arXiv:2609.12313v1 Announce Type: new Abstract: We evaluate Deep Perturbation Learning (DPL), which perturbs training images and labels along influence-derived directions, in three roles in which prior work has positioned it for machine unlearning: a direct deletion signal (the strongest claim), a utility-preserving regularizer, and a warm start for adversarial unlearning. Evidence for the weaker roles has been used to support the stronger one, so we test each role separately under a matched protocol with exact-seed retraining baselines. An audit of the public implementation identifies two correctness issues: image directions are computed on augmented, normalized tensors but applied to raw images, and the label perturbation falls below float32 resolution, leaving labels unchanged. After correcting the image-perturbation pipeline, DPL fails the direct-deletion criterion on CIFAR-10/ResNet-18 in all three paired seeds. Its utility effects are inconsistent in sign across seeds, and once direction-computation time is counted it underperforms simple warm-start baselines. A one-seed Tiny ImageNet check likewise does not favor DPL as a regularizer or warm start; preprocessing inconsistencies in the released code make the direct comparison there inconclusive. These results cover random instance deletion only and do not rule out influence-based methods in other deletion regimes. We release a role-matched evaluation protocol and an audit checklist for perturbation-based deletion claims.

arXiv:2609.12394v1 Announce Type: new Abstract: Mobile GUI agents are shifting from multi-module frameworks to native models trained end-to-end, yet industrial deployment faces three persistent gaps. Sandbox training produces a distribution mismatch with production environments; expensive real-device failures remain underutilized; and fixed benchmarks saturate, losing the power to guide iteration. We present BlueLM-GUI, a 35B-A3B mobile GUI agent built as a real-device-centric flywheel that closes these gaps through three principles. Every Sample Matters: a dual-track pipeline with Heterogeneous Triple-System Consensus evaluation and an Error Correction \& Derivation Module salvages every trajectory into usable supervision. Every Rollout Is Real: a three-stage recipe---continual pre-training, supervised fine-tuning, and agentic reinforcement learning on hundreds of real phones---grounds every rollout in real production environments, so the capability the model learns transfers directly to deployment. Every Query Evolves: a quota-driven benchmark methodology with three orthogonal axes enables precise attribution and allows the benchmark to be systematically upgraded as the model improves. BlueLM-GUI achieves 87.4 on MobileGUI-VBench, surpassing the best closed-source model by 5.1 points, and 84.9 on AndroidWorld, the best result among open-source models and competitive with closed-source models. These results demonstrate that grounding model training and iterative improvement in both real devices and the three Every principles yields strong, robust, and transferable mobile GUI capability.

arXiv:2609.12413v1 Announce Type: new Abstract: Large language models (LLMs) are rapidly evolving from conversational assistants into agentic AI systems that reason, plan, invoke tools, maintain persistent memory, communicate with other agents, and execute multi-step tasks. At the same time, modern models exhibit substantially stronger native safety alignment than earlier generations on which many jailbreak attacks and defenses were originally studied. This shift raises a fundamental question: \textit{which established jailbreak-security findings remain valid in the era of modern LLMs and agentic AI?} We address this question through a Systematization of Knowledge (SoK) that reframes jailbreak security around the full agentic execution pipeline. We develop unified taxonomies of attacks and defenses spanning user interaction, planning and reasoning, memory, tool use, and inter-agent communication, and introduce a security--utility--efficiency evaluation framework that separates native harmful-prompt safety, adversarial jailbreak robustness, and agent-level security outcomes. We further conduct a controlled empirical study of representative attacks and defenses within a common agentic framework. Our results reveal three important gaps. First, strong native alignment does not imply robustness to adversarial jailbreaks. Second, defense effectiveness is highly model-, attack-, and component-dependent and can come at substantial cost in over-refusal, utility, and latency. Third, low final-response attack success can mask severe intermediate compromise: planning, memory, and tool interactions may remain unsafe even when the final response is successfully filtered. These findings motivate a shift from response-centric jailbreak defense toward cross-layer, execution-aware security that protects agent state, component transitions, and external actions while preserving practical utility and efficiency.

arXiv:2609.12446v1 Announce Type: new Abstract: Social-deduction games such as Werewolf are increasingly used to evaluate LLM agents, but existing evaluations often rely on final game outcomes. We propose a belief-shift evaluation benchmark in Werewolf for analyzing communication skills through belief updating. Using LLM-played games, we annotate suspicion and accusation messages and measure how an observing village-side model's beliefs change after each message. We evaluate 40 open-weight LLM configurations on 1,224 annotated messages. Our results show that larger models better distinguish true wolves from villagers based on game history, but accusations still strongly influence their beliefs. Models become more suspicious of the accused target and less suspicious of the accuser, especially when the accuser is trusted, even if the accuser is wolf-aligned. Larger models better resist accusations from accusers they already distrust. Overall, our findings suggest that current open-weight LLMs up to 120B parameters still struggle to integrate accusation content with source trust in strategic communication. Our benchmark and code are available at https://rlg.iis.sinica.edu.tw/papers/werewolf-accusation-benchmark.

arXiv:2609.12482v1 Announce Type: new Abstract: We aim to characterise the value of artificial intelligence in the workplace. Current studies largely measure this value in terms of the current automation capabilities and public adoption of AI. However, such metrics ignore the greater impacts of human--agent collaboration in transforming the nature of work. To account for this, we must expand the scope of our analysis beyond atomised tasks of today, and instead focus on how AI can augment entire workflows of the future. To ground this analysis, we establish a precise definition of AI augmentation comprising six conditions, spanning durable net value, meaningful human control, accountability and recovery, and long-term human development through learning, career pathways, and job purpose. We elaborate on these conditions and apply the framework in a case study of AI-mediated social surveys. We conclude by outlining how organisations, researchers, and government leaders can use this framework to make sense of the future of work.

arXiv:2609.12749v1 Announce Type: new Abstract: Offline-to-online reinforcement learning reduces interaction cost for real-world robot learning but suffers from persistent value estimation instability. Existing methods address this through pessimistic regularization, lower-bound calibration, and architectural normalization, but an overlooked source of instability lies in the entropy formulation: the standard log-entropy term can become negative, destabilizing policy updates. We introduce SCQ (Sigmoid-Bounded Conservative Q-Learning), which replaces this term with a sigmoid-bounded formulation that stays strictly positive. SCQ retains conservative Q regularization and return-based lower-bound calibration, stabilizing policy optimization without sacrificing exploration. We evaluate SCQ on D4RL (Minari) benchmarks under both single-demonstration and standard dataset settings, as well as on simulation and real-world visual tasks. SCQ matches or exceeds baseline performance while exhibiting more stable training dynamics across state-based and visual benchmarks, and transfers to four real-robot platforms including manipulation, wheeled, quadruped, and humanoid systems. A direct clipping intervention that removes negative log-probability contributions, together with gradient-matched positive-score controls, indicates that positivity rather than a particular score shape alone drives much of the improvement. Project website: https://scq-rl.github.io.

arXiv:2609.13009v1 Announce Type: new Abstract: Low reported scores on leading physics benchmarks, including those featured in the Artificial Analysis Intelligence Index (2026), suggest that frontier language models still struggle with advanced physics, a demanding test of their scientific reasoning and quantitative problem-solving abilities. Yet this impression does not always align with domain experts' experiences using these models in their work. We revisit these reported findings by evaluating frontier models on six widely used physics benchmarks and auditing them with experts, focusing on text-only problems with verifiable final answers. For each subfield of physics, faculty and graduate researchers with relevant expertise carefully review problem statements, reference solutions, and model responses to distinguish genuine model errors from grader errors, incorrect reference solutions, and ambiguous or underspecified questions. Most audited cases initially evaluated as incorrect reflect these benchmarking issues rather than errors in the models' physics reasoning. We then ask experts to address these benchmarking issues by correcting erroneous reference solutions and repairing or excluding flawed questions. We find that GPT-5.6-Sol's measured mean@4 rises from 47.3% to 78.7% on HLE-Physics and from 61.0% to 87.2% on CMT-Benchmark, while its corrected pass@4 reaches 94.4% on the 54 retained CritPt challenges. Corrected scores are computed on the retained evaluation subsets following expert review. Scores on the audited subsets of UGPhysics, PRISM-Physics, and PHYBench also rise substantially after correction. These findings suggest that current benchmarks substantially understate frontier models' ability to solve well-posed physics problems. Near-saturation on these closed-ended tasks highlights the need for more demanding, expert-validated evaluations.

arXiv:2609.12397v1 Announce Type: cross Abstract: Multi-modal image generation, particularly subject-driven customization, has garnered growing attention in recent years. Despite the rapid advancement of generative models, their evaluation remains largely lagging. Existing methods, whether embedding-based or Multi-modal Large Language Model (MLLM)-based, evaluate alignment with each modal condition in isolation, which contradicts the simultaneous condition alignment objective of multi-modal image generation, leading to poor consistency with human judgments. To address this challenge, we propose UFO, the first unified framework for omni-condition alignment simultaneous evaluation. Specifically, UFO introduces a novel Atomized Chain-of-Evaluation paradigm, \emph{i.e.}, it first decomposes omni-condition alignment into a sequential chain of fine-grained, disentangled Atomic Evaluation Units (AEUs), categorizes them into distinct modality-relevance classes, and then employs general or dedicated functional calls for accurate verification of different AEU types. Experimental results demonstrate that UFO achieves the highest correlation with human evaluation preferences, delivering an average improvement of 15.25\%. Furthermore, we present UFO-Bench, a dedicated benchmark designed to holistically evaluate the performance of existing customization models under the diverse mutual interactions of textual and visual conditions.

arXiv:2609.12757v1 Announce Type: cross Abstract: Test-time scaling improves code generation by spending additional inference budget (e.g., calls or tokens) on direct sampling, feedback-conditioned repair, and reasoning-guided implementation. Search-based methods can allocate this budget adaptively, but two challenges remain. First, tree-structured search treats each generation history as a separate state even when trajectories converge to the same program, duplicating evaluation and preventing statistics from being shared. Second, sampling, repair, and reasoning have complementary and state-dependent payoffs, making online allocation among them difficult under a finite budget. To address these challenges, we propose an adaptive graph search method with heterogeneous actions (GraphAHA). GraphAHA organizes the test-time code generation in a typed directed acyclic graph. Equivalent programs are merged into a single code node, allowing their downstream search statistics to be reused across all discovery paths. Hierarchical Thompson sampling then selects whether to generate a new state or follow an existing successor and, for generation, chooses among the type-valid sampling, reasoning, implementation, and repair operations. Evaluated on LiveCodeBench and CodeContests with Qwen2.5-Coder and DeepSeek-Coder, GraphAHA achieves the best score in 18 of 20 cases. For Pass@1 measured using visible tests, it outperforms the strongest baseline for both models on both benchmarks by 4.1 percentage points on average, demonstrating more effective use of a fixed inference budget.

arXiv:2609.12818v1 Announce Type: cross Abstract: Long video understanding often behaves like a visual needle-in-a-haystack problem: query-relevant evidence is sparsely distributed across long temporal spans, while packing dense frames into a single VLM context incurs \textit{context rot} and high cost. Existing video agents often rely on query-agnostic offline preprocessing or ad hoc tool sets, which can miss query-specific details and waste computation. In this work, we present VideoXAgent, a purely online video-agent harness for long video understanding that starts from the given video file and user query, plans and decomposes the task, invokes specialized expert tools on demand, and aggregates multimodal evidence to produce a final answer while resolving conflicts among observations. To support this on-demand invocation, we design a suite of heterogeneous expert tools guided by a data-driven taxonomy of atomic capabilities, spanning scripts, VLMs, and domain models (e.g., detection, OCR, ASR, face recognition). The harness further enforces objective evidence prompting and budget-aware control to curb hallucination and non-termination. Across Video-MME-Long, LongVideoBench-Long, LVBench, and MINERVA, VideoXAgent is competitive with frontier LMMs and video agents under a smaller context footprint---about 50k tokens of agent context per sample, even on hour-long videos. In particular, on complex video-reasoning benchmarks such as MINERVA, it matches this level while using only about 15\% of the context of a 1,024-frame dense-packing baseline. Notably, the harness remains effective with a visually weak or even text-only orchestrator, suggesting that strong long-video understanding can emerge from progressive agentic evidence seeking rather than from packing the full video into a single context. Project page: https://go-agent-x.github.io/video_agent_harness/

arXiv:2508.16129v4 Announce Type: replace Abstract: Multimodal large language models (MLLMs) have recently demonstrated remarkable reasoning abilities under reinforcement learning (RL) paradigm. However, most existing multimodal medical reasoning models focus on basic reasoning, which refers to shallow inference based on visual feature matching. In contrast, real-world clinical diagnosis extends beyond basic reasoning, demanding complex reasoning that integrates heterogeneous clinical information (such as chief complaints and medical history) with multimodal medical imaging data. To bridge this gap, we introduce MM-Retinal-Reason, an ophthalmic multimodal dataset covering the full spectrum of perception and reasoning. Specifically, it is the first dataset in ophthalmology to encompass both basic and complex reasoning tasks with Chain-of-Thought (CoT) trajectories, aiming to enhance visual-centric reasoning and emulate realistic clinical decision-making. Building upon MM-Retinal-Reason, we propose OphthaReason, the first RL-enhanced ophthalmic multimodal reasoning model with step-by-step reasoning traces. To enable flexible adaptation to both basic and complex reasoning tasks, we further introduce Uncertainty-Aware Dynamic Thinking (UADT), which estimates sample-level uncertainty via entropy and dynamically modulates exploration depth through a shaped advantage mechanism. Comprehensive experiments demonstrate the effectiveness of our model on both basic and complex reasoning tasks, outperforming general-purpose MLLMs, medical MLLMs, RL-based medical MLLMs, and ophthalmic MLLMs by at least 15.47\%. Project Page: \href{https://github.com/lxirich/OphthaReason}{link}.

arXiv:2604.05333v4 Announce Type: replace Abstract: As LLM agents act across personal applications, web browsers, and other interfaces, their reusable skill libraries can scale to thousands of skills. This scale introduces two challenges. First, loading the full library saturates the context window, driving up token costs, hallucination, and latency. Second, semantic retrieval surfaces topically relevant skills but can miss upstream and downstream prerequisite skills, creating a prerequisite gap that leaves the retrieved bundle insufficient for execution. We present Graph-of-Skills (GoS), an inference-time structural retrieval layer for large skill libraries. GoS constructs an executable skill graph offline from skill packages, then retrieves a bounded, dependency-aware bundle through hybrid semantic-lexical seeding, reverse-aware Personalized PageRank, and context-budgeted hydration. Across SkillsBench and ALFWorld, with three model families (Claude Sonnet 4.5, MiniMax M2.7, and GPT-5.2 Codex), GoS attains the highest average reward in all six model-benchmark blocks, at a fraction of the token cost of loading the full library. On SkillsBench with GPT-5.2 Codex it raises average reward by 7.0 absolute points over full skill loading, a 25.6% relative gain, while cutting total tokens by 56.7%. Ablations isolate the mechanism: replacing reverse traversal with forward propagation costs 9.1 reward points, a larger loss than removing the graph altogether. The gain thus comes from traversing dependencies backwards, not from graph diffusion as such. A budget-matched retrieval study holding seeding, reranking, hydration, and context budget fixed reproduces the same ordering, with dependency-pair co-recovery falling from 0.654 to 0.362. Code is available at https://github.com/davidliuk/graph-of-skills

arXiv:2609.11318v2 Announce Type: replace Abstract: Deep research agents are increasingly capable of web search, tool use, multimodal evidence analysis, and information synthesis. However, existing benchmarks mainly evaluate medium-horizon exploration and rarely test whether agents can sustain long, dependency-heavy research processes. We introduce Mr. LHDR (Multimodal real-world Long-Horizon Deep Research), a benchmark for evaluating real-world deep research over long, irreducible chains of interdependent evidence across eight categories. Each question is constructed from a hidden Node-Relation graph and requires an average of 12.1 necessary intermediate conclusions with a mean dependency depth of 10.4 before reaching a short, unique, and verifiable answer. Questions incorporate multimodal evidence, including images, maps, PDFs, logos, charts, tables, and video frames, with at least one non-text element that changes the reasoning state. Mr. LHDR evaluates both final answers and the correctness of intermediate conclusions under annotated dependencies. We evaluate general models, deep research systems, and agent frameworks using Overall Accuracy (OA), Strict Accuracy (SA), Checklist Score (CS), and Dependency-Aware Checklist Score (DACS). Results show that even the strongest system achieves only 43.1% OA and 34.3% SA, indicating that final-answer accuracy substantially overestimates complete research success. Removing images reduces DACS by 12.6 points, demonstrating the importance of multimodal evidence, while SA consistently declines as reasoning chains become longer. These findings reveal sustained, dependency-consistent evidence integration, rather than isolated fact retrieval, as a key bottleneck for current deep research agents.

arXiv:2503.00785v2 Announce Type: replace-cross Abstract: How to endow aerial robots with the ability to operate in close proximity remains an open problem. The core challenges lie in the propulsion system's dual-task requirement: generating manipulation forces while simultaneously counteracting gravity. These competing demands create dynamic coupling effects during physical interactions. Furthermore, rotor-induced airflow disturbances critically undermine operational reliability. Although fully-actuated unmanned aerial vehicles (UAVs) alleviate dynamic coupling effects via six-degree-of-freedom (6-DoF) force-torque decoupling, existing implementations fail to address the aerodynamic interference between drones and environments. They also suffer from oversized designs, which compromise maneuverability and limit their applications in various operational scenarios. To address these limitations, we present FLOAT Drone (FuLly-actuated cOaxial Aerial roboT), a novel fully-actuated UAV featuring two key structural innovations. By integrating control surfaces into fully-actuated systems for the first time, we significantly suppress lateral airflow disturbances during operations. Furthermore, a coaxial dual-rotor configuration enables a compact size while maintaining high hovering efficiency. Through dynamic modeling, we have developed hierarchical position and attitude controllers that support both fully-actuated and underactuated modes. Experimental validation through comprehensive real-world experiments confirms the system's functional capabilities in close-proximity operations.

arXiv:2507.01663v2 Announce Type: replace-cross Abstract: Reinforcement learning (RL) has become a pivotal technology in the post-training phase of large language models (LLMs). Traditional task-collocated RL frameworks suffer from significant scalability bottlenecks, while task-separated RL frameworks face challenges in managing complex dataflows and resolving resource idling. Furthermore, most existing frameworks are tightly coupled with LLM training or inference engines, making them difficult to support custom-designed engines. To address these challenges, we propose AsyncFlow, an asynchronous streaming RL framework tailored for efficient post-training. Specifically, we introduce a distributed data storage and transfer module that provides panoramic data management and fine-grained scheduling capabilities in a fully streamed manner. This architecture inherently enables automated pipeline overlapping among RL tasks and dynamic load-balancing. Moreover, we propose an asynchronous producer-consumer workflow, which is engineered to minimize computational idleness by strategically deferring the parameter update process within staleness thresholds. Finally, the core capabilities of AsyncFlow are architecturally decoupled from underlying training and inference engines and encapsulated by service-oriented user interfaces, offering a modular and customizable user experience. Extensive experiments demonstrate an average throughput of 1.59x compared to the state-of-the-art baseline. The architecture presented in this work provides actionable insights for designing next-generation RL training systems.

arXiv:2603.01131v4 Announce Type: replace-cross Abstract: Clinical diagnosis is a gradual process of evidence integration, in which physicians move from symptoms and medical history to examinations, competing hypotheses, disease relations, and treatment decisions. Large language models have advanced medical text understanding and generation. Yet their clinical use remains limited by weak evidence grounding, opaque reasoning, and inconsistent links among differential diagnosis, final diagnosis, diagnostic basis, and treatment planning. We introduce MedCollab, a multi-agent framework for full-cycle clinical diagnosis and report generation. MedCollab coordinates specialist and examination agents according to patient records. It structures agent deliberation with an Issue-Based Information System (IBIS) protocol, so that each diagnostic position is supported by patient-specific evidence and medical knowledge. It also builds Hierarchical Disease Relation Chains (HDRC) to connect accepted hypotheses through progression, complication, and comorbidity relations. During multi-round deliberation, a verifier-guided consensus module evaluates evidence support, medical plausibility, and logical conflicts. It then adjusts agent contributions and filters unsupported reasoning. Experiments on ClinicalBench and MIMIC-IV show that MedCollab outperforms leading LLMs and medical multi-agent baselines in diagnostic accuracy, evidence consistency, and clinical reasoning quality. These results indicate that structured and auditable collaboration can produce more faithful and clinically coherent diagnostic reports.

arXiv:2608.00183v2 Announce Type: replace-cross Abstract: Retrieval over financial filings is difficult because queries are short and acronym-heavy while the answer-bearing evidence sits inside long, table-dense documents. We study sparse-dense hybrid retrieval on FinDER, a benchmark of expert-annotated questions over corporate 10-K filings. Our first finding is methodological: if the retrieval unit is larger than the dense encoder's input window, the dense model never sees a large share of the labeled evidence, confounding comparison against a full-text sparse baseline. We measure this directly and reduce the imbalance with windows chosen for the encoder budget. On this corpus, fusing BM25 and a compact dense encoder improves reference-level Hit@10 by roughly 28 percent over either component, and training-free, untuned reciprocal rank fusion exceeds the equal-weight blend in an exploratory comparison. We then ask whether choosing the fusion weight per query helps: an oracle over the interpolation-weight grid shows headroom of 21.8 percent, yet none of the three lightweight adaptive routers (a score-confidence heuristic, a random forest over query features, and a ridge regressor over query embeddings) establishes a statistically reliable improvement over the fixed blend under company-grouped cross-validation with cluster-robust inference. Simple fusion is a strong baseline here, and we discuss why per-query weighting does not capture the available headroom.

arXiv:2608.15424v2 Announce Type: replace-cross Abstract: The rapid adoption of large language models has enabled the development of clinical multi-agent systems (MAS) capable of integrating multimodal patient data and supporting increasingly complex clinical decision-making. However, the deployment of these systems in real-world healthcare settings raises critical ethical concerns related to safety, fairness, accountability, transparency, and patient trust. While numerous organizations, including the World Health Organization, the National Academy of Medicine, and the FUTURE-AI consortium, have proposed ethical frameworks and governance principles for healthcare AI, these efforts remain largely conceptual. To address this challenge, we present ETHOS (Ethics and Trust through Hierarchical Oversight System), a modular ethics framework designed as a governance meta-agent that can be integrated with any existing multi-agent system without requiring changes to its underlying architecture. ETHOS translates stakeholder-informed ethical requirements into executable runtime oversight through a layered governance approach consisting of deterministic checks, contextual reviews, and a final ethics critic. These components continuously evaluate intermediate reasoning steps and final outputs, enabling the system to identify ethical risks, request revisions, or suppress responses that fail predefined safety and trustworthiness criteria. We demonstrate ETHOS within a hepatology clinical decision-support MAS. Results show that ETHOS improves decision reliability by detecting incomplete, inconsistent, or out-of-scope evidence and appropriately increasing abstention when safe recommendations cannot be supported. By embedding ethical governance directly into system operation, ETHOS provides a practical and auditable mechanism for transforming high-level AI ethics principles into deployable safeguards.

arXiv:2608.18234v3 Announce Type: replace-cross Abstract: Whole-body motion tracking policies turn a humanoid into a robust control interface: the teleoperator---or an upstream model---only supplies a coarse movement intent, while the low-level policy keeps the robot balanced and physically feasible. Existing trackers deliver this interface only on flat ground: trained in empty scenes, they never learn how contact with terrain and objects reshapes their dynamics, and they attempt to teach the policy to balance under any command by continually enlarging the reference-motion corpus, which stops working once feasible behaviors become environment-dependent. We present GigaBrain-WBC-0.5, the first Behavior World Model (BWM) for humanoid whole-body control. Rather than a purely reactive tracker, we train a causal Transformer to jointly predict its next action, next state, and the distribution over its next latent behavior command, so the network that acts also models how the environment shapes what it can do next. An automatic terrain-annotation pipeline recovers full 3D contact geometry from retargeted motion, enabling terrain annotation at the scale of existing motion datasets. The predicted distribution is reused at deployment to detect implausible commands online and retract them onto learned behaviors, so the robot attempts tasks in a "best-effort" manner. The result is a unified policy that takes real-time command, interacts with environment, and stays robust to implausible commands, falls, and disturbances. GigaBrain-WBC-0.5 achieves the highest success rate across all four regimes among three large-scale tracker baselines: 81.3% on terrain interaction (4.3x the strongest baseline), 83.1% under implausible commands, and 99.3% fall recovery (16.8x the strongest baseline). Hardware trials show robust interaction under missing supports and disturbances; the Unitree G1 checkpoint transfers to the Maker L01 robot with simple fine-tuning.

arXiv:2609.09589v1 Announce Type: new Abstract: Deep neural networks exhibit regular macroscopic behavior despite highly nonlinear dynamics in vast parameter spaces. We develop a statistical-mechanical description of learning directly in function space, treating parameter configurations as microscopic realizations and functions with their dynamical operators as macroscopic variables. For mean-squared loss, the exact error dynamics are governed by the learning operator \(M=JJ^\ast\). Combining the dynamical Boltzmann weight of the conditional stochastic dynamics with the parameter-space density of states, whose local curvature defines a statistical operator \(B\), and integrating over local fluctuations yields $$ \Phi_{\mathrm{fluc}}(M;B)=\frac{\sigma_\xi^2}{2}\log\det(M^{-1}+B)+\mathrm{const}. $$ At fixed spectrum, this term is rotationally stationary when \([M,B]=0\), is minimized by pairing large eigenvalues of \(M\) with small eigenvalues of \(B\), and generates a local restoring contribution against rotational mismatch. For ReLU-type function spaces under mild stable statistical conditions, \(B=\sigma_\xi^2L^\ast\mathcal K L\), where \(L\) measures coarse-grained second-order structure. Thus the low-\(B\) sector corresponds, up to bounded anisotropy of \(\mathcal K\), to low structural curvature, implying a preference for faster relaxation along smooth, data-adaptive directions. These results identify function space as a natural macroscopic level for studying stable collective organization in learning.