Papers

We study steady solutions of the two dimensional incompressible Euler equations in the exterior of a compact domain. We assume that fluid enters the exterior domain through every point of the obstacle boundary. Under suitable assumptions on the direction and decay of the velocity at infinity, we prove that the flow is irrotational if and only if the velocity has no stagnation points.

We study deterministic and randomized midpoint discretizations of Langevin dynamics for a target $\pi \propto e^{-V}$, where $0 \prec \alpha I\preceq\nabla^2V\preceq\beta I$ and $\kappa=\beta/\alpha$. To achieve $\sqrt\alpha\,W_2\leqslant\varepsilon$, we show that deterministic Heun uses at most $\widetilde O(\kappa^{4/3}d^{1/3}\varepsilon^{-2/3})$ gradient queries, and underdamped exponential midpoint uses $\widetilde O(\kappa^{5/4}d^{1/4}\varepsilon^{-1/2})$. The proofs exploit cancellation at stationarity and smoothing using techniques from Malliavin calculus, outperforming previous upper bounds based on standard couplings. At bounded condition number, a lower bound matches the $d$ and $\varepsilon$ powers of both deterministic methods. To contrast, for the randomized midpoint methods and Poisson midpoint with at least two grid points (both overdamped and underdamped variants), a simple Gaussian calculation yields a lower bound $d^{1/3}\varepsilon^{-1/3}$ to get an $\varepsilon$-close sample despite starting at a benign initialization. This shows surprisingly that in high dimensions, deterministic discretizations can outperform their random counterparts.

Biological agents, from regulatory modules and microbial strains to organisms and populations, act on a common environment while each responds to its own costs, benefits, and limits. A dynamic game makes such interactions explicit by assigning who controls which decision, what each participant values, and which restrictions constrain unilateral deviations. We present a library of 25 biological dynamic games with 2 to 10 players spanning molecular, cellular, organismal, and population scales and competitive, antagonistic, mutualistic, and regulatory interactions. Instances cover single seasons, cycles that return to their initial state, or episodic tasks; seven include duration choice and some players' feasible sets are coupled. For each game we provide biological motivation, an executable Julia specification with player-owned controls, objectives and restrictions, a parameter instance, and a reference open-loop generalized Nash equilibrium candidate with interpreted trajectories. The solver CorleoneGame.jl computes candidates by sequential optimal-control best responses and audits them for feasibility and profitable unilateral deviations on a coarse and a refined control grid. All 25 reference computations satisfy the audit tolerances on both grids. Using a plant–herbivore game as a running example, we show how an executable specification yields an audited reference candidate that qualitatively reproduces a published prediction, how biological hypotheses become instance variants with different candidates, and how the candidates generate synthetic data for inverse games.

General purpose language models are increasingly able to control robotic hardware. Understanding the capabilities and safety of these models when embodied is therefore increasingly important for understanding their societal impact and risks. To this end, we introduce Inspect Robots, a modular, open-source framework for developing and running evaluations of embodied agents. Inspect Robots pairs customizable, reusable abstractions for specifying physical evaluations and analyzing their results with infrastructure that automates evaluation setup, execution and termination. We demonstrate Inspect Robots by using it to evaluate the capabilities and safety of six policies based on frontier language models. Inspect Robots has seen significant early uptake, receiving nearly 100,000 downloads in the three months since its release.

We consider a general smooth bounded strictly convex domain $\Omega\subset\mathbb{R}^d$ of dimension $d\geq2$ and describe dispersion for the semiclassical Schrödinger equation with Dirichlet boundary condition. Our results hold more generally on compact smooth Riemannian manifolds with smooth strictly convex boundary. More specifically, we construct a sharp local in (semiclassical) time parametrix and then proceed to obtain dispersion estimates: our fixed-time decay rate for the Green function exhibits a loss of $1/4$ in the exponent of $h/t$ with respect to the boundaryless case. The loss is sharp for dispersion, as shown earlier by the first author in the case of a model convex domain. On compact three-dimensional manifolds with smooth strictly convex boundary, the resulting spectrally localized Strichartz estimates yield global well-posedness in the energy space for the defocusing cubic nonlinear Schrödinger equation, thus matching the corresponding result for boundaryless manifolds due to Burq-Gérard-Tzvetkov. Moreover, we extend bounds on the time growth of higher Sobolev norms from Planchon-Visciglia-Tzvetkov to our setting.

Humans can often acquire and synthesize complex, recursive concepts from minimal experience. Leveraging cognitive insights, we propose the Minimalist Machine, a framework for inductive program synthesis designed to model such conceptual learning. The system uses a compact relational subset of Prolog: Programs are searched within a fixed schema of body-free facts and two-body conjunctive Horn clauses. Recursion is not defined by a dedicated metarule. Instead, it emerges when a target predicate is reused inside the body of a learned clause. Inspired by a primary school curriculum, the model is taught through a human-curated, sequential introduction of new concepts in arithmetic. Starting from initially empty knowledge base, it first acquires simple structural predicates, then successor-based state transformations, and finally recursive programs for addition, subtraction, multiplication, and division. Ultimately, this approach yields the fully transparent, inductive reasoning trace necessary for human-like conceptual learning.

In his Tau Manifesto, Michael Hartl presents a formula, credited to Jeff Cornell, for the volume of the unit $n$-ball in terms of the quarter-turn constant $\eta=\pi/2$. We read the formula as a set of clues to a geometric proof, leading through orthant geometry and a higher-dimensional version of Lambert's equal-area projection. We conclude with some additional thoughts on the circle constant.

The concept of subspace designs was introduced by Guruswami and Xing (STOC'13), and explicit constructions were given by Guruswami and Kopparty (FOCS'13). These are families of subspaces that have small intersection with any given subspace of a fixed dimension. We introduce robust subspace designs. Informally, these are a quantitative extension in which we demand that not too many subspaces of the family contain directions that lie close to any given subspace of a fixed dimension. We give a probabilistic construction of such a robust subspace design of polynomial size, as well as a non-trivial explicit construction of superpolynomial size. Our main application of this new concept is a quantum space-bounded variant of the Valiant-Vazirani theorem (Theor.Comput.Sci.'86), which shows that restricting $\mathsf{NP}$-complete problems to instances with at most one accepting witness preserves hardness under randomized reductions. For quantum witnesses, the analogous quantity is the dimension of an accepting witness subspace. We use our probabilistic construction of robust subspace designs to isolate a unique witness for space-bounded quantum Merlin-Arthur protocols with perfect completeness and an acceptance gap outside their perfectly accepting subspace. As further applications, we give a randomized reduction of well-conditioned nullity testing to space-bounded quantum Merlin-Arthur protocols with perfect completeness. Using a similar idea, we find that ordinary subspace designs allow us to recover the classical $\mathsf{C_= L}$ containment of Allender, Beals, and Ogihara (STOC'96) for general nullity testing through a simpler proof.

We construct a nonmicroscopic $G_\delta$ subset of the real line all of whose compact subsets are microscopic, answering a question of Zindulka and Nowakowski. We first obtain such a set in the standard Cantor space by coding branches of a finitely branching tree together with suitable choices of marked nodes. The coding of the branches ensures that the resulting set is not microscopic, while the marked nodes allow us to construct covers showing that every compact subset is microscopic. The example is then transferred from the Cantor space to the real line by the binary-expansion map.

In this series of articles, we study two graph classes and their various geometric representations. The first class consists of strongly involutive self-dual graphs (SISD graphs); the second consists of non-bipartite quadrangulations of the projective plane (NBQP graphs). These classes naturally arise in many geometric problems. For example, NBQP graphs are connected to diameter graphs in $\mathbb{R}^3$, tangency graphs of families of pairwise intersecting circles and pseudocircles, generalized thrackles, and graphs formed by the main diagonals of negatively self-polar polytopes. On the other hand, SISD graphs correspond to the skeletons of negatively self-polar polytopes, self-dual cones, and extremal ball-polytopes. In Part I we show a natural one-to-one correspondence between non-bipartite quadrangulations of the projective plane and 2-connected strongly involutive self-dual maps. In particular, we show that the reduced vertex-face incidence structure of a 2-connected strongly involutive self-dual map is a non-bipartite quadrangulation of the projective plane, and every non-bipartite quadrangulation of the projective plane arises that way. This allows us to translate natural properties from one class to the other. These results lay the foundation for later parts of the series.

Fast-spinning, nonaxisymmetrically deformed neutron stars (NSs) are promising targets for continuous gravitational-wave (CGW) searches with current and next-generation ground-based detectors. Ultraluminous X-ray pulsars (ULXPs), characterized by super-Eddington accretion and strong magnetic fields, provide extreme environments in which substantial nonaxisymmetric deformations may develop. In this Letter, we investigate thermo-elastic mountains generated by magnetically induced temperature asymmetries in the accreted crusts of ULXPs. We connect the crustal thermal structure to an accretion-column model and solve the full anisotropic heat-transport equation, extending previous perturbative treatments into the nonlinear, strongly magnetized regime. We find that the weak-field approximation begins to break down near $B_{\rm ref}\sim10^{13}\,{\rm G}$, beyond which nonlinear anisotropic heat transport significantly modifies the quadrupolar thermal response. The resulting thermal asymmetries generate substantial thermo-elastic mass quadrupoles at higher field strengths. Although the currently known ULXPs are unfavorable CGW targets because of their relatively slow spins and, in most cases, large distances, our illustrative spin-evolution calculations show that a NS can enter the super-Eddington accretor phase with a spin period of order $20\,{\rm ms}$. Such rapidly rotating Galactic ULXPs could lie within the projected sensitivities of next-generation detectors such as the Einstein Telescope and Cosmic Explorer. These results extend thermal-mountain physics into the strongly magnetized regime relevant to ULXPs and connect super-Eddington accretion with prospective CGW observations.

Sentiment analysis is a fundamental problem in Natural Language Processing (NLP). Standard sentiment classification for the Arabic language remains challenging due to the high volume of dialectal Arabic. To advance research in this area, this paper proposes the Shared Task on Sentiment Analysis and Swapping in Arabic Dialects (DialectSentEval), hosted with the Arabic Natural Language Processing Conference (ArabicNLP 2026). This shared task consists of two subtasks: Subtask 1 focuses on multi-class and multi-dialect sentiment analysis, requiring models to identify sentiment polarity across various Arabic dialects. Subtask 2 introduces a generative task for Arabic sentiment swap, challenging models to invert sentiment polarity while preserving core semantics. In this overview paper, we present the motivation, dataset creation, and summarize the main findings from participating models.

Recovering the forward operator of an imaging system from unpaired data avoids both calibration hardware and the clean/degraded pairs that supervision requires-pairs that, for a real lens, are often impossible to acquire. Existing unpaired approaches that explicitly estimate the forward operator by distribution matching evaluate a candidate only through the measurements it generates, whose distribution should match that of the real ones. Components suppressed by the operator are barely present in the degraded images, and therefore barely constrain the operator under such a criterion. Yet inversion precisely tries to recover them. As a result, two operators that are nearly indistinguishable as forward models can invert very differently. We address this by adding a comparison on the clean side: real degraded images, once restored, should be distributed like clean ones. Restorations are computed by a differentiable plug-and-play algorithm parameterized by the operator being learned. On spatially varying PSF calibration and blind super-resolution, the method recovers more accurate operators than degraded-side matching alone, and closes much of the gap to restoration with the true operator.

When does automation become an economically self-sustaining process? We define automation breakaway as the condition under which saving generated by physical capital and artificial labour is sufficient to finance continued automation. Artificial labour is labour-like in production but capital-like in accumulation, creating an income base that can support further investment. We characterise breakaway and its dynamic viability, showing that productivity and factor-income composition provide distinct financing channels. Artificial-labour accumulation strengthens the saving base supporting breakaway, while its effect on labour income changes sign with substitutability with human labour. Near full automation, aggregate labour income can contract through fewer human-filled jobs while average labour income in surviving jobs remains positive or rises. Optimal reliance on worker finance can begin to decline while it still provides a positive current net benefit.

Reconfigurable Intelligent Surfaces (RISs) offer a promising paradigm to mitigate blockage and extend millimeter-wave coverage in 6G multi-cell networks. However, dynamically allocating shared RIS infrastructure across competing base stations is an NP-hard problem posing severe scalability bottlenecks. In this paper, we propose a scalable framework combining Graph Neural Networks (GNNs) with Deep Reinforcement Learning (DRL) for dynamic shared RIS orchestration. By formulating allocation as a Markov Decision Process, we introduce a physical topology sparsification strategy that prunes dense channel matrices into a sparse tripartite graph. This pruning reduces edge density by 77% and removes representation noise, thereby improving global coverage probability while reducing computational complexity. Our relational message-passing architecture naturally generalizes to arbitrary network dimensions without model retraining. Furthermore, structural ablation studies reveal that physical path loss localizes surface dependencies, enabling a highly efficient localized graph design with linear computational scaling. Extensive simulations in dense urban environments demonstrate that under strict infrastructure budget constraints, the proposed GNN-DRL framework consistently outperforms greedy heuristic baselines by up to ~12% in coverage while delivering faster inference speed via GPU acceleration.

Physics-informed machine learning as recently emerged as a new paradigm for the resolution of partial differential equations, including functionalities for parametric modeling and inverse design, using neural networks to represent the solution fields. However, several difficulties have been reported concerning the training of such networks, resulting in slow convergence, low accurate solutions and sensitivity to the choice of hyper-parameters. In order to better understand the mechanisms underlying these difficulties, the present work proposes a rigourous comparison of neural networks and B-Splines representations within the same physics-informed learning framework. Four problems with closed-form solutions are numerically studied to establish a rigorous analysis of the accuracy and efficiency of the two representations, in terms of asymptotic convergence rates, sampling requirements, loss function minimization and arithmetic precision needed.

Multimodal Large Language Models (MLLMs) have demonstrated remarkable potential in video understanding, yet their reliance on retrospective summarization and text-centric priors often limits their ability to bridge unobserved causal transitions when applied to Video Event Prediction (VEP). To address this, we propose VepAgent, an agentic framework that integrates causal-transition reasoning with tool-augmented reinforcement learning (RL) for robust VEP. Unlike prior methods that passively project future trajectories from historical dependencies, our approach explicitly models the logical progression from terminal observed states to future events. Specifically, we first construct futurebench-4K, a high-quality chain-of-thought dataset for supervised fine-tuning (SFT) that effectively bridges the causal-logic gap by structuring the deduction of unobserved intermediate states. Subsequently, we develop a diagnostic tool library integrating state tracking, frame retrieval, and region magnification, enabling the agent to dynamically augment reasoning with external tools to recover missing spatio-temporal evidence and resolve visual ambiguities during inference. Moreover, we propose a composite reward mechanism that jointly optimizes prediction accuracy, causal coherence, and reliable prior, compelling the agent to rely on genuine visual grounding rather than superficial textual similarities. Extensive evaluations on FutureBench and NEPBench datasets demonstrate that our method achieves state-of-the-art performance, significantly outperforming larger MLLMs and validating the empirical effectiveness of our agentic, future-oriented reasoning paradigm.

We investigate two learning procedures and their relationship to the formation of equilibrium in potential mean field games (MFGs). The first one is a version of the fictitious play in which players observe directly the actions (controls) of other players and learn the distribution of the MFG through an update rules on controls. This leads to a new iterative procedure which converges for a wide class of potential MFGs of controls, even in the presence of common noise. On the other hand we also consider the learning procedure of a principal agent trying to solve a mean field control problem with only local gradient information. We prove that under very general assumption, the gradient descent of such agent converges to a solution of the associated potential MFG.

Heterogeneous Supercloud systems are transforming data analytics by enabling scalable and efficient task distribution across diverse resources. This paper presents computational models and performance estimation techniques tailored for accelerating Dense Cholesky (CH) Decomposition and Singular Value Decomposition (SVD)-based analytics on Heterogeneous Superclouds. Our ReDSEa tool-chain automates mapping, load balancing, scheduling, parallelism, and computation overlap. Implemented on a heterogeneous system with a Huawei Kunpeng 920 ARM CPU and an Ascend 910 AI accelerator, our LLVM compiler tool-chain employs novel performance models for recursive, iterative, and blocked computations, achieving up to 17x speedup for CH and 88x for SVD over fully optimized 48-core CPU implementations.

GAMBIT is an opening chess move in which a player sacrifices a piece, typically a pawn, to gain a positional advantage later in the game. Analogously, in multi-robot coordination, individual robots may need to forgo locally reward-maximising behaviours to improve overall team performance. Such self-sacrificial behaviours are difficult to capture with manually designed heuristics, particularly in dense, interaction-rich environments. Focusing on double-integrator continuous dynamics, this work studies how to learn such coordinated heuristics over motion primitives for multi-robot trajectory execution. Our framework, GAMBIT, first learns coordinated motion-primitive selection through imitation learning and subsequently fine-tunes the policy through reinforcement learning. We further introduce a safeguarded rollout mechanism with backup trajectories that guarantees collision-free execution at all times. Experiments demonstrate that GAMBIT substantially outperforms a range of baselines, including centralised motion planners and decentralised reactive planners, while exhibiting strong scalability. In particular, it coordinates over a thousand robots with planning latency below a few hundred milliseconds in continuous domains.

Let \(\mathbb{F}_q\) be a finite field with \(q>7\). Zhi-Wei Sun conjectured that for every \(a_0\in\mathbb{F}_q\), there is a circular permutation \((a_1,\dots,a_{q-1})\) of non-zero elements of \(\mathbb{F}_q\) such that \(a_0+a_i a_{i+1}\) is primitive for \(1\le i\le q-1\), where \(a_q=a_1\). In this paper, we confirm this conjecture for \(q>18\,888\,871\). For \(a_0=0\), the conclusion holds for all \(q>4\). For \(a_0\ne0\), we apply the Chvátal–Erdős theorem to obtain a Hamilton cycle.

Wi-Fi channel state information (CSI) enables contactless presence detection and gesture recognition. Its high-dimensional complex-valued time series require input representations that preserve informative temporal variations during compression. We propose Trajectory-Guided Tokenization (TGT), which combines complex trajectory decomposition with asymmetric attention to construct compact continuous tokens. For each antenna link and subcarrier, an orthonormal Helmert transform decomposes short, ordered temporal patches into local-center and centered-trajectory coordinates. Keys are learned from the centered-trajectory coordinates, while values retain both components. Learnable queries aggregate subcarriers into frequency slots, which are fused into temporal tokens. Trained jointly from scratch, TGT with TokenMLP achieves the highest mean accuracy of 92.83% among all evaluated frontend-backend combinations on the self-collected dataset. Experiments on EHUNAM and Widar further support the applicability of TGT to cross-domain presence detection and gesture recognition.

Industrial content moderation must process massive message streams under tight latency constraints, yet most abusive language (AL) detection systems rely on sentence-level classification (ALC), which neither localizes abusive spans nor identifies who is targeted. We define Abusive Language Identification (ALI) as a sequence-labeling task that jointly extracts AL spans and target mentions, and assess whether this approach can be used for text moderation. On a pilot corpus drawn from a production moderation pipeline, we compare ALI with ALC on cross-domain generalization and implicit abuse, and we also evaluate AL and target span detection. ALI remains competitive with ALC while providing localized outputs for moderators, with a modest and configuration-sensitive advantage on implicit abuse. Exact AL boundaries and target spans remain difficult to recover. We complement this comparison with a qualitative analysis and discuss perspectives on complete target–span linking and on structured benchmarks for ALI.

Long-context large language models (LLMs) have demonstrated strong capabilities across a wide range of tasks, but the growing KV cache introduces substantial memory and inference overhead. Existing one-shot KV cache compression methods typically commit to irreversible eviction immediately after prefill, before any signal from actual generation becomes available. Our quantitative analysis shows that early queries from the actual generation stage provide attention signals that are more consistent with subsequent decode attention, with the largest single-step gain occurring at the prefill-decode boundary. Based on this observation, we propose DeferKV, which moves the eviction decision from the end of prefill to the first real decoding step and temporally combines prompt-side and decode-side observations, thereby better aligning KV importance estimation with subsequent generation requirements. DeferKV requires no additional training, draft model, or future-query prediction module, making it simple and easy to deploy. Experiments on LongBench, RULER, and Needle-in-a-Haystack demonstrate that DeferKV consistently improves model performance under KV cache compression while maintaining low inference latency.

Concept bottleneck models (CBMs) are neural classifiers that allow to explain their decisions via high-level concepts, potentially enabling understanding, steering and debugging. However, their explanations are often derived heuristically. Building on formal explainability, we argue they should also be faithful, i.e., not misreport which concepts actually matter. We show that, for widespread CBM architectures, including recent VLM-based variants, faithful explanations must include all concepts in the bottleneck, compromising interpretability when this is large. This result applies to both heuristic and faithful-by-construction formal explanations. To encourage the existence of compact faithful explanations, we suggest i) modeling concepts probabilistically as binary or categorical random variables (rather than logits), and ii) employing per-concept training-time sparsification via group lasso (rather than regular elastic net). We also extend algorithms from formal explainability to CBMs, and show they outperform natural heuristics in terms of guarantees and explanation size. Overall, our work warns against naive interpretability claims and provides formal conditions and practical strategies for ensuring CBMs are as interpretable as advertised.

We present a receding-horizon optimal control for fixed-wing aircraft subject to state constraints. These constraints include operational constraints such as altitude, geofencing, and obstacle avoidance, as well as bounds on flight-path angle, roll angle, and airspeed. The state constraints are composed into a single soft-minimum control barrier function (CBF). We use this composite CBF in a constrained-approximate dynamic program to obtain a sequence of analytic closed-form control functions that approximately minimize a quadratic finite-horizon integral cost subject to the CBF constraint. The resulting receding-horizon control is non-myopic in the sense that it approximately optimizes the integral cost while satisfying the state constraint at all times along the entire prediction horizon. We demonstrate constraint satisfaction and performance in simulations of a fixed-wing aircraft navigating an obstacle-filled airspace under wind uncertainty. We also compare this receding-horizon control with 2 other methods.

Compact, reliable, and cost-effective gas sensors have become a highly demanding subject for safety management in the medical sector, chemical manufacturing, food quality monitoring, agriculture, and industrial safety. Hazardous gas emissions need to be controlled and monitored with fast-responsive and highly sensitive sensing devices. A first-principles study employing density functional theory (DFT) was used to investigate the adsorption behavior of Cl2, CO, CO2, NO, NO2, and HCN gas molecules with our proposed alloys, which consisted of hexagonal boron nitride (hBN) and graphene (Gr). The alloy consisting of 22% carbon (BNGr-2) was found to be most competent for sensing Cl2, CO, CO2, and HCN with sufficient adsorption energy, charge transfer, and bandgap alteration. However, NO and NO2 gas molecules showed more engagement with 33% carbon-proportioned alloy (BNGr-3) in terms of adequate gas sensing properties. NOx gases exhibited the most chemiresistive sensitivity towards the adsorbents. Other gases also showed significant chemiresistive sensitivity and distinct selectivity ratios, which would facilitate these alloys as chemiresistive sensors. Besides, noticeable work function variation (~20%) of these systems manifested potential as work function based sensors. Cl2 and NO2 showed strong physical adsorption, while the rest of the gases were weakly to moderately physisorbed, resulting in very short recovery times (10-1 ~ 10-6 seconds). Additionally, the distinctive absorption spectra observed for the gas analyte systems highlighted the potential of the proposed alloys as optical gas sensors. Temperature variation revealed that all gas molecules can be freed from the adsorbent BNGr-2 at 425 K. These findings imply hBN-Gr alloys as promising gas sensors for pollution auditing.

We introduce a novel invariance for peptide tandem mass spectrometry data, unlocking self-supervised representation learning that improves de novo sequencing of peptides. This invariance exploits the physical relationship between precursor properties (mass and charge) and fragment-ion evidence, without requiring peptide sequence labels. We introduce dIon, which adapts the DINO framework with two latent prediction tasks, both recovering a clean teacher representation: one from a spectrum mixture, using the precursor as a selection query, and one from a partial spectrum with the precursor withheld. The first associates precursor information with fragment-ion evidence; the second prevents representational collapse onto that information alone. Mechanistic probes support both effects, and ablations show that the full objective performs best. Under identical end-to-end training, dIon initialization improves de novo peptide precision over training from scratch by 5.5 and 8.4 percentage points on the held-out MassIVE-KB and Kingdoms test sets, and by 2.3 and 4.8 percentage points with a larger supervised training corpus. The resulting models surpass fully supervised state-of-the-art de novo sequencing models on the diverse, multi-species Kingdoms corpus under the same greedy-decoding protocol. Without peptide labels, dIon learns strong native peptide-similarity geometry compared with other learned models; with limited peptide-supervised adaptation, it achieves the best retrieval and pair-discrimination performance across all representation benchmarks.

In this article, we study the viscoelastic wave equation with a localized history memory term, and without frictional damping. Under the geometric control condition on the memory region, we prove exponential decay of the viscoelastic energy using the Gearhart-Prüss-Huang Theorem. We prove in high frequency that a loss of derivative is compensated by a decay in frequency.

World action models (WAMs) have emerged as a promising paradigm for robotic manipulation. They act by first predicting how a task should be performed and then decoding the actions from that future. However, the remaining actions are invalid once execution drifts from the prediction. Simply replanning from the already out of distribution state rarely restores what the task still requires; existing execution monitors decide when to stop, but not what to restore. We observe that the answer is already in hand: the future the WAM predicted before acting depicts exactly the states it intended to pass through. We introduce FAVOR (Future Anchored Verification and Online Recovery), a lightweight framework that keeps these predicted frames as anchors and uses them for verification and recovery. An Anchor Verifier compares each observation with its anchor, together with the executed actions, to flag deviations that break the task. Anchor-Guided Recovery uses a vision-language model to turn the flagged anchor into a short corrective instruction. Under strengthened instruction guidance, the WAM executes this instruction to return to the intended future. It then resumes the task. FAVOR raises the task success of the base WAM from 97.85% to 98.10% on LIBERO and from 72.60% to 72.98% on LIBERO-Plus without modifying the policy.