Fargues-Scholze developed a framework for the geometric Langlands program on the Fargues-Fontaine curve. In particular, they proved the geometric Satake equivalence on the moduli space of closed Cartier divisors on the curve. We prove the derived version of this equivalence with one leg. Namely, we show that the derived category of etale sheaves on the local Hecke stack is equivalent to the category of L-group-equivariant perfect complexes over the symmetric algebra of the shifted and (-1)-Tate-twisted Lie algebra.

The monogamy and polygamy properties of quantum entanglement characterize fundamental constraints on the distribution of entanglement in multipartite quantum systems. In this paper, we investigate tighter monogamy and polygamy relations for multipartite entanglement. By establishing a new mathematical inequality, we derive a family of improved monogamy and polygamy inequalities for tripartite quantum systems and further extend these results to general multipartite systems. Comparisons with existing results show that the obtained bounds are tighter. Illustrative examples are provided to demonstrate the effectiveness of the proposed relations.

Electron confinement within a small volume is intriguing as a realization of the particle-in-a-box system, which appears in every quantum mechanics textbook. While the electron confinement is readily imaginable in solid-state systems, it also occurs in liquids, where the local voids in the liquid serve as confining "boxes." Confinement within these flexible cavities in liquids is expected to differ fundamentally from that in solids. Here, we experimentally investigate the electrons confined in liquid water, which are called hydrated electrons, using transient two-dimensional electronic spectroscopy. Our experiment reveals the large nonuniformity of the shape and the size of hydrated electrons with significant fluctuation at the timescale shorter than 30 fs.

Economists often interpret estimates from linear regressions with log dependent variables as elasticities. However, the coefficients from log-log regressions estimate the elasticity of the geometric mean of $y_i|x_i$, not the arithmetic mean. The unbounded difference between the two is known as retransformation bias and can take either sign. We develop a specification-robust debiased estimator of the average arithmetic elasticity and re-estimate 50 results from top 5 papers published in 2020. We find that 19 are significantly different, with the median absolute difference being 65% of the OLS elasticity estimate. Furthermore, we show standard instrumental variables assumptions with log dependent variables do not identify the elasticity. We specify a control function approach and re-estimate papers that use 2SLS with log dependent variables. We find that 13 of 19 results from top 5 papers are significantly different between the two approaches. Retransformation bias arises as a result of heterogeneous responses. The geometric mean elasticity corresponds to the average response. Arithmetic and geometric means are elements of the power mean family. We show power mean elasticities are sufficient statistics for a common class of decision problems.

Biological polymers, such as intrinsically disordered proteins, play a central role in cellular biology, including mediating phase separation and controlling activity of biological condensates. The physical properties and functions of biopolymers are determined by their residue sequence. Recently, significant computational and theoretical efforts have been devoted to characterizing the combinatorially complex sequence dependence of biopolymer phase diagrams. Here, we quantitatively show that monomer accessibility is central to determining the strength of pair interactions. We formulate an analytical perturbative approach, phenomenologically precluding two polymers' centers of mass from overlapping within a correlation hole. This theory yields the correction to the strength of mean-field interactions in terms of a residue-accessibility parameter (RAP), which accounts for the limited availability of inner monomers to interactions. Despite the simplicity of the approach, RAP rationalizes the variations in critical temperatures found in extensive Monte-Carlo simulations for thousands of two-letter polymer solutions of varying length and sequence. RAP may thus be effective for deciphering the polymer-sequence dependence of phase diagrams given any polymer length, set of monomer types, and polymer mixtures.

This paper studies mechanism design environments in which the designer does not know the distribution of agents' private information a priori and instead learns from agents' behavior induced by the mechanism itself. We formalize a notion of self-confirming mechanisms and a refinement thereof, capturing the idea that an equilibrium mechanism is optimal given the designer's belief and that this belief is consistent with the information produced by the mechanism. We establish a fictitious revelation principle, showing that any incentive-compatible mechanism can be represented as a direct mechanism with filtered type reports that preserve the original mechanism's informational content. Applying the framework to a monopoly problem, we show that, subject to an equilibrium refinement, dominant-strategy self-confirming mechanisms are exactly posted-price mechanisms with locally revenue-maximizing prices.

We study a one-dimensional swarmalator model with inertia. Previous studies have focused almost exclusively on the overdamped limit. We find inertia introduces a new unsteady collective state in which the rainbow order parameters undergo multiharmonic oscillations. This "thrashing" phase wave bifurcates from the model's static phase wave state through a subcritical Hopf bifurcation that coincides with a saddle-node of limit cycles. The wave itself exists in clockwise and counterclockwise symmetric pairs. For small populations we observe attractor switching between these chiral states, while for larger systems the dynamics settle onto a single branch.

The weighted set multi-cover problem is a fundamental generalization of set cover that arises in data-driven applications where one must select a small, low-cost subset from a large collection of candidates under coverage constraints. In data management settings, such problems arise naturally either as expressive database queries or as post-processing steps over query results, for example, when selecting representative or diverse subsets from large relations returned by database queries for decision support, recommendation, fairness-aware data selection, or crowd-sourcing. While the general weighted set multi-cover problem is NP-complete, many practical workloads involve a \emph{bounded universe} of items that must be covered, leading to the Weighted Set Multi-Cover with Bounded Universe (WSMC-BU) problem, where the universe size is constant. In this paper, we develop exact and approximation algorithms for WSMC-BU. We first discuss a dynamic programming algorithm that solves WSMC-BU exactly in $O(n^{\ell+1})$ time, where $n$ is the number of input sets and $\ell=O(1)$ is the universe size. We then present a $2$-approximation algorithm based on linear programming and rounding, running in $O(\mathcal{L}(n))$ time, where $\mathcal{L}(n)$ denotes the complexity of solving a linear program with $O(n)$ variables. To further improve efficiency for large datasets, we propose a faster $(2+\varepsilon)$-approximation algorithm with running time $O(n \log n + \mathcal{L}(\log W))$, where $W$ is the ratio of the total weight to the minimum weight, and $\varepsilon$ is an arbitrary constant specified by the user. Extensive experiments on real and synthetic datasets demonstrate that our methods consistently outperform greedy and standard LP-rounding baselines in both solution quality and runtime, making them suitable for data-intensive selection tasks over large query outputs.

The fate of massive stars during the latest stages of their evolution is highly dependent on their mass-loss rate and geometry. These processes have a significant influence on stars with masses between 25 and 40 Msun, i.e., type II SN progenitors. We aim to investigate the mass-loss history, geometry, and physical conditions of the yellow hypergiant in a post-RSG stage, IRAS 17163-3907. We place it in context with another famous yellow hypergiant, IRC+10420. We combine M-band high-resolution CRIRES+ spectroscopy with VLTI/MATISSE mid-infrared L-band interferometry, and FORS2 optical spectropolarimetry to probe both the small-scale circumstellar structure and the large-scale dusty environment of IRAS 17163. The CRIRES+ spectrum provides the first M-band coverage of IRAS 17163, revealing prominent low-excitation metal lines and hydrogen recombination features, but lacking the pronounced CO absorption seen in IRC+10420. The MATISSE observations reveal the first high angular scales of the source in the L-band and spatially resolve the Brα line-emitting region, which hints at a marginally asymmetric and variable ionised wind. FORS2 spectropolarimetry points to deviations from perfect spherical symmetry also on larger scales. The data show no evidence for a binary companion within the explored parameter space, indicating that the observed clumpy and time-variable mass loss is likely intrinsic to the star rather than companion-driven. Our results demonstrate that IRAS 17163 hosts a dense, structured, and time-variable wind, coexisting with extended dusty shells. The comparison with IRC+10420 highlights diversity among post-RSG YHGs. These findings emphasise the role of clumpy and near-symmetric mass-loss in shaping the circumstellar medium of evolved massive stars, with implications for their subsequent evolution and core-collapse supernova progenitor properties.

We investigate asymptotic inference in a linear regression model where both response and regressors are functions, using an estimator based on functional principal components analysis. Although this approach is widely used in functional data analysis, there remains significant room for developing its asymptotic properties for function-on-function regression. Our study targets the mean response at a new regressor with two primary aims. First, we refine the existing central limit theorem by relaxing certain technical conditions, which include generalizing the scaling factor, resulting in incorporating a broader class of random functions beyond those having scores with independence or finite higher moments. Second, we introduce a residual bootstrap method that enhances the calibration of various confidence sets for quantities related to mean response, while its consistency is rigorously verified. Numerical studies compare the finite sample performance of both asymptotic and bootstrap approaches, demonstrating higher accuracy of the latter. To illustrate bootstrap inference for mean response, we apply it to the Canadian weather dataset.

Location-Based Services (LBS) such as ride-sharing, accommodation, food delivery, and location-driven social media platforms entangle digital systems with physical spaces, thereby generating impacts that extend beyond users to others who share the same environments. Existing design approaches struggle to address the dual challenge of value tensions that arise in shared physical spaces and the locality-specific contexts in which LBS operate. To respond, we introduce Location-Aware Value Sensitive Design (LA-VSD), a domain-specific adaptation of VSD tailored to the distinctive characteristics of LBS. LA-VSD guides designers through three heuristics to help (1) identify and prioritise stakeholders through local space-sharing scenarios, (2) adapt empirical methods to capture values and tensions in context, and (3) support value-aligned interactions across both digital and physical layers of the service. Through a case study of e-scooter sharing in Melbourne, Australia, we demonstrate how LA-VSD enables more grounded, context-aware, and actionable design of LBS.

Characterizing resonant scatterers is challenging because their poles and zeros usually lie away from the real-frequency axis, whereas most measurements sample only real frequencies and infer off-axis behavior from fitted models. Here we introduce complex-frequency chirped pulses: finite-energy analytic waveforms that probe a device continuously along a prescribed contour in the complex-frequency plane. We give a direct synthesis rule for an in-phase/quadrature (I/Q) waveform and show that finite-duration windowing deterministically distorts the realized trajectory, which makes it necessary to analyze only a central time interval where the window contribution is small. For stable linear time-invariant devices, we extract a time-local least-squares input--output ratio and identify when it follows the continued complex-frequency response, with errors that grow at higher traversal speeds and near resonant poles. Numerical tests on a coupled-mode resonator validate the method and show that closed contours enable an integer phase-winding consistency check. We also outline an implementation based on standard arbitrary waveform generation, I/Q modulation, coherent reception, and digital signal processing.

Asymptotic inference using functional principal component regression (FPCR) has long been considered difficult, largely because, upon any scalar scaling, the FPCR estimator fails to satisfy a central limit theorem, leading to the prevailing belief that it is unsuitable for direct statistical inference. In this paper, we upend this traditional viewpoint by establishing a new result: upon suitable operator scaling, valid Gaussian and bootstrap approximations hold for the FPCR estimator. We apply this surprising finding to hypothesis testing for the significance of the slope function in functional regression models and demonstrate the strong numerical performance of the resulting tests. While concise, our results yield powerful inferential tools for functional regression. We believe it paves the way for new lines of inferential methodology for more complex functional regression settings.

Earth's climate stability, characterized by the global radiative feedback parameter ($λ$), varies decadally due to changing surface temperature patterns. Recent variations in $λ$ are poorly understood as coordinated model simulations typically end in 2014. We apply a convolutional neural network trained on climate model simulations to observation-based surface temperature reconstructions to estimate variations in $λ$ up to 2025. We find that $λ$ reached a minimum (maximum stability) around the mid 1990s ($λ\simeq\SI{-3}{Wm^{-2}/K}$), but has since weakened significantly ($λ\simeq\SI{-2}{Wm^{-2}/K}$). We confirm these results with climate model simulations extended to 2022. The recent $λ$ weakening is not significantly affected by El Niño Southern Oscillation or Pacific Decadal Oscillation. Attribution reveals that warming in the subtropical Northeast Pacific is an important driver of the recently weakened feedback, confirmed by targeted experiments in E3SMv2. Our approach enables near real-time monitoring of Earth's climate stability.

Software fairness testing is a central method for evaluating AI systems, yet the meaning of fairness is often treated as fixed and universally applicable. This vision paper positions fairness testing as culturally situated and examines the problem across three dimensions. First, fairness metrics encode particular cultural values while marginalizing others. Second, test datasets are predominantly designed from Western contexts, excluding knowledge systems grounded in oral traditions, Indigenous languages, and non-digital communities. Third, fairness testing raises ethical concerns, including the reliance on low-paid data labeling in the Global South, and associated with this, the environmental costs of training and deploying large-scale models, which disproportionately affect climate-vulnerable populations. Addressing these issues requires rethinking fairness testing beyond universal metrics and moving toward evaluation frameworks that respect cultural plurality and acknowledge the right to refuse algorithmic mediation.

Learning tasks through videos is a dynamic way to acquire skills by witnessing entire processes. However, compared to in-person demonstrations, videos may omit tacit knowledge, including subtle details and contextual nuances. Users' unique circumstances, like missing ingredients in a recipe, may also require adaptation beyond the video content. To fill these gaps, many users turn to the comment section, seeking additional guidance and interactions with creators or peers to personalize their experience. Despite their importance, there is limited understanding of how users engage with and apply comments in task-learning scenarios. In our study, we explore the role of comments in video-based task-learning through interviews with 14 users, and co-watching sessions with eight. Our findings show that while comments are critical for learning, they are poorly integrated into all stages of the learning process. Based on our findings, we outline design opportunities to better utilize comments in video-based task-learning.

In this article, we investigate the effect of systematics on weak lensing beyond the standard ΛCDM paradigm. Specifically, we consider the 2- and 3-point statistics of the shear field for the set of cosmological models, including CPL dark energy, interacting dark energy (IDE), and Hu-Sawicki f (R) modified gravity. We consider two major systematics such as photometric redshift uncertainty and intrinsic alignment A IA . Our findings are derived from the Fisher matrix. These results indicate that σ z and A IA can substantially degrade constraining power, especially for f (R) gravity. Moreover, it also highlights the critical role of higher-order statistics and the need for robust systematic control for future cosmological surveys.

Structured optical waveforms are emerging as powerful control fields for the next generation of complex photonic and electromagnetic systems, where the temporal structure of light can determine the ultimate performance of scientific instruments. However, identifying optimal optical drive fields in strongly nonlinear regimes remains challenging because the mapping between optical inputs and system response is high-dimensional and typically accessible only through computationally expensive simulations. Here, we present a physics-guided deep learning framework for the inverse design of optical temporal waveforms. By training a light-weighted surrogate model on simulations, the method enables gradient-based synthesis of optical profiles that compensate nonlinear field distortions in driven particle-field systems. As a representative application, we apply the approach to the generation of electron beams used in advanced photon and particle sources. The learned optical waveform actively suppresses extrinsic emittance growth by more than 52% compared with conventional Gaussian operation and by approximately 9% relative to the theoretical flattop limit in simulation. We further demonstrate experimental feasibility by synthesizing the predicted waveform using a programmable pulse-shaping platform; incorporating the measured optical profile into beamline simulations yields a 31% reduction in the extrinsic emittance contribution. Beyond accelerator applications, this work establishes a general way for physics-guided inverse design of optical control fields, enabling structured light to approach fundamental performance limits in nonlinear photonic and high-frequency electromagnetic systems.

We investigate ferroaxial magnets, a new class of spin-order-driven multiferroic magnets in which magnetic ordering induces mirror-symmetry breaking while preserving both time-reversal and spatial-inversion symmetries. These systems exhibit a ferromagnet-like axial anisotropy that allows optical control of the ferroaxial polarization, while their macroscopic time-reversal symmetry makes them attractive for antiferromagnetic spintronics. Using spin crystallographic group analysis, we identify the candidate materials and the nonrelativistic ferroaxial nature stemming from the strong exchange splitting of magnets. Furthermore, a symmetry-based identification shows magnetic materials that host ferroaxial order and metallic conductivity, realizing the ferroaxial metal state that undergoes a ferroaxial phase transition while remaining metallic. As a direct probe for the ferroaxial metal, we propose a third-order nonlinear Hall effect originating from the transverse coupling between the electric field and Berry curvature dipole mediated by the ferroaxial anisotropy. Our results establish ferroaxial magnets as a platform for nonrelativistic multiferroicity and spintronic applications.

A normalizing-flow-based implementation of the density-of-states approach has recently been used to successfully reconstruct the partition function of (1+1)D scalar lattice field theory. In this preliminary work, we extend this framework to a lattice gauge theory by employing gauge-equivariant normalizing flows to reconstruct the density of states of pure (1+1)D U(1) lattice gauge theory, both with and without a $θ$-term. In the absence of a $θ$-term, we first demonstrate that the normalizing-flow-based reconstruction of the density of states reproduces the known analytic results for this theory. We further show that, in the presence of a $θ$-term, this formulation enables the generation of gauge-field configurations at fixed values of the topological charge.

We consider the problem of lifting a regular cluster structure on a quasi-affine variety to the ambient affine space and a similar problem of defining a regular pullback of a regular cluster structure under a dominant rational map between affine spaces. We provide sufficient conditions for the existence of the corresponding object, called an almost-cluster structure, study its combinatorics, compatible Poisson bracket and the corresponding upper cluster algebra.

We review a certain problem on covering triangles in the plane. Equivalently, it can be viewed as a family of 'isobilliard' inequalities in convex shapes, and as a special case of Viterbo's conjecture in symplectic geometry. We give an elementary overview of these topics and, using the optics of the covering problem, we establish several new special cases of Viterbo's conjecture, provide a simple explanation of the counterexample of Haim-Kislev and Ostrover, and state a few open questions. The main novel result is a proof of Viterbo's conjecture for lagrangian products $K \times Q$, where $Q \subset \mathbb{R}^2$ is any quadrilateral and $K \subset \mathbb{R}^2$ is any convex shape.

Similarity experiments were performed on the DIII-D and TCV tokamaks to explore the scaling of energy confinement in negative triangularity plasmas using non-dimensional variables. Near up-down symmetric plasmas with large top-bottom averaged negative triangularity were created in a lower single null configuration, with the shape of the separatrix being closely matched between the two devices. The normalized energy confinement is found to weakly improve at increasing collisionality and, between the two devices, shows a machine size scaling behavior between Bohm and gyro-Bohm. Engineering scaling on a large DIII-D dataset is in agreement with the non-dimensional experiment.

We give a proof of the cofreeness of the Lubin-Tate deformation ring, by generalizing earlier results by Matt Ando and Yifei Zhu about $\mathsf{H}_\infty$-orientations to the context of power operations for Morava $E$-theory.

The U.S. High-Performance Research Reactor program aims to convert high-power research reactors from highly enriched uranium to low-enriched uranium using a monolithic U-10Mo fuel design. A critical aspect of U-10Mo fuel performance is fission gas bubble behavior. These bubbles grow by trapping gas atoms (particularly Xe) but can disintegrate via irradiation-induced "re-solution". The interplay between the trapping and re-solution rates governs bubble evolution, impacting fuel performance and safety. In this study, binary collision approximation (BCA) and molecular dynamics (MD) simulations were performed to quantify the Xe gas bubble re-solution rate in U-10Mo fuel. First, the energy loss of fission fragments (FFs) through electronic and nuclear stopping was evaluated. The effect of electronic stopping on re-solution was then analyzed using MD simulations coupled with the two-temperature model. Results indicate that thermal spikes generated by electronic stopping do not contribute to gas bubble re-solution in U-10Mo. To quantify re-solution due to nuclear stopping, BCA simulations of FFs in U-10Mo were performed to obtain the average FF incidence probability, energy, and angle as a function of distance from the FF origin. Subsequent simulations assessed FF--bubble interactions in U-10Mo for different FF energies and bubble radii. From these analyses, an overall re-solution rate $b$ was calculated at equilibrium bubble pressure per unit fission rate density, yielding values ranging from $4.4 \times 10^{-26}$ m$^3$/fission for the largest bubbles to $8.8 \times 10^{-25}$ m$^3$/fission for the smallest. The effect of bubble pressure on the re-solution rate was also evaluated, revealing an inverse relationship between the two.

This note is a meditation on a generalization $\mathbb{W}_E$ of the classical p-typical Witt vectors $\mathbb{W}_p$, which arises (geometrically) from isogenies of deformations of formal groups, or (topologically) from the theory of power operations on Morava $E$-theory. For formal groups of height $1$ we have $\mathbb{W}_E=\mathbb{W}_p$, but the $\mathbb{W}_E$ are richer when height is $\geq 2$. We show that $\mathbb{W}_p$ splits naturally from $\mathbb{W}_E$. A key property of $\mathbb{W}_E$ is the isomorphism $π_0E\approx \mathbb{W}_E(π_0E/\mathfrak{m})$, the ``cofreeness of the Morava $E$-theory'' proved by Burklund, Schlank, and Yuan. This isomorphism determines a natural ``Witt filtration'' on $π_0 E$. We describe how this Witt filtration interpolates between the $p$-adic filtration and a geometric filtration on $π_0E/(p)$. We use this to give a new proof of cofreeness.

Wigner crystals are typically confined to ultralow temperatures where thermal motion is frozen out. Moiré superlattices in twisted two-dimensional materials have extended their stability to higher temperatures and densities, but rely on delicate stacking that fixes the lattice geometry and limits tunability. Here we demonstrate a lithographic approach that bypasses these constraints. Using high-resolution nanofabrication, we pattern a nanoscale triangular lattice directly into a graphene gate integrated with a monolayer MoSe2 semiconductor. This engineered potential landscape localizes electrons into generalized Wigner crystal states that persist up to 15 K and densities of 2X10^12 cm-2, representing an order of magnitude improvement over pristine monolayer MoSe2. Gate-voltage control allows real-time switching between stable and unstable crystalline states, with the latter exhibiting stochastic telegraph noise from nearly degenerate configurations. This work demonstrates the ability of this platform to transform Wigner crystals from fragile, static phases into reconfigurable quantum matter.

In a recent work, we constructed a rational map from a simple Lie group $\mathcal G$ to itself that intertwines the standard Poisson--Lie structure on $\mathcal G$ with a Poisson homogeneous one defined by a pair of quasi-triangular solutions to the classical Yang--Baxter equation (CYBE) known as R-matrices. We also showed, in the case of $SL_n$, that if the combinatorial Belavin--Drinfeld data associated with these R-matrices satisfies certain aperiodicity conditions, the map is, in fact, birational and can be used to obtain an initial cluster for an exotic cluster structure on $SL_n$ via the pullback of Berenstein--Fomin--Zelevinsky cluster variables. The same strategy was later used by the first author and D.~Voloshyn to describe generalized cluster structures compatible with the Poisson dual of the Poisson--Lie bracket defined by a quasi-triangular R-matrix. In this paper we further promote the use of birational Poisson maps in constructing generalized cluster structures by applying it in the situation when the aperiodicity condition is not satisfied. To this end, we describe a generalized cluster structure on $GL_n$ compatible with the Poisson homogeneous bracket defined by two Cremmer--Gervais solutions to the CYBE related via conjugation by the longest element of the Weyl group. The key ingredient to our construction is a birational map that connects the bracket under consideration with two other Poisson brackets: the Poisson dual to Cremmer--Gervais Poisson--Lie bracket on $GL_{n-1}$ and the bracket on a certain space of complex rational functions of one variable closely related to cluster algebraic interpretation of Coxeter--Toda flows. New notions of a regular pullback of a seed and of an almost-cluster structure whose detailed description are given in a separate note also play an important role in our construction.

GPUs play an increasingly important role in modern software. However, the heterogeneous host-device execution model and expanding software stacks make GPU programs prone to memory-safety and concurrency bugs that evade static analysis. While fuzz-testing, combined with dynamic error checking tools, offers a plausible solution, it remains underutilized for GPUs. In this work, we identify three main obstacles limiting prior GPU fuzzing efforts: (1) kernel-level fuzzing leading to false positives, (2) lack of device-side coverage-guided feedback, and (3) incompatibility between coverage and sanitization tools. We present cuFuzz, the first CUDA-oriented fuzzer that makes GPU fuzzing practical by addressing these obstacles. cuFuzz uses whole program fuzzing to avoid false positives from independently fuzzing device-side kernels. It leverages NVBit to instrument device-side instructions and merges the resultant coverage with compiler-based host coverage. Finally, cuFuzz decouples sanitization from coverage collection by executing host- and device-side sanitizers in separate processes. cuFuzz uncovers 43 previously unknown bugs (19 in commercial libraries) across 14 CUDA programs, including illegal memory accesses, uninitialized reads, and data races. cuFuzz achieves significantly more discovered edges and unique inputs compared to baseline approaches, especially on closed-source targets. Moreover, we quantify the execution time overheads of the different cuFuzz components and add persistent-mode support to improve the overall fuzzing throughput. Our results demonstrate that cuFuzz is an effective and deployable addition to the GPU testing toolbox. cuFuzz is publicly available at https://github.com/NVlabs/cuFuzz/.

We describe the mapping class group action on the cohomology of the twisted $\mathrm{SL}_n$-character variety of a surface $Σ_g$ of genus $g$. Our main tool is a relative version of the endoscopic decomposition of Maulik-Shen; this allows us to reduce the problem to the mapping class group action on the cohomology of a canonical finite cover of $Σ_g$, which was studied by Looijenga.