Papers

Training on broad, unfiltered data enables high-quality image generation, but can also teach models concepts that should not be generated; restricting the training data ensures safety, but reduces quality because examples containing unsafe concepts also provide useful knowledge that is not itself unsafe. Existing approaches largely address this trade-off post hoc through concept unlearning, which can lead to catastrophic forgetting as the number of concepts to remove grows. We instead ask whether unsafe concept knowledge can be compartmentalized during pretraining so that access to it can be disabled at inference. We introduce Pristine Models, a novel text-to-image generation architecture with an attention mechanism that encourages the separation of safe and unsafe knowledge into independently parameterized branches. The branch-specific parameters are learned during training, while the unsafe branch can be detached at inference. On ImageNet, Pristine Models enable safer generation while retaining much of the utility of full-data pretraining, achieving higher image quality and better text alignment than training only on safe data or applying post-hoc concept removal. We further scale our approach to CC12M, demonstrating that it remains effective in a substantially larger and more diverse open-world pretraining setting.

Exterior Gaussian-splat twins for real-estate listings are captured to a fixed recipe: a consumer drone orbits a single property in several overlapping passes at stepped altitudes, with occasional vertical ascents, so the imagery is dense, sequential, GPS- and time-tagged, and revisits each facade from many angles. This structure is tantalizing: it suggests camera poses could be recovered by a dense pose-graph optimization (PGO) whose only unknowns are the poses, far fewer than the joint pose-and-point problem bundle adjustment (BA) solves. PGO is mature in robotics, but almost always as post-processing: it redistributes the drift of an already-decent trajectory once loop closures tie it together. Splatting captures push that regime to its limit, every frame closing loops with dozens of others, raising a clean question: does pose-only optimization reach BA accuracy? We built a system specialized to this capture structure, not a general-purpose SfM tool, to test it. It beat COLMAP on speed but never on quality; this paper is the candid post-mortem. The shortfall arises from several interacting causes, hand-set weights, greedy merging and a noisy GPS ruler, together with two limits inherent to the formulation: SE(3) edges carry no scale degree of freedom, so independently scaled sub-maps drift when merged, and a two-view edge stops short of the coupling one 3D point imposes on every frame that sees it. A viable design must add back Sim(3) constraints, absolute GPS priors, retained structure, and uncertainty-aware weighting

Smart glasses are a promising platform for vision-language assistants, but private split inference remains expensive: protecting visual representations before transmission does not remove the cost of on-device vision encoding, wide protected tokens, or hardware-unaware execution. We present SAGE (Smart-glasses-Aware Guarded Encoding), a hardware-aware framework for efficient protected VLM inference on smart glasses. SAGE keeps raw images and unprotected visual tokens on the trusted device, reduces visual input cost, executes shape-specialized INT8 graphs, and compresses protected representations into compact 256-dimensional INT8 packets before transmission. Unlike efficiency methods optimized only for nominal model size, SAGE selects its operating point using downstream answer quality together with measured latency, memory, and thermal behavior on real Mentra Live glasses. On a matched Qwen workload, SAGE reduces glasses inference latency by 39.18%, device-sensor temperature rise by 21.91%, and protected packet size by 96.56%, while retaining 85.62% of the protected reference's relaxed utility and comparable prompted containment. Direct glasses-to-phone tests deliver all 192 packets and reduce median packet-plus-acknowledgment time from 71.3 to 10.0 ms. The compact protected interface also transfers across four additional VLM families. These results show that private wearable VLM inference must be optimized jointly across representation protection, communication, and real-device execution rather than through model compression alone.

A photon has the strange property that it travels along a null interval with zero proper time, hence its emission and absorption points are, in a sense, a singular event. For their part, the emission and absorption processes have a finite duration which determines the spectral content of the photon. The question in the title points to the conceptual difficulty in `stitching' together these three intervals, which all correspond to the same physical entity but have fundamentally incompatible temporal descriptions. A previously developed model for quantum wavefunction propagation solves this apparent dilemma by formulating the life cycle of a photon in terms of recursive Fourier transforms on dual spaces. Because it doesn't treat space as dependent on time, and it is built on the Fourier transform whose phase is an invariant in special relativity as well as the foundation of the uncertainty principle, the model is a promising candidate framework for making quantum mechanics compatible with relativity. The model proposes to serve as an improved Klein-Gordon equation, or relativistically compatible replacement for Schrödinger's equation.

Philip L. Roe, the renowned British scientist, passed away on April 26, 2026, at the age of 87, while Professor Emeritus at the University of Michigan. My association with Phil Roe began in 1984, when he joined the Cranfield Institute of Technology, UK, as Professor of Computational Fluid Dynamics, while I was a postdoctoral research fellow at the same institution. For more than four decades, his incisive ideas, physical principles and distinctive approach to research on numerical methods for solving hyperbolic partial differential equations have had a lasting influence on my own approach to research in the field. Here I recount some personal memories and anecdotes emerging from the rewarding experience of interacting with Phil during the Cranfield years and beyond. I also review, quite succinctly in places, some of the main research topics that he pursued throughout much of his academic career. These notes are intended as a modest homage to Roe's impact on the subject, as well as to his influence on the careers of generations of scientists in both academia and industry.

We study Gaussian maxima arising from isotropic families of unit vectors and show that their excess expected maximum is controlled by the distance from an orthonormal basis. We apply this stability result to correlation matrices with equal nonzero eigenvalues, obtaining an exact minimum when the matrix size is divisible by the rank and a quantitative gap otherwise. In dimension two, we determine the leading asymptotic term and describe the structure of near-optimal configurations. We also obtain covariance-block recovery results and a finite-dimensional lower bound for the higher-rank limiting problem.

We study a model of a random centrally symmetric polytope and ask about the existence of points in the polytope with non-zero, integer coordinates. Our main results are the location of a sharp threshold for the existence of these integer points, a characterization of the typical empirical distribution of their coordinates, and showing that near the feasibility threshold these integer points are almost always totally isolated — at linear Hamming distance to the nearest other integer point in the polytope. The behavior of these random integer programs is related to that of the symmetric binary perceptron (the case when coordinates of points are $\pm 1$-valued). The lack of symmetry in the set of potential solutions leads to new questions and phenomena, such as the distribution of the coordinate sizes in typical solutions, and requires the development of new information-theoretic techniques to deal with higher-dimensional optimization problems arising in the method of moments.

Let p(n,k) denote the number of partitions of n into exactly k parts. In 1942 Auluck, Chowla and Gupta conjectured that for every n the sequence p(n,1), p(n,2), ..., p(n,n) is unimodal. Szekeres proved this for all sufficiently large n, but with a threshold that was never made explicit. We prove the conjecture for every n >= 1. For n >= 10^5 we give an analytic argument with fully explicit constants: an exact probabilistic representation of the ratio p(n,k+1)/p(n,k), a second-order saddle-point main term with a rigorous uniform error bound in the transition window, and separate elementary or saddle-point estimates in the two tails. Every numerical constant in this argument is certified in ball arithmetic. For n <= 2*10^5 the conjecture is verified exhaustively by two independent programs, one of them using exact integer arithmetic only. Since log-concavity of k -> p(n,k) fails for every n >= 4, the argument works with first differences and never uses log-concavity.

We study roots of Bernstein-Sato polynomials of ideals. Our first main result gives a general partial differential equation criterion for detecting roots of the Bernstein-Sato polynomial of an ideal. For homogeneous case, we recast this criterion as an analytic condition, connecting the algebraic problem of locating roots with analytic properties of the defining data. As an application, we give a series of roots for the general determinantal ideal prove the strong monodromy conjecture for them. Keywords. Bernstein-Sato polynomial, determinantal variety, monodromy conjecture.

Robotic perception must remain reliable when extreme environmental conditions severely degrade the primary RGB observation. Although robotic platforms commonly carry complementary sensors such as depth and infrared, much of the modern perception stack remains RGB-centric, creating a mismatch between multimodal onboard sensing and pretrained downstream models. We study whether complementary sensing can be used upstream to recover a task-compatible RGB observation that preserves existing perception capabilities without modifying or retraining the downstream stack. To this end, we introduce SenseWeather, a controlled multimodal benchmark for robotic perception under severe rain, snow, dust, fog, and darkness, with registered RGB, depth, and infrared observations, clean references, and grounded high-level perception tasks. We further develop Fusion-to-RGB, with two parameter-efficient sensor-fusion realizations based on token-space fusion for a diffusion transformer and residual latent conditioning for an inpainting UNet. Across frozen open-vocabulary grounding and vision-language models, RGB+D+IR fusion improves all paired downstream comparisons over the corresponding RGB-only restorers, including AP50 gains across all ten detector comparisons. Importantly, these perception gains do not consistently follow conventional image-fidelity metrics, and increasing weather severity does not induce monotonic downstream degradation. These results show that complementary sensing can preserve reusable robotic perception under extreme conditions, while highlighting task-level perception robustness as an objective distinct from visual restoration quality.

Exact permutation covariance does not itself supply a sparse metric geometry. For permutation-covariant, full-support $k$-body interaction orbits, the incident interaction norm scales as $\binom{N-1}{k-1}|J_N|\|h\|$; such orbits cannot have uniformly finite range in metrics of diverging diameter. A permutation-invariant Hamiltonian diagonal in the graph basis has ground-state multiplicity at least $N!/|\mathrm{Aut}(G)|$ whenever $G$ is a ground graph. We give a quantitative version for approximately symmetric energies, together with fixed-radius connectivity and unitary-isometry obstructions. The trace-distance asymmetry of an arbitrary state also bounds its probability of selecting one labeled graph. We then construct two exactly solvable finite relational history Hamiltonians. A cyclic clock with a reversible incidence update has a unique ground state; gauge reduction yields a cycle Laplacian with a rank-one pin and the exact gap $\Delta_{N,L}=E_* (1-\cos\theta_L)$, where $\tan(L\theta_L/2)=1/\sin\theta_L$ and $0<\theta_L<\pi/L$. In particular, $\Delta_{N,L}/E_*=\pi^2/(2L^2)-2\pi^2/L^3+O(L^{-4})$, independently of the number of vertices. In an alternative order-free architecture, one clock qubit per unordered pair gives commuting four-local terms, $2^N$ ground states and a size-independent gap $E_*(1-1/\sqrt2)$. A fixed-weight control penalty selects $\binom Nq$ ground states without reducing the gap. These exact finite results expose clock and locality resource tradeoffs; they do not establish an emergent geometric phase.

We investigate the zeros of Leonardo polynomials and their distribution in the complex plane. We show that the structure of the zeros differs between even and odd degrees. For even degrees, all zeros are simple and purely imaginary, with imaginary parts lying in \((-2,2)\). For odd degrees, there is exactly one positive real zero, whereas no negative real or purely imaginary zeros occur. We obtain two-sided bounds for the unique positive real zero, prove that it decreases strictly to zero as the degree increases through odd values, and determine its asymptotic behavior. Using the relation between Leonardo and Fibonacci polynomials together with a complex trigonometric parametrization, we transform the zero equation into a family of fixed-point problems. Suitable invariant disks are constructed, and the corresponding mappings are shown to be contractions. This yields a constructive description of the complete zero set, proves the simplicity of all zeros, and provides geometrically convergent iterations for their numerical approximation. The method also provides explicit control over the approximation error. Finally, we establish a degree-dependent vertical strip containing all zeros and show that its width tends to zero as the degree increases. Consequently, the zeros become uniformly concentrated near the imaginary axis. Numerical examples illustrate the convergence of the proposed method, the associated error estimates, and the resulting zero distributions.

An LZ77 codec bounds every match by one sliding window, trading cheap offsets for short matches against reach for long ones. Match-length-dependent windows refine this trade-off: each match length has its own window, and the decoder infers the offset range from the decoded length. We present two such codecs: WLZ4, a byte-aligned LZ4-class format with one-byte and flagged offsets, and WZIP, an entropy-coded Zstandard-class format. Each block begins with its decoded length, from which WZIP derives its widest window and a decoder sizes its output. LZ4 blocks may expand incompressible input by 0.4%; WLZ4 adds at most 15 bytes, WZIP 2. We describe the formats, threaded match finders, decoders and parser. On Silesia, WLZ4 reaches a ratio up to 16% above LZ4HC's, decoding more slowly, and WZIP exceeds Zstandard 22's ratio while its trusted decoder runs at 91% of Zstandard's speed (87% checked; on enwik9, about as fast).

We implement (in a concrete example) a procedure of Dubrovin which gives a braid group action on spaces of solutions of ordinary differential equations such as the quantum differential equation of complex projective space. Such actions originated in physics for the tt* (topological-antitopological fusion) equations. We make the procedure systematic by utilizing an algebraic description of braid group actions on distinguished bases. We also generalize to the case of (arbitrary) solutions of the tt*-Toda equations, one of which corresponds to the quantum cohomology of complex projective space.

Embodied Audio-Visual Navigation (AVN) requires an agent to locate sound targets by jointly leveraging visual and auditory cues. While prior work has achieved notable success with static sound sources, real-world scenarios often involve moving targets, giving rise to the more challenging problem of Dynamic Audio-Visual Navigation (D-AVN). Existing approaches largely rely on semantic auditory features, which limits generalization to unheard sound categories and hinders fine-grained spatial reasoning. To address these challenges, we propose a novel Two-Stage Multi-modal Learning framework for D-AVN. In the Physical Alignment stage, the proposed Intensity-Calibrated Geometry Encoder (ICGE) performs physics-informed alignment of sound intensity with visual geometry to capture category-agnostic spatial correspondence. In the Scene Aggregation stage, a Transformer-based Cross-Modal Scene Aggregator (CMSA) dynamically integrates multi-modal features to construct a robust state representation for policy learning. Extensive experiments on Replica and Matterport3D demonstrate that our method achieves state-of-the-art success rates, while significantly improving generalization and navigation efficiency in complex environments and under unheard sound scenarios.

We study the length of one round of Uno under the official rules, for N players who each start with M cards. The fastest possible round takes at least M turns with two players and at least 2M-1 turns with three or more, and both bounds are attained for small M. An exact search over random deals shows that real deals rarely allow them. Under the official rules a round has no maximum length. Under forced play the probability that a round lasts more than t turns decays exponentially in t at a rate that does not depend on M. Our main result concerns very large hands. If every card a player ever holds is eventually played, a round lasts (27/19)NM turns to first order, and the constant depends only on the composition of the deck. The official rule that a Wild Draw Four may only be played by a player who holds no card of the current colour breaks this law. Every large hand hoards Wild Draw Fours, M/23 of them per player, and the round ends with a dumping endgame. For an even number of players this gives (53/46)NM turns. For an odd number of players the dumping cannot alternate round the table, and the endgame settles into a seating pattern whose turn rate we compute numerically. Simulations with up to 32,000 cards per hand support these limits. Finally we solve a small strategy game: with two players the Nash equilibrium over a family of 48 strategies is a single pure strategy that wins 62.8% of rounds against uniformly random play.

Multi-view perception is often treated as a safe design choice for robot manipulation: if one camera is useful, more cameras should be better. We revisit that assumption in tabletop bimanual manipulation by comparing policies trained with two large-field-of-view (FoV) wrist cameras against policies that additionally receive a head view, using controlled-variable experiments on a real bimanual platform. Across three tabletop tasks, adding a head view does not consistently improve task completion or execution quality: when the wrist cameras already observe the objects, grippers, and goal regions, wrist-only policies form a strong baseline. Restricting the wrist-camera FoV increases the potential value of a head view, but performance improves most reliably when the added view supplies concentrated, task-relevant information: region-of-interest (ROI) cropping yields more consistent head-view gains than the uncropped image, which is dominated by background and task-irrelevant regions. The data-scaling results further suggest an interaction between FoV and demonstration count: wide observations improve coverage but may require more data to stabilize, and they may reach a higher performance ceiling once enough demonstrations are available. These results suggest that camera design should be framed as a task-relevance and information-density problem rather than as a simple count of input views.

Large language models typically support different inference budgets through separate model sizes or fixed compute configurations. In multi-turn interactions, however, compute demand can vary substantially across turns. Using a high-compute model for every turn wastes computation, while routing among independent models may require rebuilding model-specific history states after switches. We therefore ask whether a single shared model can support multiple compute budgets that can be flexibly assigned across turns. We introduce Multi-Ratio ConceptMoE, which jointly supports token-to-concept compression ratios of 2, 4, and 8 (CR2/CR4/CR8) within a shared backbone. Lightweight full-token modules reduce compression-independent cost, ratio-specific boundary routers support distinct compression levels, and cross-ratio training enables mixed-ratio processing within the same context. Across PTBench and SFTBench, the three fixed modes outperform their compute-matched MoE baselines in all 6 comparisons while remaining competitive with independently trained fixed-ratio specialists. On CodeChat and CoSQL, mixed-ratio inference improves the accuracy–compute trade-off, outperforming the Large MoE by 5.87 and 2.68 percentage points while reducing FLOPs by 41.1% and 12.3%, respectively. Under the same turn-wise budget schedule, Multi-Ratio ConceptMoE further reduces FLOPs by 78.4% and 46.9% relative to routing among independent models. These results demonstrate that a single shared model can provide multiple compute budgets for flexible compute allocation across turns.

Large language models (LLMs) remain unreliable when reasoning about the future. Existing models often overlook the difference among future states, committing to specific answers even when the queried facts are inherently volatile. To capture this challenge, we introduce ChronoRegime, a future-oriented temporal reasoning task that requires models to determine whether historical temporal evidence justifies commitment to a future answer. Furthermore, we propose Confidence-Aware Temporal Reasoning (CTR), a framework that estimates future predictability from temporal dynamics and regulates answer commitment accordingly. Rather than performing unconditional forecasting, CTR commits only when the reconstructed evidence supports the target future state. Across multiple LLMs, CTR improves strict overall accuracy by 6.7 percentage points and reduces hallucination rates by 18.4 percentage points on average over the plain QA baseline. These results demonstrate that explicit predictability assessment is an effective mechanism for safer and more reliable temporal reasoning under future uncertainty.

In this article, we mainly investigate when the binary operation $T^{[n]}:[0, 1]^2\rightarrow [0, 1]$ satisfies the associative law and thus becomes a triangular norm where \begin{equation*} T^{[n]}(x,y)=\left\{\begin{array}{ll} f_{n}(T(f_{n}(x),f_{n}(y))), &\text{if}\,\, (x,y)\in{[0,1)^{2}}, \\ \min\{x,y\}, &\text{otherwise,} \end{array}\right. \end{equation*} $f_{n}:[0,1]\rightarrow\left\{0, \frac{1}{n}, \frac{2}{n}, \dots, \frac{n-1}{n}, 1\right\}$ given by $f_{n}(x)=\frac{1}{n}{\lfloor nx \rfloor}$ with $n$ a positive integer and $T:[0, 1]^2\rightarrow [0, 1]$ is a triangular norm. We present some sufficient conditions for $T^{[n]}$ to be triangular norms, and for $T^{[n]}$ to be never associative when $T$ is strictly monotonic.

Joint-embedding predictive architectures (JEPAs) learn compact latent world models without pixel reconstruction. Existing JEPA pipelines typically train visual representations and action-conditioned dynamics before adding a control policy or solver. We introduce LeWAM, an end-to-end JEPA for control with: (1) a joint training objective over next latent state prediction and action flow matching, and (2) an isotropic Gaussian constraint on the latent state, enabling a single model to support direct policy execution and efficient planning by evaluating sampled actions with its learned latent dynamics. Compared with prior JEPAs in visual planning, LeWAM improves success rate from 28.6% to 89.7% on contact-rich tasks and from 10.9% to 46.2% on long-horizon tasks, and plans 32.3 times faster. On behavior cloning tasks in simulation and physical robots, LeWAM is also competitive with established methods. Interpretability analyses show that LeWAM learns control-relevant representations and action distributions effectively guiding test-time planning and control.

We construct compact, embedded solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ that arise from various non-compact initial data and exist for a finite time.

For $\kappa > 1$, we study the reciprocal LCM-power function \[ \sigma^{[\kappa]}_{-1}(n) := \sum_{d \mid n}\mu(d)\,\sigma_{-1}(n / d)^\kappa. \] We determine its average and extremal orders. A natural Robin-scale normalization is strictly increasing in $\kappa$ and tends to $\sigma_{-1}$, yielding an RH-equivalent criterion. Unconditionally, the corresponding Robin inequality holds eventually on every bounded parameter interval. Directed interval arithmetic proves the inequality for each $n > 5040$ and $1 < \kappa \le 100{,}000$.

RDMA can reduce communication overhead in distributed machine learning systems, but realizing these benefits in production requires more than replacing TCP. Variable-size messages, concurrent connections, bursty incasts, and service failures introduce challenges that hardware offload alone cannot address. This paper reports our experience deploying ByteExpress, a software communication layer over hardware RDMA, in ByteDance's parameter-server-based inference and training systems. We present three lessons from the deployment: efficient RDMA requires buffer management across application and transport layers; scalable communication requires coordinated management of connections, receive capacity, and outstanding data; and service availability requires recovery above the transport layer. ByteExpress combines an RDMA-aware allocator, zero-copy transfers, QP multiplexing, receive-credit flow control, a software send window, and RPC-assisted recovery. In production inference, ByteExpress reduces mean and P99 model parameter fetch latency by 35.7% and 43.7%, respectively, and total CPU usage by 10.69% compared with kernel TCP. In training, ByteExpress reduces P99 model parameter push latency by up to 74.0% compared with TCP, with no PFC pauses observed at runtime.

We prove a large deviation principle for additive functionals of local Lipschitz functions of a one-dimensional symmetric zero-range process on the diffusive scale. The rate function is obtained from the dynamical large deviation principle for the empirical density and is characterized by a variational constraint involving the density at a fixed spatial point. On the microscopic side, a super-exponential replacement is needed to replace the local observable with a function of the empirical density. On the macroscopic side, the spatial regularity enjoyed by finite-rate trajectories is needed to pass from spatial averages to point evaluation.

LLM agents increasingly operate over sensitive resources, such as private documents and mutable system states, which can create serious safety and security risks. Runtime guardrails should therefore make inspectable, reproducible, and deterministic allow or refuse decisions before potentially irreversible agent actions are executed. However, safety and security requirements are rarely complete before deployment, so guardrails need to keep adapting as new risks emerge. Existing adaptive guardrails learn from deployment experience, but they usually apply safety policies through LLM inference, which can lead to inconsistent decisions and ambiguous policy interpretations. Formal guardrails, on the other hand, enforce policies deterministically, but they usually rely on pre-deployment or task-specific policies and have no mechanism to adapt to newly observed risks. We introduce VEPA, a framework for the verified evolution of formal agent guardrails. VEPA represents the guardrail as a formal policy of executable symbolic rules over typed agent events. During deployment, we use an LLM guided by deployment feedback to propose policy patches that fix newly observed failures while preserving previously correct behavior. Then, a deterministic verifier only accepts policies that are structurally valid and conflict-free. Finally, an evidence-aware authority layer activates a verified rule only when it has enough supporting evidence, so uncertain updates cannot directly affect guardrail decisions. Across ASSEBench-Safety, ASSEBench-Security, and ATBench, VEPA consistently outperforms baselines while expanding the scope of deterministic formal-policy enforcement. These gains also hold under more challenging deployment conditions, including mixed streams, corrupted feedback, and longer horizons.

We develop a class of semiparametric zero-altered models (ZAMs) for count data, estimated through the generalized method of moments (GMM). The proposed framework extends the utility of ZAMs by allowing for both under- and over-dispersion while preserving support on the entire set of natural numbers. This offers an alternative to other models that can experience computational challenges (e.g. Conway-Maxwell-Poisson) or whose supports can be truncated as functions of the parameter space (e.g. generalized Poisson). By formulating estimation through moment restrictions rather than full likelihood specification, the GMM approach provides substantial robustness against model misspecification. In particular, inference based on likelihood methods for parametric generalized linear models (GLMs), such as those built on Poisson, negative binomial, or related count distributions, can become unreliable when the assumed distributional form is violated, often leading to biased estimates and invalid inference. In contrast, our methodology requires only the first two moments of the distribution, ensuring consistent estimation even when higher-order distributional features are misspecified. We establish the asymptotic properties of the estimators, including consistency and asymptotic normality, under general regularity conditions. The flexibility of the approach is illustrated through both simulated experiments and real-data case studies, demonstrating superior performance relative to popular parametric GLMs, particularly in settings characterized by distributional misspecification. To facilitate practical adoption, we provide accompanying R functions for easy implementation, enabling practitioners to apply the proposed methods in applied settings without requiring bespoke programming.

Reward-guided inference-time alignment can improve joint audio-visual generation, but optimizing multiple alignment objectives requires many expensive model evaluations, making existing methods costly under limited compute. Particle-based search enables global exploration but incurs estimation error with a finite population, while Metropolis-Hastings (MH) can refine candidates locally but benefits from a strong initialization. We introduce Spend Where It Matters (\modelname), a two-stage, training-free, reward-agnostic framework that combines particle search with MH refinement under a fixed compute budget. In the first stage, we adaptively allocate particles across denoising checkpoints using online variance estimates, minimizing estimation variance while concentrating compute where it is most useful. This search provides a promising candidate for MH refinement, which uses local noise-and-denoise proposals on completed outputs to further improve reward. By starting from a strong candidate, MH can recover additional reward without requiring a larger particle population or costly global exploration. We show that, at matched compute, combining particle search with MH refinement achieves higher rewards than either approach alone. On JavisBench-Mini, at closely matched compute, \modelname achieves mean relative gains across seven metrics of $30.7\%$ over BoN and $14.1\%$ over EvoSearch with JavisDiT++, and $9.2\%$ over BoN and $17.0\%$ over EvoSearch with LTX-2. Beyond joint audio-visual generation, we also demonstrate the benefits of adaptive particle allocation on SD1.5 image generation, improving AestheticScore by $2.2\%$ and ImageReward by $6.0\%$ over uniform allocation.

In this paper, we study the transition threshold problem for the 2D Navier-Stokes Boussinesq equations around the Couette flow with a constant background temperature in two domains: $\mathbb{T} \times [-1, 1]$ and $\mathbb{R} \times [-1, 1]$. It was proved that if the sizes of initial perturbations of the vorticity and the temperature in Sobolev spaces are $\nu^{\fr13}$ and $\nu^{\fr56}$, respectively, then this system admits a globally stable solution. Besides, we derive the enhanced dissipation estimate for the vorticity and temperature, as well as the inviscid damping estimate for the velocity. The coupling in the equations poses some difficulties to the analysis. We overcome these challenges and establish the results via the quasi-linear method and energy estimates.

Continuous random node features increase the expressivity of message-passing neural networks (MPNNs), but existing theoretical analyses are not fine-grained enough to capture their limitations, particularly in approximating functions across graph sizes or under a fixed number of random bits. We develop a population-level framework for size-uniform approximation. For MPNNs with a graph-size-independent Lipschitz bound in their initial features, including standard mean, max, and order-normalized sum architectures, we show that continuous random features can be quantized to finitely many colors with arbitrarily small uniform error in expected output. This reduction lets us transfer limitations on finite-color distinguishability to limitations on size-uniform approximation with continuous random features. We use this framework to prove that these MPNNs are not size-uniformly universal, analyze the expressivity gains from colors and independent trials, and establish incomparability between random-color MPNNs and higher-order Weisfeiler-Leman methods. We also give finite-color MPNN constructions for tree subgraph isomorphism via color coding and a local MST task.