Papers

Time-series language models typically align numerical signals with language using labels, captions, or question-answer pairs, without explicitly requiring their representations to preserve how the underlying signals evolve. We introduce forecast grounding, a pretraining principle that requires language-aligned time-series representations to remain predictive of the future evolution before they are adapted to downstream tasks. We instantiate this principle in Lotus, which divides each time-series channel into non-overlapping patches and maps them into the LLM embedding space. It then predicts the next patch from the LLM hidden states of the time-series tokens, encouraging the learned representation to retain temporal information. To capture both temporal changes and periodic patterns, we also propose a temporal-spectral patch representation that combines each raw time-series patch with its spectral representation before mapping it into the LLM embedding space. Across three language model backbones and six downstream evaluation datasets, Lotus achieves the best average performance with minimal trainable parameters, reaching 59.91% accuracy and outperforming the strongest matched baseline by 3.36%. Our representation analyses further show that Lotus uses the numerical input and preserves temporal structure in its learned patch representations. These results establish forecast grounding as a promising pretraining principle, enabling compact time-series language models to outperform models 25-50 times their size on TSRBench. Our code is publicly available at https://github.com/DA2I2-SLM/Lotus

Professionals are increasingly turning to artificial intelligence (AI) for career advice. We tested whether large language models (LLMs) provide different advice to female and male job candidates in the context of career negotiations. Across two large-scale audit experiments, including four leading models, the LLMs not only recommended lower salary targets for female than male personas, but also constructed distinct negotiation agendas, prioritizing flexibility, caregiving, and well-being more highly for female candidates, and financial rewards, mobility, and advancement more highly for male candidates. These differences closely tracked independent human ratings of employment terms associated with primary caregiver and breadwinner roles, respectively. Our findings identify a subtle pathway through which personalized AI advice may reinforce gender inequality at work: steering otherwise equivalent users toward traditional gender roles.

Contact-rich manipulation requires a robot to be precise in free space to establish the intended contacts, and compliant once in contact to avoid excessive forces. Imitation learning has proven effective for simpler manipulation tasks, but the compliance underlying a demonstration is not directly observable and is therefore difficult to infer from demonstration data alone. We present a framework that makes compliance explicit at every stage. During data collection, the operator teleoperates the real robot while a synchronized digital twin visualizes the error between the commanded reference and the achieved pose, making the current compliance visible, and selects between two discrete compliance modes. To generalize across object poses, we combine Laplacian editing, which retargets trajectories efficiently but is not physics-consistent during contact, with simulation-based trajectory optimization, which enforces physical plausibility but offers no manual guidance. Finally, a residual reinforcement learning policy refines the interaction forces by adjusting the reference pose of the underlying admittance controller. We evaluate on three flipping and pushing tasks on a Franka FR3 based on a total of 7 demonstrations across tasks. Our method is successful on all 40 out-of-distribution trials, compared to 33/40 and 24/40 for two baselines, with our base policy alone reaching 35/40.

Foundation models are expanding robot capabilities, enabling robots to handle more diverse objects, follow sophisticated instructions, and operate in complex environments. These advances bring us to a future in which robots become part of everyday household life. However, cluttered, dynamic household environments also expose robots to unfamiliar objects, tasks, and interactions, making reliable zero-shot task completion an ongoing challenge for robotics research. From a security perspective, we argue for open-world safety: household robots should preserve specified safety requirements in unseen situations, including when they cannot complete the requested task. Safety protections must account for context and interactions that persist across actions and tasks. In this position paper, we examine why existing safety methods can fail to provide this protection. Protection can fail through misinterpreted context, overlooked past interactions, or stopping and recovery strategies that become unsafe under unfamiliar conditions. We therefore propose a research agenda to specify, learn, and ground safety requirements, enforce them with reliable recovery, and evaluate the generalization of such protections. We call for household robots whose safety protections remain reliable beyond the situations, context, and history used to train and develop them, to achieve open-world safety.

Recent progress in learned whole-body controllers for humanoid robots has enabled robust free-space motion by tracking kinematic targets such as joint or end-effector trajectories. This interface provides a useful abstraction for upstream systems, including motion capture, teleoperation, or generative models, which can specify desired motion without reasoning about forces or balance. However, this abstraction has largely been limited to robot motion, requiring task-specific controllers or retraining for object interaction. We introduce OIL, an Object Interaction Layer that extends the kinematic command interface of whole-body control to include desired object motion and contact. OIL augments a standard controller with object-centric commands and translates them into coordinated whole-body behavior while preserving locomotion and balance. This design enables a single learned policy to execute non-prehensile interactions, including pushing, pulling, two-arm pick-and-place, and locomotion-coupled transport, directly from object-level kinematic targets without task-specific retraining. In simulation, we demonstrate that object-level commands improve performance over robot-centric controllers while matching task-specific policies across variations in object properties such as mass and size. We further demonstrate zero-shot generalization to novel behavior combinations and sim-to-real transfer on the Digit humanoid robot.

Problem 73 on Green's list of open problems asks whether a smooth submanifold of codimension two in $\mathbb{R}^n$ must meet some affine two-plane in five points when $n$ is large. In this paper, we answer this question for every $n \geq 6$. Every nonempty relatively open subset of such a submanifold contains the vertices of an affinely regular pentagon, and the bound $n \geq 6$ is sharp for affinely regular pentagons. More generally, for every prime $p \geq 5$, the graph of a continuous map from a ball in $\mathbb{R}^{p-1}$ to $\mathbb{R}^2$ contains an affinely regular $p$-gon. A second argument gives six coplanar points when $n \geq 7$. Both proofs find the points by making the low Fourier modes of the graph map vanish along a circle.

Multimodal large language models (MLLMs) often answer visual reasoning questions by relying on linguistic priors rather than task-relevant visual evidence. Textual chain-of-thought reasoning can partially mitigate this issue by encouraging models to decompose visual questions into intermediate evidence-seeking steps, but generating these steps autoregressively increases inference cost. Latent reasoning avoids explicit rationale generation, but existing approaches provide limited control over what intermediate states encode, making it difficult to impose separate supervision for planning, grounding, and evidence selection. We propose Structured Latent Visual Reasoning (SLVR), a training framework that bridges explicit chain-of-thought and latent reasoning by organizing multimodal reasoning into typed latent stages for planning, grounding, evidence selection, and reasoning integration. SLVR first trains the model to rely on the image by masking answer-revealing text and contrasting the correct answer with visually plausible distractors. It then organizes reasoning into latent stages for planning, grounding, evidence selection, and integration, supervising each stage with the corresponding signal: plans, boxes, visual evidence, and final rationales. This gives latent reasoning an explicit functional structure while avoiding generated textual chains at inference time. Built on Qwen2.5-VL-7B, SLVR improves consistently across multimodal reasoning benchmarks, with absolute gains of +9.4 on MMVP and +14.2 on BLINK Relation, as well as improvements on V*, MathVista, and ChartQA. These results suggest that structured latent supervision can improve fine-grained visual reasoning without the decoding overhead of textual CoT. Project page is available \href{https://bogao-code.github.io/SLVR/}{here}.

We develop a parity-resolved spectral-connection construction for a one-dimensional Dirac particle in an attractive soft-core Coulomb potential. The square-root dependence of the potential produces algebraic branch points in the coordinate plane. We uniformize these branch points by introducing \(t=\sqrt{x^2+\beta^2}\), while retaining the full-line \(Z_2\) symmetry generated by \(\Pi=\sigma_z P_x\). In each parity sector the resulting problem is formulated as a global connection problem between a solution selected at the physical origin and the decaying solution at infinity. This defines parity-resolved spectral connection functions whose zeros determine bound-state energies. We further construct a quasi-exact Bethe-ansatz sector directly from the same Dirac equations and use its polynomial solutions as independent benchmarks of the spectral condition. The comparison clarifies the distinction between local quasi-exact solvability and the global spectral determinant. The construction is inspired by the role of discrete symmetry and global analyticity in the quantum Rabi model, without identifying the present spectral function with Braak's \(G\)-function.

We study derivations and automorphisms of arbitrary non-abelian extensions of $n$-Lie algebras through their full crossed-product data. Their relative center is a canonical coefficient module, and a third cohomology class obstructs completion of data satisfying the mixed Filippov identities. When this class vanishes, a relative second cohomology space classifies the remaining extension data. We prove a non-abelian good criterion, construct the Lie algebra and group of compatible pairs, and obtain derivation and automorphism Wells sequences with explicit lifting equations. The derivation obstruction is a Lie algebra $1$-cocycle; the additive automorphism obstruction is a crossed homomorphism. We distinguish the ordinary center from the relative center, and outer derivations of the kernel from the mixed action required. Sparse crossed products are treated, including the representation class of their second mixed operation and the additional equations governing changes of sparse sections.

AI foundation models pretrained on weather and climate data are increasingly fine-tuned to Earth science tasks well beyond weather forecasting. Their development and adoption are outpacing the scientific community's ability to evaluate them. These models are judged almost entirely by benchmark skill metrics, which measure how closely a forecast reproduces a reference product but not whether a model represents the physical processes governing the system it predicts. Forecast skill and physical reliability are therefore distinct properties. The distinction is most consequential under the nonstationary conditions of a changing climate for which these models were never trained. We identify five priorities for the physical evaluation of AI models in Earth science from task-specific emulators to foundation models, spanning training data, fine-tuning, behavioral testing, mechanistic interpretability, and output validation. We recommend three activities for the coming decade: 1) open AI-ready evaluation datasets, 2) a shared reporting standard for physics-based evaluation, and 3) a dedicated research program on the safety of these models.

Time accurate speech transcription is essential in critical appli- cations, such as medical conversation transcription, reading as- sessment, and atypical speech recognition. Encoder-decoder auto- matic speech recognition models, such as Whisper, do not produce word-level timestamps out-of-the-box for all languages. Existing approaches to word-level time stamp estimation either use an exter- nal CTC model capable of frame level phoneme/character prediction which is computationally expensive, or use the cross attention scores internal to the model which gives coarse time stamps. We consider timestamped speech transcription for Hindi in this paper. We pro- pose a CTC-based approach, in which we train a low-complexity character level CTC head directly on the Whisper encoder output, to address the complexity and granularity limitations of existing works. We evaluate the system using manually-annotated word-level times- tamps. Experiments show that the proposed internal-CTC based approach performs similar to using an external CTC model and better than the cross attention score-based approaches. Further, we study the usage of CTC-head output to detect and mitigate hallu- cinations in conversational speech. Our results show that a simple token length criterion derived from the CTC output can improve the transcription quality for short segments where hallucinations are more probable.

Contrastive Language–Audio Pretraining (CLAP) learns a joint text–audio representation space, which enables zero-shot audio understanding from natural-language descriptions. However, its English-only text encoder restricts CLAP to tasks and datasets expressed in English. We present Indic-CLAP, a multilingual text–audio model that extends CLAP to nine Indic languages spoken by over a billion people. Exploiting CLAP's dual-encoder structure, we propose to train only an Indic text encoder by distillation while keeping the audio encoder frozen, using machine-translated AudioCaps captions. We compare distillation from CLAP's text encoder, audio encoder, and a novel hybrid objective combining both. Our evaluation shows that the Indic-CLAP encoder inherits the cross-modal alignment from the CLAP teacher, as evidenced by the performance on cross-modal retrieval and zero-shot classification tasks. Further, we study the cross-modal alignment using modality gap analysis, which explains the performance trends.

Spoofing-aware speaker verification (SASV) must confirm who is speaking and that the speech is genuine, but current systems gain one at the expense of the other. Modular cascades are accurate yet run two large self-supervised encoders, whereas end-to-end models are efficient but lose accuracy when a single embedding has to serve two conflicting objectives. We show that this trade-off between efficiency and specialization is avoidable. A single frozen W2V-BERT~2.0 backbone is adapted per task by deep hybrid Wavelet Prompt Tuning (WPT), so each branch obtains its own view of the shared encoder through dedicated prompts and a task head, and the scores are combined only at inference. Training only about 7M parameters, a small fraction of those a strong two-encoder cascade requires, the system surpasses it on SpoofCeleb evaluation with 0.03% CM-EER and 0.038 min a-DCF against 0.16% and 0.047. Because the backbone is shared and frozen, new branches attach without retraining the heads and prompts already in place. On ASVspoof5, with adversarial attacks and codec distortions, it reaches 0.092 min a-DCF and 3.60% CM-EER using only the provided data, indicating that the design holds under realistic in-the-wild threats.

Detecting synthetic and voice-converted speech remains difficult for low-resource languages with dialectal diversity, where systems must generalize across regional dialects and unseen acoustic channels. We present the UZH-CL submission to the ArA-DF 2026 Shared Task on Arabic speech deepfake detection, covering Track~1 (dialect generalization) and Track~2 (acoustic robustness). We freeze a W2V-BERT-2.0 backbone and adapt it with Wavelet Prompt Tuning, updating under 1% of parameters, and aggregate multi-layer representations with cross-layer attention and a general attentive-statistics pooling head rather than a specialized graph backend. Complementary detectors are obtained by varying adaptation strategy, training data, augmentation, and encoder family. We find that the two shift types require different fusion regimes: a broad multi-window ensemble for dialect generalization, and a compact, channel-matched, center-crop ensemble for acoustic robustness. Official evaluation yields 1.96% EER on Track~1 (6th place) and 1.04% EER on Track~2 (3rd place), corresponding to 87% and 96% relative reductions over the XLS-R+AASIST baselines.

We study a three-species competitive Lotka–Volterra system with cyclic dominance, motivated by rock–paper–scissors interactions in ecological communities. The model contains a singularly strong interaction coefficient of order $1/\epsilon$, representing a regime in which one species exerts a much stronger competitive pressure on another. We show that, despite this strong asymmetry, the cyclic dominance mechanism persists. Our main result is a quantitative description of one complete dominance pattern in the singular regime $0<\epsilon\ll 1$. Starting near the state where the third species dominates, we prove that the solution successively passes near the dominance states of the second and first species before returning to a comparable configuration. We obtain logarithmic estimates for the return time and for the population densities at the return section. In particular, although one species becomes extremely small at the beginning and at the end of each pattern, it recovers to an order-one density during the cycle. We then prove that this pattern repeats indefinitely and that the omega-limit set is composed of three heteroclinic orbits on the boundary of the positive orthant, connecting the three single-species equilibria. Thus the long-time behavior is cyclic but aperiodic, with increasingly long excursions near successive single-species states.

We study moving-law random-environment matrix cocycles whose one-step quasiperiodic cocycle law varies stationarily along an ergodic environment orbit. Two fixed-law structures then cease to suffice: a single stationary projective measure does not record transport between successive environment fibers, and pooling the moving supports may create products that are not dynamically admissible along any environment orbit. We replace the first by environment-indexed stationary projective families and prove an environment-wise Furstenberg formula. When the top Lyapunov exponent vanishes, averaged stationarity upgrades to exact equivariance at the support level, \(g_*\eta^\omega=\eta^{\Theta\omega}\), which forces either an environment-wise compact reduction or a finite support-invariant family of nonzero proper measurable subbundles. Environment-wise noncompactness and environment-wise strong irreducibility therefore imply positivity. To address the second obstruction, we compare admissible products along matched environment segments with equal total torus translation. Thus their quotients are genuine returns on a common vector fiber and generate environment-matched monodromy groups. Stable noncompactness and stable strong irreducibility of these groups verify the cocycle-level hypotheses. Applied to stationarily varying random–quasiperiodic Schrödinger cocycles, the framework yields strict positivity of the top Lyapunov exponent at every real energy under the stated assumptions.

A rainbow $k$-dominating function assigns the empty set or exactly one of the colors $\{1\},\{2\},\ldots,\{k\}$ to each vertex of a simple graph in such a way that every vertex receiving $\emptyset$ sees all $k$ colors in its neighborhood. This model has several facility-location interpretations where colored vertices represent sites hosting different types of services, and uncolored vertices represent the beneficiaries of those services who must have access to all of them. In this paper we study this domination variant in regular graphs. First, we develop lower and upper bounds for this class of graphs, and explore their structural consequences. Then, we use those results to prove NP-completeness of a decision formulation of this problem. Additionally, we derive exact formulas for the rainbow $k$-domination number of cycles, Möbius ladders and prisms. Lastly, we develop a dynamic-programming algorithm for generalized Petersen graphs of the form $P(6t, t)$ and state two conjectures based on the output of its implementation and related integer linear programming computations.

Large language models (LLMs) progressively support embedded hardware development, but a plausible recommendation may pose safety risks when acting on the physical hardware. Because electrical safety can be different between every microcontroller board, a practical question is raised: can LLMs have the capability to make the right decision on whether a proposed user operation is safe for the board? Although benchmarks for embedded development exist, they primarily target code generation and hardware design tasks. This highlights the lack of tested board-specific judgment regarding electrical safety. To address this gap, we introduce the ElecSafety benchmark to evaluate whether an LLM can respond with the correct label and derive that label from the proper manufacturer constraints and rules. We extract these constraints from vendor datasheets and hardware manuals. Each scenario is defined by a decisive condition and a gold label (safe, hazardous, or cannot determine) before translating the scenario into natural language. We categorize the scenarios into violation and compliant cases, board-swap, near-boundary, and missing-information. We then evaluated the scenario on six open-weight LLMs under various conditions of rule access and token limitations. The finding shows board rules can improve accuracy, whereas limiting the token has a smaller impact. Even the best configuration remains unreliable on near-boundary and missing-information scenarios, and a correct label often rests on the inaccurate constraints.

We develop a sparse graph method for transporting probability mass toward multimodal target distributions through damped nonlocal Hamiltonian dynamics. The formulation combines logarithmic-mean mobility with symmetric Lévy-type interaction weights, coupling the evolving density to an edge momentum field. A graph constructed from nearest-neighbor connections and sampled long-range edges provides direct mass exchange between spatially separated regions. Once the graph is constructed, the density evolution is deterministic, and each update costs linear in the number of nodes and the long-range sampling budget. Linearization around the target distribution yields a damped oscillator governed by a weighted graph Laplacian. Its spectrum characterizes the interaction between nonlocal connectivity and inertia, with the Lévy exponent alpha tuning the nonlocal connectivity: the spectral gap determines the optimal asymptotic damping, while the largest eigenvalue governs the time-step stability. Experiments on synthetic multimodal distributions demonstrate improved mode balance and more stable mode coverage relative to first-order and MCMC baselines. The resulting framework provides a sparse implementation of nonlocal inertial density transport for sampling problems with low-dimensional spatial structure.

Purpose: To evaluate the performance-feasibility tradeoffs of foundation models (FMs) and task-specific convolutional neural networks (CNNs) trained from scratch for breast implant classification in 2D mammography, with emphasis on suitability for near real-time clinical deployment. Methods: We evaluated four models: two FMs (RAD-DINO and MammoCLIP) and two CNNs trained from scratch for implant prediction (ResNet18 and our lightweight ResNetLite). Using the Emory Breast Imaging Dataset, 5,000 unilateral screening mammograms were used for training/validation and 1,000 manually reviewed unilateral images were held out for testing. For the FMs, global image embeddings from the pretrained encoder were classified using a support vector machine (SVM). The CNNs were trained end-to-end on 2D mammograms, with ResNetLite optimized via grid search over depth and width to balance accuracy and efficiency. Performance was evaluated using AUROC, sensitivity, specificity, accuracy, embedding visualization, and inference-latency. Results: All models demonstrated strong performance on held-out test data (n = 1,000). MammoCLIP achieved the highest AUROC (0.999) with the quickest training time of 493 seconds. RAD-DINO achieved the highest sensitivity (0.980; accuracy 0.989) but had the slowest inference and training times. ResNet18 and MammoCLIP achieved comparable accuracy (0.985). ResNetLite showed no statistically significant difference from ResNet18 (AUROC 0.993; accuracy 0.976) despite using only 1.4% of ResNet18's parameters, and had the fastest inference time. Conclusion: FMs and task-specific CNN models reliably detect breast implants on 2D mammography. Model selection is best guided by deployment context: MammoCLIP for GPU-equipped hospital settings requiring scalable integration, and lightweight CNNs such as ResNetLite for resource-constrained or edge deployments.

The Capacitated Vehicle Routing Problem (CVRP) remains a central concern in logistics research, as route optimisation under rigid capacity and fleet restrictions directly affects operational performance. Genetic Algorithms are widely used for this problem, yet their effectiveness depends heavily on the choice of solution encoding and on the mechanisms used to preserve feasibility during the search. This paper proposes a sequential framework that separates the evolutionary process from route construction. We adapt Node Shift Encoding (NSE) to the CVRP giant-tour setting and benchmark it against Path Representation and Double Chromosome. We further propose an unsupervised machine-learning repair heuristic that restores fleet feasibility after an unconstrained linear split, and compare this pipeline against fleet-constrained split taken from the literature. Computational tests on standard benchmark instances show that NSE consistently outperforms the other encodings in solution quality and feasibility. The repair heuristic restores feasibility in approximately 72% of cases, and the Linear-Split-ML-Heuristic pipeline proves more robust than the constrained split, delivering lower costs and higher feasibility rates.

Genome-wide association studies identify genomic variants associated with traits. Mixed linear model association (MLMA) methods using a whole-genome relationship matrix, such as those implemented in GCTA, are powerful but computationally expensive. Here, we remove key memory and CPU bottlenecks in GCTA, reducing REML memory usage by nearly 75% and substantially accelerating MLMA by orders of magnitude in biobank-scale cohorts while preserving exactness. We further exploit relatedness in the mapping cohort through a reduced-rank Woodbury matrix approach, delivering further orders of magnitude performance gains with controlled genomic inflation. Native on-the-fly dominance recoding also eliminates slow I/O-operations on intermediate files, enabling efficient additive and dominance MLMA analyses in large agricultural cohorts.

For some integer $\ell\geq 2$, consider $m=2\ell-1$ voters, each of which has a preference ranking over $n$ alternatives. The Smith set is the smallest nonempty set of alternatives each of which defeats every alternative outside the set in a pairwise majority comparison. Under the standard benchmark impartial culture where each voter uniformly and independently chooses a random preference ranking over all alternatives, we study the asymptotic probability that the Smith set has cardinality $s$ for fixed $\ell$ and $n\to \infty$. First, we prove that the probability that the Smith set has constant cardinality $s$ goes to zero at a rate of $\Theta_{\ell,s} (n^{-s(\ell-1)/\ell})$. Next, if $\min\{s,n-s\}\to\infty$, we show that this probability decays superpolynomially. Further, when $s$ and $n-s$ are both $\Theta(n)$, we prove that the probability decays exponentially and determine the exact exponential base. Finally, we prove that the probability that the Smith set contains all alternatives approaches one at a rate of $\Theta_\ell(n^{-(\ell-1)/\ell})$. These theorems resolve the conjectures in [Liu et al, Ann Stat 2026] under impartial culture.

We investigate whether the geometry of a curved conductor–dielectric interface can modify the effective mass and propagation of scalar surface plasmons (SSPs). In the hydrodynamic limit, we describe SSPs as density perturbations in a semiclassical jellium electron gas. Using a thin–layer dimensional reduction procedure, we derive a massive Klein–Gordon equation for SSPs on the curved interface with a geometric potential. We find that under specific conditions, the geometric potential allows SSP propagation with a vanishing effective mass. We then apply the formalism to a well known analogue black hole configuration, determining the corresponding quasinormal mode frequencies. Our results show that interface curvature provides a geometric handle on plasmon propagation and localization, suggesting new possibilities for plasmon transport and analogue gravity experiments with surface plasmons.

A time-series forecast is scored on a future value of the series. A representation loss that regresses the next latent, as in LeNEPA, is a different least-squares problem on the same bottleneck. We write both programs down. The forecast program minimizes the error of a decoded latent on the coordinate that will be reported. For a scalar target and a linear decoder, every latent rank of at least one matches ordinary least squares, and an isotropy constraint is only a rescaling: after the decoder is refit, the forecast does not move. The other program fits the whole next vector at a fixed rank, then freezes the encoder and attaches a head. On a four-dimensional series whose last three coordinates are the same autoregression, that rank-1 fit puts mass $0.9998$ on the repeated coordinate and forecasts the remaining signal at the marginal variance $2.794$. The forecast program puts mass $1$ on the signal and matches the innovation variance $0.992$. Rank $2$ gives the vector fit a second direction, and the two programs agree. Iterating the fitted one-step coefficient $0.803$ raises the open-loop error from $0.992$ at one step to $2.700$ at eight steps.

We study a divisor-weighted version of Golomb's self-description: the nondecreasing sequence $A$ in which each positive integer $m$ occurs $\sum_{d\mid m}A(d)$ times. This rule has a unique solution, and $$ A(n)\sim Cn^{\varphi-1},\quad C=\left(\frac{\varphi}{\zeta(\varphi)}\right)^{2-\varphi},\quad \varphi=\frac{1+\sqrt{5}}{2}. $$ The golden-ratio exponent of Golomb's sequence survives, while divisor weighting changes the leading constant. We prove the asymptotic without assuming regular variation, by a contraction of upper and lower power envelopes. The argument applies to every nonnegative integer Dirichlet-convolution kernel $w$ with $w(1)=1$ and $\sum_q w(q)q^{-\varphi}<\infty$. For the divisor kernel, a quantitative version gives relative error $O_\gamma((\log n)^{-\gamma})$ for every $0<\gamma<1$. Although the global growth is smooth, the normalized run lengths retain arithmetic fluctuations. A uniform comparison with the divisor sum $\sum_{d\mid m}d^{1-\varphi}$ transfers them to an explicit random Euler product. We prove that its law is singular continuous with full support $[1,\infty)$, determine all real Mellin moments, obtain the sharp maximal order of the fluctuations, and show that sampling sequence positions instead of block labels produces the size-biased law.

The new approach is develop to determine the hyperfine structure of the rotationally excited $1 s_{\mu} n \ell_{e}$-states in the three-body helium muonic ${}^{3}$He $\mu \; e$ and ${}^{4}$He $\mu \; e$ atoms. The formulas derived in this study allow one to evaluate the hyperfine structure splittings for an arbitrary rotationally excited (bound) $1 s_{\mu} n \ell_{e}$-state in the three-body helium muonic atoms. Similar formulas have never been produced in earlier studies and this explains why these hyperfine structure splittings are still unknown for these bound states.

Higher-order multivariable Hermite-Kampé de Fériet polynomials provide a unified construction of umbral representations for canonical catastrophe integrals. Expanding the unfolding exponential in these polynomials and integrating against the leading catastrophe kernel gives convergent moment expansions on suitable decay contours. A universal formal umbral operator then expresses the resulting fields through explicit moment ground states. We develop this construction for the fold, cusp and swallowtail, and relate the Euclidean Pearcey field to quartic scalar partition functions. The distinction between formal and analytic umbral frameworks becomes essential when integration is performed before umbral evaluation. For Euclidean Pearcey, the integrated linearized kernel is a rational function whose action is not specified by the original formal rules. The Analytic Umbral Framework supplies a convergent Mellin pairing for this kernel and recovers the independently obtained bivariate Pearcey representation. This establishes, in this example, an exchange of integration and analytic evaluation and identifies a specific extension beyond the formal framework.

A wiring diagram is an arrangement of $m$ pseudolines, every two of which cross exactly once. Reading its crossings from left to right encodes it as a reduced word of the permutation $w_0\in S_m$ that reverses $1,\dots,m$, unique up to commutations, and a braid move $\sigma_i\sigma_{i+1}\sigma_i\leftrightarrow\sigma_{i+1}\sigma_i\sigma_{i+1}$ flips one of its triangles. We prove that deciding whether two wiring diagrams are within a given braid-move distance is NP-complete. Equivalently, deciding flip distance is NP-complete in the higher Bruhat order $B(m,2)$, for triangle flips of marked arrangements of pseudolines, and for rhombic tilings of a $2m$-gon; this answers an open question on flip graphs of pseudoline arrangements (SODA 2024). The reduction is from Vertex Cover: for every graph $G$ we construct two wiring diagrams at braid-move distance $|D|+2\operatorname{VC}(G)$, where $\operatorname{VC}(G)$ is the vertex cover number of $G$ and the Hamming distance $|D|$ is the number of triples of wires whose triangles are oriented differently. The construction glues copies of a seven-wire gadget whose braid-move distance exceeds its Hamming distance by $2$. Adding a line at infinity transfers the construction to rank-$3$ uniform oriented matroids, where two oriented matroids are adjacent in the mutation graph if their chirotopes differ in one basis; so deciding distance in the mutation graph is NP-complete in rank $3$. An iterate of Rambau's expansion of signotopes amplifies the excess and gives NP-hardness in every fixed rank $r\ge4$, and by duality in every fixed corank at least $3$. The same instances show that flip distance in the higher Bruhat orders $B(n,k)$ is NP-complete for every fixed $k\ge2$.

Lattice constellations can provide significant coding and shaping gains over traditional quadrature amplitude modulations (QAMs), but have not been widely adopted due to the high implementation complexity of modulation, demodulation, and decoding. Voronoi constellations (VCs) enable an efficient modulation map that achieves shaping gain using the Voronoi cell of a lattice. Multilevel coset codes (MLCCs) are a recently proposed coded modulation scheme for VCs that realizes more efficient decoding. However, MLCCs still suffer from a soft demodulation complexity of $O(d2^{sd})$ for a $d$-dimensional lattice with spectral efficiency $s$ bits per real dimension. In this paper, we propose fast quantize-walk demodulation (FQWD), an efficient demodulation scheme for MLCCs that reduces the complexity to $O(sd2^d)$. Our strategy utilizes novel radial projections onto the Voronoi cells of the $D_4$ and $E_8$ lattices. Experimental results demonstrate that FQWD has only negligible loss in block error rate (BLER) relative to maximum likelihood demodulation under the additive white Gaussian noise (AWGN) channel. Using FQWD, we simulate MLCCs on higher-dimensional lattices with $s>2$ bits per dimension under AGWN for the first time, achieving BLER gains over multilevel polar codes on QAMs of up to 0.6 dB for the $D_4$ lattice and 1.2 dB for $E_8$.