for modeling developmental branching. We propose KoopCell, a unified generative framework based on Koopman-Mori-Zwanzig theory that jointly learns representations and predictive linear latent dynamics. Theoretically, using the weak continuity equation, we derive a closed-form least-squares estimator for the Koopman generator from distribution snapshots and establish convergence guarantees under suitable assumptions. To model branching dynamics, we further develop KoopCell-M, which incorporates non-Markovian memory into the latent Koopman dynamics through a Markovian embedding. Experiments on synthetic systems and three scRNA-seq datasets demonstrate the ability of our framework to recover Koopman spectra, model branching through memory, and scale to predicting high-dimensional gene expression distributions, achieving state-of-the-art performance among the evaluated methods.

Streaming data, characterized by high volume, rapid arrival rates, and evolving distributions, have become increasingly prevalent in modern applications. Developing efficient and reliable estimation procedures is therefore essential for real-time statistical analysis. However, most existing online estimation methods rely on the assumption of batch homogeneity, which can be violated in practice due to abnormal batches, distributional shifts, or other forms of batch heterogeneity. To address this challenge, we develop renewable online expectile regression procedures for heterogeneous streaming data. Specifically, we propose two complementary strategies for handling abnormal batches: (1) a detection-based approach that employs a sequential monitoring mechanism based on score test statistics to identify and remove potentially abnormal batches; and (2) an adaptive-weighting approach that assigns data-driven weights to incoming batches, reducing the influence of abnormal or drifting batches while retaining information from reliable observations. Both strategies rely solely on score test statistics and can be seamlessly integrated into existing renewable estimation and inference frameworks without requiring additional structural assumptions. Furthermore, to enhance robustness against heavy-tailed errors and outliers, we replace the conventional l2 loss with the Huber loss and develop a robust extension of renew?able online expectile regression. Extensive simulation studies and analyses of clinical datasets demonstrate that the proposed methods achieve improved estimation accuracy and robustness in the presence of batch heterogeneity. Overall, the proposed framework provides a flexible and effective solution for renewable expectile regression in complex streaming data environments.

We prove that the simple affine vertex algebra $L_k(\mathfrak{sl}_2)$ is classically free at the boundary admissible level, which confirms a conjecture of Andrews, van Ekeren, and Heluani.

World models can be studied as experimental systems with response laws of their own. We introduce MultiEcho, a framework for estimating these laws through controlled counterfactual interventions, delimiting their applicability, and separately testing their physical correspondence. Across nine simulated physical systems and seven frozen model configurations, three-reference estimators predict complete intervention responses and recover intervention parameters. Estimator selection uses discovery data only; frozen fits are evaluated on validation and confirmation contexts. The experiments distinguish response predictability, intervention readability and physical accuracy. Responses can be locally describable yet poorly match physical effects in the same target coordinates. Event-window, visibility and camera interventions reveal conditional applicability, and paired generator configurations show reduced readability under a scene prompt with stronger guidance. Magnitude sweeps expose small image errors alongside large relative effect errors. An exact-reset material experiment separates registered visible-response success from fixed-readout failure on material-dependent futures at matched positions and velocities. Exact finite-scale identities resolve odd and even response errors; first-order remainder bounds specify when refined calibration converges. MultiEcho provides an experimental basis for studying learned-world laws independently of, and in relation to, physical laws.

Sign-aware recommender systems have recently been developed to leverage negative feedback for a deeper understanding of user preferences. However, our empirical diagnosis reveals that state-of-the-art graph-based sign-aware recommender systems are paradoxically valence-blind. Even though they explicitly incorporate sign information during training, they consistently fail to differentiate liked items from disliked ones at the ranking stage, frequently infiltrating top-K recommendations with disliked content. Through linear probing, we show that while valence information exists in the learned embeddings, it remains inaccessible to the inner-product scoring function. This widespread failure remains entirely undetected because conventional evaluation metrics, such as Recall, HR, and NDCG, assign a uniform utility of zero to both negative and unobserved items, creating a systematic evaluation blind spot. To bridge this gap, we propose a family of signed metrics, Signed Recall, Signed HR, and Signed NDCG, that explicitly penalize the recommendation of disliked content. Systematic re-evaluation under our proposed metrics fundamentally reshapes the established performance landscape, revealing that methods ranked highly under conventional metrics often fail to protect users from disliked content. Finally, through a proof-of-concept auxiliary loss, we confirm that the proposed metrics provide actionable training signals, guiding models toward valence-aware behavior without sacrificing conventional relevance. For transparency, our source code is available at: https://anonymous.4open.science/r/signed-rec-benchmark-07E4

Multilingual self-supervised speech models can benefit from sharing information across languages, but under a matched total pretraining data budget they still fall short of monolingual models. We show that strengthening the model's ability to discriminate languages during pretraining reduces and, on some measures, closes this multilingual gap on continuous phonetic and higher-level linguistic measures, while preserving substantial cross-language sharing. Using a controlled English/French HuBERT setting, we test two interventions which strengthen language discrimination: an auxiliary language classifier and per-language k-means targets. Across interventions, continuous-feature phone discrimination error (phone-ABX, lower is better) decreases from 11.6% in the bilingual baseline to 10.4% (monolingual: 10.8%), while lexical performance (sWUGGY, higher is better) increases from 52.1% to 56.7% (monolingual: 58.5%) and prosodic performance (ProsAudit, lexical subtask, higher is better) from 68.9% to 72.9% (monolingual: 72.6%). Across HuBERT training stages, the strongest gains on most linguistic measures occur when language discrimination is introduced in the first iteration, whereas later or repeated interventions yield smaller improvements and are accompanied by increased language-wise segregation. These results support a causal role for language discrimination in reducing the additional cost of multilingual learning.

Scene Coordinate Regression (SCR) has recently emerged as a promising approach for LiDAR-based localization, achieving accurate localization without requiring an explicit 3D map. Despite their effectiveness, existing SCR methods rely on scene classification-based global embedding that struggles to provide fine-grained discrimination among nearby locations. Moreover, their reliance on uniform sampling of local features during training assigns equal importance to all points, thereby inadvertently propagating features from dynamic objects or unstable regions and potentially degrading training stability. In this paper, we present ReLoc, a revamped SCR architecture that can effectively address these limitations. First, we redesign the global embedding module by combining learnable context tokens with a feature aggregator to capture richer and more discriminative scene context. Second, we introduce an attention-based local feature enhancement module to mitigate the impact of noisy local features while encouraging context-consistent structures, yielding more robust local feature representations. Experimental results on two large-scale outdoor datasets demonstrate that our approach achieves state-of-the-art accuracy over previous SCR-based methods while maintaining real-time inference performance.

Faithfulness evaluation of abstractive summaries remains an open challenge, with existing metrics addressing only isolated hallucination types: factual entity errors, relational inconsistencies, or numerical fabrications, without capturing their co-occurrence or interaction. We introduce CHI (Composite Hallucination Index), the first unified hallucination metric that decomposes faithfulness errors into three orthogonal dimensions: entity hallucination (EHI), relation hallucination (RHI*), and quantity hallucination (QHI). Each dimension employs a shared softmax-normalized architecture over Venn diagram-derived factors representing extractiveness, positive hallucination, over-focus, negative hallucination, and lost focus. The novel QHI component introduces tolerance-aware numerical matching with exact, epsilon, derived, and temporal comparison modes. We fuse the three dimensions via harmonic mean to produce a single composite score that penalizes weakness in any dimension. We validate CHI on 800 source articles spanning four domains (news, medical, legal, financial) with summaries from five generation systems. Empirical results demonstrate that: (i) the three dimensions are statistically orthogonal (mean rho = 0.148), confirming they capture distinct error types; (ii) CHI achieves the highest system-level correlation with human judgments (rho = 0.66, p = 0.006) on SummEval, outperforming ROUGE (rho = 0.53), EHI (rho = 0.58), and all individual components; and (iii) ablation studies confirm that all three dimensions contribute unique variance, with the full composite outperforming any individual component while providing decomposable error diagnostics unavailable from single-score baselines. CHI provides practitioners with a decomposable, interpretable, and efficient faithfulness metric suitable for both offline evaluation and online monitoring of summarization systems.

Molecular property prediction is central to drug development and materials discovery, but experiments are costly and labeled data are scarce. Context-aware methods use auxiliary assay labels to support few-shot prediction, and recent work supervises property relations with label agreement. However, label agreement is sensitive to class marginals and does not directly capture dependence between properties. We propose CalibHyper, a chance-corrected relational hypergraph method based on the joint label distribution. CalibHyper subtracts an independence baseline from the ordered four-state label distribution and shrinks the residual according to the number of joint observations. A swap-equivariant relation head estimates these residuals, which choose the auxiliary properties for each molecule and set the sign and weight of their hyperedge messages. On thirteen datasets from five benchmarks, in both 1-shot and 10-shot settings, CalibHyper and its ablation settings achieve ROC-AUC competitive with the strongest reported results.

The tokamak, a toroidal magnetic device confining a hot plasma, is one of the most advanced approaches to fusion energy. A central obstacle is turbulent transport driven by the ion-temperature-gradient (ITG) mode. Operation scenarios that rely on a weak or zero magnetic shear core, such as the ITER hybrid scenario, have achieved markedly improved confinement, yet the underlying mechanism has remained unclear. Here we show that the radial width of ITG poloidal harmonics is constrained not only by the familiar parallel Landau damping, but also by the isotropy of micro-turbulence in the plane perpendicular to the magnetic field. In the weak-shear limit, micro-isotropy dominates and constrains the radial width at one poloidal wavelength, yielding a critical magnetic shear $s_{\text{crit}} \approx 1/(2\pi)$. Above this threshold, the extended Type II ballooning modes prevail; below it, the localized Type I modes emerge, comprising only two or three harmonics. Global gyrokinetic simulations spanning DIII-D, JET~and ITER parameters confirm this geometric criterion. Nonlinear simulations with sustained heating show that weak-shear plasmas spontaneously form internal transport barriers with the turbulence suppressed in the radial region determined by $|s| < s_{\text{crit}}$. This geometric criterion defines the weak-magnetic-shear regime and offers a fresh perspective on turbulent transport in fusion plasmas.

Contextual specialization can improve forecasting accuracy, but a correction selected on one historical interval may become unreliable under temporal distribution shift. To address this issue, we propose RAEGL, a Risk-Aware Evidence-Gated Learning framework for selective contextual forecasting. RAEGL retains a validated global predictor by default and activates a contextual residual only when pre-deployment evidence supports its use. The framework separates candidate selection from gate calibration and jointly evaluates randomization significance, practically meaningful gain, and temporal stability. Experiments on real-world audits and controlled panels show how RAEGL can prevent harmful contextual deployment while making conservative opportunity costs explicit. In a reconstructed Our World in Data audit, exact fallback avoids RMSE degradations of 0.0960 and 0.0239 caused by two validation-selected corrections. In a sealed World Development Indicators evaluation, a region-based correction passes the randomization test but is withheld because its gain is only 0.000092, its country-clustered 95% confidence interval crosses zero, and only 0.02% of bootstrap replicates reach the practical threshold. In controlled panels, the stability- and support-aware extension activates in 97.2% of strong, stable-context runs while rejecting all high-drift settings. These results support RAEGL as an auditable, evidence-based mechanism for managing contextual deployment risk and as a conservative alternative to validation-driven contextual selection.

Sign Language Production (SLP) aims to generate sign motions from text. Conditional Flow Matching methods have achieved strong performance in SLP by constructing conditional paths that transform a source distribution into a target distribution. However, existing methods construct these paths via linear interpolation, whereas the rotational geometry of human joints confines valid joint rotations to a manifold embedded in Euclidean space. Consequently, linear interpolation between two sign motions leaves this manifold and ignores the motion distribution on it. In this paper, we revisit SLP from the perspective of manifold transport and propose a Naturalness-guided Manifold Flow Matching framework, termed SignNMFlow, which constructs conditional paths directly on the motion manifold by jointly considering geometric efficiency and the motion distribution. Specifically, we exploit the intrinsic geometry of the manifold and introduce a motion naturalness measure to characterize the motion distribution. By minimizing the kinetic energy under this measure, we learn a naturalness-guided interpolation that couples a closed-form geodesic, which provides geometrically efficient transport, with a learnable deviation that incorporates the motion distribution, thereby significantly improving the fidelity of generated sign motions. Extensive qualitative and quantitative evaluations demonstrate the effectiveness of this work.

Referring video segmentation with heterogeneous multimodal queries—spanning text, audio, and reference images—demands both robust cross-modal understanding and precise spatio-temporal reasoning. We propose OPERA (Omnimodal Progressive spatio-tEmporal Reasoning Agent), a unified reasoning agent built on a single MLLM that performs dual-axis progressive reasoning via three specialized stages. Along the temporal axis, a Temporal Reasoning Agent narrows the frame search space through coarse-to-fine filtering to identify the most informative key frame. Along the spatial axis, a Distillation Agent establishes what to locate via cross-modal semantic distillation, and a Grounding Agent enhanced with GRPO determines where the target appears, with dense mask propagation completing the pixel-level output. OPERA sets a new state of the art on OmniAVS and Ref-AVS and transfers zero-shot to standard referring video segmentation benchmarks.

Online safe reinforcement learning (RL) seeks policies that maximize reward while satisfying safety constraints. A popular line of research in safe RL relaxes safety to a soft expected-cost constraint and solves the resulting Constrained Markov Decision Process via primal-dual Lagrangian updates that only enforce safety on average. To address this limitation, hard, state-wise constraints are introduced and often imposed through Hamilton-Jacobi (HJ) reachability. Yet such constraints require solving different objectives in the feasible and infeasible regions: reward maximization in the former, recovery toward the feasible regions in the latter. The resulting target action distributions are inherently multimodal, and this structure poses a fundamental challenge for the Gaussian or deterministic actors used in existing HJ-based safe RL, which often collapse onto suboptimal modes. Diffusion policies provide the expressiveness needed to represent such distributions, and recent work on Q-score matching offers a route to training them for online RL by score regression – but has been applied only to reward maximization. We propose Safe Score Matching (SSM), an off-policy actor-critic method that adapts Q-score matching to hard-constrained safe RL by gating a two-branch score target with HJ reachability: inside the feasible set, the denoising process degenerates to Q-score matching on actions classified as viable by the HJ critic; outside, a recovery branch biases denoising toward regions with lower worst-case violation. On quadrotor and fixed-wing trajectory-tracking and stabilize-and-avoid benchmarks, SSM attains the best or near-best task performance with low false-safe rates, whereas the primal-dual baseline admits more unsafe behavior and reachability-based baselines tend to be more conservative; on Safety-Gymnasium velocity tasks, SSM attains the lowest cost with competitive reward.

Model-based imitation learning (MBIL) improves real-environment interaction efficiency by optimizing policies on imagined rollouts from a learned world model. However, the gap between model-induced and real-environment occupancies makes policy learning sensitive to model error. Conservative MBIL mitigates model exploitation during policy optimization, but when real-environment interactions are collected by the same conservative policy, uncertain regions around the expert distribution remain insufficiently sampled. Generic uncertainty-driven exploration, on the other hand, may allocate interaction to novel but task-irrelevant dynamics. We propose REcoverability-CONditioned Exploration for Model-Based Imitation Learning (RECON). RECON separates conservative policy learning from active data collection by maintaining a main policy for task execution and an explorer for real-environment interaction. The explorer is optimized based on epistemic uncertainty conditioned on recoverability estimated from multi-step main-policy imagination, focusing data collection on unknown states from which the main policy can still return toward expert behavior. Experiments on locomotion, navigation and manipulation show consistent gains in interaction efficiency, imitation performance, and robustness, indicating that RECON directs real-environment interaction toward recovery regions around the expert distribution that are underexplored by prior methods, and thereby learns a world model better suited for imitation.

Predicting how an environment will change before acting is a natural route to better decision making for agents. Recent post-training methods therefore require agents to predict the next observation and turn that prediction into a reward or a direct supervision signal, which is called world model. Existing next-observation training methods help the agent to learn the environmental content. However, they additionally involve an optimization process, which may introduce several effects other than learning to predict the world. Consequently, where the performance gain comes from during the training process remains an open question. We answer this research question through replacing true next-observation targets with in-distribution mismatched observations during the training process. Across two interactive text environments, mismatched targets lower prediction accuracy by 15.3-61.6% relative to ground-truth targets, yet retain substantial task gains over the base model. Compared with the base model, trained models consider more candidate actions and exhibit less looping. We also introduce a setting that replaces prediction-based rewards with independent random signals. This training expands task coverage (pass@64) even when the reward carries no environment information. We also generalize this finding to VisualWebArena, where random-reward training raises pass@64 by 14.3% relative to the base model, without observation-matching rewards or an external multimodal teacher for reward construction.

Training-free KV-cache compression methods such as SnapKV, H2O, and PyramidKV evict tokens at the end of prefill, aiming to preserve the attention mass that future queries are expected to use -optimizing what to keep. We show that this objective fails in two distinct ways. (1) Compensation: restoring the evicted attention mass can recover the attention-level target without recovering task quality. (2) Selection: covering more of the true decode-query mass can hurt quality when the recovered mass is fragmented rather than concentrated in coherent spans. These failures share a common cause: eviction occurs before the queries that determine the answer trajectory exist. We propose Draft-Guided Eviction (DGE), which defers eviction until after drafting the first k=2 answer tokens using the full cache - just one decode step beyond prefill. Because the draft is generated from the answer's own prefix, no cache entries are discarded before this trajectory signal becomes available. The per-head cache budget remains unchanged, and DGE can be applied directly to SnapKV, PyramidKV, H2O, and StreamingLLM without modifying their eviction scores. Unlike extra-pass methods, DGE changes when eviction occurs rather than what cache entries are selected. Extensive experiments demonstrate that DGE outperforms prior methods at every evaluated budget on five of six instruct-tuned backbones, achieving 44.2 on LongBench, nearly matching FullKV at 44.3. The timing-only control DGE-W achieves the same score, demonstrating that the gain comes from when eviction occurs rather than what is selected - an effect we term trajectory anchoring.

An outerplanar graph is maximal if no edge can be added without losing outerplanarity. Lin and Ning determined the outerplanar graph with the largest spectral radius, and the maximizer is a maximal outerplanar graph. We determine the minimizer. In this paper, we prove that every $n$-vertex maximal outerplanar graph $G$ satisfies $\rho(G)\ge\rho(F_n)$, where $F_n$ is the zig-zag triangulation of the $n$-gon, that is, the square of the path on $n$ vertices, with equality if and only if $G=F_n$. The proof uses three local operations on maximal outerplanar graphs, each of which strictly decreases the spectral radius: the first reverses the way a piece is attached along a chord, and the second and third move a piece from one vertex to its twin across a chord when the twin carries nothing or a single ear, respectively. A graph at which no operation applies is $F_n$, or has spectral radius greater than $4$, or consists of a central triangle with three zig-zag blades of at least three triangles each and has at most $15$ vertices; in the last case it contains one of two explicit graphs on $12$ vertices whose spectral radius exceeds that of $F_{15}$. Since $\rho(F_n)<4$ for all $n$, this completes the proof. The numerical inequalities used along the way are certified by explicit integer vectors with small entries.

Leading multi-bit watermarking methods for language models encode messages by biasing the model's next-token probabilities, creating a trade-off between message recovery and text quality. Their decoders typically return the highest-scoring candidate from accumulated token-level evidence, without a certified abstention rule that bounds the probability of outputting an incorrect message. We introduce CertMark, a distribution-preserving multi-bit watermark with certified decoding. Rather than modifying probabilities, CertMark uses the embedded message to seed an exact Gumbel-max sampler, thereby preserving the model's original sampling distribution. We propose two scalable decoders: a model-agnostic, text-only decoder and a model-aware variant that leverages the original next-token distributions for stronger recovery. Both support certified abstention with mathematical bounds on the probability of returning an incorrect message. Across text completion, summarization, and story generation, CertMark matches the perplexity of unwatermarked text while reliably recovering multi-bit messages. The model-aware decoder further achieves higher bit accuracy than probability-biasing baselines. Our code is publicly available at https://github.com/Batorskq/CertMark.

We give a direct, self-contained construction of the three sporadic Fischer groups $\mathrm{Fi}_{24}'$, $\mathrm{Fi}_{23}$, and $\mathrm{Fi}_{22}$ from the $783$-dimensional Conway-Parker algebra. We prove that its distinguished roots define involutory algebra automorphisms whose projective actions generate the full Fischer $3$-transposition group $\mathrm{Fi}_{24}$. Its commutator subgroup gives $\mathrm{Fi}_{24}'$, while $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{22}$ arise as centralizer quotients associated with one and two commuting transpositions. The root and frame geometry determines the group orders and leads to elementary proofs of simplicity, as well as natural rank-three actions and nonsplit central extensions. The construction uses standard facts about the Golay code, Parker's loop, and $M_{24}$. It does not use the Monster or previously known existence or order results for the Fischer groups. Fischer's classification and later recognition theorems are used only for the final identification.

Feed-forward 3D Gaussian Splatting (3DGS) enables efficient novel-view synthesis from sparse multi-view images, yet its representations remain costly to store and transmit. Existing approaches compress either the input images, incurring heavy receiver-side reconstruction, or the reconstructed Gaussian primitives, which are difficult to compress due to their heterogeneous and irregular attributes. We instead compress compact intermediate features, providing a better balance between compression efficiency and receiver-side complexity. Based on this paradigm, we propose FeCoSplat, a feedback-guided compression framework for feed-forward 3DGS. FeCoSplat first compresses multi-view features to obtain an intermediate 3DGS, whose rendered views are used as feedback to guide a second-stage compression for further refinement. The resulting bitstreams are decoded into a compact implicit state, from which the final Gaussian primitives are reconstructed with a lightweight predictor. Experiments demonstrate that FeCoSplat achieves favorable rate–distortion performance, particularly at low bitrates, while requiring only 3.45M parameters for receiver-side Gaussian reconstruction. Code will be released soon.

We develop a discrete Morse-Bott theory for Lefschetz complexes with real-valued incidence functions and finitely many cells in each dimension. Our main result is a reduction procedure, based on four elementary operations called Moves, for computing the remainder series $R_t$ associated with the Morse-Bott inequality. These Moves preserve $R_t$, and the procedure recursively constructs, using finitely many Moves in each dimension, a disjoint union of two-cell elementary blocks with vanishing Poincaré series. The resulting decomposition computes the coefficients of $R_t$ by counting the blocks in the corresponding dimensions. The same procedure also computes the Betti numbers of the original complex by counting the cells removed as isolated cells. As consequences, we obtain the nonnegativity of the coefficients of $R_t$ and the Morse-Bott inequality for Lefschetz complexes. The method applies in particular to CW complexes, where the nonnegativity is obtained by identifying each coefficient of $R_t$ with the number of elementary blocks in the corresponding dimension.

Whether orthogonalising the moment basis changes a central-moment lattice Boltzmann scheme is decided by the geometry of the velocity set. The basis enters the collision only through the conjugate $\mathit{T}^{-1} \Lambda \mathit{T}$, so two bases related by a constant matrix $\mathit{A}$ define exactly the same scheme if and only if $[\Lambda,\mathit{A}]_{ij}=\mathit{A}_{ij}(\lambda_i-\lambda_j)$ vanishes, and on physical states only its columns for the non-conserved moments matter. We prove that on lattices with velocities in $\{0,\pm1\}^d$ and symmetric weights orthogonalisation reaches the shear moments only through the face and body diagonals: whatever the orthogonalisation, the shear rate is free on D2Q9 and D3Q15, whereas on D3Q19 and D3Q27 the scheme changes whenever the fourth-order moments do not relax at the shear rate. On rectangular lattices, where the symmetry is broken, an independent bulk rate yields an orthogonal counterpart that does not share the transport coefficients of the published scheme, which is itself orthogonal in the inner product of the rest equilibrium up to the one relaxation entry that consistency requires. Where the formulations differ but share their hydrodynamics, a basis orthogonal in the inner product of the rest equilibrium, which guarantees linear stability at rest, was never less robust in the least stable flow direction by more than a few per cent, often much more robust, and the more accurate against a Taylor-Green DNS. Computed through the non-orthogonal transforms with a conjugated relaxation matrix, it costs at most $2\%$ more on D3Q27.

We prove an incidence bound for bipartite graphs on finite subsets of $\mathbb{F}^2\times \mathbb{F}^2$ defined by Boolean combinations of polynomial equations of bounded degree. If such a graph is $K_{k,k}$-free and its vertex classes have sizes $m$ and $n$, then it has $O_{t,k}((mn)^{2/3}+m+n+mn/p)$ edges, where $t$ bounds the description complexity, $p$ is the characteristic of $\mathbb{F}$, and $1/p=0$ in characteristic zero. We also prove this bound for incidences between points and distinct geometrically irreducible components of a two-parameter polynomial family, allowing singular and nonreduced members. The proof extends Lewko's interpolation and contact-multiplicity method from lines to algebraic families. Applications include rich components, polynomial values on difference sets, and polynomial expansion.