Papers

Envelope regression is a crucial part of multivariate linear modeling, leveraging separation into material and immaterial parts to provide parsimonious data reduction. Rekabdarkolaee et al. (2020) construct spatial envelope, a novel multivariate Gaussian process, extended by May et al. (2022) to linear coregionalization envelope. Our proposal (TEASE) combines spatial envelope and full-scale basis graphical lasso (LeDuc et al., 2025), leveraging targeted graphical elastic net (Kovács et al., 2021) for immaterial precision matrices. Material part is a multivariate multiresolution Gaussian Markov random field (Kleiber et al., 2019; Caringi and Secchi, 2026). Enhanced envelope penalty (Kwon and Zou, 2025) regularizes regression coefficients. Demonstrations are on climate data.

Automated building design must comply with seismic and wind codes and satisfy structural mechanics constraints, yet most existing agents produce visually plausible models without verification grounded in mechanical analysis and code compliance. We introduce the Vibe Building task and propose PE-Loop (Physics-Engine-in-the-Loop), an agent in which a deterministic physics engine is the sole source of evaluation signals, mapping code constraints to a physics process reward, while the language model is confined to proposing discrete revisions (section menu, topology, and lateral system). Designs are verified by held-out seismic and wind time-history checks and a constructability gate. On VB-Bench, 3,577 physics-adjudicated building instances across six code families, PE-Loop achieves the highest verified success rate under three of four backbone LLMs, the highest held-out seismic pass rate under all four, and the highest held-out wind pass rate under three. Replacing the physics verdict with a language-model judge, all else fixed, leaves 58.43% of accepted designs noncompliant. These results suggest that reliable structural design rests less on a stronger LLM proposer than on an adjudicator the proposer cannot influence, a division of labor for agents whose outputs must hold up in the physical world.

Large language model (LLM)-based multi-agent systems (MAS) have become a promising paradigm for complex information-seeking and reasoning tasks by enabling collaborative problem solving among specialized agents. However, existing MAS frameworks tightly couple task reasoning with coordination operations, including task selection, role assignment, message routing, and context management. As interactions grow, using powerful LLMs for these bounded control decisions introduces substantial token overhead and latency, limiting the scalability of agentic Web services. In this paper, we investigate whether coordination can be decoupled from expensive reasoning without compromising collaborative performance. We propose S1-MAS, a token-efficient multi-agent framework based on System One-guided computational division of labor. S1-MAS assigns bounded coordination decisions to lightweight System One models while reserving open-ended reasoning for capable LLM workers. Specifically, a lightweight controller selects inspection conditions, chooses subsequent tasks, and determines termination, while a compact reader retrieves condition-relevant evidence from authorized sources to support these decisions. Through a decision-evidence loop, selected tasks dynamically determine worker roles and source access, enabling adaptive collaboration without task-specific training. Extensive experiments on seven diverse benchmarks demonstrate that S1-MAS achieves superior accuracy while substantially reducing the inference cost. Across individual comparisons with AgentVerse, DyLAN, and SelfOrg on seven benchmarks, S1-MAS reduces GPT-4o token consumption by 44.9%-97.2% and measured end-to-end latency by 37.8%-93.0%. These results highlight its potential for scalable and cost-effective agentic Web applications.

We study size fluctuations of chemically active droplets in the regime of weak reactions and strong phase separation. By combining a thin-interface model for the deterministic droplet growth with stochastic noise based on fluctuation-dissipation arguments, we report excellent quantitative agreement between our theory and numerical field simulations. Finally, we use our model to infer the effective non-equilibrium potential that governs droplet growth, and show that active reactions that enable size control suppress nucleation.

Multiple-choice question answering (MCQA) is commonly used to evaluate large language models under the assumption that one of the provided options is correct, typically using answer-selection accuracy. However, in real deployments, users or retrieval systems may provide invalid option sets in which none of the listed choices is correct, and selecting one of them may incur downstream cost. We study this setting as penalty-framed no-valid-option MCQA. Using the mathematics subset of MMLU-Pro, we remove the labeled correct option, allow models to either choose a remaining option or output ABSTAIN, and penalize invalid forced-choice responses. We further introduce correct-conditioned analysis, evaluating abstention only on instances that the model originally answered correctly. Experiments show that high MCQA accuracy does not fully guarantee abstention reliability: even under explicit no-valid-option-aware instructions and penalty-based scoring, models still produce invalid forced-choice responses for a subset of originally correct instances. These results show that penalty-framed no-valid-option MCQA reveals an aspect of model reliability not captured by standard answer-selection accuracy.

In time-varying D-optimal subset selection, the number of replaced coordinates does not measure physical displacement: when the optimal selection for a set of mobile sensors changes, a single sensor may have to cross the entire graph. We therefore measure movement by a bottleneck matching distance on a fixed mobility graph, and define the distance margin at radius P as the smallest old-objective loss among configurations farther than P from the current maximizer. Our main result shows that if the projector drift is small and the distance margin at a chosen radius exceeds an explicit threshold, then every maximizer of the new objective lies within that radius, without tie-breaking rules or uniform conditioning assumptions over subsets. Moreover, we show that separation information of this kind is genuinely needed: a weighted-path construction shows that, at a fixed baseline eigengap, an arbitrarily small perturbation can move an optimal sensor from one end of the graph to the other, so no bound depending only on perturbation size and eigengap can vanish with the perturbation. We also give a finite certificate, conditions under which local and Voronoi restrictions retain a global optimum, and oversampled and regularized extensions, and we identify the learned dictionary criteria to which the results apply.

We give a general principle that converts fixed-uniformity bounds for forbidden traces into bounds with polynomial dependence on the uniformity. More precisely, let $H$ be a fixed set system on $h\ge1$ vertices, and suppose that, for some $\alpha\ge0$, $\operatorname{ex}_j(m,\operatorname{Tr}(H))=O_{H,j}(m^\alpha)$ for every fixed integer $j\ge1$. Then, for every $\varepsilon>0$, there is a constant $C_{H,\alpha,\varepsilon}$ such that \[ \operatorname{ex}_r(m,\operatorname{Tr}(H)) \le C_{H,\alpha,\varepsilon} r^{h-1-\alpha+\varepsilon}m^\alpha \] for all $m\ge r\ge2$. In particular, for trace-$C_4$-free hypergraphs and every $\varepsilon>0$ there is a constant $C_\varepsilon$ such that \[ \operatorname{ex}_r(m,\operatorname{Tr}(C_4)) \le C_\varepsilon r^{3/2+\varepsilon}m^{3/2} \] for all $m\ge r\ge2$. We also construct trace-$C_4$-free $r$-graphs showing that \[ \operatorname{ex}_r(m,\operatorname{Tr}(C_4)) \ge c r^{1/2}m^{3/2} \] for an absolute constant $c>0$, for every fixed $r\ge3$ and all sufficiently large $m$ (depending on $r$).

Vision-Language-Action (VLA) models have become a central paradigm for robot policy learning, which predict actions in three forms: raw action chunks, discrete action tokens, or continuous action latents. However, existing action representations primarily model action trajectories, with limited consideration of the visual dynamics induced by these actions. We introduce ViDAL, a Visual Dynamics-grounded Action Latent Space that anchors continuous action latents in the future visual dynamics of the scene. Specifically, ViDAL learns action latent space by training an Action Variational Autoencoder (Action VAE) to reconstruct action chunks while aligning its latent with future scene dynamics. When integrated into downstream robot policies, the proposed Action VAE serves as a plug-in action interface compatible with multiple VLA architectures and enables optional future-video prediction as an additional capability. Empirically, ViDAL outperforms competitive baselines on LIBERO with 98.1% average success, improves a multi-task $\pi_{0.5}$ policy on RoboTwin 2.0 from 54.3% to 65.5% (Clean) and from 33.2% to 43.1% (Random) success rates over 50 dual-arm tasks, and yields 20.0% and 23.4% absolute success-rate gains on real-world single-arm Franka and dual-arm ARX robot platforms.

We prove that, on a Galton-Watson tree with almost surely finite offspring and infinite offspring mean, the random interchange process has critical parameter zero for infinite cycles, conditionally on survival. Combined with the finite-mean strict inequality established in arXiv:2503.03319, this gives a dichotomy for locally finite Galton-Watson trees with offspring mean in $(1,\infty]$: the loop and link critical parameters coincide exactly in the infinite-mean case. The proof reduces the tree problem to a finite permutation estimate derived from peeling estimates for random surfaces.

We study linear Maxwell's equations on a cuboid that is split into two parts by a planar interface, across which the tangential magnetic field jumps proportionally to the tangential electric field. Conditions of this type arise in the modeling of atomically thin metamaterials, where the sheet is replaced by an effective surface current. Working in a state space that admits a surface charge on the interface, we show that the problem is governed by a contraction semigroup. To this end, we provide density results, Weber-type inequalities, and trace estimates for the underlying function spaces. We further prove a hidden regularity result, stating that the fields in the domain of the Maxwell operator are piecewise Sobolev regular of first order. Since the damping acts on the interface only, the energy does not decay for all initial fields. We decompose the dynamics into a unitary part, carried by the eigenmodes that are invisible to the damping, and a strongly stable part. In the non-refracting case with a non-resonant interface position, we characterize the decaying initial fields by two moment conditions.

We derive the density matrix of a nonstationary system in the adiabatic representation and obtain the corresponding Liouville equation. This representation is particularly suitable when the standard interaction representation is inapplicable. We illustrate the approach with two identical, spatially separated spin-1/2 particles coupled by a magnetic dipole-dipole interaction in a slowly varying external magnetic field. Specifically, we analyze the spin polarization of pseudo-qutrit states in regimes where the states become entangled. The application of the adiabatic representation to calculate the dependence of observed physical quantities on time for quantum systems is illustrated. Our results show that entangled states can significantly modify the behavior of these observables and provide additional opportunities for their control.

In this paper, we introduce the concept of quasi non-signaling in time (QNSIT), which extends non-signaling in time (NSIT) to quasiprobability distributions. QNSIT requires that the marginals of the quasiprobability distribution under consideration at intermediate times remain unperturbed by the measurement back-action induced by a sequence of quantum measurements. Associated with QNSIT, we propose a quasiprobabilistic characterization of macrorealism, ${\rm MR}_{\rm qp}$, which requires that both the QNSIT conditions are satisfied and the quasiprobability distribution constructed from the measurement outcomes is positive. We prove that ${\rm MR}_{\rm qp}$ implies NSIT, whereas the converse does not hold because of hidden superposition paths generated or preserved by the system dynamics. We also prove that, for a two-level system, the ${\rm MR}_{\rm qp}$ and NSIT conditions coincide. ${\rm MR}_{\rm qp}$ therefore provides a more sensitive tool for testing macrorealism and can be implemented experimentally through a quantum non-demolition measurement scheme.

Let M be a compact orientable simple 3-manifold, and let F be a component of the boundary of M with genus at least two. Suppose that alpha and beta are separating slopes on F and that both M(alpha) and M(beta) are reducible. We prove that the distance between alpha and beta is zero. Thus the reducible-reducible case of the conjecture of Qiu and Zhang on separating degenerating slopes holds in arbitrary boundary genus.

Autoformalisation is increasingly used to verify mathematical texts, including those generated by AI, as in OpenAI's announced proof of blow-up of solutions to the Navier-Stokes equations. In this process, an AI system translates the text from a natural language (NL) into a formal language such as Lean. Once this translation is done, the argument expressed in the formal language can easily be mechanically verified. The purpose of this article is to demonstrate why this process may offer no confidence in the original NL argument, owing to the various difficulties in performing the translation semantically faithfully. In particular, we highlight that the problem of resolving ambiguities in mathematical NL text, which is necessary in order to provide semantically faithful translation, is arbitrarily high up in the Solvability Complexity Index (SCI) hierarchy/arithmetical hierarchy (the SCI $= \infty$). Hence, informally, providing semantically faithful AI autoformalisation is harder than any computational problem including the Halting problem (which has SCI $= 1$). To demonstrate the effect of this result we provide several examples of AI mistranslations of NL statements and proofs into Lean in practice, resulting in mismatches between NL proofs and their Lean `verifications'. These include OpenAI's announced Navier-Stokes proof. In particular, we show that the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations.

In this study, we propose a kinetic trap–diffusion model to describe the emergence of Pareto distributions in money-exchange systems. Using kinetic Monte Carlo simulations, we show that the Pareto exponent depends explicitly on temperature and takes values in the range $0.5 \leq \nu(T) \leq 1.5$. In the present framework, the temperature $T$ acts as a control parameter that regulates the exchange dynamics through thermally activated diffusion. Unlike conventional kinetic exchange models, where the Pareto exponent is fixed by microscopic rules, the proposed model generates a range of Pareto exponents as a function of $T$. The temperature-dependent Pareto exponent constitutes the central novelty of the proposed kinetic trap–diffusion framework. This feature provides a natural explanation for the empirically observed variations in Pareto exponents across different countries and economic conditions. In addition, the fraction of zero-wealth agents and the Gini index exhibit a non-monotonic dependence on temperature, revealing distinct dynamical regimes arising from the competition between trapping and money mobility. The persistence of the Pareto-like stationary distribution under strongly nonuniform initial conditions further supports the robustness of the proposed mechanism. These results show that different Pareto-tail and inequality regimes can emerge from the same trap–diffusion dynamics through changes in a single control parameter

Zero-shot coordination in embodied settings requires acting while the partner is intermittently out of view, leaving existing methods with ambiguous partner representations and uncertainty over hidden partner states. We propose Predicting Intention of Partner (PIP) to jointly address these challenges. PIP uses a Joint-view VAE to distill richer training-time evidence from the union of both agents' local observations into a partner representation available from local observations alone. Partner-state Belief networks further infer the partner's hidden location and behavioral tendencies from the ego agent's interaction history. We evaluate PIP in Burrito-PO, Overcooked-PO, and a Melting Pot substrate, together with a human evaluation in Burrito-PO. PIP attains the highest mean performance among the compared methods across all three benchmarks. Human evaluation and diagnostic analyses further support coordination with unseen partners and the contributions of both components under partner occlusion.

Non-Gaussian features in femtoscopic observables have been observed in high-energy collisions, yet their origin remains not fully understood. This study focuses on the phenomenological approach to femtoscopy, going beyond the Gaussian approximation by utilizing Lévy-stable source distributions. Using UrQMD simulations, three-dimensional analyses of pion sources are done in central Ar+Sc collisions, and the extracted parameters are studied as functions of transverse mass and collision energy. This analysis contributes to a hadronic baseline motivated by past and future research work in femtoscopy at the NA61/SHINE experiment. A direct methodological comparison is presented between parameters extracted from the projections of three-dimensional pair-source distribution and those obtained from the three-dimensional correlation function, reconstructed via Fast Fourier Transform.

The stochastic gravitational-wave background (SGWB) is a major science target of future space-based interferometers because it encodes information about unresolved astrophysical populations and physical processes in the early Universe. Its stochastic nature, however, means that its detection and parameter inference rely on distinguishing its statistical contribution to the data from that of instrumental noise. In particular, both the SGWB and instrumental noise contribute to the data covariance, so an incomplete noise model can be misinterpreted as part of the SGWB and thereby bias the inferred spectrum and its parameters. Here we investigate this effect in the presence of correlations between the acceleration (ACC) noises of adjacent test masses (TMs) in each spacecraft. We simulate the correlated noise at the single-link level, propagate it through the moving-orbit response into the time-delay interferometry (TDI) observables, and analyze the resulting TDI auto spectra with two Bayesian frequency-domain templates: the Corr. Template includes the correlated-noise covariance, whereas the Non-Corr. Template neglects it. We find that, when correlated ACC noise is present in the mock data, the Non-Corr. Template exhibits systematic parameter offsets that generally become more pronounced with increasing correlation amplitude. In contrast, the Corr. Template recovers the injected SGWB and instrumental-noise parameters substantially more accurately. These results demonstrate that correlations in test-mass ACC noise can constitute a relevant systematic for space-based SGWB searches and motivate their explicit inclusion in mission-realistic noise models and inference pipelines.

Despite several decades of searches, the neutrino magnetic moment remains unknown. If it is large, the neutrino evolution can be affected by its coupling to strong magnetic fields like the ones present in at least some core-collapse supernovae. We explore these effects for Majorana neutrinos. We first discuss a schematic model often employed and show the impact of resonant spin-flavor precession on the neutrino signal for a future galactic supernova. Then, we consider detailed simulations of a neutrino-driven supernova with 17 ${\rm M}_{\odot}$ leaving a neutron star and a magnetorotational supernova with 26 ${\rm M}_{\odot}$ with a black-hole remnant. We present predictions for the supernova neutrino fluxes and the event rates in presence of a non-zero neutrino magnetic moment, for the running JUNO and upcoming Hyper-Kamiokande experiments. Our results show a significant impact for transition magnetic moments of $10^{-14} ~\mu_B$ during the neutronization burst and the accretion phase for the neutrino driven supernova, and as small as $10^{-15} ~\mu_B$ at the early times for the magnetorotational supernova.

Legal intelligence aims to support reliable decision-making across long-horizon legal processes involving evolving case states and multiple roles. However, real-world legal deployment exhibits substantial case heterogeneity in facts, evidence, and procedural contexts, exposing the limitations of static agent strategies. Moreover, legal reasoning is inherently interdependent across roles and procedural stages, making global reliability fundamentally different from isolated role competence. To address these challenges, we study training-free test-time agent adaptation, where agents continuously exploit deployment-time signals from preceding cases and ongoing interactions without updating model parameters. We propose \method, which introduces Test-Time Memory Evolution to retrieve reusable experience from previous cases, adapt it to the current factual and procedural context, and consolidate accumulated experience for subsequent decision-making. Further, Rubric-Aligned Collaboration verifies and revises role-specific actions according to behavioral and procedural requirements, enabling coordinated decision-making across roles and stages. Extensive experiments on J1-EVAL and LegalWorld across five backbone models demonstrate consistent improvements over representative reasoning and agent baselines with reasonable interaction and computational costs. Ablation and case studies further show that the two components provide complementary benefits in experience adaptation and cross-role coordination, improving the reliability and efficiency of long-horizon legal reasoning.

AI systems create images and videos with image/video generation models or by writing code and graphics descriptions that are then rendered. These routes can produce similar visible artifacts but expose different representations, intervention points, and provenance evidence. We develop a production-centered framework that compares detection and watermarking across both routes. An explicit verification specification distinguishes passive inference, message recovery, and authenticated provenance. We organize image, video, source-code, and rendering-aware watermarks by production stage. We examine the different requirements of generated images and video, plots and SVG, programmable video, and agent-composed workflows. Documented Claude, OpenAI, and rendering-tool interfaces connect the framework to concrete systems. We pose ten scoped research questions on identifiability, observability, fair comparison across stages, recoverable payload, reconstruction, synchronization, composition, hybrid local contribution, and private production-event authentication. The result is a conceptual research agenda grounded in published methods, inspected interfaces, and elementary boundary examples. It reports no experiments and claims no new theorems; its appendix results are elementary calculations, and documentation and source inspection establish interfaces, not empirical robustness.

Generative recommenders decode items from a user's interaction history, but offer no way for users to correct a recommendation that misses their current intent. Adding conversation is natural since items and words share same output space, yet training the model to converse may overwrite the history-to-item mapping it relies on. We introduce INTEGER (**INTE**ractive **GE**nerative **R**ecommendation), which extends generative recommendation to multi-turn interaction with a learned routing token that lets the model decide when to recommend, history re-anchoring that conditions each item on both past behavior and the dialogue, and behavioral replay with instruction-data rehearsal that prevents forgetting during adaptation. Users can thus give feedback on recommendations within the dialogue, while recommendations stay grounded in behavioral history and accuracy is not traded for fluency. On Amazon Beauty and Toys, INTEGER matches or exceeds the strongest baselines in accuracy with competitive conversation quality, improving Hit@10 by 13.3% on Amazon Beauty, and significantly outperforms the generative recommender it starts from. Our analyses show that INTEGER learns behaviors that naive adaptation fails to acquire, recommending once the user's intent is clear and staying attentive to behavioral history at the moment of recommendation. INTEGER also learns an intent-agnostic replacement over the item space, which suppresses rejected items but points to attribute-aware feedback as the next step.

We study a one-dimensional multispecies TASEP model with each particle type having an associated speed parameter and faster particles overtaking slower ones with the rate being the positive part of their speed difference. After expressing the sorting time of a finite system given a reverse-speed initial condition as the maximum of sorting times for all different fast-slow cuts, we prove that forbidding fast-fast and slow-slow overtakes does not alter the law of the sorting time. This allows for an LPP formulation and an analytic expression of the sorting time.

We design an almost instance optimal algorithm for the sum estimation problem using weighted sampling. We show that the sample complexity for sum estimation is closely related to the $\ell_2$ norm of the weighted sampling distribution. We show an almost instance optimal lower bound for this problem as well. We also study the moment estimation problem and design an algorithm that has better instance-wise sample complexity bounds.

Vision-Language-Action (VLA) models achieve strong robotic manipulation performance but incur high computational costs from processing long token sequences at every control step, limiting real-time deployment. Visual token pruning offers a direct solution, as visual patches dominate the input sequence and contain considerable redundancy. Existing approaches, however, either rely on indirect training-free heuristics, such as attention scores and motion thresholds, or require costly fine-tuning of the base VLA model. We introduce VLA-ACL (Action Consistency Learning), which learns a lightweight visual token pruning policy through action-level supervision while keeping the base VLA model entirely frozen. The training objective encourages actions produced from pruned visual contexts to remain consistent with the full-context teacher, with ground-truth actions as auxiliary supervision. This directly ties token selection to its effect on the downstream control output. Experiments on LIBERO and real-world manipulation tasks show that VLA-ACL prunes up to 87.5% of visual tokens while retaining competitive performance, reduces computation by up to 75%, and achieves a 1.5x inference speedup. These results establish a stronger performance-efficiency trade-off than existing frozen-VLA pruning methods and demonstrate the value of action-level supervision for visual token selection. Code is available at https://github.com/du-owen/VLA-ACL.

Concept drift threatens production machine learning, yet the empirical behavior of multivariate two-sample drift detectors at scale remains under-characterized. Existing benchmarks rarely address the hundreds of millions of rows and high-cardinality features typical of industrial-operational datasets. We evaluate five multi-column two-sample tests (marginal, projection-based, and kernel embedding methods) across three complementary environments: the Harvard Dataverse, a validated Failing Loudly reproduction (mean absolute error between 0.030 and 0.053), and a novel synthetic-injection benchmark on the 137.5-million-row Trendyol collection-ranking feature table. Testing four drift types across two severity-scope regimes, we demonstrate that distributed Maximum Mean Discrepancy with Random Fourier Features on Apache Spark scales robustly. Averaged over the four drift types in the strong regime and under a calibrated threshold, it achieves a Pearson correlation of r = 0.940 with expected drift magnitude, an 80.4% true positive rate, and a 3.2% false positive rate. Conversely, the per-dimension Kolmogorov-Smirnov test failed due to statistic saturation from ID-like columns under asymmetric sampling, establishing a critical constraint for large-scale sampling design. At weak configurations (realized-flip fractions of at most 0.57%), detectors struggled to reliably discriminate, highlighting the need for future intensity-grid power analyses to distinguish fundamental sensitivity bounds from scalable threshold shifts.

Achieving compositional concept generalization (CoCoGen), the ability to understand novel situations by recombining learned primitives, remains a fundamental challenge in artificial intelligence. Compositional semantic models such as Compositional Distributional Semantics (DisCoCat) offer solutions by generalising vectors to tensors, but suffer from scaling bottlenecks when learning the tensors. Mapping DisCoCat onto Variational Quantum Circuits (VQCs) resolves this limitation for text, yet the methodology has not been expanded to multimodal situations such as the ones involved in CoCoGen. This paper introduces Mu-DisCoCat: a multimodal variational quantum learning framework for DisCoCat that achieves CoCoGen. The framework first learns stable object representations from single-object image-text pairs, then fixes these and uses them to learn the relations between them in multi-object situations. In classical simulations, the model used Uhlmann state fidelity to compute the overlap between the multimodal circuit representations and achieved higher relational OOD accuracy than the evaluated CLIP baseline. Its deployment was evaluated using the destructive SWAP test across noisy quantum emulators, including a range of IBM fake backends, IQM FakeAphrodite, and the IBM Marrakesh quantum processor. Despite real-world device noise, the hardware-executed models maintained a strong positive correlation with simulated fidelities, reliably distinguishing unseen similar and dissimilar pairs. Our work establishes a framework for executing CoCoGen on VQCs, demonstrating a viable use case for near-term quantum hardware.

We show that the worst-case combinatorial discrepancy of $n$ points in the plane with respect to axis-parallel rectangles is $\Theta(\log^{3/2}n)$. The known bounds were $\Omega(\log n)$ and $O(\log^{3/2}n)$; we prove the matching lower bound. It holds for random point sets: for every $A>0$, there is a constant $c_A>0$ such that, with probability at least $1-e^{-An}$, every coloring of $n$ independent uniform points in the unit square has an anchored rectangle with imbalance at least $c_A(\log_2n)^{3/2}$. The proof is surprisingly simple and elementary. It reveals one coordinate digit by digit. With overwhelming probability over the points, the conditional gains of an oscillation potential add up to $\Omega(\log^{3/2}n)$ over $\Theta(\log n)$ digits. Bounded differences control the fluctuations well enough for a union bound over all colorings.

Cooperation with unfamiliar partners requires adapting to communication conventions that are not known in advance. We study this problem in a controlled Hanabi-derived environment with scripted hint generation, LLM-controlled receiving decisions, and frozen model weights. Across eight LLMs, linear probes recover intent conventions substantially more accurately than target conventions, yet receiving choices do not consistently agree with the sender's convention. We compare probe-predicted and ground-truth conventions presented either as general rules or as externally computed action recommendations. Rule statements yield modest and model-dependent changes in cooperation, whereas action translation produces larger gains on average. In a Qwen3-8B case study, matched-state statement reversals reveal much greater sensitivity to action recommendations than to rule statements. Activation transfers from oracle-action and non-oracle hint-restatement donors improve intent accuracy on both action classes, but the tested alternatives do not reliably reproduce these benefits. Together, these results distinguish convention decodability, sensitivity to convention information, and cooperative performance, and highlight limitations in turning available partner information into receiving decisions.

Ariel's Tier 3 observing strategy offers a unique opportunity to explore planetary and stellar variability in the spectral and time domain. However, without the correct toolkit, temporal variability of the host star may complicate combined analysesof observations obtained at different epochs. We investigate the impact of stellar variability on Ariel Tier 3 observations and establish three different mitigation approaches of increasing physical complexity. Using simulated repeated observations of a favourable system contaminated by different spot and facula configurations, we compare a naive, unmodified stacking approach, separate analysis of the infrared region through progressive removal of the most strongly contaminated optical wavelengths, and the correction of individual visits using combined star-planet retrievals. We find that stacking inconsistent visits can produce substantial biases in the retrieved planetary properties under strong stellar variability, although a single activity correction can still recover the planetary parameters reasonably well. Applying progressive optical data cuts substantially improves the accuracy of retrievals up to a point, without requiring any in-retrieval correction and causing only a modest loss of precision. Correcting individual visits prior to stacking produces more consistent spectra and generally smaller uncertainties, but requires greater computational effort. These results demonstrate that stellar variability should be considered when interpreting repeated Ariel observations, but need not preclude atmospheric characterisation of active systems. A flexible suite of mitigation approaches, supported by preparatory and contemporaneous stellar monitoring, can allow the treatment of stellar variability to be adapted to individual targets while maintaining the efficiency required to maximise Ariel's scientific return.