Science & Mathematics

We study the fully dynamic exact distance problem in unweighted directed graphs, where the graph undergoes edge insertions and deletions, and the distance between two query vertices must be reported. Under the combinatorial Boolean matrix multiplication conjecture, any subquadratic-time algorithm for this problem must use algebraic techniques. All prior work uses the polynomial matrix inverse as its algebraic tool, but within subquadratic time, this technique can only maintain distances up to a small-polynomial hop bound. So larger distances must be reassembled from short-range pieces by a combinatorial layer. This hybrid algebraic-combinatorial construction has led to a substantial slowdown for dynamic distances compared to other dynamic graph problems relying on algebraic techniques. We give the first subquadratic-time dynamic distance algorithm built on dynamically maintaining a different algebraic object than the matrix inverse. By maintaining the Frobenius form $A=KCK^{-1}$ of a dynamic generic (i.e., cyclic) matrix $A$, we can query all the powers $A^\ell = K C^\ell K^{-1}$ efficiently. For adjacency matrix~$A$, the smallest $\ell$ with $(A^\ell)_{s,t}\neq 0$ is the $st$-distance in the associated graph. So, unlike dynamic matrix inverse, we get distance information directly without hop truncation, and we no longer need to perform any combinatorial reassembly. This purely algebraic approach to dynamic distances leads to the fastest dynamic distance algorithm with $O(n^{1.58})$ update time. The earlier use of the Frobenius form maintained the dense change of basis $K$ explicitly and required $\tilde O (n^2)$ time per update [KS23]. Our key observation is that this change of basis is highly structured, which lets us maintain it implicitly. This gives the first subquadratic dynamic algorithm for the Frobenius normal form and characteristic polynomial for generic matrices.

Energy-efficient deployment of large language models requires understanding how model architecture, hardware, and serving settings shape energy per generated token. We introduce the Token-Energy Law, a compact, interpretable model of GPU-board energy per output token for dense, full-history decoding. It relates token energy to two core dimensions: active bytes and realized decode time. Active bytes describe logical byte demand per output, separating dense weights shared within a decode iteration from private key--value (KV) histories read and updated for each request. Realized decode time is the elapsed decode duration divided by the number of output tokens produced. We combine the two byte coordinates and time in a common affine form, with coefficients calibrated for each model and hardware/runtime configuration. This formulation explains how batching amortizes shared-weight work, longer attended histories increase private KV demand, and throughput determines the time-associated cost per output. Our experiments cover three models under BF16 decoding on NVIDIA Hopper and Blackwell GPUs, including two-GPU tensor parallelism on Hopper. Using fixed coefficients and observed decode time, held-out condition-mean energy estimates differ from measurements by at most 3.73%.

Large Language Model (LLM) agents can handle increasingly complex tasks, navigating long trajectories of observations and actions. As these trajectories grow, key-value (KV) cache I/O and storage become the dominant cost. Trajectories can also exceed the LLM's maximum context length, requiring control over active cache growth. Prior work uses context engineering to reduce cache size by trimming or summarizing history. However, modifying the trajectory prefix invalidates subsequent cached states and triggers costly re-prefilling of retained context. Task accuracy can also suffer when too little context is retained to limit re-prefill costs. We propose cache engineering as a new scheme. It allows agents to directly control their active KV cache without modifying the trajectory or re-prefilling retained context. Specifically, agents can proactively use drop to exclude selected cache from future attention and repos to adjust retained keys' position encodings. We formalize these cache operations and identify history dependence as a serving challenge: identical token sequences can yield different cache states under different cache operation sequences, making prefix matching insufficient for correct cache reuse. Thus, we co-design the AKV serving engine. It provides a stateless interface compatible with existing serving APIs, while internally managing stateful cache reuse, eviction, and restoration. Experiments show that a tool-response rolling-drop cache policy improves average benchmark scores by 1.7 points and reduces average cost by 27.7%, compared with conventional threshold-triggered summarization. With AKV, it achieves up to 1.2x the output throughput and a 1.5x speedup in P95 request latency. Recursive self-improvement of cache policies further reduces cost by 7.1% over expert design. Our code is at https://github.com/thu-nics/AKV.

In this report, we introduce Sage-A1.5, a 35B Mixture-of-Experts agentic model that builds on A1 to advance long-horizon problem solving through adaptive capability composition. Compared with A1, Sage-A1.5 learns not only to solve individual tasks, but also to organize work and revise its approach during execution. Sage-A1.5 highlights three improvements: agentic orchestration, verification, and open scientific discovery. First, agentic orchestration extends single-agent execution through task delegation and parallel collaboration among subagents with isolated contexts, enabling complex tasks to be decomposed and addressed concurrently and supporting efficient execution while reducing interference across concurrent subagents. Second, agentic verification strengthens long-horizon reliability through progress checking, evidence validation, and reflective strategy revision, allowing the agent to recognize unproductive search paths and pursue alternative approaches. Third, open scientific discovery extends agentic capabilities beyond answering predefined questions to conducting exploratory research, connecting scientific inquiry with interactive analysis and inspectable research artifacts. We showcase these capabilities through complicated agentic tasks and multidisciplinary scientific applications. Besides, we hope Sage-A1.5 provides the community with an accessible foundation for open scientific discovery, offering a practical path from long-horizon task execution to collaborative, verifiable, and inspectable scientific exploration.

Cellular states are shaped by both intrinsic gene expression programs and the spatial microenvironment in which cells reside. Spatial transcriptomics makes this context measurable, but effective representation learning requires capturing both spatially informative molecular signals and intercellular organization. Existing approaches address these challenges only partially: conventional gene selection can overlook spatially structured signals, while methods for encoding cell-cell relationships often either reduce a cell's microenvironment to proximity alone, discarding direction, or rely on absolute coordinates tied to an arbitrary tissue frame. Consequently, cells with similar neighbor distances but distinct spatial organization may receive overly similar representations, despite experiencing different biological contexts that shape their cellular states. We introduce Tessera, a spatial representation learning framework that addresses both molecular selection and cellular context. Tessera combines spatially informed gene selection with a geometric inductive bias that contextualizes each cell within its local tissue environment. By modeling relative cell-cell relationships, Tessera is translation-invariant while preserving direction, allowing neighboring cells to contribute differently according to their spatial context and cellular state. Across held-out tissues and technologies, Tessera outperforms existing spatial foundation models in representation quality and generalizes zero-shot to contexts up to $16\times$ larger than those seen during training.

Continuous-time reinforcement learning seeks an optimal policy under unknown dynamics driven by stochastic differential equations using observations collected at discrete times. We develop a policy gradient framework that combines rigorous convergence analysis with learning from discrete data. We first establish a performance difference lemma and derive policy gradient formulas for randomized and parametrized policies. We then prove exponential convergence of an entropy-regularized policy flow to the regularized optimum. For discrete parametrized updates, we derive a convergence bound that explicitly separates the contributions of critic error, policy-regression error, and the learning step size. To enable learning under unknown dynamics, we integrate this framework with PhiBE, which constructs a continuous-time surrogate generator from discrete transition data. The resulting on-policy and off-policy model-free actor-critic algorithms preserve the continuous-time structure of the control problem. Numerical experiments on an entropy-regularized linear-quadratic regulator and a nonlinear wheeled robot demonstrate the performance and stability of the proposed methods with both known and data-driven dynamics.

Program optimization aims to improve performance while preserving behavior. Coding agents can already make programs faster by changing algorithms, data structures, and execution strategies. However, existing equivalence checks mostly rely on tests, which agents have been shown to exploit, passing the tests without preserving the intended behavior. We propose agentic verified optimization, in which an agent produces a faster implementation together with a machine-checked proof of equivalence to the existing one. Because the existing implementation serves as the specification, the task avoids the specification-writing bottleneck of verified code generation, and recent advances in theorem proving make the proofs attainable. We formulate the task at the function and repository levels in Lean 4, where programs can be both executed and reasoned about. Few real-world applications are written in Lean, so we develop Aqueduct, an agentic framework that migrates C, C++, Go, and Rust programs into native Lean, validates each port by differential testing against its original. On the 189 reference implementations in Verina, agents produce faster proved rewrites for 156, including algorithmic changes such as replacing exponential recursion with dynamic programming. On 17 migrated applications, their 92 proved rewrites speed up 47 of 51 application scenarios. These application-level gains are smaller than the function-level ones, making the choice of functions to optimize an important problem. Current agents can thus already carry out verified optimization on both standalone functions and applications migrated from real programs. We discuss the remaining challenges and the steps toward verified optimization in production.

Collisions are a basic probe of unknown distributions, used for testing and for counting substructures in graphs, and they underlie Rényi entropy, a fundamental measure of uncertainty with applications in cryptography, machine learning, physics, and biology. Motivated by this, we study the natural birthday estimator. We prove Bernstein-type concentration for it and show that it is minimax optimal, uniformly and jointly in alphabet size, order, accuracy, and confidence, with no sample splitting, clipping, or median aggregation. This, in particular, settles the sample complexity of integer-order Rényi entropy estimation up to absolute constants in all regimes, closing the gap between the best known upper and lower bounds. Our concentration bound rests on a sensitivity analysis of collision count, and the matching lower bound on a sharp analysis of detecting a hidden spike in a plateau distribution.

Autoregressive models (ARMs) face a "Sequentiality Bottleneck" in reasoning, while diffusion language models (DLMs) offer parallelism but lack semantic coherence. To addresses both limitations, we propose Collaborative Reasoning Asymmetric Learning (CoRAL), a hybrid framework via an Asymmetric Planner-Executor (APE) architecture: a DLM Planner generates K diverse reasoning trajectories in parallel, while a large ARM Executor selects the optimal plan and synthesizes the final answer. To train the Planner effectively, we introduce Reward-Guided Denoising for CoT (RDC), which integrates outcome-based rewards into the diffusion process to steer it toward goal-directed reasoning. Experiments on vision-language benchmarks show that CoRAL significantly outperforms prior diffusion-based methods and achieves competitive performance against strong ARMs, achieving both superior efficiency and quality.

Simulation is a powerful tool for robotics, but building diverse, simulation-ready scenes at scale remains difficult. Reconstructing scenes from single images is a promising alternative, but visual similarity alone is insufficient: objects must also be recovered, accurately placed, and physically grounded. Existing methods that rely on post-hoc physics settling often misplace objects, producing scenes that look correct but behave poorly under manipulation. We present SceneRig, an agentic system that turns a single image into an editable, simulation-ready scene. SceneRig combines geometric, visual, and physical tools to reconstruct a faithful 3D scene. An agent then performs edit-and-validate loops inside a physics simulator to achieve physical stability. Across diverse inputs, SceneRig improves instance segmentation, object-centric appearance, and 3D placement while keeping the reconstructed scenes physically stable. Furthermore, SceneRig is effective for downstream robotics: its scenes predict real-robot policy outcomes more accurately than the baseline method and more reliably support replay of recorded manipulations without trajectory adaptation.

Agentic on-policy distillation (OPD) trains a student agent on its own trajectories while a teacher scores every token. A natural explanation of where it goes wrong is compounding error: the student's mistakes carry its trajectories beyond the region where the teacher is reliable, so recent methods keep training inside that region, through episode-length curricula, teacher-guided turns, prefix replay, or teacher intervention where the two disagree. We show this explanation is incomplete. From the states where the student departs most from the teacher, the teacher's own takeovers still finish the task, yet the student trained on its token scores keeps running out of turns. We attribute this to the myopia of next-token scoring. The teacher rates each token without seeing where the episode ends, so its scores let through the safe, delaying moves that outcome-based RL would punish with zero reward, a penalty agentic OPD lacks. We call this difference between what the teacher does and what it scores the rollout-scorer gap. We propose Rollout-to-Scorer Distillation (R2S), which repairs the scorer before distillation by mining high-disagreement states from the untrained student, running teacher takeovers from them, keeping the states where enough takeovers pass a reward-free check such as consensus or an LLM judge, and fine-tuning the teacher to give its takeover-informed scores on every assistant token from the divergence turn onward, without the takeover in context. The repaired teacher is frozen as the sole scorer for standard OPD, requiring no task reward during repair or training. Across three environments, two teachers and four students, R2S outperforms three state-of-the-art agentic-OPD methods in all twelve teacher-student-environment cells, by 12.8 success points on average over the best of them in each cell. Project page: https://tree-shu-zhao.github.io/r2s/.

We prove that every $\operatorname{ReLU}$ network can be exactly simulated by a $\operatorname{ReLU}$ network with two hidden layers. We present two proofs: one that is based on fillings in homological algebra and a direct combinatorial proof.

Agentic robotics systems combine planners with robot harnesses to perform complex manipulation tasks. Recent work improves these systems through better harnesses and external memory, but relies on powerful closed-source models as planners and does not improve the planners' own capabilities. In contrast, coding agents have improved substantially by post-training the model itself. We therefore investigate whether post-training can also improve robot planners. However, applying it across robot harnesses requires accommodating different execution mechanisms, including VLA skill invocation and control-code generation. To this end, we introduce AgenticRoboRL, a framework for post-training robot planners across different harnesses. The training recipe combines supervised fine-tuning with multi-turn reinforcement learning. To accommodate different execution mechanisms, harness-specific adapters connect recorded planner interactions with task outcomes, enabling a common RL training pipeline while keeping execution components fixed. On LIBERO-PRO, AgenticRoboRL raises the success rate of a 9B-parameter planner from 18.1% to 71.0% without external memory, approaching the 72.1% achieved by Codex (GPT-5.5) with external memory. These results demonstrate post-training as an effective approach for improving robot planners. We will release the framework and trained planner checkpoints.

Camera-trap imagery presents substantial variation in animal appearance, illumination, background clutter, and visual degradation. Masked autoencoders (MAEs) offer a label-free framework for modeling such data through reconstruction of randomly hidden image patches. We present WildMAE, a decoder-side extension of MAE that combines local structure-aware refinement with masked-to-visible cross-attention. A zero-gated convolutional residual path introduces a two-dimensional neighborhood prior, while gated cross-attention allows masked positions to retrieve contextual information from visible-position features without replacing the original Transformer decoder. The associated training recipe progressively introduces normalized-pixel, Sobel-edge, and perceptual objectives. In the original Snapshot Serengeti study, covering 4,744 test images at 75% masking, the reported reconstruction scores were 27.90 versus 24.33 dB PSNR, 0.840 versus 0.780 SSIM, and 0.160 versus 0.320 LPIPS for WildMAE and vanilla MAE, respectively. Additional historical records document favorable average differences under four synthetic degradation conditions and describe component variants. We present the method, experimental observations, and their scope together: the retained metrics are historical protocol-dependent results, rather than a newly reproduced, harmonized comparison or a validation of restoration from genuinely unavailable pixels. WildMAE provides a practical architectural case study for combining global token mixing with spatial and targeted contextual refinement in wildlife masked-image modeling.

First-person humanoid teleoperation ties access to side and rear targets to operator rotation. We present coupled visual--motor remapping, pairing a panoramic azimuth warp with its inverse upper-limb yaw transform. A working prototype implements Global Zoom (GZ) and Focus+Context (F+C). In a within-subject study (N=8); 72 trials), GZ had the largest completion count (23/24; Fixed: 20/24; F+C: 21/24), whereas F+C was preferred by 6/8 participants. Therefore, completion and preference favored different mappings in this sample.