Papers

Frequency inhomogeneity can obscure the metrological advantage of squeezed spin states, making the choice of readout crucial. We study interacting spin-1/2 ensembles with static local frequency disorder and optimize a common collective readout over independent disorder realizations. Using Bayesian optimization of the full measurement-outcome distribution, we identify observables combining dipole and quadrupole operators that yield greater classical Fisher information than the optimized dipole readout considered here. For ensembles of up to 17 spins, their disorder-averaged CFI surpasses the standard quantum limit at both disorder strengths studied, including a regime where the dipole readout remains below it. For the larger ensembles studied, fixing the encoding axis to the direction that maximizes the disorder-averaged quantum Fisher information retains nearly the same performance. These results identify a route to extracting metrological information from frequency-disordered spin ensembles through the choice of collective observable.

A set $A$ of integers is an $h$-basis for $n$ if every integer in $[0,n] = \{0,1,2,\ldots,n\}$ belongs to the sumset $hA$, that is, can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. The classical postage stamp problem asks for the largest integer $n_h(k)$ such that $[0,n_h(k)] \subseteq hA$ for some set $A$ of nonnegative integers of size $k+1$. We consider the effect of allowing negative elements. Let $n^{\flat}_h(k)$ be the largest integer such that $[0,n^{\flat}_h(k)] \subseteq hA$ for some set $A$ of integers of size $k+1$ containing at least one negative integer. For $k=2$, we have $n^{\flat}_h(2) < n_h(2)$ if $h$ is odd or $h=4$ and $n^{\flat}_h(2) = n_h(2)$ if $h$ is even and $h \neq 4$. Proofs of these comparison results were autonomously discovered and formally verified in Lean by https://aristotle.harmonic.fun/}{Aristotle}, a formal reasoning agent developed by https://www.harmonic.fun/}{Harmonic}. In this paper we give new proofs of these results.

Let $M$ be a complete Riemannian manifold and define a magnetic Schrödinger operator $\LL = ({\bf d}+ iA)^*({\bf d}+iA) + V$ where $A$ is a real $1$-differential form in $L^2_{loc}$ and $V \in L^1_{loc}$ is a nonnegative electric potential. We prove that the magnetic Riesz transform $({\bf d} + iA) \LL^{-\einhalb}$ is of weak type $(1,1)$ with bound at most $2$. It extends to a bounded operator on $L^p$ for $p \in (1, 2]$ with norm bounded by a constant $c_p$ depending only on $p$. It is then independent of the dimension of the manifold and independent of $A$ and $V$. We also prove weak type $(1,1)$ bounds for symmetric elliptic operators subject to various boundary conditions on rough domains of $\RR^d$. The proofs are based on a new weak type $(1,1)$ criterion for operators.

In this paper, we establish sparse bounds for singular integral operators with rough non-translation-invariant Calderón-Zygmund kernels. Our approach verifies the localized hypothesis in the sparse domination framework introduced by Conde-Alonso, Di Plinio et al. The proof refines Seeger's microlocal decomposition using a Littlewood-Paley analysis adapted to the non-translation-invariant setting. As applications, we obtain quantitative weighted estimates for Calderón-type commutators and other singular integrals with rough kernels.

Motivated by the relative-slice problem in $4$-manifold topology, we define for a knot $R$ the notion of a relative knot concordance group $\mathcal{C}_R$ and analyze the structure of its algebraic analogue $\mathcal{AC}_R$ modulo torsion. Using Milnor signatures, we show that $\mathcal{AC}_R$ has infinite rank for every $R$. We also show that any prescribed finite set of natural numbers can be realized as orders of elements in a single relative concordance group.

Chest X-Ray (CXR) is a primary modality for diagnosing pulmonary diseases, yet the overlap of bony structures often obscures critical radiographic findings, increasing the risk of diagnostic uncertainty. While Dual-Energy Subtraction (DES) is the clinical benchmark for bone suppression, its widespread adoption is primarily hindered by specialized hardware requirements and increased radiation exposure. Recently, diffusion-based generative models have shown promise for this task, yet they are still constrained by reconstruction-biased latent compression, prior-induced domain gap, and a scarcity of high-quality paired datasets. To address these challenges, we propose DeBoneDiT, a depth-driven conditional bridge diffusion Transformer architecture for efficient high-resolution bone suppression. We first introduce Depth Auto-Encoder (DAE), a structure-aware vision tokenizer for latent compression. By incorporating a multi-level depth feature loss derived from the pretrained Depth Anything V2 encoder, DAE preserves both visual fidelity and structural consistency. Second, we formulate bone suppression as a source-conditioned Brownian bridge diffusion process, leveraging CXRs as both the diffusion prior and structural guidance to narrow the source-to-target domain gap and maintain anatomical plausibility. Furthermore, we have constructed and released SZCH-X-Rays, the largest publicly available high-quality paired dataset for bone suppression to date, comprising 741 pairs of CXR and DES soft tissue images. Extensive experiments and downstream evaluation demonstrate that DeBoneDiT achieves superior bone suppression performance with reduced computational overhead.

In initialized spacecraft tracking, a short-horizon motion prediction remains available as event support changes. Observation-Aligned Event Innovation (OA-EI) uses events to correct selected components of this prediction. A constant-body-twist model propagates the complete pose; the event encoder proposes rotational and normalized image-plane translation corrections, retaining propagated axial depth. Candidate magnitudes come from a bounded Scale Gate and a deterministic activity factor, with empty windows returning the prediction unchanged. Evaluation uses four FRESH physical-event trajectories, twenty rollout starts, and three training seeds. Each rollout starts from two reference poses and then uses self-estimates. At 7.7 seconds, rotation and translation errors fall from 41.63{\deg}/57.28 mm for prediction alone to 15.73{\deg}/22.41 mm for OA-EI, reductions of 62.2% and 60.9%. Failure rates fall from 20.0% to 1.67%. Matched ablations favor the rotation-bearing interface for overall pose accuracy; combining scaling with activity support gives the lowest errors at every tested event-retention level. On Jetson Orin Nano, the measured in-memory update takes 44.56 milliseconds at the median and 58.77 milliseconds at P95.

We prove a folklore conjecture on exponential sums with multiplicative coefficients: if $f$ is multiplicative with $|f(n)|\le1$, $(a,q)=1$, $|\alpha-a/q|\le q^{-2}$ and $q\le N/(\log N)^2$, then \[ \sum_{n\le N}f(n)\operatorname{e}(n\alpha)\ll\frac N{\log N}+\frac N{\sqrt q}. \] Complete multiplicativity is not required. We also prove two related conjectured bounds: the refinement in which $q$ is replaced by $q(1+N|\alpha-a/q|)$, when this quantity is at most $N/(\log N)^2$, and the corresponding bound for the sum with logarithmic weights. Finally, we show that the bound can fail for multiplicative $f$ satisfying only $|f(p)|\le A$ at primes and $\sum_{n\le x}|f(n)|^2\le A^2x$, for every fixed $A>1$.

Let $\LL = -\Delta + V$ be a Schrödinger operator with a non-negative potential. It is known that its associated Riesz transform $\nabla \LL^{-\sfrac{1}{2}}$ is bounded on $L^p(\RR^d)$ for $p \in (1,2]$, with a bound independent of $V$. We prove more, namely that the set \[ \{ \nabla \LL^{-\sfrac{1}{2}},\ 0 \le V \in L^1_{\mathrm{loc}}(\RR^d) \} \] is ${\mathcal R}$-bounded on $L^p(\RR^d)$ for $p \in (1,2]$. We also prove the ${\mathcal R}$-boundedness on $L^p(\RR^d)$, for $p \in (1,\infty)$, of the spectral multipliers \[ { m(\LL),\ 0 \le V \in L^1_{\mathrm{loc}}(\RR^d) } \] for any $m:[0, \infty) \to \CC$ satisfying a Hörmander type condition. Our approach is based on the analysis of Schrödinger operators $-{\bf \Delta} + {\bf V}$ on vector-valued spaces $L^p(\RR^d,\ell^2)$. In particular, we prove heat kernel estimates, Riesz transform bounds, and spectral multiplier results.

Emerging data-intensive space applications generate large volumes of data that must be processed, stored, replicated, and delivered efficiently. However, relying on terrestrial cloud regions can expose satellite data to high delivery latency and loss of timeliness due to intermittent satellite-to-ground visibility and limited access to ground infrastructure. Space Data Centers (SDCs), which organize collaborative satellites into logical compute and storage clusters, offer a promising approach for in-orbit data management and service continuity. In this paper, we first evaluate how satellite clustering with different sizes affects satellite visibility and gateway outage. We then propose BIER-DC, a stateless one-to-many communication protocol tailored for satellite-based SDCs. Unlike terrestrial multicast protocols designed for stable data center fabrics, BIER-DC decouples inter-SDC forwarding from intra-SDC delivery and encodes compact cluster-level forwarding information in packet headers. Comparative evaluations against BIER, BIER-TE, and Yeti show that BIER-DC significantly reduces packet-header length and byte-hop overhead, improving effective goodput and scalability for next-generation LEO space data centers.

We prove explicit positive corrections to the Dirichlet and Neumann Pólya inequalities at every positive eigenvalue index of every annular sector. The coefficients depend only on the radii and are uniform in the aperture. Real-order angular integration preserves the exact area term, while directional rounding estimates and radial trials control all frequencies. Optimized bounds attain the sharp circular-arc coefficient for the continuous count. We identify the full sector perimeter asymptotic and a low-index obstruction to using its coefficient throughout the spectrum. The strict Neumann inequality also passes to products with bounded Lipschitz domains.

We study learning-augmented streaming algorithms for estimating the value of Boolean Maximum Constraint Satisfaction Problems (Max-CSPs). Specifically, we consider streaming algorithms equipped with an $\varepsilon$-accurate oracle: for each variable, the oracle outputs a label in $\{-1,1\}$ that agrees with a fixed optimal assignment with probability $1/2+\varepsilon$, and disagrees otherwise. Prior work by Dong, Peng, and Vakilian [ITCS 2025] showed that, for the special case of Max-Cut, such an oracle enables a $(1/2+\Omega(\varepsilon^2))$-approximation using $\operatorname{poly}(1/\varepsilon)$ words of space in insertion-only streams, and $\operatorname{poly}(1/\varepsilon,\log n)$ words of space in dynamic streams. We substantially generalize and strengthen their result, provided that a highly accurate estimate of $\varepsilon$ is available. For any Boolean Max-$2$CSP and any $\eta>0$ independent of $\varepsilon$, we obtain, with high constant probability, a single-pass $(1-\eta)$-approximation using $\operatorname{poly}(1/\varepsilon,1/\eta)$ words of space in insertion-only streams, and $\operatorname{poly}(1/\varepsilon,1/\eta,\log n)$ words of space in dynamic streams, where $n$ is the number of variables. We further extend our techniques to general Boolean Max-$k$CSPs for $k\ge 3$, with slightly worse space complexity.

We construct quantum codes from codewords of a cyclic code and the minors of a polynomial matrix. The construction uses a two-row, three-column matrix whose minors yield a second matrix satisfying the CSS orthogonality condition. We derive the code dimension from the greatest common divisors of the entries and minors, with a simplified formula for odd lifting factors. A residue characterization identifies additional common factors of the minors and leads to a method for controlling the dimension through polynomial selection. We also give a condition for changing the minor generator without changing the code. Explicit polynomial constructions yield 13 quantum codes with verified minimum distances, including a $[[255,80,12]]$ code of rate $80/255$.

Federated Multi-Agent Systems (FedMAS) offer a promising paradigm for collaborative reasoning among LLM-based agents while keeping raw client data decentralized. However, reliable multi-round collaboration remains challenging. Agent selection must take into consideration evolving task requirements and the agent's historical reliability, while privacy rules must adapt to changing collaboration contexts and meet the client's personalized needs. To address these challenges, we propose PFedMA, a federated multi-agent framework for feedback driven multi-round collaboration under heterogeneous privacy requirements. Across rounds, PFedMA maintains historical capability and reliability states to guide agent selection, evaluates task responses before aggregation, and refines subsequent queries according to accumulated collaboration feedback. PFedMA enables bidirectional local to global and global to local private rule evolution across heterogeneous clients. Clients generalize locally updated privacy rules and selectively incorporate globally maintained rules, thereby facilitating cross-client privacy rule evolution while preserving personalized constraints. Extensive experiments across reasoning, privacy sensitive, and adversarial settings show that PFedMA improves task performance, reduces measured privacy leakage, and improves system robustness.

Expected-reward maximization in reinforcement learning concentrates probability mass on a few outcomes, even in tasks that require many distinct, high-quality solutions, a failure known as mode collapse. Inverse Probability Scaling (IPS) counters this in group-based policy gradients such as GRPO by dividing each reward by the outcome's probability, estimated from its frequency within the sampled group. When the outcome space is large relative to the group, outcomes rarely repeat and these estimates become uninformative. Since an outcome can be constructed along many trajectories, its probability is a sum over all of them, and computing it directly is intractable. We propose Marginal IPS (MIPS), which performs marginal evaluation by combining the sampled trajectory with a learned model of the trajectories that lead to each outcome. On Hypergrids, where the exact sampling distribution is computable, MIPS stays near the reward-distribution target as the outcome space grows relative to the rollout group, while frequency-based IPS degrades. In molecular synthesis and phylogenetic inference, MIPS sustains diverse, high-quality sampling where GRPO and frequency-based IPS collapse, showing that MIPS serves as a reliable solution to mode collapse in RL.

The problem of Private Information Retrieval (PIR) across graph-based replicated systems has seen a surge of interest in recent literature. In such systems, servers are modeled as the vertices of a graph, and each file is represented by an edge connecting the pair of servers that store it. While previous research on Graph-PIR (G-PIR) has focused mostly on perfect information-theoretic privacy, relaxing this constraint has the potential for significantly higher retrieval rates. In this work, we consider the setting of Graph-based Weak Private Information Retrieval (G-WPIR), under an $\epsilon$-privacy metric. We adapt an existing G-WPIR scheme for arbitrary graphs to analyze its performance under the $\epsilon$-privacy metric. Additionally, we establish a lower bound on the expected download cost for arbitrary graphs under the $\epsilon$-privacy metric and derive from it an explicit rate upper bound for complete graphs. Finally, we show the achievable rate-privacy trade-offs for two specific graph classes—complete graphs and complete bipartite graphs.

We introduce the Enhanced Neumann Pólya inequality, which is stronger than Pólya's conjecture, and verify it for many classes of domains, including strip-tiling domains, Euclidean balls, thin product domains, and certain higher-dimensional tiling domains. We develop a geometric removal approach for the Enhanced Neumann Pólya inequality. In particular, our results provide new classes of domains satisfying the Neumann part of Pólya's conjecture.

Digital advertising is an important means for platforms to reach potential users and drive business growth. Advertising delivery is the core link between marketing objectives and actual returns. Existing methods have achieved partial automation in individual tasks such as bidding and budget allocation, but they still rely on human operators for overall coordination, making it difficult to support large-scale operations. They also often fail to achieve optimal performance, and individual experience remains difficult to turn into reusable operational capabilities. In addition, as advertising markets change, advertising delivery systems need timely strategy updates to remain effective in dynamic markets. To automate the entire advertising operations process and improve adaptability to market changes, we propose AdCoEvo, a co-evolving multi-agent advertising delivery system. Specifically, the system connects delivery decisions across different timescales and operational granularities through role specialization. Diagnostic results guide account planning and strategy adjustment in the next period. The multi-agent system updates its model parameters based on actual delivery outcomes and continually improves delivery strategies through mutual feedback among planning, adjustment, and diagnosis, thereby achieving co-evolution. Experiments show that, relative to the rule-based operational baseline (Rule), AdCoEvo increases advertiser value by 14.02% while reducing cost per acquisition by 7.07%. Compared with a multi-agent system with fixed parameters, AdCoEvo increases advertiser value by 3.96% and reduces cost per acquisition by 1.24%, indicating that co-evolution can yield greater delivery returns.

We study a Patlak–Keller–Segel–Navier–Stokes system on $\mathbb T\times\mathbb R$ with density-dependent diffusion and chemotactic sensitivity growing like $n^{1+\alpha}$, where $0<\alpha<1$. The coefficients satisfy the constitutive relation of Painter and Hillen (Can. Appl. Math. Q. 2002; 10(4): 501–543). Near a sufficiently strong Couette flow, we prove global classical solvability and uniform density bounds for smooth initial data with arbitrary finite cell mass. A key ingredient is a joint energy estimate for the full density and the non-zero-mode vorticity, which preserves the positive nonlinear diffusion and avoids the mixed diffusion terms arising from separate estimates of the density modes. Bounded Fourier corrections yield enhanced dissipation within this energy, allowing the non-zero-mode density–fluid interactions to be controlled. For the density zero mode, which is not mixed by Couette transport, we observe that $\alpha<1$ allows one-dimensional interpolation to absorb its chemotactic production into diffusion for every finite mass. Together, these estimates control the nonlinear diffusion and fluid coupling without imposing a smallness condition on the initial cell mass.

Several recent studies propose detecting prompt injections from a language model's internal state. However, these methods do not measure a single security property. Residual-stream probes, attention scores, role projections, exposure probes, and concept readouts may capture injection presence, recognition, information routing, or behavioral influence. It is therefore unclear what differences in prompt-injection detection performance reveal about the internal signals these methods read. We study how five representation-reading methods vary under a common structured indirect prompt-injection setup on Qwen3.6-35B-A3B and Gemma-4-26B-A4B-it. We evaluate each method by how well it distinguishes injected prompts from benign controls. We then further analyze each detection method's ability to detect attacks in cases where they are successful and in cases where they are not. This helps us better assess what properties each detection method might rely on. Methods trained to detect injection presence or attack-related concepts generally separate unsuccessful injections more strongly from benign inputs, whereas attention diversion, exposure, and role-confusion-based methods separate successful injections more strongly on both models. Attention Tracker provides the strongest consistent successful injection separation, reaching 0.949 AUROC on Qwen and 0.887 on Gemma. These results suggest that signals closer to response formation may better encode information about an attack's success, and that detection performance can depend strongly on the prompt template. We share the data associated with this study at https://anonymous.4open.science/r/paper-evaluation-data-7877

We study two explicit constructions of binary Calderbank–Shor–Steane quantum low-density parity-check (QLDPC) codes with check weight six. The first uses a fixed circulant shift pattern and yields parameters $\code{6N}{2N+2}{6}$ with Tanner graph girth twelve for $N=31$ and every $N\ge37$. A graph incidence interpretation gives the ranks and girths, and a uniform weight six logical operator establishes the exact distance. The second construction uses fixed matrix polynomials with circulant and affine permutation coefficients. Here, a polynomial commutation identity guarantees orthogonality for every cyclic lift. Four realizations give $[[192,16,8]]$, $[[288,16,12]]$, $[[384,16,14]]$, and $[[576,32,20]]$, with Tanner girth eight and computationally certified distances. We give reproducible construction data, distinguish classical cycle distance from quantum distance, and identify the limitations of both families.

In 2018, Goddard and Henning conjectured that every cubic graph has four total dominating sets covering each vertex at most twice, and asked the analogous question for transversals of $3$-uniform $3$-regular hypergraphs. We prove the conjecture, which also answers the question in the affirmative.

For a Fourier-Mukai transform whose kernel is a family of Gieseker stable sheaves, we shall give a condition for the preservation of Gieseker stability of sheaves.

In this work, we calculate the binding energies of an equal-mass fermion–antifermion system in the $0^{-+}$ and $1^{--}$ channels from the ladder Bethe–Salpeter (BS) equation with scalar, pseudoscalar, and vector exchange. The results are compared with those of the Schrödinger equation with the corresponding leading static interaction in two regimes: pointlike massless exchange at weak coupling and finite-mass exchange with monopole form factors at stronger coupling. For the BS calculation, we use two independent numerical methods based on momentum-space integral equations and coordinate-space differential equations. The differential-equation method gives estimated relative binding-energy errors below $10^{-9}$ in both regimes. For massless vector exchange in Coulomb gauge, the numerical results are consistent with the uncrossed-ladder perturbative spectrum through the $O(m\alpha^5)$ terms. The results for massless scalar exchange and vector exchange in Feynman gauge are consistent with the common Wick–Cutkosky expansion through order $m\alpha^3$. For finite exchange masses, the differences from the leading static Schrödinger spectra at the same masses, couplings, and form factors depend strongly on the exchange and channel. Scalar exchange gives nearly degenerate channels with much weaker binding than the Schrödinger reference. In contrast, vector exchange with only the metric term retained gives stronger binding than the Schrödinger reference in $0^{-+}$ and weaker binding in $1^{--}$. Notably, retaining the full Proca numerator substantially increases the vector binding in both channels. Pseudoscalar exchange also supports bound states and favors $1^{--}$, although its leading static central interaction vanishes.

While Large Language Models (LLMs) show promise for Automatic Program Repair (APR), existing benchmarks suffer from data contamination, in which test samples appear in training sets, artificially inflating performance. To address this, we propose BenchChecker, a framework compatible with both open-source and closed-source models that supports more credible evaluation by detecting such contamination. Grounded in the observation that persistent patches likely remain in recent repository snapshots used for training, BenchChecker employs a forensics-based two-stage detection process using only textual model outputs together with public repository history. The first stage conducts a repository presence test by probing the model with code prefixes to infer training inclusion; the second stage performs a patch presence test to verify patch persistence via similarity alignment and test-suite verification. Evaluations using StarCoder-15B, StarCoder2-15B, and CodeGen2-16B trained on The Stack v2 demonstrate that BenchChecker achieves robust detection with AUC scores around 0.9 across six benchmarks. Furthermore, applying BenchChecker to SWE-bench Verified and Multi-SWE-bench reveals that filtering contaminated samples reduces the reported resolution rates of most LLMs by more than 20% relative to the original rates on medium-difficulty tasks.

Recommendation systems, systems that predict a user's preference for items in order to curate that user's feed, are a ubiquitous and central feature in online social networks. While they are a central piece of our modern digital infrastructure, many modern approaches are opaque black boxes, making them difficult to audit, understand, and assess the trustworthiness of. In this paper we present an exact, graph-based approach for generating explanations for decisions made by recommendation systems. We argue that in the risk-sensitive sociotechnical context of user-facing AI such as recommendation systems, explainability is a key pillar for establishing the trustworthiness of models and validating their effectiveness. We propose a technique built on LightGCN like architectures, an influential and popular class of recommendation models. We describe our approach and argue that it provides an appropriate assurance mechanism for these AI systems leading to specific advantages, including interpretable explanations and a theoretically guaranteed measure of explanation fidelity. We conclude by discussing limitations and directions for future research.

Neural instrument synthesizers can clone the timbre of a solo instrument, letting music producers make music with the sound they desire. However, an instrument of interest is often embedded in a fully produced, multi-instrument recording, beyond the reach of existing methods. Source separation before timbre cloning is an approach, but it may suffer from separation artifacts and is also likely a detour for the task. In this work, we propose Point-and-Play, a system that clones the timbre of a cued source directly from a polyphonic mixture and uses it to synthesize new pitches and velocities. To achieve this, we introduce a Cue-Gated Statistics Pooling (CGSP) module that extracts the timbre code of the cued instrument from adapted CLAP embeddings of the music mixture. This extracted timbre code is then used to condition a separately trained note synthesizer together with pitch and velocity inputs. Experiments show that our proposed system outperforms separate-then-clone baselines in terms of timbre fidelity and synthesis quality, achieving comparable performance with that cloned from a clean solo instrument.

Random reshuffling stochastic gradient descent (RR-SGD) is widely used for finite-sum optimization in machine learning. For convex $L$-smooth components, recent work establishes convergence guarantees for an averaged iterate with component stepsizes as large as order $1/L$. However, the final iterate is a natural output in practice, and existing last-iterate guarantees in this setting require stepsizes that scale inversely with the number of components $n$. We establish an expected objective-gap bound for the last iterate of standard RR-SGD on unconstrained problems with a constant component stepsize satisfying $0<\eta\lesssim 1/(L(1 + \log(nK)))$, where $K$ is the number of epochs. By retaining the sampling bias at random comparison points, we remove the explicit $1/n$ factor from the stepsize restriction. Combined with existing small-step bounds, our result matches the corresponding averaged-iterate convergence rate up to logarithmic factors.

Native multimodal pre-training trains a single model from scratch on text and multimodal data, enabling it to understand both language and visual inputs. However, how visual signals shape scaling behavior and support capabilities beyond visual understanding remains unclear. We investigate this question through controlled visual-content ablations across model sizes and training budgets. Specifically, we replace image embeddings with content-free padding token embeddings. Retaining visual content improves text prediction on image-text examples without materially changing the lowest text-only loss at a given compute budget, the optimal balance of model size and training data, or aggregate language scores. The relative multimodal loss benefit grows approximately as a power law of compute. To test benefits beyond understanding images, we evaluate spatial reasoning from text descriptions alone. Both conditions improve with model and data scaling. During training of the largest model, visual gains are most sustained in localization, navigation, and symbolic planning, which involve maintaining object configurations or composing spatial updates; gains in spatial relations, geometry, and physical commonsense are less consistent. These findings identify visual content as a scalable anchor for multimodal learning, with benefits that extend beyond visual inputs while preserving aggregate language performance.

In this paper, we develop a Koopman operator framework for modeling and control of traffic flow governed by the Lighthill–Whitham–Richards (LWR) conservation law. We show that the nonlinear dynamics induced by the Greenshields flow–density relation naturally motivate a dictionary of polynomial observables, while the controlled dynamics become bilinear in the lifted Koopman coordinates. A reduced-order Koopman model is identified from spatio-temporal traffic-density snapshots and used as the prediction model in a model predictive control (MPC) framework for ramp metering. The resulting Koopman-MPC is compared with two predictive control baselines that use, respectively, a fixed linearized LWR model and the nonlinear finite-volume LWR model. In a one-hour freeway experiment, the Koopman-based controller maintains the traffic density below the critical Greenshields value and closely reproduces the behavior of the nonlinear PDE-based MPC, with a maximum-density difference below 1%. The Koopman predictor also reduces the mean online model-evaluation cost by approximately one order of magnitude relative to the nonlinear PDE predictor. In contrast, the linearized LWR predictor is computationally cheaper but exhibits substantially larger prediction errors and yields a more conservative metering policy. These results demonstrate that the analytical structure of the nonlinear traffic conservation law can be exploited to construct an interpretable data-driven surrogate for efficient traffic prediction and control