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arXiv2026

Universality in the Transition from Inspiral to Plunge for Extreme-Mass-Ratio-Inspirals: High-Accuracy Analytic Solutions and Catastrophe Theory

We revisit the transition from inspiral to plunge for extreme mass-ratio inspirals on quasi-circular, inclined orbits in Kerr spacetime from the perspective of catastrophe theory. Our goal is to uncover the mathematical structures underlying the universality of the transition dynamics, which remains governed by the same Painlevé I differential equation as for equatorial inspirals despite the additional complexity. We first analyze the solution of the Painlevé I equation selected by the physical boundary conditions of slowly evolving quasi-circular inspiral at early times. We argue that these conditions uniquely select the tritronquée solution of Painlevé I. We then compare existing high-accuracy analytic approximations of the tritronquée solution with direct numerical integrations of the Painlevé I equation, finding comparable accuracy and improved stability under differentiation and integration for the analytic solution. In the second part of this work, we show that the equilibrium structure of the Kerr radial effective potential admits a natural interpretation in terms of catastrophe theory. Equatorial orbits are associated with the fold catastrophe, while inclined orbits are described by the cusp catastrophe. In both cases, the transition to plunge corresponds to slow evolution across fold lines of the catastrophe manifold, providing a geometric explanation for the universal appearance of the Painlevé I equation in the transition dynamics.

gr-qc↗

arXiv2026

Short Spike, Long Story: Episode-Dependent Shifts of Long-Duration Type-I GRBs on $E_{\rm p,z}$--$E_{\rm iso}$ Plane

Observations of peculiar GRBs have challenged the traditional $T_{90}$-based classification, demonstrating that duration does not map uniquely onto progenitor type. A striking class is long-duration Type~I GRBs -- merger-origin events whose prompt emission lasts far longer than the canonical 2 s boundary, typically comprising an initial short hard spike followed by softer extended emission. Identifying the physical origin of such bursts requires diagnostics beyond duration alone, among which the Amati relation, linking rest-frame spectral peak energy $E_{\rm p,z}$ and isotropic-equivalent energy $E_{\rm iso}$, is widely used as a complementary classification tool. We analyze a sample of eight long-duration Type~I GRBs and merger candidates by separating the initial spike from the extended emission and examining their episode-dependent locations on the $E_{\rm p,z}$--$E_{\rm iso}$ plane. We find that the initial spike generally lies within, or close to, the empirical Type~I region, consistent with a compact-merger-like prompt-emission component. In contrast, the extended-emission episode systematically occupies a region closer to Type~II GRBs, and could be misidentified as collapsar-like if analyzed in isolation. This episode-dependent Type~I-to-Type~II transition is further supported by time-resolved spectral analysis, although its magnitude and trajectory vary among bursts, suggesting diversity in central-engine evolution or outflow properties between the two phases. Our results caution that the Amati relation alone can lead to misleading empirical classification when the initial hard spike is weak, outside the instrumental bandpass, or missed entirely, leaving only the extended emission to be analyzed. Broad temporal and spectral coverage, and independent multi-wavelength diagnostics, is therefore essential for identifying the physical origin of these events.

astro-ph.HE↗

arXiv2026

On-Policy Distillation with Curriculum Turn-level Guidance for Multi-turn Agents

Multi-turn agents that plan, invoke tools, and interact with environments offer a promising paradigm for solving complex tasks, yet their capabilities typically rely on very large models whose inference cost is prohibitive in practice. On-Policy Distillation (OPD) is a natural recipe for transferring such capabilities to smaller students, but we find that it suffers a characteristic failure mode in this setting: small student errors compound across turns and push the trajectory out of the teacher's familiar state distribution, so the teacher's supervision becomes least reliable precisely where the student needs it most. We propose Guided On-Policy Distillation (Guided-OPD), a simple yet effective algorithm that mixes teacher- and student-generated turns within each rollout and schedules the teacher's intervention probability along a curriculum that decays to zero. Strong guidance keeps early trajectories close to the teacher distribution and is then gradually withdrawn to recover the purely on-policy regime used at inference. On ALFWorld, ScienceWorld, and WebShop, distilling Qwen3 students from a Qwen3-30B-A3B teacher, Guided-OPD yields average relative gains of 21.1\% in Score and 25.5\% in Success Rate over vanilla OPD, with larger gains on smaller students.

cs.LG↗

arXiv2026

Coherent Control of an Embedded Bound State Without a Spectral Gap

Bound states in the continuum (BICs) can confine photonic excitations in open systems without conventional cavities or band gaps, making them natural candidates for long-lived quantum storage and single-photon control. Their use is limited, however, by two obstacles: they are dark to incident photons, and they lack spectral-gap protection from the surrounding continuum. We overcome both limitations in a giant atom coupled to a one-dimensional waveguide using two temporal control knobs. Atomic-frequency modulation breaks and restores the destructive-interference condition, enabling deterministic capture and release of mode-matched single photons. Coupling modulation instead preserves the BIC condition while tuning the atomic and photonic weights of the stored state. A key result is that this embedded state can nevertheless be controlled adiabatically despite the absence of a spectral gap, with an intrinsic leakage probability linear in the ramp rate. By separating radiative access from BIC-preserving deformation, the protocol turns a dark BIC into a single-photon memory whose fidelity is set by the intrinsic continuum-induced leakage law, providing a route to embedded-state control in open photonic platforms.

quant-ph↗

arXiv2026

Starter-Iterator Neural Operator: A Unified Architecture for High-Fidelity Forward and Inverse PDE Problems

Operator learning is an emerging field at the intersection of machine learning and scientific computing. By learning mappings between function spaces, neural operators provide data-driven surrogate models for families of partial differential equations (PDEs). Once trained, these models can evaluate solution operators efficiently, making them suitable for many-query applications such as real-time prediction and parameter sweeps. However, maintaining high approximation accuracy and stable long-term predictions remains challenging for complex forward and inverse problems. To address these challenges, we propose the Starter-Iterator Neural Operator (SINO), which incorporates the initialization and residual-correction structures of classical iterative solvers into neural operator learning. The frequency-domain Starter captures dominant global spectral features and provides an informed initial approximation, while the latent-space Iterator applies successive residual-based corrections to refine local and multiscale solution structures. Experiments on representative time-dependent PDEs, including the Navier-Stokes and acoustic wave equations, together with applications to image super-resolution and weather forecasting, show that SINO achieves competitive accuracy and stable performance across the benchmarks considered in this work.

math.NA↗

arXiv2026

Towards Entanglement-Enhanced Atom Interferometry Using Bow-Tie Cavities

Atom interferometers are among the most sensitive instruments for precision measurements and tests of fundamental physics. Their performance, however, is ultimately limited by quantum projection noise when uncorrelated atomic ensembles are employed. Cavity-assisted generation of entangled states has proven to be a promising route toward quantum-enhanced interferometry beyond the standard quantum limit. In this work, we present the realization and characterization of a monolithic bow-tie cavity developed to achieve a strong collective atom-light coupling with strontium atoms. Unlike conventional standing-wave Fabry-Pérot resonators, the traveling-wave geometry of the bow-tie cavity provides homogeneous atom-light coupling over the entire atomic ensemble, making it particularly suitable for entanglement-enhanced atom interferometry with freely falling atoms. The monolithic cavity architecture presents several scientifically relevant features such as high mechanical stability, high finesse, robustness against mirror misalignment, optical and atomic access and the option of generating squeezed states through different strategies. The cavity was realized for operation on the strontium $(5s^2) ^1S_0-(5s5p) ^3P_1$ transition at 689 nm and achieves a finesse of $\mathcal{F}=5.7\times 10^4$ while keeping the transmission of a single mirror sufficiently large to allow for efficient atomic information extraction. In this geometry, the cavity supports two foci with waists of 164 $μ$m and 31 $μ$m which gives access to different regimes of atom-cavity coupling. For ensembles containing up to $10^5$ atoms, the cavity is expected to enable a Wineland squeezing parameter $ξ^2$ of 24 dB through cavity-feedback squeezing, and 28 dB through quantum non-demolition measurements, demonstrating its potential as a platform for next-generation quantum-enhanced atom interferometers.

quant-ph↗

arXiv2026

Three-particle di-light-cone distribution amplitudes of the $B$-meson in heavy-quark effective theory

We present a systematic study of the three-particle di-light-cone distribution amplitudes (DLCDAs) of the $B$-meson. They are defined through $B$-meson--to--vacuum matrix elements of trilocal HQET operators, in which the light antiquark and the gluon field-strength tensor are located on two back-to-back light rays. In this sense, the DLCDAs generalise the conventional $B$-meson light-cone distribution amplitudes to configurations where soft fields couple to collinear degrees of freedom in two distinct directions. As such, they parametrise the non-perturbative dynamics associated with non-factorisable soft-gluon contributions in rare and non-leptonic exclusive $B$-meson decays. We derive the complete Lorentz decomposition of the matrix elements of generic trilocal operators, identify eight independent DLCDAs, and organise them in a basis of definite twist. Using local operator identities and equations-of-motion constraints, we obtain tree-level relations for their normalisation integrals and first moments in terms of a minimal set of hadronic parameters. These relations allow us to construct simple momentum-space models for all independent DLCDAs. For the leading-twist distribution, we further incorporate the perturbative radiative tail at order $α_s$ and discuss its impact on the resulting parametrisation.

hep-ph↗

arXiv2026

Restart and Adaptive Acceleration in Stochastic Gradient Methods

We study restart schemes in stochastic optimization problems for non-smooth and weakly convex functions that satisfy a Kurdyka-Łojasiewicz (KŁ) inequality. Using restarts allows us to leverage KŁ inequalities to achieve optimal convergence rates, with acceleration depending explicitly on the KŁ exponent. Furthermore, for stochastic gradient descent (SGD), optimal restart schedules correspond to piecewise constant (step decay) step sizes. While regularity constants such as the KŁ exponent are typically unknown in practice, we prove that restart schemes are robust to significant misspecification of these constants, hence nearly adaptive. We detail numerical experiments on a number of problems where the KŁ exponent is controlled.

math.OC↗

arXiv2026

Black Hole Occupation Fraction: Dependence on Black Hole Mass Threshold, Environment, Resolution and Redshift

We take advantage of the state-of-the-art semi-analytic model \texttt{FEGA25} \citep{contini2025}, run on merger trees extracted from three dark matter-only cosmological simulations, to study the relation between the black hole (BH) occupation fraction, $f_{\rm BH,occ}$, and galaxy stellar mass as a function of BH mass threshold, galaxy type, simulated volume, numerical resolution, sampled galaxy population, and redshift. \texttt{FEGA25} includes an improved treatment of active galactic nucleus feedback and does not impose a pre-existing BH seed population: BHs grow naturally through quasar and radio modes. Starting from the prerequisite that \texttt{FEGA25} reproduces the observed BH mass function from at least $z=2$ to the present day, our analysis leads to several results. We find that $f_{\rm BH,occ}$ increases with stellar mass, but that its normalization and shape depend strongly on the adopted BH mass threshold and on the relative contribution of central and satellite galaxies. The relative behavior of central and satellite galaxies depends on the simulation box and BH mass threshold, while the global relation should be interpreted as a population-weighted quantity. We also find significant box-to-box variations, reflecting the combined impact of numerical resolution, simulated volume, and sampled galaxy population. The redshift evolution is not universal: YS50 and the \texttt{NewCluster} zoom-in simulation show a trend qualitatively similar to that reported by \citet{tremmel2024}, whereas larger-volume boxes show the opposite behavior. Finally, comparison with other studies shows that the inferred occupation fraction is highly sensitive to BH mass threshold, simulated volume, numerical resolution, and sampled galaxy population.

astro-ph.GA↗

arXiv2026

CHEX-MATE: AMALGAM weak-lensing analysis of 41 Planck Sunyaev-Zel'dovich-selected galaxy clusters

We present a weak-lensing shear analysis of 41 Planck SZ-selected galaxy clusters at $0.11\le z\le 0.55$ from the CHEX-MATE sample, using wide-field Subaru/Suprime-Cam and CFHT/MegaPrime imaging from the AMALGAM project. We detect the azimuthally averaged weak-lensing signal around the X-ray peak of each cluster, achieving a median S/N of 6.5 per cluster. The $45^\circ$-rotated component has a median S/N of -0.1 and ranges from -1.8 to +1.8, consistent with zero. We model the excess surface mass density profile of each cluster with an NFW profile to infer weak-lensing mass and concentration constraints. The total systematic uncertainty in the weak-lensing mass calibration is assessed to be $8\%$. Using a hierarchical Bayesian framework, we then derive weak-lensing-calibrated scaling relations for the halo concentration, $c_{200}$, as a function of $M_{200}$ and redshift, and for the Planck SZ mass proxy, $M_{SZ}$, as a function of $M_{500}$ and redshift, while accounting for sample selection effects, weak-lensing modelling biases, and residual calibration uncertainty. At $M_{200}=10^{15}M_\odot$ and $z=0.25$, we find $c_{200}=3.53\pm0.71$ with an intrinsic scatter of $0.22\pm0.04$ dex. The inferred normalisation and scatter are consistent with recent $Λ$CDM predictions for massive haloes, with no significant mass or redshift dependence over the probed range. For the Planck mass proxy, our baseline regression yields $M_{SZ}/M_{500}=0.83\pm0.09$ at $M_{500}=7\times10^{14}M_\odot$ and $z=0.25$, with an intrinsic scatter of $0.10\pm0.02$ dex. A restricted model with fixed unit mass slope and no redshift evolution gives $1-b=0.72\pm0.11$. We also provide weak-lensing-calibrated posterior estimates of $M_{500}$ for the sample based on the baseline $M_{SZ}$--$M_{500}$--$z$ relation. These results provide an initial weak-lensing mass calibration for CHEX-MATE multi-probe cluster studies.

astro-ph.CO↗

arXiv2026

Diffusion Models for Radio Map Estimation: Theoretical Performance Analysis and Sampling Rate Guideline

Radio maps, which characterize the spatial distribution of radio frequency metrics, such as the received signal strength, are essential for a wide range of wireless applications. The problem of radio map estimation involves constructing a radio map from a limited set of radio samples measured by sparsely distributed sensors.Recently, diffusion models have been increasingly adopted for this problem, yet their theoretical performance remains largely unexamined. Consequently, it is difficult to evaluate their performance when ground-truth radio maps are unavailable, as is often the case in practice. To bridge this gap, we first formulate radio map estimation as a non-linear matrix completion problem. We then derive a theoretical expression for the estimation error of a diffusion model, capturing the effects of the mismatch between the training and deployment environments, the model design, and the sampling strategy on the quality of the estimated radio map. Moreover, considering that the derived error expression depends on certain information that is difficult to obtain in practice, we propose an empirical approximation that is readily computable from observable data. Finally, extensive simulations demonstrate that the empirical formula closely approximates the theoretical error expression, validating its effectiveness for practical deployment. Our results provide guidelines not only for evaluating the performance of a diffusion model when ground truth is unavailable, but also for determining how many sensors are required to spatially sample the region to achieve a given estimation accuracy. They further reveal a critical sensor count beyond which the estimation performance of diffusion models converges.

eess.SP↗

arXiv2026

Structured Spectral Step-Sizes and Rebound-Aware Ordering for Gradient Methods

The performance of gradient methods depends critically on step-size selection. Spectral step-size design involves constructing candidates from finite history and ordering them during iteration. We address both through a finite-moment spectral framework. For candidate construction, we relate Huang--Dai--Liu determinant pencils, limited-memory steepest-descent Krylov--Ritz extraction, pseudo-memory moment realization, and Gu--Du recovery of spectral nodes and component weights. The framework distinguishes generalized-eigenvalue recoverability, finite-window implementability, and component-weight interpretability. Under explicit positivity, rank-one, regularity, and exact-recovery assumptions, these properties become compatible in a common finite-moment regime, with the long, short, and mean LMSD constructions as representatives. For ordering, we identify spectral rebound: a step targeting a lower spectral scale may reactivate previously reduced high-frequency components. Idealized one-target, block, and recomputed memory-$m$ models yield phase-count recurrences motivating a high-frequency-first strategy with spectral refreshes. We propose a rebound-aware spectral gradient method (RASG) that ranks Ritz candidates by frequency and estimated strength, and sets phase duration via a Ritz-uncertainty estimator. Two variants, RASG-KT and RASG-LMSD, use a Kato--Temple-inspired estimator and a long/short LMSD discrepancy. For strictly convex quadratics, global $R$-linear convergence holds under an admissibility safeguard. Experiments on ill-conditioned quadratic problems and two large-scale convex problems illustrate the behavior.

math.OC↗

arXiv2026

Joint Residual Reweighting for Classifier Free Guidance in Flow-Matching Zero-Shot TTS

Classifier-free guidance (CFG) is widely used in flow-matching-based zero-shot text-to-speech (TTS), where generation is conditioned on text content and a speech prompt. Standard CFG uses a single guidance weight for their joint conditional effect, while branch-selective guidance emphasizes text or speaker conditioning and can introduce a trade-off between text accuracy and speaker similarity. In this paper, we revisit CFG under independently masked conditions and decompose the guidance field into text, speaker, and joint residuals. We show that condition-specific branch differences couple the joint residual with the corresponding text or speaker residual under a shared weight. Trajectory analysis further shows that the joint residual varies over flow time and contains information that cannot be represented by reweighting the text and speaker residuals alone. Based on these observations, we propose joint residual reweighting, which assigns independent weights to the three residuals. Experiments on F5-TTS, CosyVoice2, and GLM-TTS across three evaluation sets show overall improvements in speaker similarity and text accuracy over the default CFG settings without retraining.

eess.AS↗

arXiv2026

TOPS: First-Principles Visual Token Pruning via Constructing Token Optimal Preservation Sets for Efficient MLLM Inference

Multimodal large language models (MLLMs) have achieved strong multimodal reasoning capabilities, but their efficiency is limited by the large number of visual tokens, which introduces substantial computational overhead. Visual token pruning offers a natural solution, yet existing methods are imperfect: attention-based criteria tend to retain redundant tokens, while diversity-based criteria are often agnostic to user instructions. Even methods that combine multiple criteria still lack a principled formulation of the intrinsic objective of token pruning. In this paper, we revisit visual token pruning from a first-principles perspective and formulate it as constructing Token Optimal Preservation Sets. Through a top-down information-theoretic analysis, we identify three fundamental principles for effective token selection: Task Relevance, Information Coverage, and Semantic Diversity. Based on these principles, we propose TOPS, a training-free and model-agnostic pruning module that can be applied to various MLLMs. Extensive experiments on 7 MLLM backbones and 14 benchmarks demonstrate that TOPS outperforms prior methods under diverse pruning settings. Notably, on LLaVA-NeXT, TOPS removes 77.8% of visual tokens while preserving 100.0% and 100.6% performance on its 7B and 13B models, respectively, suggesting that pruning redundant visual tokens can sometimes mitigate hallucination and inspire future lightweight MLLM design.

cs.AI↗

arXiv2026

Algorithmic Unverifiability of Safety for Fixed and Recursively Self-Improving Systems

We establish mathematical limits of algorithmic safety verification for Turing-complete self-modifying systems, the class in which recursive self-improvement takes place, both for a fixed system and across its own modification. Statically, no verifier is sound, complete and tractable: over unbounded domains by Rice's and Gödel's theorems, over all finite configurations by Trakhtenbrot's theorem, and over succinctly described finite environments because verifying a policy against an adversary is coNP-complete and synthesising one is PSPACE-complete. Dynamically, we model one step of self-modification as a computable transformation of code and ask whether a safety property survives it. If the transformation depends only on behaviour, this is Rice's theorem one level up; if it reads the code, as self-modification does, the question is no longer semantic, yet the same s-m-n reduction works inside a class of behaviourally identical programs and inherits the halting degree. One step is never harder than the property; persistence along the whole trajectory can be $Π^0_2$-complete. Certification by a total algorithm is possible only for transformations of restricted expressivity, not merely for systems that stop changing. No tower of supervisors helps, and every total supervisor errs on an undecidable set of systems. For effectively pointwise properties, every faithful bounded scheme that certifies on finite behavioural evidence admits evolution traces certified at every stage while the property is violated. What survives is exact: a monitor that raises an alarm on violation semidecides it, and comparison against a frozen reference keeps the full theory.

cs.LO↗

arXiv2026

An FPT algorithm for cycle rank on semi-complete digraphs

Cycle rank is a depth parameter for digraphs introduced by Eggan in 1963. Gruber (DMTCS 2012) and Giannopoulou, Hunter, and Thilikos (DAM 2012) asked whether the problem of determining if a given digraph has cycle rank at most $w$ is fixed-parameter tractable parameterized by $w$. We provide such algorithms for semi-complete digraphs, and for digraphs of bounded directed clique-width. Specifically, we show that given an $n$-vertex semi-complete digraph~$G$ and an integer $w$, one can in time $2^{\mathcal{O}((w+1)4^{w})} n^2$ determine whether $G$ has cycle rank at most~$w$. The proof is reduced to the case of bounded directed clique-width, and we then show that given an $n$-vertex digraph $G$ with a directed clique-width $k$-expression and an integer $w$, one can in time $2^{\mathcal{O}((w+1)4^k)}n$ determine whether $G$ has cycle rank at most $w$. Additionally, we consider the \textsc{Minimum Feedback Arc Set} problem on semi-complete digraphs, and show that it can be solved in time $n^{\mathcal{O}(w)}$, when a cycle rank decomposition of depth $w\ge 1$ is given.

cs.DS↗

arXiv2026

Warranted Attention: Learning What to Pass from Attention to Prediction

Relevance of information read by attention does not guarantee that its contribution benefits the current prediction. We propose Warrant, which learns how strongly attention-derived item contributions should be transmitted under the current query. Warrant applies learned item-wise permission before aggregation without renormalization, jointly controlling relative allocation and total transmission mass. Across backbones, we construct interfaces connecting these con- tributions to prediction scores or states and compare against ungated models on the same paths. On three CyGNet datasets, ungated paths reduce MRR, whereas learned permission partially or almost fully recovers the losses. In a 5-seed HotpotQA/RoBERTa experiment, distractors receive lower permission than gold support, while mean Support MRR rises from .9111 to .9141 and the unsupported selection rate falls from .2271 to .2215. Performance comparisons and contribution- level interventions across five task families reveal both the effects and limits of selective control. The results support learning the strength of contributions transmitted to prediction separately from attention relevance

cs.AI↗

arXiv2026

The Conway knot has infinite concordance order

We examine how the Rasmussen invariant, satellite operations, and null-homologous twists can be used to establish infinite order of knots in the smooth concordance group. As an application, we show that the Conway knot has infinite concordance order.

math.GT↗