Papers

We prove that the Füredi-Hajnal limit and the Stanley-Wilf limit of a uniformly random permutation matrix of order $k$ are at most $\exp\bigl(O(\sqrt{k}(\log k)^{5/2})\bigr)$ with probability tending to one as $k\to\infty$. This improves the bound $\exp\bigl(O(k^{2/3}(\log k)^{7/3}/(\log\log k)^{1/3})\bigr)$ of Cibulka and Kynčl. Together with the lower bound due to Fox, these bounds show that the logarithms of both limits are $k^{1/2+o(1)}$ for almost all permutations.

Visuomotor policies trained via imitation learning often inherit the unnecessarily slow timing of teleoperated demonstrations. Yet uniform speedup is unreliable because different phases of a manipulation task tolerate acceleration differently. In this work, we introduce AdaTempo, a self-supervised method that accelerates visuomotor policies by exploiting shared relative-tempo structure in demonstrations. AdaTempo establishes phase correspondence, aggregates the aligned relative tempo into a consensus, and maps it to a continuous speedup profile used to resample demonstrations into accelerated training trajectories. Training standard policies such as ACT or Diffusion Policy on these resampled trajectories directly embeds the desired tempo in the learned behavior, without runtime tempo selection or online retiming. Extensive evaluations show that AdaTempo achieves up to a $3.57\times$ speedup and yields a stronger success–speed trade-off than the original policies and representative acceleration baselines.

The Muon optimizer derives its update rule for hidden linear layers by solving a local linearization of the loss penalized by the spectral norm, motivated by an RMS-stability argument for dense linear layers. Standard Muon implementations, however, exclude the input (embedding table) and output (language model head) layers from this principled treatment, for which they use AdamW instead. We present MuonIO, a single Muon-style update for both of these layers. For the language model head $\mathbf{L} \in \mathbb{R}^{V \times d}$, we motivate the use of the $2\to\infty$ operator norm, due to the Lipschitz continuity of the softmax output geometry, while for the embedding table $\mathbf{E} \in \mathbb{R}^{d \times V}$, we draw on the $1 \to 2$ operator norm, based on the one-hot input geometry identified by Bernstein & Newhouse (2025). The identity $\lVert\mathbf{L}\rVert_{2\to\infty}=\lVert\mathbf{L}^\top\rVert_{1\to2}$ then puts both matrices in the same vocabulary-oriented geometry: MuonIO applies a single normalized-vector rule, which appears as column normalization for $\mathbf{E}$ and row normalization for $\mathbf{L}$. Empirical evaluations demonstrate the effectiveness of our approach, with MuonIO reducing I/O optimizer state memory by 50% and I/O update FLOPs by $\sim$46% compared to Muon for 1B LLaMA pretraining on C4, while also improving validation perplexity.

Time series are produced continuously at enormous scale by industrial equipment, wearables, power grids, and clinical monitors, yet annotation remains manual, expensive, and expert-dependent. The binding constraint in large-scale time series analytics is therefore not data volume but label volume, and the question facing a practitioner is concrete: how many examples per class must be labeled before a classifier becomes usable? We study this question directly, in a regime where the label space is fixed and known in advance and the decision rule must be constructed from only K labeled examples per class. We propose Dual-Stream OSSE-LSTM, an episodic metric-learning framework that pairs an Omni-Scale CNN with Squeeze-and-Excitation recalibration, for multi-scale motif extraction without per-dataset kernel tuning, with a Bidirectional LSTM for global temporal context. The two streams are independently normalized and fused into a prototype-oriented embedding. Because decisions taken from a few labels must also be explainable, we introduce Counterfactual Integrated Gradients (C-IG), which attributes the prototype margin between target and opposing classes rather than an isolated classifier logit, and reuses the resulting maps as soft masks for test-time prototype refinement without updating the encoder. On 19 univariate UCR datasets, OSSE-LSTM attains the highest average accuracy and per-dataset win count at every support size, and its accuracy remains within a 0.36-point band (96.36-96.72%) across that range. Its weakest configuration still exceeding the best result any compared baseline achieves at any K (93.99%).

On-policy distillation (OPD) improves large language model reasoning by training students on their own rollouts with dense token-wise supervision from the teacher. However, token-wise OPD does not explicitly provide a coherent alternative reasoning step showing how the student's step could be revised to improve subsequent reasoning. Furthermore, this paradigm can become less effective when the student produces a degenerate reasoning prefix, as subsequent teacher supervision remains conditioned on that prefix and may reinforce poor reasoning patterns. In this work, we focus on learning reasoning revision with segment-wise OPD to rework intermediate reasoning steps and better support subsequent reasoning. Through controlled reasoning interventions, we find that replacing student segments with teacher redrafts improves subsequent reasoning accuracy. Therefore, we address the problem of turning teacher redrafts into explicit supervision for learning to revise reasoning. We propose Segment-wise On-Policy Distillation (Seg-OPD), which selects student segments based on an uncertainty metric and obtains corresponding teacher redrafts. Seg-OPD trains the student to prefer teacher redrafts over their paired student segments while retaining dense token-wise OPD supervision. Extensive experiments on mathematical reasoning and competitive programming tasks show that Seg-OPD-trained students achieve higher revision success rates than baselines. Seg-OPD consistently outperforms the compared state-of-the-art baselines in reasoning accuracy with an average relative improvement of 5.22% across diverse models and tasks. Code is available at https://anonymous.4open.science/r/Seg-OPD.

Multi-agent debate is increasingly used to reach consensus among LLM agents, yet agents often yield to a unanimous majority. When an agent changes its answer, has it changed its mind or only its statement? We study this with two-hop factual questions whose intermediate entity (the bridge, e.g. the country in "the capital of the country where the Sagrada Familia is located") is never stated by anyone. Scripted peers, in the role of Asch's confederates, unanimously assert a wrong answer taken from another fact with a different bridge. At the moment the agent answers, we read the bridge from its residual stream with the Jacobian lens (J-lens) and, for comparison, the logit lens. In pre-registered tests on held-out facts with four open-weight models, agents of Qwen3.5-4B, Qwen3.6-27B and Gemma-4-E4B-it that gave in still represented their original bridge in the pre-registered layers below the output (hit@100 above a control entity: 0.85, 0.22 and 0.24), where the logit lens rarely ranked it among the top 100 tokens (0.00-0.06). These agents also represented the bridge behind the peers' answer, beyond a mention baseline. A pre-registered addendum hid the agent's earlier answer or removed it: agents that gave in still represented their original bridge in all four models (0.43, 0.29, 0.37 and 0.25 with the answer hidden), including Llama-3.1-8B-Instruct, which barely did so with its answer in view (0.03). The premise can thus be computed from the question alone while the agent states the majority's answer. Hiding the earlier answer also changed conformity: Qwen3.5-4B gave in on 89% of questions instead of 8%. In exploratory interventions, injecting the bridge's J-lens direction brought agents back to their original answer only in the two Qwen models. Stated consensus in multi-agent debate can thus overstate agreement. We also report the negative results of our pre-registered program.

Mixture-of-Experts (MoE) models can increase parameter capacity without proportionally increasing active computation, but it is unclear how this trade-off behaves in particle-physics transformers. We study dense and MoE Particle Transformers on 188-class JetClass-II, varying expert count, routing capacity, top-K, and auxiliary loss. We find that, when token dropping is avoided, top-1 MoE models improve over the dense baseline at nearly unchanged nominal forward compute, while further increasing the number of stored experts produces little additional accuracy gain. Activating multiple experts per token yields additional predictive improvements at higher computational cost. Routing analyses show that expert assignments become more strongly associated with particle identity and kinematics in some configurations, but this structure does not increase monotonically with classification performance. These results highlight the need to distinguish stored parameter capacity, active computation, routing capacity, and routing organization when evaluating sparse expert models for jet classification. Code and experiment configurations are available at https://github.com/kpendiyala/MPT.

On-policy self-distillation (OPSD) supplies dense token-level feedback on trajectories sampled from the student's own policy, a richer training signal than the outcome-level rewards of reinforcement learning. This feedback comes from a teacher conditioned on a full reference solution unavailable to the student. The reference solution specifies the target but not how to move from the student's current error toward it, creating a solution-conditioned shortcut risk. We introduce AIR-OPD, an adaptive iterative repair framework for on-policy distillation that provides error-to-repair supervision. Given a failed response, a guidance generator synthesizes repair guidance for the current error. The student samples an on-policy retry with this guidance. If the retry remains incorrect, the generator produces new repair guidance for the newly observed error. At each round, a fixed teacher receives the guidance as privileged context and supervises the student on an error-aligned region of its latest failed response. Outcome-aware stage weighting favors early repair stages and credits stages whose immediate retry passes verification. We train AIR-OPD on the DAPO-Math-17K dataset and evaluate on AIME24, AIME25, and HMMT25, alongside out-of-distribution tests on MMLU-Pro and GPQA. We examine two guidance sources, self-guidance from the current student policy and external guidance from a larger model. For both Qwen3-4B and Qwen3-8B, AIR-OPD attains the best mathematical-reasoning averages, improving over the strongest baseline by up to 3.6 points, while preserving base-model performance on the out-of-distribution benchmarks.

AI can help adults produce plausible workplace artefacts while leaving their independent reasoning difficult to inspect. LearnAdapt Praxis addresses this design problem through a five-stage learning workspace: Frame, Learn, Build, Validate and Transfer. The application stores an initial response, versioned specifications, coaching records, validation observations, a separate transfer response and facilitator feedback. Project-level server controls restrict coaching, access to earlier work and learner export during an active transfer attempt; they do not establish that a learner avoided assistance outside the application. This technical report describes the implemented architecture and evaluates selected workflow, authority, failure-recovery and provenance properties. A reproducible synthetic rehearsal exercised 120 serial project episodes across three fictional workplace contexts and four injected provider conditions. All 3,990 recorded checks met their specified expectations. The resulting records contained 240 specification versions, 1,200 activity events and 60 stored synthetic hints; 60 injected provider failures produced no fabricated hints. Separate authentication regressions exposed and resolved a session-expiry defect, and limited staging and production checks verified account access and live model connectivity. The evidence supports the tested engineering properties of the recorded release. It does not establish learning gains, model-output quality, unaided assessment validity, population usability or production capacity. The contribution is an implemented arrangement for separating assisted work, project-level assistance withdrawal and inspectable evidence, accompanied by executable tests and an explicit account of what remains to be validated with adult learners.

This article proposes and articulates structure-driven methodology as an emerging cross-domain paradigm. In contrast to the objective-driven paradigm that has dominated inverse-problem solving for over two centuries, structure-driven methodology advocates identifying and exploiting the intrinsic structure of a problem to construct the solution method directly, rather than approaching the solution through iterative search. Drawing on examples from the Liouville-variant Goldbach conjecture, the three-dimensional Kakeya conjecture, Noether's theorem, and the author's nearly four decades of geophysical inversion practice, the article illustrates the common logic of structure-driven thinking in number theory, pure mathematics, physics, and seismic imaging. The author also systematically proposes the Structure-Driven Inversion (SDI) paradigm, comprising Mathematical Structure-Driven Inversion (MSDI) and Physical Structure-Driven Inversion (PSDI). Structure-driven methodology does not negate objective-driven methodology but complements it: objective-driven is a general search strategy, whereas structure-driven is a custom direct strategy. The article concludes that the significance of this shift lies not only in efficiency gains but also in cognitive deepening—when facing complex problems, the first question is no longer “how to solve the problem faster” but “what is the structure of the problem.”

Recent vision-and-language navigation (VLN) systems increasingly adopt streaming Video-LLM policies that map egocentric RGB observations and instructions directly to low-level actions. Yet these policies inherit weak 3D geometric priors from 2D pretraining. Existing geometry-aware extensions charge a persistent inference-time price: depth sensors, 3D encoders, or per-step perception tool calls. We propose GeoScaffold, a geometric supervision framework that pays this price once, at training time, by internalizing geometry into the policy itself. It first learns a compact depth tokenizer on depth maps from the training trajectories and freezes it. It then fine-tunes the policy with a handful of learnable geometry query tokens, training their hidden states to reconstruct navigation-critical geometry such as depth, connectivity, and traversability. This supervision turns the query states into compact geometric latents for action decoding, and through the shared weights also internalizes geometry into the backbone's own representations. Like a scaffold, the tokenizer, target generators, and reconstruction heads are discarded after training, leaving the backbone and action interface unchanged. Extensive experiments show that GeoScaffold consistently outperforms leading vision-only navigators on continuous VLN benchmarks, offering a practical paradigm for lightweight edge deployment of spatially aware embodied navigation models.

Given a group $G$, the prime graph of $G$, denoted $\Gamma(G)$, is the graph with vertex set the set of primes $p$ such that there exists an element of order $p$ in $G$, and such that there is an edge between primes $p$ and $q$ if and only if there exists an element of order $pq$ in $G$. Let $T$ be nonabelian finite simple group. A $T$-solvable group $G$ is a group such that there exists a finite subnormal series of $G$ such that each factor is either abelian or isomorphic to $T$. In this paper, we continue the classification of the prime graphs of $T$-solvable locally torsion groups and classify prime graphs of locally torsion metanilpotent groups using tools from model theory. We also classify the prime graphs of ultragroups with certain first-order properties and prove results on the number and type of certain classes of groups that can realize a given graph as a prime graph.

Reliable large language models (LLMs) must not only produce accurate answers but also express confidence that faithfully reflects their probability of being correct. Such calibration is essential for identifying uncertain predictions and supporting reliable decision-making in real-world deployment. Recent studies incorporate calibration learning into reinforcement learning (RL), jointly optimizing answer correctness and verbalized confidence using ground-truth correctness supervision. However, their reliance on labeled data limits their applicability in practical test-time settings, where ground-truth labels are unavailable and calibration may need to adapt to newly encountered target tasks. To address this challenge, we propose Test-Time Calibration Learning (TTCL), a label-free framework that jointly adapts reasoning accuracy and verbalized confidence directly on unlabeled target-task data. Specifically, TTCL derives self-supervision signals for both correctness and calibration from multiple model-generated responses, enabling calibration learning at test time without ground-truth labels. Theoretical analysis further establishes TTCL as a bounded surrogate for the ideal calibration objective. Extensive experiments on mathematical reasoning and factual question answering demonstrate that TTCL consistently improves both accuracy and calibration across diverse models and tasks. On base models, TTCL achieves an average relative accuracy improvement of +40.13% and an ECE reduction of +70.80% across eight benchmarks. Moreover, TTCL can further improve both accuracy and calibration for already calibrated models under domain shift, particularly when source-domain calibration transfers poorly to target tasks. In the math-to-factQA setting, TTCL achieves an average relative accuracy gain of +20.35% and reduces ECE by +53.83%. The source code is released at https://github.com/tmlr-group/TTCL.

Future planetary surface missions are likely to involve multiple independently operated rovers sharing the same deployment region, raising the need to verify compliance with operational constraints such as Lunar Safety Zones. Because continuous in-situ observability is rarely available, such verification requires post-hoc reconstruction of rover trajectories from sparse telemetry, including odometry, pose priors, and relative inter-rover detections. We introduce the problem of forensic trajectory analysis for non-cooperative planetary rovers in the presence of Byzantine agents: rovers that provide miscalibrated or deliberately falsified measurements to support an incorrect trajectory. We show that standard outlier-robust pose graph optimisation methods are vulnerable in this setting, because Byzantine rovers can generate measurements that are internally consistent and numerous enough to make truthful incriminating measurements appear as outliers. To address this, we propose an attribution-aware trajectory estimation method that reasons over rover credibility rather than individual measurement validity. The method evaluates candidate credible rover subsets by comparing the statistical consistency of their internal and boundary relative detections against provided priors, and then estimates trajectories using only measurements attributed to credible agents. Across synthetic simulations and real planetary-analogue trajectory data, the proposed method identifies Byzantine rovers and produces significantly more accurate trajectory estimates than existing robust pose graph optimisation baselines.

BN products formed on Fe–B sources under flowing N$_2$ were investigated primarily at 1250–1400~$^\circ$C over nominal B contents of 1–10~wt% to determine how rBN-related Raman character depends on growth condition and history. Spatially resolved Raman mapping across 44 growth conditions identified a restricted temperature–composition region in which the integrated rBN-related low-frequency response over 750–835~cm$^{-1}$, $A_{750-835}$, is strongly developed together with the integrated total response of the main $\sim$1367~cm$^{-1}$ BN band, $A_{1367}^{\rm{total}}$. In addition, the asymmetric high-wavenumber extension of the $\sim$1367~cm$^{-1}$ band was quantified by subtracting a symmetric fitted core and integrating the far high-wavenumber residual, $T_{\rm{high}}^{\rm{far}}$. The independently defined $A_{750-835}$ and $T_{\rm{high}}^{\rm{far}}$ responses exhibit a strong correspondence across the full dataset, supporting their use as complementary rBN-related Raman observables. In contrast, $A_{750-835}/A_{1367}^{\rm{total}}$ varies systematically among growth conditions, indicating that the relative prominence of the low-frequency response changes with temperature and composition rather than tracking the overall BN response through a fixed proportionality. Growth-duration experiments at 1350~$^\circ$C further show that this ratio also depends on high-temperature N$_2$ dwell time, establishing growth history as an additional kinetic variable. Powder X-ray diffraction on selected products independently confirms the presence of rBN-related stacking, while second-harmonic generation provides complementary symmetry-sensitive evidence. These results define a processing window favorable for strong rBN-related Raman response and introduce a quantitative framework for tracking both the low-frequency and band-asymmetry signatures associated with rBN-related BN structure.

Let $\eta_{1},\cdots,\eta_{d}$ be nonconstant one-variable inner functions satisfying $\eta_{i}(0)=0$ for all $1\leq i\leq d$. Denote by $M_{\eta}$ the submodule of the Hardy space $H^{2}(\mathbb{D}^{d})$ generated by the function family $\{\eta_{i}(z_{i})-\eta_{i+1}(z_{i+1})\mid 1\leq i\leq d-1\}$. For the corresponding quotient module $N_{\eta}=H^{2}(\mathbb{D}^{d})\ominus M_{\eta}$, we define the compressed multiplication operators $S_{z_{i}}=P_{N_{\eta}}M_{z_{i}}\vert_{N_{\eta}}$, where $P_{N_{\eta}}$ is the orthogonal projection from $H^{2}(\mathbb{D}^{d})$ onto $N_{\eta}$. In this paper, we characterize the Hilbert-Schmidt property of the self-commutators $[S_{z_{i}}^{\ast},S_{z_{i}}]$ and cross-commutators $[S_{z_{i}}^{\ast},S_{z_{j}}]$ with $i\neq j$. Moreover, explicit formulas for their Hilbert-Schmidt norms are established whenever these commutators are Hilbert-Schmidt.

Interactive world simulators can provide scalable environments for robot planning, policy training, and evaluation by predicting action consequences while reducing reliance on repeated physical rollouts. To serve these applications, they must generate future image sequences that respond faithfully to robot actions and preserve the dynamics of robot-object interactions over long horizons. However, existing world models typically predict the entire next latent state and often fail to capture subtle changes induced by robot actions. Such omissions can produce physically implausible outcomes, including object interpenetration and excessive deformation. To address this limitation, we propose DeltaWorld, a physically consistent interactive world simulator for robotic manipulation. Our method introduces the Delta Latent Transition Model (Delta-LTM), which predicts action-induced latent feature changes and adds them to the current latent state to obtain the next state, rather than predicting the next latent state directly. To mitigate object interpenetration and excessive deformation in predicted future frames, Interaction-aware Latent Alignment is introduced to construct counterfactual interaction regions and supervise interaction-related latent changes. DeltaWorld is evaluated on the IWS manipulation benchmark and a self-collected cross-robot dataset covering multiple robot embodiments and manipulation tasks. On the cross-robot dataset, DeltaWorld reduces FVD by 46.6% and LPIPS by 31.1% relative to the IWS baseline. These results highlight the potential of DeltaWorld for long-horizon action-conditioned video prediction in robotic manipulation.

Assuming a weak form of resolution of singularities in positive characteristic, we show several at-the-characteristic numerical invariants, including Hodge numbers, are given by homomorphisms out of the Grothendieck ring of varieties. Under the same assumption, we obtain two consequences: (i) $\widehat K$-equivalent varieties have the same Hodge numbers, answering a question of de Fernex–Mere; and (ii) universally homeomorphic varieties need not have the same class in the Grothendieck ring of varieties, answering a question of Nicaise–Sebag. In the appendix, we provide unconditional evidence for (i) by proving that birational Calabi–Yau varieties of dimension at most four have the same Hodge numbers.

For a smooth projective $3$-fold $V$ of intermediate Kodaira dimension, we establish Noether-type inequalities relating the Iitaka volume $\operatorname{Ivol}(V)$ and the geometric genus $p_g(V)$. Specifically, for a smooth projective $3$-fold $V$ of Kodaira dimension $2$, we show that $\operatorname{Ivol}(V)\geq \frac{3}{4}p_g(V)-\frac{5}{4};$ while if the Kodaira dimension of $V$ is $1$, then $\operatorname{Ivol}(V)\geq p_g(V)-1.$ Both inequalities are sharp. In the Kodaira dimension $2$ case, we prove that the equality holds only if the bicanonical system of $V$ induces the Iitaka fibration and the canonical system of $V$ is composed with a pencil of surfaces whose general member has Iitaka volume $\frac12$ provided that $p_g(V)\geq 4$.

Technology-facilitated abuse (TFA) is an emerging gendered public health crisis affecting millions of people across the globe. TFA coincides with other forms of gender-based violence (GBV), such as emotional, psychological, and physical abuse by an intimate partner. Survivors often seek support from GBV advocates who specialize in helping survivors navigate abusive situations and potential pathways for healing and remediation. With the rise in TFA, however, survivors and GBV advocates experience barriers and gaps in knowledge in identifying and mitigating this newer, ever-evolving form of abuse. To address this, we developed trauma-informed training as a critical capacity-building intervention to improve GBV advocates' knowledge and skills for responding to survivors and to encourage multi-stakeholder collaboration toward a structural response. We facilitated 8 training workshops at local, state, and national US-based GBV organizations. Our study demonstrates that the trainings increased advocates' perceived understanding of TFA abuse vectors and their impacts on survivors, along with advocates' confidence in providing support, safety planning, and referrals through collaboration with community stakeholders. We then conducted retrospective reflections through autoethnography to surface key lessons, contestations, and tensions that emerged from our experiences in designing and coordinating trainings with host GBV organizations. Our analysis of autoethnographic notes revealed three key design decisions: coordinating logistics flexibly, reducing reflexive distance from host organizations and attendees, and using our field experience to tailor content for accessibility. We contribute a concrete checklist to guide future HCI scholars in conducting training interventions for stakeholders in high-stakes contexts.

Large language models (LLMs) are increasingly deployed in enterprise, scientific, and medical applications, where agents must incorporate domain-specific knowledge and adapt from experience. Context engineering offers a practical alternative to weight updates by improving model behavior through instructions, strategies, and evidence supplied at inference time. However, adapting context online typically requires a costly trial-and-error process, while queries are often processed independently, preventing useful experience from carrying forward. Memory systems address this limitation by retaining information across interactions, but approaches that continually append information to a shared context face increasing token costs, context-window limits, and performance degradation as the context expands. We introduce a unified formulation of context optimization and show that an agent memory system update can be interpreted as an optimization update procedure over the model's context. This perspective attempts to provide a principled framework for studying memory design and its efficiency. We then propose GraphMemory, a lightweight graph-based memory that accumulates, refines, organizes, and connects reusable strategies. For each query, GraphMemory retrieves only the relevant subgraph, enabling online context adaptation without exposing the model to the entire memory. Under bounded retrieval, the amount of retrieved memory remains constant as the number of processed examples grows. Experiments show that GraphMemory achieves competitive downstream performance while using approximately 81-85% fewer memory-construction tokens than our baselines.

We develop a rigorous justification for a parameter estimation algorithm which couples continuous data assimilation to generic optimization. For finite dimensional systems, we provide a rigorous justification of an asymptotic approximation of the sensitivity for the underlying modeled dynamical system, prove that the $L^2$ loss function satisfies an approximate Polyak-Lojasiewicz inequality, and use that result to justify convergence of gradient descent for the proposed algorithm. Numerical examples are provided that demonstrate the precision of the rigorous results.

A bounded time-dependent drift together with an arbitrarily small real potential can produce double exponential decay of the full spatial $L^2$ norm. We prove matching quantitative two-time lower bounds for energy solutions of $\partial_tu-\Delta u=vu+w\cdot\nabla u$ on $\mathbb{R}^n$, for $n\geq3$, when $w\in L^\infty_{t,x}$ and either $v$ is sufficiently small in $L^\infty_tL^{n/2}_x$ or $v\in L^\infty_tL^p_x$ for some $n/2<p\leq\infty$, without a smallness assumption. In particular, a nonzero global solution satisfies a double exponential lower bound at large times. This decay scale is optimal for these coefficient classes: In every dimension $n\geq3$, we construct a real bounded drift, a real potential arbitrarily small in the critical space and in any prescribed subcritical $L^p$ space, and a nonzero real energy solution with double exponential decay. The construction localizes Rowan's torus example by absorbing the localization error.

Large language models (LLMs) are increasingly explored for clinical reasoning, but whether they appropriately revise judgments as patient evidence evolves remains unclear. We evaluated longitudinal belief updating using matched intensive-care trajectories from electronic health records. Across diverse LLMs, conditioning on a preceding judgment more often increased than reduced prediction error when estimates changed, replicated for a second endpoint. Controlled interventions revealed two failure modes. First, with preceding assessment fixed, models responded more strongly to worsening than matched improving respiratory evidence; this asymmetry persisted after headroom normalization at moderate and strong evidence levels. Second, with current evidence fixed, increasing prior risk from 10% to 90% shifted estimates by 26.2 percentage points, demonstrating causal influence of prior model beliefs. Prompting did not restore reliable updating. Evidence-Validated Longitudinal Update (EVLU) identified fewer, more reliable revisions, revealing a reliability-coverage trade-off. These findings establish longitudinal belief updating as a distinct dimension of LLM reliability.

Can programmable radio propagation serve as computational depth rather than only as a communication channel or one-shot analog transform? We introduce the Radio Alternating Operator Ansatz (RAOA), a recurrent computing architecture that alternates an energy-derived problem update with a mixing update over a persistent latent state. Recomputing the problem field after each mix makes repeated passes compositional even when the same learned controls are reused across depth. We evaluate this idea through exact discrete optimization, constrained programmable-propagation simulation, and pretrained-model adaptation. On discrete objectives, repeated execution can improve solution quality without increasing the learned-control count, and the same formulation handles higher-order interactions directly. A passive phase-only free-space model further shows that the required operators can be approximated by programmable propagation while retaining useful downstream behavior despite realization error. When inserted as a zero-initialized residual adapter, RAOA adapts pretrained language models with WikiText performance close to a matched shallow MLP across three model families, while reasoning-task transfer remains model-dependent. Together, these results connect alternating-operator computation, programmable radio propagation, and neural adaptation within one recurrent framework. The RF realization evidence is simulation-based rather than a hardware demonstration.

Boolean factor models, including prominent subfamilies such as Boolean matrix decompositions and cognitive diagnosis models, find broad applications ranging from social sciences to engineering. Despite their flexibility, a key challenge lies in establishing the identifiability of their graphical structures, which specify how latent variables influence observed variables. Existing identifiability conditions typically rely on the strong assumption of pure nodes, which may be unrealistic in many applications. We develop a novel approach leveraging the Hasse diagram to represent the distribution of observed variables and transform identifiability into a graph isomorphism challenge. Based on this, we establish sufficient and necessary graphical identifiability conditions that do not require pure nodes. We further derive equivalent algebraic conditions and develop an efficient Boolean satisfiability (SAT)-based verification algorithm. We extend the analysis to probabilistic Boolean factor models, establishing conditions for jointly identifying the graphical structure and additional model parameters without requiring pure nodes. Our results substantially broaden the class of identifiable and interpretable Boolean factor models by removing the pure-node requirement, yielding new theoretical insights, while also providing practitioners with a concrete and easily implementable tool to assess model identifiability.

We prove that, for every $\eps>0$, any $n$-vertex graph that needs at least $\eps n^2$ edge changes to become a Turán graph has positive discrepancy at least $c_\eps n^{5/4}$. Consequently, every such regular graph has second eigenvalue at least $c'_\eps n^{1/4}$. These results prove two conjectures of Räty, Sudakov and Tomon.

Multiparametric optimal control in continuous time partitions the space of initial states into critical regions, each with its own control law involving a set of time-varying active constraints. The boundaries between two neighbouring regions are either hyperplanes, which are given in closed form by one linear equation, or curved boundaries, which must be computed. The sequence of time-varying active constraints plays a key role in defining the linear or nonlinear shape of a critical-region boundary. In this paper we show how to determine that shape prior to any computation. In particular, we show that the boundary is a hyperplane under two conditions. First, the new constraint becomes active at the same instant at every point of the boundary. Second, the durations for which the other constraints stay active are the same at every point of the boundary. Otherwise, the boundary is curved. Both conditions can be read from the sequences of active constraints of the two neighbouring regions, which gives five rules, each yielding a hyperplane or a curved boundary. We illustrate the rules on the Newell-Lee evaporator, where they classify the eight boundaries of a seven-region partition into four hyperplanes, given in closed form, and four curved boundaries, computed exactly.

Data journalism, the practice of using data analysis to surface newsworthy stories, depends increasingly on the ability of reporters and investigative journalists to uncover trends, disparities, and accountability narratives. In practice, exploring large structured datasets remains slow and brittle: journalists must navigate hundreds of variables across many datasets over years, understand data coding conventions, and write non-trivial analysis code while hypotheses evolve. Although LLMs are often touted as "ask in English, get SQL/answers," real newsroom workflows expose recurring failures, e.g., schema mismatches and drift, misread domain semantics and units, and silent assumptions. We present DataWeave, a system that addresses these needs by combining conversational interaction, schema grounding, analytical planning, and executable query generation to support exploratory analysis over structured data. Rather than treating LLMs as autonomous answer engines, DataWeave frames them as interactive partners whose outputs can be inspected, corrected, and steered as hypotheses shift. We present a case study with professional journalists using our system to analyze the U.S. Department of Education's Integrated Postsecondary Education Data System (IPEDS), a high-stakes public dataset with substantial domain semantics and frequent schema updates. We also report how deployment experience and iterative refinement shaped the current DataWeave architecture and its analytical workflow. Our findings distill design principles and deployment lessons for trustworthy human-LLM collaboration in structured data analysis.

On-policy distillation (OPD) combines student-generated rollouts with dense token-level supervision from a teacher, but providing such supervision for every rollout requires substantial teacher computation. We introduce Success-Referenced On-Policy Distillation (SR-OPD), which reduces this cost by selecting which prompts and rollouts receive teacher supervision. When the student produces both successful and failed rollouts for the same prompt, a successful rollout can serve as a natural reference for selecting failed rollouts. SR-OPD therefore focuses on such prompts and prioritizes failed rollouts whose hidden-state trajectories show sustained divergence from a successful reference, while accounting for estimated teacher-input cost. Across three teacher-student pairs and six mathematical reasoning benchmarks, SR-OPD uses only 3.46-5.02% of the teacher-input tokens required by Vanilla OPD in the one-pass setting while maintaining comparable reasoning performance. Under a controlled setting matched to 5% of Vanilla OPD's teacher-input budget, further experiments support both key design choices: focusing supervision on prompts with both successful and failed rollouts, and using successful rollouts to guide failure selection. These results indicate that a student's own successful behavior can serve as a useful reference for allocating teacher supervision under a fixed teacher-input budget.