Flood and Harvest: The Provable Necessity of Trivia for Generating Valuable Mathematics via the Lens of Language Generation in the Limit

AI systems coupled to proof assistants now generate formal mathematics at scale, and the gap between what a checker can verify and what a mathematician would value has become the binding constraint. We model the generation of valuable mathematics as nested language generation in the limit: a verifiable formal language $F$, accessed through a membership oracle (the proof checker), contains an unknown valuable language $H \in \mathcal{H}$ revealed only through an adversarial enumeration of a core $C \subseteq H$ of exact density $α$ (the literature). Every output is valuable ($\in H$), trivial ($\in F \setminus H$), or a hallucination ($\notin F$). We settle four questions. First, the verifier is not taste: the collections admitting generation with breadth are exactly those of the oracle-free model, characterized fiber-wise by Angluin's condition. Second, the verifier does buy sound coverage, covering all unseen valuable statements while asserting only valid ones: possible with it, impossible without it; it relocates unavoidable errors from false to trivial. Third, and centrally, a sharp dichotomy on the tight family: generators emitting finitely many trivia achieve optimal coverage $α/2$, while any infinite trivia allowance, even at vanishing rate, jumps the optimum to $1-α/2$ (both tight, for cores presented as the candidate intersection), and one generator attains both ends. The transition is in trivia count, not rate; the gap $1-α$ is the unrecorded mass. Fourth, both regimes instantiate in a compression model of mathematics. A perfect verifier cannot substitute for taste: the unbounded stream of correct-but-worthless statements is not an engineering accident but a provable necessity, since covering unrecorded valuable mathematics requires an infinite, but asymptotically negligible, stream of certified trivia.

Learning Manifold and Itô Dynamics with Branched Neural Rough Differential Equations

Neural rough differential equations (NRDEs) stay accurate under irregular sampling while taking far fewer integration steps than standard neural differential equations, summarising a finely sampled driver by its log-signature and advancing the hidden state over coarse intervals using the log-ODE method. This efficiency rests on the shuffle algebra, the algebraic counterpart of Stratonovich calculus. This reliance means NRDEs cannot expose the quadratic-variation terms Itô dynamics require, nor the ordered covariant derivatives that govern Itô flows on connection-equipped manifolds. Ameliorating this, we introduce Branched Neural Rough Differential Equations (B-NRDEs), a Hopf-algebraic framework that recasts the NRDE log-ODE step as geometric numerical integration on the state-space manifold, matching the driving algebra to the governing calculus: Grossman--Larson rooted trees for Euclidean Itô dynamics, Munthe-Kaas--Wright planar rooted trees for ordered covariant derivatives on manifolds, and the shuffle algebra in the classical Stratonovich case. This yields intrinsic coarse-step dynamics that exactly preserve manifold constraints. Finally, we introduce a branched signature-kernel objective to enable Itô-consistent law matching by making quadratic-variation terms visible during training. On rough Bergomi volatility, sim-to-real $\mathrm{SO}(3)$ dynamics forecasting, and SPD covariance dynamics, B-NRDEs offer a unified, effective approach to stochastic and manifold-valued dynamics beyond the Euclidean--Stratonovich setting.

S$^3$GNN: Efficient Global Mixing and Local Message Passing for Long-Range Graph Learning

Message-passing neural networks (MPNNs) often suffer from an information bottleneck when capturing long-range dependencies, leading to the oversquashing (OSQ) phenomenon. Alongside spatial connectivity enrichment (e.g., rewiring), recent studies have shown that spectral filtering can yield strong long-range learning outcomes, as spectral operators enable global information mixing that alleviates OSQ. These approaches achieve this either by stabilizing the Jacobian energies in deep propagation or by guaranteeing OSQ mitigation under strong theoretical assumptions. We revisit these conclusions and show that the associated Jacobian sensitivity lower bound is generally difficult to achieve in practice. We then propose S$^3$GNN, which mitigates OSQ without such restrictive assumptions by lightweightly reintroducing omitted components with substantially lower computational complexity, while standard stability constraints on feature transformations remain effective under our new dynamics. Extensive experiments across diverse domains (e.g., long-range benchmarks, KGQA, and mesh-based fluid dynamics) demonstrate that S$^3$GNN achieves up to an order-of-magnitude error reduction with up to 50\% fewer parameters. Our code can be found in https://github.com/EEthanShi/S3-GNN.git.

Cognifold: Always-On Proactive Memory via Cognitive Folding

Existing agent memory remains predominantly reactive and retrieval-based, lacking the capacity to autonomously organize experience into persistent cognitive structure. Toward genuinely autonomous agents, we introduce Cognifold, a brain-inspired "always-on" agent memory designed for the next generation of proactive assistants. CogniFold continuously folds fragmented event streams into self-emerging cognitive structures, bootstrapping progressively higher-level cognition from incoming events and accumulated knowledge. We ground this by extending Complementary Learning Systems (CLS) theory from two layers (hippocampus, neocortex) to three, adding a prefrontal intent layer. Emulating the prefrontal cortex as the locus of intentional control and decision-making, CogniFold achieves this through graph-topology self-organization: cognitive structures proactively assemble under the stream, merge when semantically similar, decay when stale, relink through associative recall, and surface intents when concept-cluster density crosses a threshold. We evaluate structural formation using CogEval-Bench, demonstrating that CogniFold uniquely produces memory structures that match cognitive expectations and concept emergence. Furthermore, across 7 broad-coverage benchmarks spanning five cognitive domains, we validate that CogniFold simultaneously performs robustly on conventional memory benchmarks.

LOFT: Low-Rank Orthogonal Fine-Tuning via Task-Aware Support Selection

Orthogonal parameter-efficient fine-tuning (PEFT) adapts pretrained weights through structure-preserving multiplicative transformations, but existing methods often conflate two distinct design choices: the subspace in which adaptation occurs and the transformation applied within that subspace. This paper introduces LOFT, a low-rank orthogonal fine-tuning framework that explicitly separates these two components. By viewing orthogonal adaptation as a multiplicative subspace rotation, LOFT provides a unified formulation that recovers representative orthogonal PEFT methods, including coordinate-, butterfly-, Householder-, and principal-subspace-based variants. More importantly, this perspective exposes support selection as a central design axis rather than a byproduct of a particular parameterization. We develop a first-order analysis showing that useful adaptation supports should be informed by the downstream training signal, motivating practical task-aware support selection strategies. Across language understanding, visual transfer, mathematical reasoning, and multilingual out-of-distribution adaptation, LOFT recovers principal-subspace orthogonal adaptation while gradient-informed supports improve the efficiency-performance trade-off under matched parameter, memory, and compute budgets. These results suggest that principled support selection is an important direction for improving orthogonal PEFT.

Wiener Chaos Expansion based Neural Operator for Singular Stochastic Partial Differential Equations

In this paper, we explore how our recently developed Wiener Chaos Expansion (WCE)-based neural operator (NO) can be applied to singular stochastic partial differential equations, e.g., the dynamic $\boldsymbolΦ^4_2$ model simulated in the recent works. Unlike the previous WCE-NO which solves SPDEs by simply inserting Wick-Hermite features into the backbone NO model, we leverage feature-wise linear modulation (FiLM) to appropriately capture the dependency between the solution of singular SPDE and its smooth remainder. The resulting WCE-FiLM-NO shows excellent performance on $\boldsymbolΦ^4_2$, as measured by relative $L_2$ loss, out-of-distribution $L_2$ loss, and autocorrelation score; all without the help of renormalisation factor. In addition, we also show the potential of simulating $\boldsymbolΦ^4_3$ data, which is more aligned with real scientific practice in statistical quantum field theory. To the best of our knowledge, this is among the first works to develop an efficient data-driven surrogate for the dynamical $\boldsymbolΦ^4_3$ model.

Expanding the Chaos: Neural Operator for Stochastic (Partial) Differential Equations

Stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) are fundamental tools for modeling stochastic dynamics across the natural sciences and modern machine learning. Developing deep learning models for approximating their solution operators promises not only fast, practical solvers, but may also inspire models that resolve classical learning tasks from a new perspective. In this work, we build on classical Wiener chaos expansions (WCE) to design neural operator (NO) architectures for SPDEs and SDEs: we project the driving noise paths onto orthonormal Wick Hermite features and parameterize the resulting deterministic chaos coefficients with neural operators, so that full solution trajectories can be reconstructed from noise in a single forward pass. On the theoretical side, we investigate the classical WCE results for the class of multi-dimensional SDEs and semilinear SPDEs considered here by explicitly writing down the associated coupled ODE/PDE systems for their chaos coefficients, which makes the separation between stochastic forcing and deterministic dynamics fully explicit and directly motivates our model designs. On the empirical side, we validate our models on a diverse suite of problems: classical SPDE benchmarks, diffusion one-step sampling on images, topological interpolation on graphs, financial extrapolation, parameter estimation, and manifold SDEs for flood prediction, demonstrating competitive accuracy and broad applicability. Overall, our results indicate that WCE-based neural operators provide a practical and scalable way to learn SDE/SPDE solution operators across diverse domains.

ACT as Human: Multimodal Large Language Model Data Annotation with Critical Thinking

Supervised learning relies on high-quality labeled data, but obtaining such data through human annotation is both expensive and time-consuming. Recent work explores using large language models (LLMs) for annotation, but LLM-generated labels still fall short of human-level quality. To address this problem, we propose the Annotation with Critical Thinking (ACT) data pipeline, where LLMs serve not only as annotators but also as judges to critically identify potential errors. Human effort is then directed towards reviewing only the most "suspicious" cases, significantly improving the human annotation efficiency. Our major contributions are as follows: (1) ACT is applicable to a wide range of domains, including natural language processing (NLP), computer vision (CV), and multimodal understanding, by leveraging multimodal-LLMs (MLLMs). (2) Through empirical studies, we derive 7 insights on how to enhance annotation quality while efficiently reducing the human cost, and then translate these findings into user-friendly guidelines. (3) We theoretically analyze how to modify the loss function so that models trained on ACT data achieve similar performance to those trained on fully human-annotated data. Our experiments show that the performance gap can be reduced to less than 2% on most benchmark datasets while saving up to 90% of human costs.

MAGI-1: Autoregressive Video Generation at Scale

We present MAGI-1, a world model that generates videos by autoregressively predicting a sequence of video chunks, defined as fixed-length segments of consecutive frames. Trained to denoise per-chunk noise that increases monotonically over time, MAGI-1 enables causal temporal modeling and naturally supports streaming generation. It achieves strong performance on image-to-video (I2V) tasks conditioned on text instructions, providing high temporal consistency and scalability, which are made possible by several algorithmic innovations and a dedicated infrastructure stack. MAGI-1 facilitates controllable generation via chunk-wise prompting and supports real-time, memory-efficient deployment by maintaining constant peak inference cost, regardless of video length. The largest variant of MAGI-1 comprises 24 billion parameters and supports context lengths of up to 4 million tokens, demonstrating the scalability and robustness of our approach. The code and models are available at https://github.com/SandAI-org/MAGI-1 and https://github.com/SandAI-org/MagiAttention. The product can be accessed at https://sand.ai.

Graph Pseudotime Analysis and Neural Stochastic Differential Equations for Analyzing Retinal Degeneration Dynamics and Beyond

Understanding disease progression at the molecular pathway level usually requires capturing both structural dependencies between pathways and the temporal dynamics of disease evolution. In this work, we solve the former challenge by developing a biologically informed graph-forming method to efficiently construct pathway graphs for subjects from our newly curated JR5558 mouse transcriptomics dataset. We then develop Graph-level Pseudotime Analysis (GPA) to infer graph-level trajectories that reveal how disease progresses at the population level, rather than in individual subjects. Based on the trajectories estimated by GPA, we identify the most sensitive pathways that drive disease stage transitions. In addition, we measure changes in pathway features using neural stochastic differential equations (SDEs), which enables us to formally define and compute pathway stability and disease bifurcation points (points of no return), two fundamental problems in disease progression research. We further extend our theory to the case when pathways can interact with each other, enabling a more comprehensive and multi-faceted characterization of disease phenotypes. The comprehensive experimental results demonstrate the effectiveness of our framework in reconstructing the dynamics of the pathway, identifying critical transitions, and providing novel insights into the mechanistic understanding of disease evolution.