Papers

Let $I=(a,b)$ be an open interval with finite or infinite endpoints. For $1<p_1,p_2<\infty$ and $0<q<\infty$, we study the weighted bilinear Hardy operator $ \mathcal H_2(f,g)(x)=(\int_a^x f)(\int_a^x g) $ from $L^{p_1}(v_1;I)\times L^{p_2}(v_2;I)$ to $L^q(u;I)$, including the quasi-Banach target range $0<q<1$. With $\vartheta=\min\{1,q\}$, the target and bounded-bilinear-map spaces are treated through their complete powered metrics. Under explicit local integrability assumptions on the weights, compactness is equivalent to boundedness and vanishing endpoint-tail quasi-norms. Whenever the full operator is bounded, the $\vartheta$-power of the directed combined-tail limit equals exactly the powered essential distance, finite-rank distance, and Hausdorff measure of noncompactness in the target metric. Equivalently, the unpowered quantities agree under the stated quasi-norm-radius normalisation. Piecewise-constant middle-window approximants give quantitative bounds for the bilinear approximation numbers. The finite-rank-distance identity then implies that these numbers converge to the essential (quasi-)norm. Applying published norm-equivalent boundedness characteristics to endpoint-truncated output weights yields explicit compactness criteria in the convex, mixed, moderately subcritical, and deeply subcritical regimes. Algebraic companion exponents are used to make the negative powers arising for $q<1$ explicit. Four worked examples include an asymmetric mixed-range threshold and a nonlocally convex $L^{1/2}(0,1)$ example for which $a_{m+1}^{(2)}(\mathcal H_2)=O(m^{-1/2})$.

We study how well simple mechanisms approximate gains from trade (GFT) in two-sided matching markets with independent buyer values and seller costs, where feasible outcomes form an arbitrary downward-closed family of matchings. This model includes bilateral trade and double auctions as special cases. We focus on the Generalized Random-Offerer (GRO) mechanism, which is an equal mixture of the Generalized Sellers-Offering Mechanism (GSOM) and the Generalized Buyers-Offering Mechanism (GBOM). We determine GRO's exact worst-case approximation ratio with respect to first-best GFT, showing that it is $e$. This improves the previous $3.15$ approximation guarantee (Babaioff et al. STOC 2026) for the same mechanism. In bilateral trade, the result implies a $1/e$ guarantee for the random-offerer mechanism, improving the previous $1/\pi$ bound (Jo 2026). Our analysis uses a two-dimensional inequality in quantile space to compare first-best GFT with optimal one-sided auction profits. For each potential trade, we identify the region of buyer values and seller costs for which the first-best allocation selects that trade, and then randomly shrink this region in quantile space, yielding posted-price rules whose expected profits can be evaluated exactly. We establish tightness of the $e$ approximation by constructing markets with regular type distributions, pairwise disjoint trading edges, and a single knapsack constraint. On these instances, GRO's expected GFT approaches a $1/e$ fraction of first-best GFT, therefore ruling out any better guarantee even under these restrictions.

Structured additive distributional regression flexibly relates all parameters of a conditional response distribution to covariates, but multivariate extensions remain challenging when dependence is complex. We propose a multivariate structured additive distributional regression model based on regular vine copulas. Different vine edges may use different pair-copula families, while every pair-copula parameter may vary with covariates through a structured additive predictor. The model therefore accommodates heterogeneous marginal distributions together with pair-specific asymmetric, tail-dependent, and covariate-dependent dependence structures. For scalable inference, we develop a tree-wise stochastic variational inference procedure based on component-specific Gaussian variational approximations. Marginal models are estimated first, followed by pair-copula regressions sequentially along the vine trees. We also adapt sequential vine selection by fitting candidate covariate-dependent pair-copula regressions and using information criteria based on effective degrees of freedom for both family and tree selection. In simulations, the tree-wise estimator remains close to an oracle using the true recursive conditional inputs, whereas global refinement yields more concentrated approximations and lower frequentist coverage for upstream components. An application to six-dimensional meteorological data from the Netherlands yields a vine combining Gaussian and non-Gaussian pair copulas, with pronounced nonlinear spatial and temporal variation in dependence.

IoT terminals served by LEO satellite constellations transmit data to passing satellites in discrete uplink windows. That data is delivered to buyers only when the satellite reaches a ground station. This store-and-forward structure makes the achievable price a discontinuous function of the delivery completeness threshold (a phenomenon we call a pricing cliff) and creates a geographic pricing asymmetry between polar and equatorial terminals that depends on constellation size. We present an integrated pricing model combining a full SGP4 orbital simulation of the deployed Kinéis constellation with a buffer flow model and a value decay framework with decay constants derived from operational decision-window timescales in the application literature and anchored to observed satellite imagery market prices. We show that polar and equatorial terminals are dominated by structurally different latency components requiring different infrastructure interventions, formally characterize the pricing cliff phenomenon and its commercial significance, and derive a geographic pricing premium that is exactly independent of the residual value assumption. Going from 1 to 25 satellites reduces the polar–equatorial premium by 25x, with diminishing returns beyond n=10; at full constellation size the premium falls below 1% of peak data value and remains below 1.5% across the full plausible range of buyer urgency.

The classical information bottleneck (IB) measures the relevance of a representation $U$ of $X$ to a target $Y$ by $I(U;Y)$, which does not directly characterize the error of downstream decisions. For a binary hypothesis $Y$ inferred from many separately encoded observations, the optimal error exponent is the Chernoff information between the two conditional distributions of $U$ given $Y$. We study the mutual information constrained Chernoff bottleneck, which seeks an encoder that maximizes this Chernoff information subject to a rate constraint $I(U;X) \leq R$. We show that its optimal value $C(R)$ increases strictly up to $R = H(V)$, where $V$ merges the symbols of $X$ with equal likelihood ratio, remains at the uncompressed exponent beyond, and, unlike the IB curve, need not be concave. We further show that $k+1$ outputs suffice to attain $C(R)$, where $k$ is the cardinality of $V$. We propose an alternating algorithm that updates the encoder via a generalized Blahut–Arimoto algorithm and the Chernoff parameter $s$ via a nonlinear equation, and prove that its iterates remain feasible, with nondecreasing and convergent Chernoff information. Comparing with the classical IB for $n$ samples, we show that, whenever the two objectives disagree, the IB encoder has the smaller equivocation $H(Y|U^{n})$ at $n = 1$, whereas the Chernoff bottleneck has the smaller error probability and equivocation for all large $n$. Numerical experiments confirm the theory, and on real topic-detection data from the 20 Newsgroups corpus, compressing each word to only $17\%$ of its entropy retains $90\%$ of the error exponent and nearly the accuracy of the uncompressed classifier.

The BITEM team entered both subtasks of the NTCIR-19 R2C2 task with a single agentic pipeline, in which a model searches, reads and records evidence over a movie corpus while an orchestrator holds the record and rules on what may be submitted. A claim is admitted only once an entailment cascade has checked it against the passage it cites, and an answer is released only once enough checked evidence stands behind it. Each question is run three or four times, every pass retrieving from a corpus stripped of what the earlier passes have already seen. The confidence score predicted with each answer is computed by the orchestrator from what the run leaves behind and is never asked of the model, which is offered no way to rate itself. The two retrieval runs placed 4th and 5th of 22, pooling the passes was worth 0.0709 nDCG@20, and the gain was largest on the multi-hop and post-processing-heavy questions, where the organisers rank the pooled run top of the field. Sixteen of the 25 answer runs were built on passages these two runs supplied, 12 of them submitted by other teams. HMR rewards a system whose confidence is high where it answers right and low where it answers wrong. The pipeline reached an accuracy of 0.9219, 6th of 25, while the confidence scores predicted for those answers gave an HMR of 0.4915, 13th. A few rules crafted over those same recorded signals, with no further model call and no further retrieval, raise that to an accuracy of 0.9375, 5th, and an HMR of 0.6985, 9th. Ranking on HMR alone can reward a system for answering wrongly with low confidence, so we propose accHMR, the accuracy multiplied by HMR, which reports the reward in proportion to the accuracy, and on which the revised rules would have scored 0.6549, 5th. For future work, fitting a model on the numbers the pipeline already produces, rather than writing such rules by hand, would be a real step forward.

This paper gives an order-by-order characterization of linear functional observers between the minimum dimension permitted by the algebraic observer condition and the upper endpoint determined by functional observability. Functional observability indices determine the minimum algebraically admissible order $$ q_0=\sum_{j=1}^r\eta_j, $$ where $r$ is the number of independent functionals to be estimated and $\eta_j$ is the functional observability index associated with the $j$th functional. For an $n$-dimensional system with $p$ independent measured outputs, let $n_0$ denote the rank of the observability matrix. Under functional observability, the upper endpoint of the functional-observer order range is $n_0-p$. The algebraic and spectral observer conditions are characterized at each prescribed order between these two endpoints. A nullspace representation in the original state space yields rank and matrix-pencil characterizations of the algebraic and spectral conditions, respectively. The minimum observer order is obtained as the smallest spectrally feasible order in the resulting order spectrum. The two endpoints include the classical Darouach and Luenberger orders as special cases: the lower endpoint reduces to the Darouach order when $q_0=r$, while, under complete observability, the upper endpoint reduces to the Luenberger order $n-p$.

Brain-AI alignment is often interpreted as a sign that model and brain perform similar computations. Whether the aligned units are causally involved in model computation is rarely checked. On an abstract pattern-completion task (AAABAAA $\rightarrow$ B), we compare LLM attention-head representations with human EEG and test how ablating those heads affects task performance. Alignment and causation dissociate: brain-aligned heads contribute to performance, but their removal is substantially less disruptive than removal of heads selected via attribution patching. We compare two head sets that prior interpretability work defines without reference to the brain: concept vectors (CVs), which represent abstract patterns across formats, and function vectors (FVs), selected for their contribution to correct-answer prediction. Brain alignment shows little association with FV scores, while its association with CV scores varies across models. Among brain-aligned heads, we find recurring attention profiles: one emphasizes distinctive elements (novelty heads), the other repeating elements (repetition heads). The novelty family tracks salience and attends to the same elements that humans look at, yet its removal is less damaging than random ablation on average. Repetition heads contribute modestly to performance and are associated with abstract-pattern representation (CVs). Across 17 models spanning 3B-72B parameters, FV-ranked removal is substantially more disruptive than brain-ranked removal. Brain alignment thus captures how the model reads the stimulus, and only faintly captures how it represents the pattern and solves the task.

We consider a planar and a non-planar two-loop Feynman integral family arising in non-factorisable quantum chromodynamics corrections to Higgs-boson production via vector boson fusion. We construct basis integrals which fulfil canonical differential equations with respect to seven variables for the external kinematics and an internal mass. One of the integral families involves nested square roots. We present strategies for deriving canonical master integrals, $\mathrm{d}\log$ forms and analytic continuation for such cases. Including also families with crossed massless legs, we express the $\varepsilon$-expanded integrals in terms of an algebraically independent function basis. For their evaluation, we employ numerical integration of the differential equations.

Tabular foundation models achieve strong zero-shot accuracy on structured data by pretraining on synthetic tables, but they ignore the column names, task descriptions, and auxiliary files that carry dataset semantics. Meanwhile, self-evolving machine learning engineering (MLE) agents train models from scratch on each dataset, yet jointly searching over features, architectures, and hyperparameters is noisy and prone to overfitting. We introduce TabFM-Auto, which pairs a tabular foundation model, TabFM, with a language model agent that evolves the data pipeline around it. Guided by dataset metadata and validation feedback, TabFM-Auto iteratively refines data cleaning, feature engineering, context selection, and post-processing to reduce TabFM's error. Across all 51 datasets of the TabArena benchmark, five TabFM-Auto configurations with different agents and language models take the top five overall positions, and the best raises TabFM from 1785 to 2013 Elo. The discovered pipelines also transfer to other frozen tabular foundation models (+69 to +143 Elo) with no further search. On the 8 tabular competitions of MLE-Bench, TabFM-Auto ranks first overall among MLE agents.

As the context size of text processed with an LLM grows, the size of KV caches can outstrip the memory allocated for the original model weights. This impacts LLM throughput negatively, since decoding is memory-bound and decode cost grows with cache size. Recent work alleviates this bottleneck by discarding the least relevant tokens. Eviction introduces a tension, since a one-off decision to discard content may prove detrimental later. Instead, we focus on alternative choices that can lead to cache compression without evicting tokens. We achieve this by learning a selector that is able to produce, based on context, a per-layer cache configuration towards an overall compression budget. The selector operates along three axes: sharing one cache across layers (depth), caching at fewer bits (precision), or truncating the low-rank latent cache representations (rank). We call the resulting method KV-Kaizen, for the many small per-layer choices it compounds. We observe that these interventions taken independently and uniformly over all layers limit achievable compression because they degrade accuracy. Crucially, composing them locally and adaptively to the context can instead preserve accuracy while achieving large memory savings. At inference, the selector runs once, before pre-fill. In evaluations on instruction following and reasoning tasks, our selectors reach the Pareto frontier of accuracy against cache size, against learning-free and post-hoc baselines. On long-context tasks, KV-Kaizen improves on eviction and can be composed with it, reaching a 32x smaller decode-time cache on a 14B model while preserving accuracy. A 4x cache size reduction incurs no accuracy degradation from 7B parameters up, and a compressed model is more accurate than a smaller uncompressed one with the same cache size. Together, these findings support pre-training large models and compressing them only afterwards.

The Erdős-Rogers function $f_{\ell,s}(n)$ is the largest order of a $K_\ell$-free induced subgraph guaranteed to exist in every $K_s$-free graph on $n$ vertices. While this function is well understood for $s=\ell+1$, the case where $s$ is much larger than $\ell$ has remained wide open. A long-standing lower bound of Sudakov states that $f_{\ell,s}(n)\geq n^{\frac{\ell}{2s}+O_\ell(s^{-2})}$, while a recent result of Bradač shows that $f_{\ell,s}(n)\leq n^{\frac{\ell-1}{s-1}+o(1)}$. In this paper, we close this gap asymptotically by proving that $f_{\ell,s}(n)= n^{\frac{\ell}{2s}+O_\ell(s^{-2})}$. More precisely, we prove that for all $2\leq \ell<s$, we have $f_{\ell,s}(n)\leq n^{\frac{\ell}{2s-\ell}+o(1)}$. Our proof builds on Bradač's recent tight construction for off-diagonal Ramsey numbers, which can be viewed as the $\ell=2$ case of our result.

Recovering articulated 4D representations of animals from monocular videos remains challenging due to the large diversity of quadruped morphologies and lack of animal 4D supervision data. Existing learning-based reconstruction methods operate on individual images and rely on synthetic or model-fitted 3D supervision, which inherits the constraints of strong parametric priors and limits generalization to out-of-distribution species. When applied to out-of-distribution animals, they often recover a plausible pose while producing inaccurate geometry because the underlying shape model cannot faithfully represent the observed instance. We present ORMA, a training-free reconstruction framework that decouples articulation from shape, using the predicted pose as reference for optimization while leveraging generative 3D priors for accurate shape reconstruction. Given a reference image, we reconstruct the animal geometry and register it to the parametric model SMAL+, yielding an articulated shape adapted to the observed instance. We then combine per-frame articulated pose estimates with globally consistent camera poses to recover animal motion in a shared world coordinate frame, and further refine the reconstruction using self-supervised DINO correspondences and temporal consistency. To enable quantitative evaluation, we introduce PAW4D, a synthetic multi-species benchmark with ground-truth 3D geometry and camera motion. Experiments on PAW4D, PFERD, and challenging in-the-wild videos demonstrate that ORMA improves reconstruction accuracy while recovering globally consistend animal motion across diverse quadruped species.

Coding agents open pull requests (PRs) that claim to speed up software, but studies of human performance fixes say little about how maintainers respond to such a fix or whether its claim holds. From the 71,677 agent PRs of AIDev v4, a text filter and codebook coding by language models and by the authors select 1,262 performance issues fixed by six agents in 582 repositories. We code each issue and its tests and re-execute 23 rejected and 30 merged fixes. (1) 57% of closed fixes are merged, 61% of rejections give no stated reason, and only 6 of the 23 re-executed rejected claims held under our three-run pilot on mostly agent-built workloads. (2) Acceptance rises with the agent's track record in the repository (31-37% to 70%) and with the repository's pre-opening merge rate on its other agent PRs (33% to 84%). Merged fixes delete a larger share of the lines they change (0.26 versus 0.15), a difference that holds within agent and within repository, with no such difference detected in the coded content, description, tests or measurements. (3) Repeated computation and redundant data processing cause 44% of the issues, and 46% of fixes are architectural-level. (4) Agents change tests in 37% of fixes and 11% carry a performance test or benchmark; of the 30 merged fixes, 18 met our delivery criterion, 3 fell short of the claim, 9 showed no significant gain or regressed, and 14 change behavior on untested inputs. The outcome tracks the repository's history with the agent rather than the coded content of the fix, and a merge does not show that the fix delivers what it claims.

In the first paper of this series, we showed that gravitational instability (GI) in protoplanetary disks can produce fragments of planetary rather than brown-dwarf mass, leaving the subsequent evolution and ultimate fates of such fragments an open question. Here, we follow every bound object in a global three-dimensional radiation hydrodynamic simulation of a fragmenting $0.196\,M_\odot$ disk around a $1\,M_\odot$ star: seven surviving fragments, one disrupted clump, and one merged clump, followed for $1.2$\,kyr. The fragments form at $1.4$–$3.0\,M_\mathrm{J}$, consistent with the normalized initial-mass distribution of Paper~I. Mass growth follows a single Hill-limited scaling $\propto \Sigma \Omega R_\mathrm{H}^2$, regulated by the delivery of gas into draining feeding zones, while the latest-forming fragments experience supply starvation. Migration is bidirectional and governed by gravitational interactions with both the disk and neighboring clumps. Accretion supplies most of the fragments' spin, which aligns with their orbits and provides roughly a tenth of the support against gravity. The interiors are entropy-stratified, largely convectively stable, and accumulate mass far faster than they can radiatively cool: compression is quasi-adiabatic, and central entropies near $12\,k_\mathrm{B}$ per baryon favor hot-start initial conditions for fragments that evolve into gas giants. Integrating the measured growth law against the measured gas budget, we predict that the disk produces gas giants, brown dwarfs, and a possible low-mass stellar companion, while one scattered fragment may become a free-floating planet: the outcome of disk fragmentation is determined not merely at birth, but by the subsequent mass supply and dynamics.

Turbulent flows are highly chaotic, which makes their instantaneous states difficult to predict. Data assimilation aims to reduce this predictive uncertainty by synthesizing the predictions of a physical solver with partial observations of the system. Traditional data assimilation methods such as ensemble and variational approaches can incur substantial computational cost through evaluations of high-fidelity solvers and, for variational methods, adjoint calculations. Recently, generative diffusion models have been explored in the solution of inverse problems in fluid mechanics, demonstrating their potential for data assimilation. However, existing diffusion-based methods often use free parameters instead of the prescribed observation-error covariance to control the balance between the prior and likelihood, departing from the Bayesian formulation of data assimilation. We propose a framework for principled generative data assimilation in which observations are assimilated in the clean-state space and the prescribed observation-error covariance enters explicitly through an ensemble Kalman update. In the Lorenz-63 system, the proposed method yields a lower trajectory error and equation residual than the other diffusion-based posterior sampling methods considered, although its ensemble underestimates the posterior uncertainty. In two-dimensional Rayleigh-Bénard convection, the generated fields recover the statistics of the direct numerical simulation reference at large scales, with deviations confined to the smaller scales. These results suggest that the framework can produce physically plausible turbulent flow fields, indicating its potential for more challenging turbulent flow applications.

The thermal emission spectra of terrestrial exoplanets provide a window into their surface compositions and corresponding geological processes. Close-in, rocky planets orbiting M dwarfs are ideal targets for these studies, as observations have shown that most have little to no atmosphere. LHS 3844 b is an ultra-short period super-Earth orbiting a nearby M dwarf, and is the most favorable target that does not have a dayside magma ocean. We present an updated JWST MIRI/LRS emission spectrum ($5-12$ $\mu$m) of this planet, derived from eight new and three archival eclipse observations. We combine these visits to obtain the highest SNR emission spectrum for any rocky exoplanet thus far, with a median SNR of 29 across 12 wavelength bins. Our data are sensitive to the predicted Si-O stretching feature (the strongest mid-infrared silicate feature) for a wide range of common materials and the transparency feature, which is relatively insensitive to composition and constrains the grain size and porosity of surface materials. We find that LHS 3844 b's emission spectrum resembles a blackbody shortward of 10 $\mu$m disfavoring the presence of a transparency feature, and identify a tentative feature ($1.9-3.5\sigma$) between $10-12$ $\mu$m. If this feature is astrophysical and not instrumental in nature, it can be matched by a fayalite (the iron end-member of olivine) surface, but might also be consistent with other candidate planet-forming materials like spinel or silicon carbide. We discuss possible biases introduced by the wavelength-dependent instrumental ramp, and outline several strategies to mitigate this effect in future observations of LHS 3844~b that would establish the surface composition of this rocky exoplanet.

An independent set of a graph G is a subset of VG, no two of which are adjacent. The cardinality of a maximum independent set in a graph G is called the independence number of G, denoted by alpha(G). The essential connectivity kappa'(G) of a graph G is denoted as the minimum number of vertices of G whose removal produces a disconnected graph with at least two non-trivial components. In this paper, we determine the n-vertex connected graphs with given independence number and essential connectivity that attain the minimum distance spectral radius, and fully characterize the corresponding extremal graphs.

We prove a sharp quadratic inequality between the largest two eigenvalues $\lambda_1 \ge \lambda_2$ of a graph with $n$ vertices. We also prove a quadratic inequality between the second and third largest eigenvalues $\lambda_2 \ge \lambda_3$. These results in particular imply the bounds $\lambda_1 + \lambda_2 \le \frac{8}{7} n - 2$, $\lambda_3 \le \frac{n}{3} - 1$ and $\lambda_2 + \lambda_3 \le \frac{2}{3} n - 2$. In fact we determine the closure of the set of possible $(\frac{\lambda_1+1}{n}, \frac{\lambda_2+1}{n}) \in \mathbb{R}^2$ and $(\frac{\lambda_2+1}{n}, \frac{\lambda_3+1}{n}) \in \mathbb{R}^2$. More generally, we prove quadratic bounds in the case of symmetric matrices in $[0,1]^{n \times n}$, and we also give a quadratic bound for two eigenvalues of a symmetric matrix in $[-1,1]^{n \times n}$. These bounds are proved by transforming the problem into extremal geometric questions in $\mathbb{R}^3$ and $\mathbb{R}^2$. We use the method of Lagrange multipliers to reduce to special cases with at most five points, and we deal with these special cases directly.

If we run a heavy-duty computation on prior data, can we avoid repeated computation for similar future inputs? Inspired by this question, we introduce a new computational model for graph problems called algorithms with isomorphic priors. Solving a graph problem $\Pi$ in this model involves two phases: (i) The preprocessing phase quickly analyzes prior graphs $G_1, ..., G_k$ along with the (previously computed) exact optimal values OPT$(G_i)$. (ii) Subsequently, given a new graph H, a fast query phase must either (a) output the exact solution OPT(H), or (b) correctly report that H is not isomorphic to any $G_i$. Can we avoid computing OPT(H) from scratch when H is isomorphic to some $G_i$? We show that this is the case for a number of problems; for many others, we establish conditional lower bounds. $\textbf{(1)}$ Some NP-hard problems, including Constrained Shortest Path and $\ell_p$-Shortest Path and Constrained Spanning Tree, admit polynomial preprocessing and query times in our model. In contrast, almost all of Karp's 21 NP-complete problems and $(2-\varepsilon)$-approximate $k$-Center, for every fixed $\varepsilon>0$, admit no such algorithms unless Graph Isomorphism (GI) is in P, even with O(1) priors. $\textbf{(2)}$ In contrast to conditional $n^{3-o(1)}$ fine-grained lower bounds, our framework achieves an $O(n^\omega)$ query time for Negative Triangle and a near-linear query time for Replacement Path. It also achieves near-linear query time for Maximum Flow. $\textbf{(3)}$ While it remains a major open problem whether infinite-duration games (Paritiy Game, Mean Payoff Game, Energy Game, and Stochastic Game) admit polynomial-time algorithms, they can be easily solved in near-linear time within our model. Our proofs rely on a simple combination of existing tools and are accessible to readers without specialized background.

A graph is $K_3$-irregular if its vertices belong to pairwise distinct numbers of triangles. We prove that no $8$-regular $K_3$-irregular graph exists, settling the last unresolved case. Following the initial discovery of such graphs for regularities $r \in \{10,11,12\}$ (Stevanović et al., 2024), our previous work (Hak et al., 2025) showed that no such graphs exist for $r \le 7$, provided the first example for $r=9$, and proved that any $8$-regular candidate must have between $17$ and $22$ vertices. We exclude these possible orders for $r=8$ by combining careful analysis of triangle degrees with integer linear programming techniques. Meanwhile, a recent construction (Zhang, 2026) established that regular $K_3$-irregular graphs do exist for all $r \ge 9$. Together with our results, this establishes that an $r$-regular $K_3$-irregular graph exists if and only if $r\geq 9$.

Motivated by anisotropic viscosity in geophysical fluid dynamics, where vertical momentum diffusion is often much weaker than horizontal diffusion and may be negligible in the interior yet remains essential near a solid wall, we study the vanishing vertical viscosity limit for the two-dimensional incompressible Navier–Stokes equations in the upper half-plane, with horizontal viscosity fixed at one and vertical viscosity \(\varepsilon^2\). For arbitrary divergence-free \(H^4\) no-slip initial data, with no time-differentiated compatibility conditions required, we construct the limiting horizontally viscous flow and the boundary-layer profiles on every prescribed finite interval \([0,T]\). The leading-order corrected approximation has \(O(\varepsilon)\) error in \(L^\infty\), and the full finite-order expansion reduces this error to \(O(\varepsilon^{3/2})\), uniformly down to \(t=0\). We further describe the development of the layer from data that vanish initially at the wall. At the initial–boundary corner, we determine the first two self-similar coefficients and characterize matching through exact profile tails, thereby separating the fixed-time scale \(\varepsilon\) from the short-time scale \(\varepsilon\sqrt t\). When the initial wall acceleration is nonzero, both the leading correction rate and the first normalized corner remainder are sharp. Finally, continuity estimates for fixed data yield joint small-time and small-viscosity limits for the exact solution, as well as finite-\(L^p\) asymptotics.

LLMs trained with Chain-of-thought excel in reasoning capability, but often come with excessive token cost. We find that the core of reasoning capacity lies in the Thinking model's weight component within the null space of a projection defined by the corresponding Non-thinking model's dominant singular directions, and removing the subspace component can largely improve reasoning efficiency without hurting the accuracy gained during thinking-mode post-training. Unlike existing efforts that mostly operate within the dominant subspace, we are the first to unveil the critical role of the null space and harness it for model optimization. Motivated by this finding, we propose Spectral Null-Space Swap ($S^3$), a training-free composition of paired Non-thinking and Thinking checkpoints. Our method keeps the Non-thinking model inside its own dominant subspace and takes the Thinking checkpoint outside it, improving reasoning efficiency while maintaining accuracy. We extensively evaluate $S^3$ on 2B-30B dense and mixture-of-experts (MoE) architectures spanning 28 evaluation environments across mathematical, multimodal, and audio reasoning domains. $S^3$ establishes new empirical Pareto Frontiers among training-free model composition strategies: across all settings, it reduces inference token overhead by an average of 27.4% compared to full Thinking models while simultaneously improving overall task accuracy by 1.0 percentage point (e.g., yielding +8.3% accuracy on HMMT25 alongside a 33.0% token speedup). We further use attention entropy for explanation and find that the retained component produces more concentrated attention, and we use a simplified analytical model about optimization to demonstrate why null-space can effectively reduce attention entropy, thereby improving the efficiency of reasoning.

Let $M$ be an imaginary quadratic field. Let $A$ be a polarised abelian threefold defined over $M$ with geometric endomorphism algebra isomorphic to $M$. We study residual Galois representations attached to $A$. In particular, we describe the endomorphism field and the natural restrictions placed on the residual representations by their endomorphisms. We then devise criteria for the image of residual Galois representations to be large, and provide efficient algorithms suitable for large scale calculations. Subsequently, we apply this algorithm to millions of curves in several families, whose Jacobians are abelian threefolds with imaginary multiplication, and to Sutherland's dataset of $7$-smooth Picard curves. We produce an explicit example of a Picard curve which appears to have an isogeny of degree $13$ defined over $\mathbb{Q}(\zeta_3)$. The Jacobians of all other curves, in the range of our computation with endomorphism algebra $M$, have mod-$\ell$ image as large as possible for any prime $\ell>7$. This allows us to realise the group $\Gamma\mathrm{U}_3(\ell)$ of unitary semisimilitudes as a Galois group over $\mathbb{Q}$ for all $\ell \not\equiv 1, 25, 121 \pmod{168}$. Our algorithms also led us to the discovery of several interesting rational families of Picard curves whose generic members appear to have endomorphism algebra of dimension $6$ which is not a CM field. These represent curves on the Picard modular surface and appear to be Shimura curves. Some of them appear to parametrise non-principally polarised abelian surfaces with quaternionic multiplication.

Masked diffusion models (MDMs) generate text by unmasking several tokens per step, but they are trained and sampled under different conditions. The model is trained on randomly masked sequences, whereas inference follows a trajectory shaped by the model's own predictions. Additionally, each step has no access to what the previous one computed. Recent methods narrow these limitations from separate angles, leaving open how these choices interact. We introduce PUMBA, a unified framework for trajectory-aware training that trains the denoiser on consecutive steps of policy-induced trajectories, passes information between steps, and optimizes them jointly by backpropagation through time. A controlled study of this design space shows that i) exact train–inference alignment fails due to local overfitting, whereas a looser alignment still brings training masks closer to those seen at inference; ii) passing continuous information outperforms discrete gradient estimators through the commitment at each step; and iii) performance improves as backpropagation through time spans more steps, which we support theoretically. Combined, these components match the best checkpoint of a same-size autoregressive model. Building on these findings, we scale PUMBA to supervised fine-tuning of LLaDA-8B, where it improves the trade-off between performance and number of function evaluations (NFEs) in both full-canvas and block diffusion generation. At matched performance, it needs up to 22% fewer NFEs than standard fine-tuning with twice the budget in full-canvas generation, and up to 26% fewer than standard fine-tuning for the same number of steps in block diffusion.

A majority edge-colouring of a graph $G$ is a colouring of the edges of $G$ such that, for every vertex $v$ of $G$, at most half of the edges incident with $v$ receive the same colour. This notion was introduced by Bock et al. in 2023, who proved that every graph of minimum degree at least $2$ has a majority $4$-edge-colouring. We investigate an on-line variant of majority edge-colouring in which the graph is revealed by the Presenter edge-by-edge and the Algorithm must colour each edge immediately and irrevocably. In particular. we prove that the greedy strategy, which uses at most five colours, is optimal among on-line algorithms if $\delta = O(\frac{\log n}{\log\log n})$. We further extend our results to $1/k$-majority edge-colourings.

As Graph Neural Networks (GNNs) are widely deployed as Machine Learning-as-a-Service (MLaaS) APIs, model stealing attacks have emerged as a critical security threat. By querying a victim model's black-box API, an adversary can construct a functionally equivalent surrogate model, compromising proprietary intellectual property and downstream security. Existing GNN stealing attacks, however, rely on overly permissive assumptions, such as soft-label outputs, large query budgets, full-graph query access, and prior knowledge of victim backbones that rarely hold in real-world deployments. In this work, we formalize a strictly constrained black-box, hard-label and backbone-agnostic threat model for GNN stealing attacks under a tight query budget. Given these realistic restrictions, we identify four fundamental challenges: sparse local structures and isolated nodes that degrade victim label quality, insufficient supervision signals, systematic imbalance with incomplete class coverage, and backbone mismatch. To address these interlocking barriers, we propose Dagger, a novel two-phase decoupling-based attack framework. Specifically, in Phase 1, Dagger pre-trains a surrogate using decoupled information propagation to preserve structural context over sparse local subgraphs while handling isolated nodes, combined with manifold-level node mixup to synthesize continuous supervision signals and smooth decision boundaries. In Phase 2, Dagger freezes the encoder and fine-tunes the classifier head via class-balanced sampling paired with logit adjustment to rectify severe query imbalance without requiring extra victim queries. Extensive experiments across four benchmark graphs and four GNN backbones demonstrate that Dagger consistently outperforms state-of-the-art GNN stealing attacks, achieving up to 18.16% higher fidelity while only utilizing 12.23$\times$ fewer queries than the strongest baseline.

Accurate human motion prediction is crucial for robotic systems operating around people, particularly in complex indoor spaces. In this study, we assess the relative importance of different sources of information in indoor motion prediction with a human motion diffusion model. We collected a dataset of nine naive human subjects conducting simulated indoor daily activities while wearing a pair of Meta Aria glasses. This dataset includes ten buildings from a university campus, and encompasses 238 minutes of navigation between daily tasks. Overall, we demonstrate a 42% improvement beyond a constant velocity baseline. Including body motion, scene representation, and eye gaze fixation data significantly reduced prediction error. Semantic information was found to be useful for indoor motion prediction, but to a lesser degree than in outdoor navigation. Providing explicit intent information reduced error beyond any other addition, suggesting that incorporating explicit intent estimation or user input are fundamental for finer prediction of indoor motion. One of the few indicators of intent, eye gaze fixation, was found to be especially useful in predicting deceleration, and provided basic spatial information to the model in the absence of an occupancy map. These results are a first step towards predicting human motion in highly ambiguous indoor scenarios. Code will be made public upon acceptance. Project page: https://human-motion-diffusion.github.io/

World-action models (WAMs) jointly predict how a scene will evolve and how an agent should act, however joint generation alone does not necessarily impose a shared geometric constraint on these predictions. We present PhysWAM, a unified world-action model for autonomous driving that co-denoises multiview video, metric depth, and ego motion within a single flow-matching transformer. To ground world and action generation in measured scene geometry, we introduce Coupled Point Projection (CPP) that unprojects the generated depth into 3D points, transforms them using the generated $\mathrm{SE}(3)$ ego motion, and minimizes their distance to LiDAR points transformed using the recorded ego motion. This geometric constraint promotes physical consistency with the measured scene by jointly supervising generated depth and motion alongside their standard flow-matching objectives. At inference, trajectory selection relies only on a simple label-free consensus rule, with no learned scorer or simulator feedback. We evaluate PhysWAM across NAVSIM v1 and v2 planning, zero-shot closed-loop transfer, and future video and metric-depth prediction. Despite PhysWAM's simple selection procedure, it achieves strong planning performance and transfers zero-shot to unseen driving environments. It also generates accurate metric depth and temporally coherent video, with CPP improving both planning and depth prediction. Together, these results demonstrate that the geometric relationship between scene depth and ego motion provides a direct way to couple world and action generation within a simple unified model.

High-resolution video generation is expensive, as its cost grows rapidly with the number of spatiotemporal tokens. A practical alternative first generates a lower-resolution video and then applies a refiner, but conventional multi-step refinement introduces a second sampling bottleneck. We present SoL-Refiner, a one-step video refiner that transforms low-resolution model outputs into 4K videos with a single denoising step. Our three-stage recipe combines high-resolution continual training, reinforcement learning (RL) post-training, and a final one-step distillation. We introduce Refiner-Bench, a video refinement benchmark constructed from the outputs of different video generators, and use a shared-input protocol to compare refiners at approximately 2K output resolution. With LTX-2.3 Refiner as the base model, the one-step SoL-Refiner outperforms all external refiners at 2K on the VBench and UniPercept averages, while at $3840\!\times\!2176$ it improves both metrics over the three-step LTX-2.3 Refiner. With the complete acceleration stack, SoL-Refiner achieves an $8.91\times$ speedup in refinement latency over the same baseline in our 2K latency setting. With MiniMax H3 as the base model, SoL-Refiner produces 2K videos in three refinement steps and is preferred over the official MiniMax H3 2K API in 87% of user-study judgments.