Papers

Discrete-event simulation is a standard technique for modelling and analysing the performance of computer systems, networks, and services. Although simulation tools are widely used, reasoning about the correctness and performance guarantees of the models they implement remains largely ad hoc: simulation outputs are interpreted statistically, but there is no logical foundation for deductive reasoning about their behaviour. We present a core imperative calculus that captures the essential constructs common to discrete-event simulators: asynchronous execution, continuous and discrete sampling from distributions, and time-based event scheduling through a global event queue. On top of this calculus, we develop a proof system for reasoning about almost-sure reachability and expected reaching time properties. Our main result is a sound and complete proof rule for these properties. Our framework generalizes deductive reasoning for discrete-time probabilistic programs to the setting of performance models, in which continuous time and continuous probability distributions are central. We have implemented the proof rules in a tool embedded in Lean. We demonstrate the applicability of our proof rule by deriving proofs of almost-sure reachability and expected reaching time for a number of case studies, including client-server examples that go beyond analytic solutions from queueing theory as well as convergence behaviours in network routing protocols. Establishing the soundness and completeness of our proof rules requires significantly more complex arguments than in the discrete-time setting. This is due to the fundamentally discontinuous nature of the operational semantics and the measure-theoretic challenges of continuous time and probability distributions.

We begin a systematic study of neutrinoless muon-to-positron conversion, $\mu^-+(A,Z)\to e^++(A,Z-2)$, which violates lepton number by two units and changes charged-lepton flavor. This first paper in a series constructs nuclear operators for local dimension-nine interactions containing a scalar or pseudoscalar lepton bilinear and two contracted vector or axial quark currents. The covariant nucleon currents are built with all six vector and axial form factors. For the working nuclear operator we neglect second-class currents and adopt a pion-pole-improved reduction: non-pole recoil terms generated by the displayed one-body currents are retained through second order in inverse nucleon mass, whereas the pion-pole longitudinal axial current is kept intact before the two currents are contracted. We preserve the two independent momentum transfers at the nucleon vertices and the Coulomb-shifted spatial phase of the outgoing positron. Starting from the spacetime transition element, we show explicitly how the interaction-coordinate integral fixes the sum, but not the sharing, of the two current momenta. Unequal form-factor arguments then generate relative-coordinate multipoles absent in the usually used common-momentum approximation. We give an exact recoupling into two-body nuclear tensors and specialize it to the phenomenologically important $0^+\to0^+$ transition. In this case the external nuclear rank is zero and all magnetic sums collapse, although finite-momentum internal orbital and spin tensors remain. The result is expressed in a form directly contractible with charge-changing two-body transition densities. General six-form-factor currents, arbitrary nuclear angular momenta, and power-counting qualifications are collected in appendices.

Although significant research has focused on the theory and implementation of data-based coordination languages, the critical aspect of their automated verification using model-checking techniques remains underexplored, which is essential for ensuring reliability and correctness in distributed systems. Existing tools, such as Anemone, provide a solid foundation for reachability-based verification of Bach programs. While they effectively analyze properties expressed in terms of state attainability, extending support to more expressive temporal specifications-such as liveness properties or invariants over shared space contents-remains an open opportunity. Such extensions are important to capture comprehensive system behaviors, for instance, ensuring that "a request is always matched by a response" or that "no message is silently lost". Addressing this limitation, we propose an automated translation from Bach to mCRL2, which explicitly represents the shared space, thereby enabling the use of mCRL2's mu-calculus model checker to verify complex properties beyond simple reachability. Complementing this translation, we introduce a systematic method to analyze the shared space in mCRL2 concerning data reachability, by directly expressing properties over the shared space's contents in the mu-calculus, thus providing a clearer framework for verification. This approach enables a novel verification process that combines action-based properties with state-based properties over the shared space contents, a largely unexplored area in current coordination-language verification approaches, offering a new dimension of analysis

We introduce Context Language Models (CLMs), language models that natively manage their own context. We implement this by treating the context as a file and allowing the model to make unrestricted updates to this file. This allows the model to learn what is most important to maintain in context, and naturally extends to multi-agent systems where multiple agent contexts coexist as files. Building CLMs zero-shot with existing models outperforms SOTA context management strategies across a variety of tasks: 11.4% higher accuracy with 21.5% fewer FLOPs on BrowseComp-Plus, 5% higher scores with 59% fewer FLOPs on 12-hour EdgeBench, and 65% greater improvement with the same compute on a 24-hour multi-repository agent-swarm task. Moreover, by shifting context management from external harness control to intrinsic model behavior, CLMs naturally enable both in-context and parametric learning of context-management strategies. We show that CLMs can be steered with natural-language instructions evolved through a standard skill-optimization loop, improving held-out accuracy by up to 35.9 points on a context-management task while reducing compute. We also introduce an online reinforcement learning method for CLMs, improving Qwen3.5-9B performance on BrowseComp-Plus by 47.6% while using 12% fewer FLOPs. Finally, we co-design Suffix Cache Reuse for CLM serving, further reducing server-side compute by 35% relative to standard SGLang at matched performance.

Scattering theory is a key tool for the investigation of quantum systems. Universal approaches are of interest, particularly in the framework of Random Matrix Theory. Dyson's Threefold way classifies systems according to their behavior under time reversal. In these three cases the distribution of off-diagonal scattering matrix elements and cross sections were recently calculated. However, in many applications systems neither display full nor fully broken time reversal invariance, such as systems in tuneable external magnetic fields. The study of these systems has a long history, also in Random Matrix Theory. We investigate the influence of time reversal invariance breaking on the universal behavior of the mentioned distributions. Using supersymmetry we succeed in deriving exact expressions for the distribution of off-diagonal scattering matrix elements and cross sections in systems with half-integer or integer/no spin in the absence of further symmetries. We find that the distributions are symmetric while the scattering matrix elements are not. Furthermore, our results might considerably facilitate the data analysis of time reversal invariance breaking.

Constructing hadron creation operators which properly sample the finite-volume energy spectrum is a fundamental step in lattice QCD spectroscopy and scattering calculations. Operators enabling access to spectrum information from correlation functions at early temporal separations are extremely advantageous, particularly when the signal-to-noise problem is severe. We present improved operator constructions for charmonium, glueball and baryon spectroscopy which exhibit these advantages. Novelties of our work include the extension of the distillation profiles framework to baryon and multi-meson calculations, as well as an efficient implementation of glueball-like operators which retain angular momentum information from the continuum.

Transmission spectroscopy of rocky exoplanets orbiting late M dwarfs is limited by stellar contamination rather than photon noise. We present a framework for this contamination-dominated regime, fitting all transit epochs jointly with a Gaussian process per epoch for the time-varying stellar contamination and a single atmospheric model shared across epochs. We use pure CO$_2$, H$_2$O, and CH$_4$ models scaled by their peak-to-trough amplitude, allowing us to measure how large a signal must be to be recovered. We calibrate this framework using injection-recovery tests and 17 archival JWST/PRISM transits of the airless sibling TRAPPIST-1 b to quantify the spurious signals that stellar contamination alone produces. Applied to four JWST/NIRSpec PRISM transits of TRAPPIST-1 c, our framework recovers injected CO$_2$ signals without bias and reaches detection thresholds of 146 (171) ppm for moderate (strong) evidence, compared with 96 ppm if there were no stellar contamination. CO$_2$ and CH$_4$ produce no spurious signals on the airless control planet, whereas H$_2$O shows a persistent stellar floor of $+55 \pm 34$ ppm. TRAPPIST-1 c is consistent with a featureless, airless baseline for all three molecules. Our 95 per cent upper limit of 90 ppm for CO$_2$-shaped signals effectively rules out clear, isothermal pure-CO$_2$ atmospheres ($\geq$0.1 bar) and N$_2$-dominated atmospheres with 1 per cent CO$_2$ ($\geq$1 bar) at the equilibrium temperature. An empirical scaling study using 2 to 10 transits of TRAPPIST-1 b shows the CO$_2$ detection threshold decreasing with the number of transits $k$ as $k^{-0.6}$, reaching 80 ppm at ten transits, while water searches become limited by the stellar floor beyond about six transits. CO$_2$ atmospheres, by contrast, are detectable when we utilize our framework to account for stellar activity at the cost of roughly 1.5 to 2 times as many transits.

Robot policies increasingly incorporate semantic reasoning and future-world prediction, yet combining these capabilities does not guarantee that local predictions and actions remain aligned with task progress. We introduce CogWAM, a cognition-guided world-action model that establishes an explicit semantic interface between task reasoning and world-action learning through a persistent Semantic State, which stores completed task events and the active subtask. CogWAM updates this state only when observations indicate semantic transitions, allowing task-level context to persist across multiple action chunks. To bridge semantic context with physical prediction and control, CogWAM employs progress-conditioned WORLD and ACTION queries that selectively extract task-relevant information for future-world prediction and action generation. During training, the Semantic State provides shared task-progress context for both branches, while inference removes the future-prediction branch and directly generates actions from observations and the maintained state. We further introduce semantic training strategies to improve transition learning and closed-loop conditioning. Without additional robot-action pretraining, CogWAM achieves 15.56 / 11.70 % Score/SR on RoboDojo and state-of-the-art performance on BiCoord, while real-world experiments demonstrate closed-loop dual-arm manipulation with 16.4 fewer Semantic State regenerations than step-wise updating.

Cation distribution plays a critical role in determining the properties of crystalline oxides. Understanding the chemical factors that govern cation distribution is therefore essential for the rational design of cation-ordered materials, particularly frustrated magnets in which structural disorder can strongly affect the magnetic ground state. Here, we investigate the origin and evolution of cation distribution in the extensively studied M-type hexaferrite structure using the first Mn-based M-type hexaferrite, KSb3Mn9O19, as a model system. A systematic Cr3+-doped series, KSb3(Mn1-xCrx)9O19, was examined using powder and single-crystal X-ray diffraction, electron microscopy, and energy-dispersive X-ray spectroscopy. Cr3+ exhibits a pronounced preference for the Mn Kagome sublattice, accompanied by the evolution of Mn vacancies and Mn3+/Sb3+ disorder on neighboring sites. Analysis of the local coordination environments and their evolution with Cr content suggests that crystal-field effects, chemical bonding, and local structural strain collectively govern this site selectivity. The magnetic properties of the Cr-doped compounds show similar behaviors as the undoped parent compound, other than the doping-induced spin-glass state at the highest Cr concentration, supported by heat capacity measurements. These results establish a chemical picture of cation site distribution and selection in M-type hexaferrites and demonstrate how substitution at one crystallographic site can induce coupled redistribution and disorder across neighboring sublattices. Moreover, this work establishes local coordination chemistry as a route toward understanding and ultimately controlling cation distribution in complex oxides, providing chemical design principles for structurally well-defined frustrated magnetic materials.

High velocity absorption lines of Ca II and Si II (high-velocity features, HVFs) are common in the early spectra of Type Ia supernovae (SNe Ia), but their physical origin is still poorly understood. We used 181 optical spectra of 56 SNe Ia that show HVFs in at least the Ca II NIR triplet to better understand the nature of HVFs. To model the profiles of HVFs we applied Gaussian fitting as well as spectrum synthesis modeling with the SYNOW code. We found that Gaussian fitting usually underestimates the velocity of HVFs compared to SYNOW, and derived a linear formula for correction. In accord with many previous studies, the observed properties of HVFs (velocity, strength, duration) were found to correlate with the general observables of SNe Ia (decline rate, spectral subtype). Brighter, slow-declining SNe usually exhibit stronger, higher velocity HVFs, while fainter objects typically show weak, low velocity HVFs. SNe Ia in early-type galaxies showed weaker and slower HVFs compared to events in late-type galaxies in our sample. Comparing these observational findings with theoretical predictions, we conclude that the overall physical picture of HVFs is quite complex, and could be explained by multiple models. A double detonation explosion model is promising as it is inherently capable of explaining the creation of high-velocity shells that may produce HVFs, but an asymmetric delayed detonation model or interaction with circumstellar material cannot be ruled out.

I argue for significant stochastic behavior in the common envelope evolution (CEE) of cool giants and main-sequence companions to the degree that it is almost impossible to know the exact CEE outcome of an individual binary system from the average properties of the stars and their initial orbit. I start with two observational properties of post-CEE planetary nebulae: many have jet-shaped morphologies, and they show a wide variety of morphologies. The first suggests that the standard CEE model should include jets, and the second shows that the CEE has a wide variety of evolutionary routes; for example, jets can be launched before, during, and/or after the main CEE phase. I show that the properties of the disk with which the main-sequence companion enters the CEE depend sensitively and stochastically on the pulsations and convection of the giant star. Pulsations and giant-envelope convection stochastically determine the evolution of the accretion disk inside the envelope. The stochastic nature of the accretion disk properties implies that jet power is also stochastic. Adopting the view that jets play a major role in CEE outcomes, I conclude that the CEE outcome is stochastic. I conclude that the CEE efficiency parameter alpha(CE) has little merit for individual systems, although it still has meaning in population synthesis and other studies of large populations.

Decoder-only transformers are trained only through a terminal next-token prediction loss, yet this loss constrains every intermediate hidden state through the fixed downstream computation. We formalize this constraint by studying layerwise loss-to-go functions: the terminal loss obtained by continuing a candidate hidden state through the remaining transformer blocks. Around successful validation trajectories, we show that the local second-order geometry of these functions is governed, up to low-loss residual terms, by a pullback Fisher operator on hidden-state space. Its spectrum identifies output-sensitive directions and approximately prediction-null directions, yielding a local observable subspace of the residual stream. For causal transformers, the same geometry induces a tokenwise curvature score: a Fisher-weighted sensitivity of the target logits to perturbations of each token's hidden state. This score vanishes outside the causal ancestor set of the target and is controlled by downstream Jacobian couplings, making it a loss-aware alternative to attention magnitude. We estimate these quantities using matrix-free Jacobian-vector and vector-Jacobian products and evaluate them across decoder-only language models on WikiText, OpenWebText, and FineWeb. Empirically, the induced geometry predicts perturbation sensitivity, supports nonuniform layerwise rank allocation, yields competitive structured token-pruning signals, and improves low-rank student recovery when added to stronger autoregressive distillation objectives such as reverse KL and skew KL. These results support a predictive-geometric view of transformer computation: near successful trajectories, the terminal loss induces a thin, anisotropic set of output-relevant hidden-state directions that can be measured and exploited for compression and distillation.

The relativized quark model of Godfrey and Isgur (1985) remains a reference for meson spectroscopy four decades after its publication, yet despite its impact the calculation behind it has never been publicly available. I present an independent reproduction of the whole paper, released as the open-source Julia package GIModel.jl with companion packages for transition observables and for the comparison with the original. The published parameters are used without refitting, and the paper's three-stage harmonic-oscillator algorithm is cross-checked with an independent finite-difference solver. All 209 levels of the paper's spectrum figures are reproduced within the precision of the printed diagrams. The strong-decay, radiative, annihilation, and electromagnetic tables are reproduced with median model-to-paper ratios between 0.97 and 1.06, and the $1P$ mixing angles agree with the values the authors revised after 1985. With the calculation open, the model's own wavefunctions become available for decay amplitudes and structure visualization. The package, its teaching material, and its agent-accessible interface make the model a baseline for experimental questions and theoretical extensions.

Time-series foundation models are increasingly adapted to new domains through fine-tuning on target data, under the implicit assumption that more target data yields better forecasts. We show that this assumption can fail in financial forecasting, where individual price changes are difficult to predict, but large moves tend to cluster, creating alternating calm and turbulent periods. Using financial foundation models trained on price bars of open, high, low, close, and volume, we argue that adapting to financial domains requires training signals beyond next-token prediction. We introduce Volatility-Clustering Adaptation (VCA), which augments next-token cross-entropy with a differentiable penalty on the autocorrelation of squared returns, the standard statistical signature of volatility clustering. This additional objective provides a multi-step training signal by matching the resulting dependence structure of autoregressive rollouts to those of the realized future. Across three asset sets and two evaluation conventions, VCA improves adaptation over the pre-trained model, with the strongest gains under the primary evaluation (fore), driven primarily by reduced variance error. Overall, our results suggest that effective financial adaptation requires objectives that capture domain-specific temporal structure beyond token-level prediction.

We intend to prove that for defocusing nonlinear Schrödinger equations, solutions with finite pseudo conformal energy must be global and scatter. This fact is first proved by Bourgain when $0<s_c<1$: the equation is locally well-posed in $H_x^{s_c}$, and if one further assumes that the initial data satisfies $xu_0\in L_x^2$, then the solution is global and scatters. The $s_c<0$ case remains not studied, and the best result is given by Beceanu, Deng, Soffer, and Wu: If the initial data is in $H_x^{s_c}$, radial, and compactly supported, then the solution is globally well-posed. In this paper, we extend Bourgain's result to the $s_c<0$ case. We prove that if the initial data satisfies $|x|^{-s_c}u_0\in L_x^2$ and $xu_0\in L_x^2$, or if the radial initial data satisfies $u_0\in \dot{H}_x^{s_c}$ and $xu_0\in L_x^2$, then the solution is global and scatters. Bourgain's result is based on the subcritical a priori control on $L_x^{p+2}$, while our argument relies on the supercritical bound on $\mathcal{F}\dot H_x^1$.

Existing product matching benchmarks primarily contain English-language product data and are often dominated by a single product category, such as electronics. This paper introduces Billiger.de Products, a bilingual German and English entity matching benchmark covering thirteen consumer product categories, including difficult-to-handle categories such as clothing and furniture. The benchmark data originates from the German price comparison platform billiger.de. Following the design of WDC Products, the benchmark offers multiple variants that differ in the fraction of corner cases, the size of the development set, and the fraction of entities unseen during training. An aligned English translation of every offer keeps all pairs, splits, and labels fixed, while cross-language test sets combine German and English records within individual pairs. We validate the benchmark using six supervised matchers and zero-shot GPT-5.2 on both language versions and the cross-language test sets. The validation shows the difficulty of the benchmark. The comparison of the results on the English version of the benchmark to the results on the German version shows that most matchers score on average higher on the English version. The difference is largest for RoBERTa and HierGAT, while the zero-shot LLM runs are largely insensitive to the language. Comparing the F1 scores achieved by PLM-based matchers on the English version of Billiger.de Products with their performance on existing English-language benchmarks, such as WDC Products and Abt-Buy, shows that Billiger.de Products is more difficult than these benchmarks.

Optical Character Recognition (OCR) is evolving from plain-text transcription toward general visual intelligence, requiring models to recognize, localize, and reason over textual information in complex visual environments. However, existing OCR systems often excel at only some tasks and struggle to balance recognition, parsing, and reasoning across scenarios. In this report, we present PolyOCR, a family of unified OCR foundation models of varying scales. PolyOCR combines a shared instruction-following framework with a large-scale data engine that converts heterogeneous visual resources into quality-verified OCR supervision. We introduce Competence-Guided Policy Optimization, which combines verifier-based Group Relative Policy Optimization with on-policy distillation through sample-wise routing based on teacher reliability and the teacher–student competence gap. We also introduce OCRBench v2.1, our revision of OCRBench v2 with manually verified annotation corrections and task-aligned scoring metrics. Extensive experiments across OCRBench v2.1, CC-OCR, in-house KIE Benchmark, OmniDocBench v1.6 and MDPBench demonstrate that PolyOCR achieves state-of-the-art or highly competitive performance.

In this work we look to neuromorphic computing to solve the power consumption problem that audio denoising neural networks face on edge devices like smartphones, wireless headphones and hearing aids. Spiking neural networks (SNNs) have the potential to solve this problem due to their high activation sparsity and low complexity, however many SOTA SNNs require hardware that supports a mixture of operations to be able to fully perform inference. To solve this problem, we convert SOTA audio denoising neural network Spiking-FullSubNet to a hardware friendly version showing that via QAT and activation function simplification we can achieve $\approx28\times$ improvement in power consumption to 52.9nJ per 32ms audio frame when calculated for custom digital hardware in a 45nm process node. We then propose a digital circuit which by means of a sparsity-aware flexible PE array can perform inference of the heterogeneous compute load of Spiking-FullSubNet, and validate this circuit on a PYNQ-Z1 FPGA achieving a real-time factor of 0.727 at 100MHz.

Current music generative models can produce high-quality music, but does this ability imply that they “understand” the musical qualities of their outputs, and is that understanding aligned with human evaluation? Previous attempts to use the likelihood of a generative model to evaluate music, an approach commonly used in text, have proven unsuccessful, leading researchers to rely on standalone supervised music evaluation models. In this paper, we answer this question affirmatively: we show that a model's intrinsic signals—derived from its hidden representations and predictions—are strongly correlated with human ratings. In particular, we study MusicGen and consider three types of features: (1) prediction loss, (2) prediction entropy, and (3) concepts extracted from the model using a sparse autoencoder (SAE). Using these features, we train a lightweight prediction model to estimate subjective ratings. We evaluate these features both individually and in combination. We hypothesize that these signals parallel the listening process: the temporal and frequency-domain structure of loss and entropy reflects listeners' expectation and surprise, while gradient directions in SAE latent space predict perceived quality. Experiments on five human-evaluation benchmarks spanning continuous ratings and pairwise preferences confirm this hypothesis, with SAE latents carrying most of the predictive signal.

Recent multimodal image generation models can take multiple images and textual instructions as input, enabling reference-based generation guided not only by text but also by visual instructions such as layouts, arrows, and pose cues. However, existing benchmarks do not evaluate the joint setting in which multiple references must be composed under multiple and heterogeneous visual-instruction images. To address this gap, we introduce VIF-Bench, a benchmark of 1,241 tasks designed to assess the edge of model capabilities in this joint setting by covering: (i) multi-reference generation (up to 7) under multiple heterogeneous visual instructions (up to 6), (ii) cases where reference images can potentially compete with visual instructions (e.g., a strongly posed subject vs. a target pose), and (iii) controlled comparison of visual instructions with text descriptions at different levels of specificity. Using these capabilities, we uncover three findings: (1) models face an adherence-artifact trade-off: once models reach stronger visual instruction adherence, stronger adherence tends to coincide with more instruction artifacts in generated images, (2) visual instruction adherence tends to be lower on tasks whose reference images carry a salient state of the controlled attribute (e.g., a neon-lit subject under a light-direction instruction), most consistently for light and wind, and (3) for models that can understand visual instructions, it is often better to provide visual constraints directly rather than describe them in text; when using text, a moderate level of detail works better than an exhaustive description. VIF-Bench is released as an open benchmark to establish a basis for fair comparison in controllable multi-reference image generation.

Human social behaviour is not a collection of independent motions, but a jointly organised process in which group dynamics and individual variation continuously shape one another. Yet existing social motion models often prioritise plausible trajectories while leaving interaction state implicit, limiting their ability to transfer across groups, tasks, and partial-observation regimes. To address this gap, we introduce Bilevel Representations for Agent Interaction Dynamics (BRAID), a hierarchical sequential latent-variable model for generative multi-person interaction. BRAID explicitly formulates social motion generation as a meta-transfer learning problem: shared interaction priors are learned across datasets and adapted through arbitrary context sets of observed people and joints. The model represents each scene through a group-level latent state that captures shared interaction dynamics and person-level latent states that capture individual behaviour conditioned on the evolving group context. This modelling choice enables coherent generation under full, sparse, or partial observations while exposing compact social-state vectors that can serve as an interface for downstream embodied-agent systems. We evaluate BRAID under a unified SMPL-based representation on social forecasting, tracking and in-filling, and response generation, using metrics that assess not only reconstruction accuracy but also realism, diversity, temporal alignment, and interpersonal coordination. We further analyse the hierarchical latent space, showing that it captures separable group- and individual-level structure.

Flat-band electronic states can be highly sensitive to chemical perturbations, offering opportunities to access a broader range of electronic behaviors beyond that of parent materials. Here, we report the discovery of a new series of compounds, Mo4TGa16Ge (T = Co, Rh, or Ir), derived from the strongly correlated flat-band superconductor, Mo4PtGa17, via nominally preserving the total valence electron counts. All three materials crystallize in a noncentrosymmetric cubic space group, F-43m, with Ge selectively occupying one of the Ga sites in the parent compound. Although the total electron count remains the same, Mo4TGa16Ge exhibits significantly distinct properties from Mo4PtGa17. Theoretical calculations predict metallic electronic structures with narrow, dominant Mo-d states at the Fermi energy, while low-temperature heat capacity results demonstrate finite Sommerfeld coefficients. However, electrical transport measurements show predominantly non-metallic behaviors with small observed activation energies, excluding the possibilities of conventional semiconducting behaviors. Combining chemical bonding analysis, electronic structure and compositional determination from X-ray crystallography and spectroscopy, we propose that the coexistence of the non-metallic transport features and the thermodynamically metallic behaviors can be attributed to the high sensitivity of the narrow-band materials to chemical disorders. Thus, the discovery and investigation of Mo4TGa16Ge provide a chemically tunable platform for studying disorder-controlled transport in flat-band intermetallics.

Let $K$ be an algebraically closed field of odd characteristic $p$, let $G= {\rm SL}_2(K)$, and let $V$ be the natural representation of $G$. Let $b_k$ denote the number of $G$-indecomposable factors of $V^{\otimes k}$, counted with multiplicity, and let $\delta_p=1-\log_{p^2}\!\bigl(\tfrac{p+1}{2}\bigr)$. Then there exists a smooth, strictly positive, multiplicatively $p^2$-periodic function $\omega(t)$ such that $b_k$ is asymptotic to $\omega(k)k^{-\delta_p}2^k$. We also show that $t^{-\delta}\omega(t)$ arises as the limiting density of renormalized convolutions of rescaled copies of a positive weight $3/2$ theta function, obtained from the boundary heat flux of the Dirichlet heat kernel on $(0,p)$.

Goodness-of-fit testing is a basic tool for assessing whether a fitted procedure has captured the systematic information contained in the covariates. While traditional theory has largely focused on parametric regression models, modern data analysis increasingly relies on flexible black-box learners, whose predictive success alone is insufficient to assess model accuracy. In this paper, we propose SPARK, a general framework for goodness-of-fit testing that applies to traditional statistical models and general black-box learning procedures, continuous and binary responses, and low- and high-dimensional predictors. Based on a debiasing strategy, the residuals from an initial fit of a learning procedure are projected onto nearly orthogonal directions to extract any remaining signal. To capture information across all projection directions, we propose a kernel-based projection method and establish both its asymptotic properties and the consistency of a bootstrap procedure. Comprehensive simulations and real data analyses illustrate the effectiveness and flexibility of our proposed method.

At the critical density, the classical density theory of the Bargmann–Fock space no longer determines exact zero sets. We show that this failure is substantial: the complete radial counting function together with any prescribed finite collection of angular moments does not determine exact realizability, and the finite-dimensional relation between the dimension of the vanishing space and the zero-insertion capacity breaks down in infinite dimension. For arbitrary locally finite multisets, we establish a variational characterization of exact zero sets in terms of a renormalized logarithmic potential and relative entropy. This characterization yields sharp criteria for several classes of critical configurations.

Phenotypic plasticity, the ability of cells to switch between states, is central to development, differentiation, and therapy resistance. Although it is modeled at several scales, from compartmental ODEs to phenotype-structured PDEs and single-cell stochastic equations, a framework connecting these descriptions is missing. We present such a framework. Starting from a general $n$-compartment ODE model encompassing nonlinear growth and linear transitions between phenotypes, we show, with a new, elementary and generalizable proof of a recent theorem, that under uniform competition, the long-term population distribution is solely governed by transition rates. All phenotypes become selectively neutral at saturation, and the imprint of fitness differences during growth fades at an explicit rate. Restricting transitions to neighboring states transforms the model into a discretization of a phenotype-structured reaction-diffusion-advection PDE. In this continuum model, diffusion and advection are identified from switching rates, and fitness remains asymptotically irrelevant. Interpreting the advection velocity as the negative gradient of an effective epigenetic potential transforms the PDE into a Fokker–Planck equation and converts single-cell trajectories into stochastic gradient descent (SGD) in the Langevin sense on the phenotypic landscape. Non-local transitions, such as mutations, are incorporated via an integro-differential term, yielding a unified model with reaction, gradient flow, diffusion, and jumps. State-dependent noise reshapes the effective landscape without altering the underlying potential. This gives cancer a route to elevated plasticity that static, single-cell snapshots cannot distinguish from a changed landscape. This framework provides a physical interpretation of Waddington's landscape, where cells perform SGD, and cancer corresponds to a corrupted landscape.

Class Incremental Learning (Class-IL) requires models to learn new classes over time while preserving previously acquired knowledge without access to past data or task identity. This setting intensifies the stability-plasticity dilemma and makes catastrophic forgetting a central challenge. Existing approaches include regularization, knowledge distillation, replay, and architectural expansion. However, many expansion methods rely on explicit task identifiers or predefined growth strategies, limiting their applicability when task boundaries are unavailable at inference time. This work proposes a dynamic width expansion method that increases the number of neurons within existing layers according to a normalized loss criterion, without requiring task-specific information. An attention mechanism with persistent key-value memory is also incorporated to stabilize feature representations and reduce interference between previously learned and newly introduced classes. The approach is evaluated on Split MNIST and Split CIFAR-100 under the standard Class-IL protocol. Experiments compare fixed-capacity and dynamically expanding architectures, both with and without attention, combined with established continual learning methods including EWC, LwF, and A-GEM. Results show that progressive width expansion consistently improves performance over fixed architectures, particularly when combined with functional methods and A-GEM. The combination of width expansion and attention provides the most consistent gains. Overall, dynamic width expansion based on representational demand provides an effective and flexible strategy for Class-IL, although uncontrolled growth may increase overfitting and computational cost.

This work constructs explicit closed-form Green's functions for the Helmholtz equation with Robin boundary conditions on balls in all dimensions $d\ge 2$. Despite the canonical geometry, such kernels have remained unavailable because the Robin condition couples the field and its normal derivative, preventing the method of images and obstructing standard eigenfunction expansions. By a decomposition–expansion method, the free-space fundamental solution is expanded via Graf's addition theorem in two dimensions and the hyperspherical addition theorem in higher dimensions; the regular correction is then determined algebraically by matching Robin boundary data mode by mode. The resulting series converge absolutely on compact interior subsets and are computable to machine precision. These explicit kernels yield a complete characterisation of the resonance spectra: each spectral branch increases strictly with the impedance parameter, interpolating between Neumann and Dirichlet eigenvalues. We establish a low-frequency spectral gap with an explicit cutoff estimate, and a universal high-frequency mode spacing with a second-order correction that distinguishes the Robin condition from the Dirichlet and Neumann extremes. The closed-form expressions furnish exact benchmark solutions for wave simulations in impedance-matched cavities, eliminating geometric discretisation error and providing reference data for the validation of finite-element and boundary-element algorithms in acoustics, electromagnetism and quantum mechanics.

Language models expose internal signals that predict whether an answer is correct, readable from a single forward pass of a frozen model without additional generations. Yet existing probes often commit to one signal family or layer and can be brittle under distribution shift; in retrieval-augmented settings, many specialized detectors instead target passage faithfulness, which can diverge from correctness when retrieved evidence is unhelpful or conflicting. We therefore ask where answer correctness is readable, which internal signal families carry it, and how they should be combined. We search over hidden states, token probabilities, residual-stream features, attention, and their fusion, treating the selected readouts as a predictive measurement rather than a mechanistic localization. We run this analysis separately in closed-book and with-context settings, since context can change which readouts are informative. A consistent anatomy emerges: correctness concentrates in the answer span, recovered from the answer tokens even under retrieval, and the families carry it complementarily, so fusing them helps most out of distribution, where a single signal is weakest. The protocol is effective across two backbones and gates a retrieval controller as one downstream use.

Low-space-overhead fault-tolerant quantum computation requires not only high-rate quantum error-correcting codes but also space-efficient implementations of logical operations. Transversal and fold-transversal gates are promising since they limit error propagation and require no additional qubits. However, the logical operations they enable are typically restricted, and a central challenge is to construct codes that combine a complete set of logical Clifford gates with favorable code parameters. In this work, we introduce a family of quantum codes with an asymptotically constant encoding rate and sublogarithmically growing stabilizer weight, while supporting the entire logical Clifford group using only transversal and fold-transversal gates. Our construction is based on classical codes whose code spaces are absolutely irreducible Steinberg modules of their Tanner-graph automorphism groups. Taking hypergraph products of these classical codes yields quantum codes for which all logical Clifford operations can be synthesized from a fixed set of transversal and fold-transversal gates. Moreover, the slow growth of the stabilizer weight enables a high error-correcting performance in small instances. These results provide a path toward fault-tolerant quantum computation with low space overhead using transversal and fold-transversal gates.