Papers

We prove that the $Q$th symmetric product of an absolute $\lambda$-Lipschitz retract, equipped with the bottleneck matching metric, is an absolute $(3Q-2)\lambda$-Lipschitz retract. This answers a question of De Pauw about the target spaces of Almgren's $Q$-valued maps. We also prove that the space of nonempty subsets with at most $n$ points, equipped with the Hausdorff metric, is an absolute $(6n-5)\lambda$-Lipschitz retract. The main tool is a reconstruction theorem for bounded products of metric stars with finitely many rays, modulo a finite permutation group: these quotients are absolute $3$-Lipschitz retracts, independently of the group and the number of coordinates. A variant of this theorem, in which unordered sets of sign vectors take the place of the group action, handles finite subset spaces.

Muon has motivated a growing family of optimizers that compose multiple matrix normalizations, but these methods remain fragmented and lack a unified perspective. We introduce chained linear minimization oracles (chained LMOs), which cast these methods as compositions of LMOs. Despite their empirical success, many chains fall outside the standard LMO framework and can diverge on smooth convex objectives. To explain why composition can nevertheless help, we turn to linear associative memory and show that chaining can improve over Muon under anisotropic embeddings. Empirically, we propose TensorChain, a novel optimizer within the framework that stacks compatible weight matrices across different layers and normalizes the 3d tensor across its axes. In Qwen3 0.6B and 1.7B pretraining, TensorChain outperforms all chained baselines in average token efficiency, with average token savings of 9.6% over Muon at matched validation loss.

Off-policy evaluation (OPE) estimates the value of a target policy from logged data, but limited behavior-policy coverage can force high-variance reweighting or reward-model extrapolation. Counterfactual annotations can add evidence about unobserved actions, yet practical sources, including domain experts and large language models (LLMs), may be costly, biased, or noisy. We study budgeted acquisition of such annotations for contextual-bandit OPE. Given source-specific costs and error profiles, we formulate an integer allocation problem over context-action pairs and annotation sources to minimize the component of estimator variance that depends on the annotation plan. We characterize when annotations are valuable through a first-annotation threshold and local annotation-value regimes. For the coupled multi-source problem, we develop a majorization-minimization algorithm with dynamic-programming subroutines that monotonically improves the objective. Experiments in synthetic clinical and LLM-annotated education bandits show that our allocation method reduces fixed-profile mean squared error (MSE) by 20.58% and 10.77%, respectively, relative to no annotation.

Let $k>t\ge 2$ be fixed integers, and let $G$ be a permutation group on a set of $v$ points. A $G$-invariant packing is free if every block has $|G|$ distinct images under $G$. We give two conditions on $G$ under which, as $v\to\infty$, there is a free $G$-invariant $t$-$(v,k,1)$ packing with $(1-o(1))\binom{v}{t}/\binom{k}{t}$ blocks. This is a $G$-invariant form of the Erdős–Hanani theorem on asymptotically optimal packings. For semiregular groups $G$, packings with these properties exist if and only if all but $o(v^t)$ of the $t$-subsets of points have trivial setwise stabiliser. This property is satisfied by every semiregular group for $t\ge 3$, and by every cyclic semiregular group for $t=2$. As a consequence, the Johnson bound is asymptotically attained at all lengths by optical orthogonal codes of constant weight $w$ and constant correlation $\lambda$ with $w\ge\lambda+2$, by their multidimensional and signature pattern versions, and by cyclic and quasi-cyclic constant-weight codes. The same method gives a form of the theorem for group divisible packings that are invariant under a cyclic group. This determines the asymptotic maximum size of two-dimensional optical orthogonal codes with at most one pulse per wavelength.

Assessing useful reuse in neural PDE solvers is challenging: final accuracy can reflect source learning and target-time computation. Our reuse contract separates solution accuracy, learning contribution, and numerical utility through paired state comparisons, matched target information and budgets, and cost accounting. A literature audit extracts 18 version-specific protocol records from 12 papers, documenting retained states, target-time resources, and reported controls. For a fixed linear system and residual tolerance, we construct two initial guesses with identical solution-error, energy-error, and residual norms, reaching the same solution with different conjugate-gradient (CG) iteration counts. Across 240 source-training trajectories, two linear PDE families, Fourier neural operators and convolutional networks, a fixed predictor's benefit reverses across correction algorithms. Among pairs with both relative prediction errors less than or equal to 5 percent on 64 in-distribution tasks (63 by 63 interior grids), reductions in all three norms accompany more CG iterations, at mean taskwise rates of 23.5 percent and 23.9 percent in two libraries. Work-based selection saves 2.50-3.33 CG iterations on held-out in-distribution tasks; matched adaptation demonstrates finite-budget pretraining value. Independent batches confirm a 0.73 percent complete online saving for one physics-trained Fourier neural operator against zero-initialized Poisson-preconditioned CG. Reuse requires matched state comparisons and downstream computational evidence.

Large language models are increasingly used in legal research and drafting, but they can still produce claims that sound convincing without being supported by the cited source. We introduce PARCEL, a benchmark for checking whether a legal claim is supported by the underlying authority. Using recent New York State Court of Appeals decisions, we build a dataset of 3,396 parenthetical-style claims labeled as Supported, Refuted, or Not Found. We cast this task as a three-way natural language inference problem and evaluate several state-of-the-art LLMs in a zero-shot setting. Although the strongest models reach up to 0.97 accuracy, the results also show an important weakness: models still incorrectly mark unsupported claims as supported, even when the full opinion text is provided. Across models, missing support is harder to detect than direct contradiction, and fabricated but plausible citations cause the largest drop in performance. Overall, PARCEL provides a practical benchmark for testing claim-level groundedness in legal RAG systems.

Finite photon momentum allows second-order photocurrents in centrosymmetric space groups, where the conventional bulk photovoltaic effect is forbidden. Here we study the photon-drag response of altermagnets and show that its angular dependence reveals both the symmetry of the magnetic order and the Néel domain of the sample. For example, when a crystal rotation exchanges the two domains, the domain-even and domain-odd parts of the current appear in different angular harmonics, $4n$ and $4n+2$ for tetragonal $d$-wave altermagnets, so that two measurements rotated by $90^\circ$ on a single domain isolate the domain-odd current. We also find that these microscopic contributions exhibit opposite domain parities depending on light polarization. Under linear light, only the shift term is domain odd, while under circular light, the domain-odd channel switches to the injection, Fermi-surface, and contact principal-value terms. Mirror symmetries fix the angular phase of each harmonic and distinguish $d_{x^2-y^2}$, $d_{xy}$, and $g$-wave order. In contrast, conventional collinear antiferromagnets show no domain-odd photon-drag current, and in tetragonal ferromagnets the two parts share the same harmonics. Although the altermagnetic spin splitting does not require spin-orbit coupling (SOC), SOC is essential to observe the domain-odd charge current. Our work uncovers distinct signatures of altermagnetism in photon drag and provides a route to all optical magnetic readouts of altermagnets.

Predicting thermally activated diffusion and defect evolution with event-driven models requires identifying atomic rearrangement mechanisms and their activation barriers. Discovering the associated saddle points is a major computational bottleneck: multiple rearrangements may originate from one metastable state, while costly local searches can fail or repeatedly converge to the same saddle. To address this challenge, we introduce ASPIRE (Atomistic Saddle-Point Inference with Refinement for Events), a framework that predicts a set of saddle candidates from a single initial atomic environment and refines them through Dimer searches on the original interatomic potential. The framework's equivariant set predictor, Ev-Quiformer, integrates (i) geometry-conditioned scalar-vector event slots for generating multiple saddle-point proposals and (ii) a decoder that maps each slot to a full atomic displacement field by combining atom, slot, and anchor-relative vectors with invariant coefficients. We also contribute two datasets: (i) BCCFE4VACAV-4000, comprising 4,000 four-vacancy body-centered cubic iron configurations and 65,450 reference events grouped by initial state for set supervision and post-refinement evaluation; and (ii) BCCFE-1TO4VAC, comprising 5,372 configurations with one to four vacancies each. Theoretically, we establish conditions for proposal equivariance. Experimentally, ASPIRE achieves 77.20% reference-event coverage on this benchmark, compared with 75.73% for a conventional Dimer baseline, while requiring approximately half as many Dimer force evaluations. In a timing evaluation on 50 configurations, ASPIRE reduces wall time per configuration from 478.8 s to 176.3 s under the stated hardware settings.

This work is devoted to the study of blow-up phenomena for solutions of \[-\Delta u=\lambda V(x)|x|^{2N}e^v,\quad \mbox{in}\quad B_1,\quad v=0\quad \mbox{on}\quad \partial B_1. \] where $B_1$ denotes the unit ball in $\mathbb R^2$ centered at the origin, $V(x)$ is a positive smooth function, $N\in \mathbb N$ and $\lambda>0$ is a small parameter. The main goal is to determine conditions on the potential $V$ guaranteeing the existence of families of solutions that blow up at the origin as $\lambda\to 0^+$. Moreover, we establish higher order criteria that distinguish between simple and non-simple blow-up behavior at $x=0$. In particular, for $N-1\in 2^{\mathbb N}$, we construct a non-simple blow-up sequence $v_k$, corresponding to coefficient function $V_k$ whose derivatives vanish up to the maximal order allowed by the upper bound of Wei-Zhang, thereby proving that this bound is sharp.

In this paper we study Koopman operators associated with finite periodic realizations of elementary cellular automata. The spectra of these operators encompass various properties of the underlying dynamical systems, independently of any initial condition. A characterization of these spectra is proposed, leading to a spectral classification of all elementary cellular automata. This work is thought as a preliminary step for studying the spectra of Koopman operators for infinite realizations of elementary cellular automata.

In this paper, we prove a conditional result that yields an asymptotic formula with an explicit main term and a power-saving error term for the generalized shifted ternary additive divisor sum $D_{\mathcal{I},\mathcal{J}}(X,1)=\sum_{n \leq X} \tau_{\mathcal{I}}(n) \tau_{\mathcal{J}}(n + 1)$, where $\mathcal{I}=\{\alpha_1,\alpha_2,\alpha_3\}$ and $\mathcal{J} = \{\beta_1, \beta_2, \beta_3\}$, with $|\alpha_i|, |\beta_i|\leq \delta$ for $i=1,2,3$, and $0<\delta<1/44$. The proof of our main theorem relies on assuming an averaged level of distribution for the generalized ternary additive divisor function $\tau_{\mathcal{J}}$ in arithmetic progressions up to the level $2/3$.

The Muon optimizer orthogonalizes each update matrix, setting all of its singular values to one, and has proven highly effective for training large language models. It remains unclear, however, whether this success comes from amplifying small singular directions that gradient descent neglects or from suppressing large, degenerate directions that disrupt training. We introduce Spectrally Targeted Muon, which orthogonalizes only the singular values above or below a threshold $\tau$, so that varying $\tau$ interpolates between normalized SGD and Muon. It isolates the relevant singular subspaces with projections computed by Newton-Schulz iteration on a shifted Gram matrix, so no SVD is needed. We evaluate these variants on the CIFAR-10 and NanoGPT speedruns, tracking the effective rank of gradient, update, and weight matrices and a new metric, the alignment of updates with the tangent space of the weight matrix's isospectral manifold. We find three things. First, the small singular values of the momentum are not noise. Orthogonalizing everything except the few largest singular values of each matrix nearly matches Muon while touching only a small fraction of the momentum, whereas orthogonalizing only the top falls well short even though it holds almost all of it. On language models every momentum singular value is far below one, so targeted orthogonalization can only amplify, and Muon wins by making directions that are too small to train on at their raw scale trainable. Second, shrinking the largest singular values is what keeps the parameter spectrum flat; this is cheap, and it is not what drives the loss. Third, AdamW differs from Muon mainly in how slowly it builds structure, which explains its slower start and why warmup helps AdamW but only hurts Muon.

Massive discrete Dirac-type operators on periodic planar graphs are investigated. Using Floquet–Bloch decomposition, we first show that neglecting the faces leads to spurious flat bands that are unrelated to the graph's geometry. Then, for the correct Dirac–Hodge operator $D_m$, we establish a spectral reduction expressing $\sigma(D_m)$ through the graph Laplacian on vertices $-\Delta_V$ and the Laplacian on faces $-\Delta_F$. As a consequence, $D_m$ carries a flat band precisely when one of these scalar Laplacians does. We finally apply this reduction to the study of flat bands of the operator $D_m$ on Archimedean lattices and contrast the outcome with that on the Lieb lattice.

Natural gas networks must accommodate variable demand and supply while maintaining pressure across pipelines whose stored gas couples operating decisions through time. Probabilistic surrogates can simplify compressor scheduling, but their predictive uncertainty must remain reliable where optimization uses it. We combine direct learning of the minimum pressure margin with assessment at optimizer-selected schedules. Using a warped Gaussian process, we express the posterior pressure requirement as one inequality in its predictive mean and standard deviation. We then parameterize block and Fourier schedules so scalar predictions and analytic derivatives replace trajectory states and time integration in whole-day optimization. In steady state, a surrogate that passes held-out accuracy and coverage checks nevertheless falls below the minimum pressure on 71% of its own dispatches at a nominal 1% tolerance, as the optimizer is drawn to schedules where the margin is overestimated. This diagnostic motivates an enrichment loop that audits selected dispatches against the physics and updates the surrogate by exact conditioning, with fitted quantities held fixed. The loop restores steady-state minimum-pressure risk control at every studied tolerance, with conditioning about four orders of magnitude faster than refitting. In transient operation, we complement this enrichment with empirical threshold calibration to address residual hourly optimism, bringing hourly violation rates closer to their declared tolerances with little compressor work overhead.

Agentic AI based on Large Language Model generalization capabilities offers a wide range of potential applications, including planning for embodied tasks. For example, embodied agents based on Foundation models can generate plausible plans in autonomous robotics scenarios. Due to limited context windows or hallucinatory phenomena in the next-token prediction formulation, behaviors may be generated without establishing whether the deployed robot and the observed environment actually support the requested operation, in what we call a "grounding failure". Thanks to the recent improvements in reasoning capabilities of foundation models, autonomous robot behavior generation problem can be formulated as a code generation problem. We present iAm.md, a Markdown standard and generation framework, that allows anchoring this process in complementary forms of deployment evidence. Through open-vocabulary semantic mapping, we combine local vision-language detections and object segmentation and refer them to persistent object records in this intermediate standardized representation, allowing agentic introspection. We then study this new technique on a simulated TIAGo, on navigation-and-manipulation tasks, showing how this standardized representation jointly supports skill self-assessment and executable task generalization.

Coding agents increasingly share a workstation while drawing on separate providers and subscription allowances. A second agent can inspect a completed answer, but the call spends another pool and may provide no substantive finding. We describe an advisory cross-provider review contract: distinct resource pools, bounded execution, restricted reviewer capabilities, complete input delivery, usable semantic output, explicit failure states and durable per-attempt evidence. In a controlled, agent-authored pilot of 20 paired development turns, eight had a material reviewer finding (95% exact interval 19.1-63.9%). A boundary-condition scan across both reviewer backends reproduced a previously discovered false success on partial input: four truncation levels passed historically and failed after repair. The scan also found and repaired cancellation during process reaping. In real CLI probes, Claude had no writing tools; Codex attempted writes in five of five read-only trials, each write tool failed, and no disposable repository changed. These tests cover specified paths and versions, not field reliability. A preregistered shadow study of metadata-only review allocation accrued 25 formal observations before an exact-runtime regression found a third defect: a reviewer exiting nonzero with a well-formed verdict was counted as complete. Exit status was not recorded per attempt, so exposure cannot be resolved retrospectively. The 25 formal and two pending records remain an audit cohort; the measurement-valid cohort restarted at zero and collection has begun. No gate result is reported.

We present LVSPM, a generalizable model that jointly estimates camera poses and synthesizes novel views from uncalibrated image collections. Trained with only RGB images and pose supervision, LVSPM avoids dense 3D ground truth and employs test-time training (TTT) layers to scale seamlessly to hundreds of input views. On RealEstate10k, Co3Dv2, and DL3DV, LVSPM surpasses VGGT in pose estimation across 16-256 views, with especially large margins at strict thresholds. For novel view synthesis under a practical protocol where more views cover larger scenes, LVSPM achieves state-of-the-art pose-free quality—surpassing even pose-dependent models in PSNR—and still maintains high quality as scene scale grows, while baselines collapse. The code is available at https://burningdust21.github.io/Projects/LVSPM .

In recent years, persistent homology has seen rapid adoption in deep learning, yet its computation remains a major bottleneck in network training. This paper introduces TopoGPU, a GPU streaming pipeline that computes persistence diagrams of cubical complexes induced by 2D and 3D images. TopoGPU streams the input image chunk by chunk, processing each chunk with massively parallel GPU kernels on a grid of GPU blocks; the resulting boundary relations are accumulated in host memory, where the CPU performs the boundary matrix reduction. TopoGPU introduces a stratification-aware discrete Morse matching that provably preserves persistent homology under streaming, together with a parallel topological sorting algorithm and a parallel V-path parity algorithm for deriving Morse boundaries on the GPU. TopoGPU outperforms Cubical Ripser, a state-of-the-art method for persistent homology computation, on every benchmark evaluated, achieving an average end-to-end speedup of 53.24x and a maximum of 198.01x. We further integrate TopoGPU into a topology-preserving deep network, demonstrating that it substantially reduces the cost of persistent homology computation during network training. TopoGPU is open source, with pre-built binaries, Google Colab notebooks, and Docker images available at the project's GitHub page: https://github.com/seravee08/GPU-Computation-of-Persistent-Homology-for-Image-Data.

Electrification is planned as a network problem: more demand, more upstream capacity, local resources waiting for reinforcement. We argue that a material part of that bill, in money and time, follows from architecture, not electrification itself, and propose the Short Grid as the deployment unit: a locally controlled building, block or campus whose demand, generation, storage and local market operate behind Energy Routers enforcing an approved connection envelope to the Long Grid. Because the envelope governs exchange, not installed capacity, local resources can grow and many sites can build in parallel at the pace of PV and EV installation, not grid reinforcement. PV and EV batteries dominate today, but the architecture is general: any generation or storage, fixed or mobile, fits behind the same boundary and interfaces. EnergyNet Operators coordinate exchange among Short Grids and with the Long Grid; protection stays local; Vehicle-to-EnergyNet (V2EN) and Mobility-Based Energy Peering (MEP) make mobility-financed batteries accessible within and between them; priority-based operation protects critical services in shortage. We test it in models, not yet the field: a ten-building neighbourhood in Lund, Sweden; a bus-linked microgrid pair at Stanford and Half Moon Bay; and a proposed 20-site network in Rockaway, New York. Households and services are the largest demand category in Sweden, Denmark, Norway, California and New York (484 TWh/yr); halving their grid imports would be about 242 TWh. Europe's network investment to 2040 is estimated above EUR 1.2 trillion; modelling shows it depends on system design. Negagrid value, following the negawatt, counts network investment avoided, downsized or deferred, net of local substitute costs. "Don't pay twice": build coordinated local capability first; reinforce the Long Grid for what remains.

Production AI systems deployed in high-stakes domains accumulate a governance liability that existing monitoring frameworks fail to detect: the progressive inability to explain individual decisions when regulators, auditors, or affected individuals demand accountability. We introduce TRACE (Transparency, Risk, Accountability, Compliance, and Explainability), a seven-instrument governance framework for measuring, tracking, and remediating Explainability Debt in production AI systems. The foundational instrument, the Explainability Debt Score (EDS), quantifies the proportion of production decisions falling below a governance-defined explainability confidence threshold at any point in time. Complementary instruments include DART (Debt Accumulation Rate Tracker for breach forecasting), SHIV (Scenario Health and Integrity Validator for daily governance), FDE (Feature Drift Evaluator for causal attribution), HVE (Human Validation Engine), AIDE (Audit Intervention Decision Engine), and ZERO (Zero Explainability Risk Optimiser for remediation). Through a twelve-month longitudinal case study of a production fraud detection system processing 50,000 daily financial transactions, achieving 98.46% accuracy and ROC-AUC of 0.9990, we demonstrate that an EDS of 0.23 on audit day was statistically predictable six months in advance using DART trajectory analysis (beta = 0.008/week, R-squared = 0.94, 95% CI: [0.006, 0.010]), and that 78% of Explainability Debt was concentrated in the highest-regulatory-risk decision category (transactions above $10,000), a risk asymmetry completely invisible to system-level metrics. TRACE provides the first quantitative operational architecture for EU AI Act Article 13 compliance in production AI deployment, establishing a new subdiscipline of explanation governance distinct from explanation generation.

We prove that a smooth Dunkl harmonic function in the upper half-space whose Dunkl gradient satisfies a Carleson measure condition is the Poisson integral of a function in $\mathrm{BMO}(dw)$, defined using Euclidean balls. The result holds for arbitrary finite root systems and nonnegative multiplicities. The proof uses an estimate for increments from the origin, obtained from a finite-dimensional system indexed by the reflection group, together with the atomic characterization of the Dunkl Hardy space.

Conversational Query Rewriting (CQR) turns a context dependent user turn into a standalone query for a retriever, and most methods do this in a single step from the dialogue history before retrieving once. The rewrite is therefore fixed before any corpus evidence is available to correct its reference resolution or its vocabulary. We recast CQR as a sequential retrieval problem: an agent rewrites the current turn, retrieves, and conditions its next rewrite on the returned passages. The agent acts in a typed space of three rewriting operations, resolving conversational intent into a standalone query, generating lexical reformulations, or synthesizing pseudo-documents for document-to-document matching, together with a stop action that ends the episode. We train the policy with supervised fine-tuning followed by reinforcement learning against a single retrieval-quality reward, using no human rewrite annotations. Across TopiOCQA and QReCC, the agent outperforms several retrieval-aligned baselines, while remaining effective across retrieval backends and generalizing to the CAsT benchmarks without additional training. Further analysis shows that, through retrieval-reward optimization alone, the learned policy develops a behavior of grounding pseudo-documents in passages retrieved by earlier steps, substantially improving retrieval.

Heuristic search for a plan can store exponentially many states, even when its heuristic is almost perfect. We instead learn search control, one specification per domain, written as an indexical policy: a generalized policy with registers that hold objects and modes that sequence its rules. We add the choose rule, which loads an object into a register and marks a backtracking point, where one candidate suffices; every other rule must work for all of its outcomes and needs no search. Our main result is that structural termination, which rules out infinite executions, also bounds every execution by a polynomial in the number of objects. A depth-first procedure then finds a plan in polynomial space, however large the state space, with no list of visited states. The cost is time, exponential only in the choice depth, the number of real choices along an execution. Any class that such a policy solves therefore lies in NP, and in P at constant choice depth. We learn these policies with a language model in a counterexample-guided loop that certifies termination, verifies the training tasks, and keeps the choice depth small. With the learned policies, the procedure solves 1,709 of 1,890 test tasks of the IPC 2023 Learning Track and the Autoscale Agile suite, more than LAMA, BFWS, and Levitron, and most of them within one second and 100 MiB.

We propose that the path to realizing the full potential of two-dimensional (2D) electronics lies in understanding and mastering their energy band tail states (BTS). Unlike silicon, which benefits from sharp band edges (Urbach energy $E_\rm U$ ~ 10 meV), today's monolayer 2D semiconductors have rough band edges ($E_\rm U$ ~ 100 meV) caused by spatial potential inhomogeneity. Comparing a physics-based electrical model to experiments, we demonstrate that BTS cause reduced effective mobility and "slow" transistor turn-on with excessive gate voltage in monolayer 2D transistors, which sets a fundamental limit for their energy-efficient operation. We also show that BTS effects in 2D transistors could be mitigated with higher gate capacitance, with strain engineering, and by using bilayer instead of monolayer 2D channels, as long as $E_\rm U$ is reduced below ~ 50 meV. This work provides a theoretical framework and design guidelines to navigate these fundamental limits, and to enable successful integration of 2D semiconductors into low-power nanoelectronics.

Brain signal decoding is challenging because neural recordings are noisy and vary across individuals, while labeled data are often limited. Recent attention-based models on the symmetric positive definite (SPD) manifold have nevertheless achieved strong performance using covariance and connectivity representations, yet the contribution of learned token weighting remains unclear. We examine two representative architectures, MAtt (based on log-Euclidean geometry) and GBWAtt (based on generalized Bures–Wasserstein geometry), and find that their learned attention weights remain close to uniform after training. We relate this behavior to bounded similarity parameterizations that, under the original softmax scaling, limit attention-weight contrast. Moreover, replacing learned weights with uniform weights, throughout training and evaluation, has little effect on mean predictive performance while preserving each model's original aggregation geometry. Motivated by these findings, we introduce SPD-MetaFormer, an attention-free architecture built on uniformly weighted Fréchet aggregation under log-Euclidean geometry. Its backbone uses a geodesic residual to update a summary token and a shared spectral feedforward map to transform all tokens, followed by a learned weighted readout. Token states remain SPD-valued until tangent-space classification. Across three EEG benchmarks, SPD-MetaFormer achieves competitive results relative to published Euclidean and manifold baselines. Separate matched reproductions test learned versus uniform weighting within MAtt and GBWAtt. These results suggest that, in the short-sequence and limited-data regimes studied, carefully designed SPD architectures can provide a simpler and effective alternative to adaptive manifold attention.

We present a systematic X-ray spectral analysis of the 2019 and 2022 outbursts of the accreting millisecond pulsar SAX J1808.4-3658 with extit{NICER} with a particular emphasis on the nature of the longstanding $\sim$1\,keV spectral feature. We report here the first clear evidence that the 0.9-1.0 keV feature varies systematically with source brightness, exhibiting strong correlations with both continuum and blackbody flux. This establishes that the feature is directly coupled to the accretion process. The scaling and energy distribution of the feature differ between outbursts, corresponding to a $\sim$2\,dex change in reprocessor density (log\,$n_e \approx 19.1$ in 2019 vs. $\sim$17.3 in 2022), which naturally explains the stronger, broader feature in 2019 and the weaker, more localized $\sim$1\,keV feature in 2022, in agreement with high-density reflection models. These results provide the strongest constraints to date on the origin of the $\sim$1\,keV feature in SAX J1808.4-3658.

String attractors provide a compact way of representing the complete factor structure of a word: a set of positions is an attractor if every distinct factor has at least one occurrence crossing one of the selected positions. Although the minimum attractor size is known for several classical families of words, describing all optimal attractors is typically much more difficult, since it requires understanding the geometry of all factor occurrences rather than constructing a single optimal solution. We give a complete description for the finite Thue–Morse words. Earlier work proved that four positions are necessary and sufficient for every order $n\geq 4$, but the collection of all smallest attractors remained unknown. For every $n\geq 6$, writing $h=2^{n-3}$, we prove that the smallest attractors are exactly the two reflected families where the four offsets are chosen independently from $\{0,1\}$. Hence there are exactly $32$ smallest attractors for every $n\geq6$. The initial cases are genuinely exceptional: $t_5$ has $40$ smallest attractors and $t_4$ has $87$. We also study the attractor condition independently of optimality. For every $n\geq5$, we characterize the complete antichain of inclusion-minimal factor coverages of $t_n$. It consists precisely of the coverages of $aa$, $bb$, and the eight minimal unique substrings of every generation $t_m$, $4\leq m\leq n$. Thus there are exactly $8n-22$ canonical constraints, forming an irredundant exact certificate for attractors of arbitrary cardinality. When attention is restricted to four-position sets, this linear-size system collapses to a constant one: it is enough to test the $24$ minimal unique substrings coming from three consecutive generations, and sixteen of these already force the two optimal families. We also show that three generations are necessary within this natural consecutive-generation hierarchy.

In this note, we study the problem of designing a feedback law that globally steers a system to a prescribed target configuration. Even if the system is fully actuated, topological obstructions generally prevent the existence of globally asymptotically stabilizing continuous feedback laws. We revisit this problem in a stochastic setting by allowing noise to enter through the control channels. Using a criterion for asymptotic stability in the large that combines local Lyapunov stability with positive recurrence, we constructively show that one can construct elementary feedback laws that achieve global asymptotic stability in the large, of the target equilibrium point in connected Euclidean domains with obstacles and manifolds without boundary. For Euclidean domains with obstacles, we also show that the method extends naturally to the problem of finding the minimizer of a strongly convex function with non-convex constraints. Numerical experiments illustrate the effectiveness of the approach for Euclidean domains with circular obstacles and the two dimensional sphere. Additionally, we study the role noise strength when there is a non-convex obstacle, in which case the system might show metastability.

Quantum fault tolerance is a prerequisite to harness the full potential of quantum computers. However, achieving fault tolerance remains a major challenge since it requires optimizing tradeoffs between several intertwined theoretical and experimental parameters, including the encoding rate, code distance, parity-check weights, code planarity, and routing. Surface codes remain strong candidates for fault tolerance due to their simple geometric structure, established decoding methods, locality, and experimental attainability. In this work, we present finite families of hyperbolic surface codes that leverage the optimal efficiency scaling $kd^2/n = C (\log{k})^2$ attainable by surface codes with bounded local geometry. We show that optimizing the periodic identifications can double the code distance and thereby increase the code's efficiency by a factor of four, at fixed qubit count, encoding rate, and parity-check weight. We also address the routing problem of hyperbolic surface codes by introducing a topology-aware algorithm that partitions a hyperbolic code into bounded planar modules while reducing long-range communications demand. Circuit-level Monte Carlo simulations yield finite-size threshold estimates for both modular and monolithic layouts, showing that tripling the long-range gate error probability mildly lowers, but does not eliminate, the estimated error-correction thresholds. Together, these results strengthen the case for hyperbolic surface codes as structured and highly efficient large-scale quantum error correction codes for future fault-tolerant architectures.

Reward hacking, where a model exploits an evaluation signal without completing the intended task, threatens the reliability of deployed language model systems. Existing datasets use different labels, response formats, and metadata conventions, making detector results difficult to compare. We present RH-Detect, a benchmark that combines reward-hacking-relevant subsets from eleven public datasets, comprising 92,761 rows and six behavior categories, into a common schema. On 5,021 open-ended evaluation units, each comprising a task prompt and a free-form model continuation, including multi-turn tool-use trajectories, we evaluate six off-the-shelf language models from five families as reward hacking detectors without additional training. The best model achieves a pooled AUROC of 0.962, with accuracy above 93%. For the four strongest models, however, accuracy on the two multi-turn tool-use datasets, MALT and TRACE, is 10.7-15.9 percentage points lower than on the other sources at a common decision threshold, highlighting a key gap for deployment-time monitoring. We find that different input formats have different effects across models. Removing thinking raises Qwen3.5-4B AUROC from 0.779 to 0.849, but lowers Qwen Flash from 0.977 to 0.950. We also evaluate the benchmark as a training dataset for detectors. Holding out each source in turn, single-token SFT improves average AUROC on five of six held-out sources. A GRPO follow-up on that failure case yields a slight improvement in detection performance. Our results show that a single pooled score can conceal variation across data sources, detector inputs, and training procedures.