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arXiv2026

Puncture-Forgetting Maps for Measured Foliations and Applications in Teichmüller Space and Complex Dynamics

We introduce puncture-forgetting maps for measured foliations and investigate their relations with mapping class groups, Teichmüller spaces, and extremal length. To this end, we develop the notions of cube complexes of pre-homotopic multicurves and tree coordinate systems on CAT(0) cube complexes. As an application to the dynamics of post-critically finite rational maps on the Riemann sphere, we obtain a partial result toward the finite curve attractor conjecture posed by Kevin Pilgrim. We also apply these methods to uncover a relation between horospheres and geodesic flows in the universal curve over Teichmüller space.

math.DS↗
arXiv2026

A MATLAB Tool for the Stable Generation of Matrix Polynomial Evaluation Schemes with Two-Product Savings

Computing numerical approximations of matrix functions frequently relies on the efficient evaluation of high-degree matrix polynomials. Although computational bounds are historically governed by the Paterson--Stockmeyer (PS) method, recent theoretical developments have demonstrated the viability of evaluation schemes that eliminate two matrix products ($2M$). Existing literature documents stable instances of this $2M$ reduction only for isolated cases, such as specific degrees of Taylor approximations for the matrix exponential and the matrix logarithm. However, a generalized approach for arbitrary polynomials remains unestablished. To address this limitation, this work presents a software-driven procedure that extends these computational savings to polynomials of degrees $m \in \{18, 21, 24, 26, 27, 28\}$ and all $m \ge 30$, requiring primarily a non-zero leading coefficient. Since the underlying evaluation coefficients must be determined by solving systems of nonlinear equations (SNEs), selecting a numerically stable solution set is critical. We introduce an automated verification routine designed to filter and validate robust coefficient sets for floating-point execution. The primary contribution is a MATLAB implementation leveraging variable precision arithmetic to handle the underlying SNEs, verify stability, and project precision bounds. Numerical experiments involving various matrix functions verify that the developed implementation preserves or, in some instances, enhances the numerical accuracy of the classic PS method, while systematically achieving the theoretical reduction of $2M$.

math.NA↗
arXiv2026

Hard unknots are often easy from a different perspective

Recent attempts to train AI models to recognize knots have produced millions of ``hard'' unknot diagrams resistant to simplification by Reidemeister moves, pass moves, or random walks on the Reidemeister graph. Most are easy for the methods based on triangulations of the knot complement found in Regina and SnapPy, but more difficult for simplifiers based on diagrammatic moves. We present ReAPR (Re-embedding And Pass Rerouting), a semi-diagrammatic simplifier which alternates pass-move reduction on a knot diagram with a geometric re-embedding step. The re-embedding minimizes the total variation of a height function on the diagram subject to crossing constraints. We show that for an n-crossing diagram, the minimum total variation is 2(n-k), where k is the least number of crossings one must virtualize to make the diagram virtually alternating; this is a combinatorial invariant of the diagram. Reprojecting the resulting embedding from a new viewpoint reveals previously hidden simplifications. ReAPR successfully simplifies every published hard-unknot example we are aware of, as well as several new collections (~2.6 million examples in total) in under 30 seconds of total CPU time, making it as effective as the best non-diagrammatic unknot recognition method (SnapPy) and about 60x faster. ReAPR is just as effective at simplifying diagrams of random knots, where it is regularly used to simplify diagrams with tens of millions of crossings.

math.GT↗
arXiv2026

Recovery of a Null Form in the Wave Equation from Scattering Data

We use highly oscillatory geometric optics solutions to solve the inverse problem for the system $$ \square \begin{bmatrix} u^{(1)}\\ u^{(2)}\\ \vdots\\ u^{(n)}\end{bmatrix} = \sum_{k\geq l=1}^n \left(Q_0(u^{(k)},u^{(l)}) \begin{bmatrix} q_{1kl}\\q_{2kl}\\\vdots\\q_{nkl} \end{bmatrix} + \sum_{i>j=1}^d Q_{ij}(u^{(k)},u^{(l)})\begin{bmatrix} p_{1ijkl} \\ p_{2ijkl} \\ \vdots \\ p_{nijkl} \end{bmatrix}\right), $$ where $Q_0$ and $Q_{ij}$ are the symmetric and anti-symmetric bilinear null forms. We present solutions in both the linear and weakly nonlinear regimes. In the linear regime, we show that the coefficients of order $h^3$ determine an injective light-ray transform of a vector field which depends on the coefficients $q_{rkl}, p_{rijkl}$. In the weakly nonlinear regime, we see that the coefficients of order $h$ determine the non-abelian light ray transform for matrices associated with the coefficients. While we do not have an injectivity result for this case, we do have one if we assume the coefficients do not depend on the time variable $x_0$, as our coefficients instead determine an injective non-abelian X-ray transform.

math.AP↗
arXiv2026

Critical Flicker Fusion Frequency As An Experience-Restricted Constraint On Visual Temporal Resolution: What Does And Does Not Change It

Experience-dependent plasticity is fundamental to adaptive behaviour, yet the conditions under which basic sensory timing can be modified in adulthood remain poorly specified. Critical flicker fusion frequency (CFFF), the threshold at which flicker is perceived as continuous, is unusually informative here: what fails to change it is as well documented as what does. Reviewing the stability and training literature, we argue CFFF is best characterised neither as non-plastic nor as held near a physiological ceiling, but as modifiable only by a specific class of experience, not by amount. Repeated testing, cognitive training without temporal content, and incoherent-flicker exposure leave the threshold unchanged regardless of duration. By contrast, a narrow class of perceptual-learning paradigms -- pairing coherent directional motion with a task-relevant target -- reportedly raises the threshold substantially, with gains retained at one year in a small subsample. Gains are largest below typical values, as in amblyopia, and absent in normally sighted observers under the same protocol. Three qualifications apply: the evidence rests on small samples; the threshold-raising paradigms have been assessed almost exclusively with heterochromatic flicker photometry rather than luminance-defined CFFF, leaving construct equivalence unestablished; and a minority of untrained controls show comparable changes. We examine whether the restriction originates at peripheral, thalamocortical, or cortical levels; available data do not adjudicate between them. Functional arguments for why such a restriction might be adaptive are offered as rationales, not evidence. Proposed links to working-memory precision and metacognition are stated as predictions; the first test of them was largely negative.

q-bio.NC↗
arXiv2026

Pólya's conjecture for higher-dimensional Neumann balls

We prove Pólya's conjecture for the Neumann eigenvalues of the Laplacian on Euclidean balls in dimensions three and higher. The proof further develops the approach introduced in our earlier work on the two-dimensional case and on Dirichlet eigenvalues in arbitrary dimensions. The main difficulty in the higher dimensional Neumann case is that one has to estimate zeros of the derivatives of ultraspherical Bessel functions, rather than of the usual Bessel functions. For low-lying eigenvalues, we use variational estimates involving dimension-dependent test functions, which is a novel ingredient allowing us to control a larger dimension-scaled frequency range. Other components of the proof include phase-function bounds, lattice-point counting techniques, and computer-assisted arguments.

math.SP↗
arXiv2026

Predicting Startup Exit from Textual Descriptors - A Computational Linguistics Framework

This study shows that textual descriptors alone can predict early-stage startup success, defined as Exit, without relying on contextual, financial, or human capital variables. Using venture capital-curated datasets covering 7,419 startups over 20 years, the research isolates text-based framing variables and engineers 850 features via startup narrative mapping. Data subsets and vector embeddings are evaluated for statistical significance, followed by supervised machine learning experiments across six models. Binary Exit prediction using Logistic Regression attains an F1 of 0.48 with 0.55 recall using all features (excluding embeddings), and an F1 of 0.26 with 0.59 recall using textual descriptors only (including embeddings). Feature analysis indicates that optimized densities of hyping markers such as adjectives, jargon, and buzzwords are associated with higher Exit probability, while excessive statement or name length is associated with lower probability. The study also introduces a quantifiable Hyping Score for potential application in venture screening. Findings indicate that startup framing can serve as standalone predictor of economic outcomes, in high-information-asymmetry investment environments.

cs.CL↗
arXiv2026

From Eigenvalues/Eigenvectors of Hypermatrices to Canonical Form of Tensors

The rings of non-square matrices based on the dimension-keeping (DK-) semi-tensor product (STP) are considered, where a virtual identity is introduced to make each ring possess an identity element. Using this ring structure, four kinds of eigenvalues/eigenvectors (EEs) of hypermatrices, namely, the ordinary EE (OEE), the universal EE (UEE), the diagonal EE (DEE), and the horizontal diagonal EE (HDEE), are proposed with respect to preassigned matricizings. The Kronecker canonical form (KCF) of non-square pencils is used to calculate the OEEs; the monic decomposition algorithm (MDA) is then applied to extract the UEEs, DEEs, and HDEEs from them. Finally, the KCF of non-square pencils is further used to construct the KCF of a tensor, which reveals all the EEs of the tensor. The KCF of a tensor not only shares the main properties of the Jordan canonical form of matrices but also includes the latter as a special case. Consequently, all the EEs of a hypermatrix are straightforwardly computable via its KCF.

math.RA↗
arXiv2026

Output-Aware Rotation for INT2 KV-Cache Quantization

The key-value (KV) cache has become a major memory and bandwidth bottleneck in long-context large language model inference, making ultra-low-bit quantization increasingly important. However, existing rotation-based INT2 methods optimize cache statistics or proxy errors before the complete attention readout, even though the model is ultimately affected by the error propagated through attention and the output projection $W_O$. To address this mismatch, we propose \textit{OptR}, an output-aware rotation method that minimizes post-$W_O$ attention-output error. OptR decomposes the post-$W_O$ attention-output error into key- and value-induced terms and learns per-head orthogonal corrections through the full INT2 quantization and attention path. OptR further applies an attention-equivalent key reparameterization to reduce large channel-wise offsets without changing the softmax distribution. Across three models and five reasoning and coding benchmarks, OptR consistently improves both QuaRot and OSCAR and strengthens long-context retrieval, while preserving the paged KV-cache format with negligible inference overhead.

cs.LG↗
arXiv2026

Test Time Adaptation Methods for Point Cloud Registration in Laparoscopic Surgery

3D point cloud registration in laparoscopic surgery estimates the transformation between an intraoperative organ reconstructed from video and its preoperative mesh. Because ground-truth transformations are unavailable for real data, supervised networks are trained on synthetic organ pairs. At test time, real reconstructions differ from synthetic data and are noisy, sparse, and occluded, which degrades correspondence estimation. Test-time adaptation (TTA) can reduce this domain shift, but existing methods mainly rely on logits, entropy, class prototypes, or cache memories unavailable in registration. Registration also involves paired inputs with an asymmetric shift that primarily affects the intraoperative cloud. We analyse and modify state-of-the-art TTA methods from three families to 3D registration: model, normalization, and input adaptation. We analyze four representative approaches based on auxiliary-task model updates, backpropagation-free token purging, feature alignment, and layer-normalization calibration. We modify them to handle asymmetric shifts between preoperative and intraoperative point clouds and replace classification-based entropy objectives. Using a correspondence-based model trained on clean synthetic source data, we evaluate adaptation to corrupted synthetic and real target data on P2P and P2ILReg. For synthetic targets, we apply eight corruptions, including uniform noise and global density reduction, at five severity levels. All methods improve registration on P2P, whereas on P2ILReg only input adaptation reduces the error, while normalization adaptation degrades it. Considering the computational overhead of backpropagation-based adaptation, input adaptation is the most promising option for laparoscopic surgery, providing low inference latency and consistent error reductions across datasets. Code: https://github.com/ninaa-git/survey_pc_registration_tta

cs.CV↗
arXiv2026

TACT: Taxonomy-Aligned Post-Training for Pedagogically Adaptive English Tutoring

Large language models (LLMs) are increasingly used to provide conversational practice for English-as-a-second-language (ESL) learners. Effective ESL tutoring, however, requires more than fluent response generation: a tutor must select an appropriate pedagogical action based on learner behavior and dialogue context. Human-tutoring research offers principles for adaptive support, but they are often task-specific and remain insufficiently integrated into LLM-based ESL tutor training and evaluation. We present TACT (Taxonomy-Aligned Conversational Tutor), a human-grounded framework for post-training and evaluating pedagogically adaptive ESL tutors. Drawing on established literature, we develop two complementary taxonomies: the Tutor-Strategy Taxonomy with 13 tutor response strategies and the Student-Move Taxonomy characterizing learner behavior by move type and status. Using these taxonomies, we construct TACTCorpus, which enriches 260 authentic teacher-student conversations with 32,379 annotations and quality-controlled augmented training data. We then post-train Qwen3.5-4B through supervised fine-tuning followed by taxonomy-aligned Group Relative Policy Optimization, producing TACTutor and optimizing it for scaffolding quality rather than reference imitation alone. On TACTBench, a strategy-balanced diagnostic benchmark comprising 78 authentic tutoring contexts, TACTutor improves over its backbone by 20.30% and outperforms all evaluated proprietary baselines under the same protocol, while maintaining backbone performance on established external educational benchmarks; in a blinded study with 50 learners, it also receives the highest overall mean rating among the evaluated tutors. We release the data, benchmark, and model weights, providing an open foundation for developing pedagogically adaptive ESL tutors.

cs.AI↗
arXiv2026

Development of Thomson parabola spectrometer for diagnostics of ions driven by ultrahigh intensity laser: Simulations and numerical analysis

A Thomson parabola ion spectrometer (TPIS) has been designed and developed for diagnostics of laseraccelerated ion beams in the MeV energy range. The TPIS has been validated by ion acceleration experiment at petawatt laser facility. Necessary simulations to evaluate the electric and magnetic field distributions have been performed with the help of a numerical simulation software to aid the selection of the spectrometer geometry while minimising fringe-field effects. Analytical dispersion expressions have been formulated from the simulations that take into account the spatial variation in the electromagnetic field profiles. The ion deflections obtained from these expressions demonstrate an improved agreement with experimentally measured proton trajectories compared to the case when constant fields are considered. The TPIS hence fabricated in-house has been subject to magnetic field measurements, which are in excellent agreement with the simulated field profile. The TPIS has the provision to change the field-free drift region, showcasing flexibility to be employed over a broad energy range and with different experimental setups. The spectrometer has been subsequently used for detecting laser-accelerated ion beams from thin aluminum foil targets. These experiments have demonstrated the capability of the spectrometer to resolve multiple ion species with sufficient separation between them. The developed TPIS provides a compact, flexible and accurate diagnostic for high-energy laser-plasma experiments.

physics.plasm-ph↗
arXiv2026

How a shared state is described determines whether AI agents synchronize

Language-model agents increasingly act in populations, where the outcome that matters is collective: whether they align, split or fail to coordinate. Each acts not on the world but on a text description of it, a choice usually fixed in software. Using synchronization, the canonical probe of how interaction rules produce collective order, we show that this choice can decide the outcome. Agents on a circle chose to advance, stay or move back after reading the others' relative positions, in 507,112 valid responses across matched populations, controlled inputs and three model families. In GPT, numerical summaries aligned every matched population at both positive couplings, whereas histograms aligned none; Claude showed the reverse at the stronger coupling. Re-describing identical states shifted action probabilities in all three families, even between histograms carrying the same information. No single directional coefficient explained the outcome: state descriptions are part of the interaction rule that turns individual responses into collective order.

physics.soc-ph↗
arXiv2026

Liouville Rigidity and Universal Spacelikeness Estimates for a Lorentzian Prescribed Mean Curvature Equation

We prove a Liouville theorem for nonnegative entire strictly spacelike solutions of \[ \operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+u^p=0 \qquad\text{in }\mathbb R^n. \] If $n=2$ and $p\geqslant1$, or if $n\geqslant3$ and $1\leqslant p\leqslant\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfying $|\nabla u|<1$ vanishes identically. This resolves, in the classical strictly spacelike setting, the nonexistence conjecture of Byeon, Ikoma, Malchiodi, and Mari, including the critical endpoint. No symmetry, decay, integrability, or uniform spacelike gap is assumed. A key ingredient is a universal bound, valid for every $n\geqslant2$ and $p\geqslant1$, for both the height $u$ and the Lorentz factor $(1-|\nabla u|^2)^{-1/2}$. Then a weighted trace-free tensor identity from the invariant-tensor approach, combined with a common cutoff estimate, a core-counting argument and Souplet-type feedback inequality, yields a unified proof in the subcritical and critical ranges. The upper endpoint is sharp for $n\geqslant3$, as supercritical radial solutions exist. The theorem also gives half-space rigidity for complete spacelike hypersurfaces, including at the critical exponent.

math.AP↗
arXiv2026

Wisdom in Unity: The Role of Multilingual Training in Figurative Language Identification in Proverbs

Although multilingual approaches to figurative language identification are not new, the shift beyond language-homogeneous training data requires a clearer understanding of the contribution of translated multilingual supervision. We examine this question using 742 proverb concepts across 6,787 translated instances for seven languages. We evaluate five models including multilingual encoders and instruction-tuned LLMs through progressively increasing levels of multilingual supervision. Moreover, we introduce multidimensional annotation framework for proverbs that characterizes proverbs through four complementary figurative forms: Metaphorical, Moral/Advisory, Cause-Effect, and Culture-Specific. Our findings show that overall, adding multilingual training data beyond 50% provides only limited additional improvement, although the best supervision level varies across models and languages. Also, we show that combining diverse figurative forms yields the strongest overall performance. A notable finding is that the least frequent figurative form culture-specific exhibits the largest performance gains under multilingual supervision. Furthermore, the moral/advisory and culture-specific forms of proverb contribute more to instruct tuning LLM overall figurative identification performance. These findings motivate multilingual figurative identification to move beyond metaphor-centric taxonomies toward concept-level multidimensional frameworks that explicitly model complementary forms of figurative meanings that are context representative.

cs.CL↗
arXiv2026

VectraYX-Vision-1B: A Sub-2B Spanish/LATAM Cybersecurity Vision-Language Model with Structured Visual Reasoning and Native Tool Use

We build VectraYX-Vision-1B, a sub-2B Spanish/LATAM cybersecurity vision-language model coupling a frozen SigLIP-so400m encoder to a 1.04B-parameter decoder via a two-layer MLP projector, and report a diagnostic negative result: not that visual grounding failed, but why. After repairing five silent fine-tuning defects, grounding on a nine-field extraction gate with a shuffled-image control is 2/9, invariant across every configuration that leaves the encoder alone; 2x2 tiling, the one that changes it, loses a field and gains none. Resolution is not the operative variable: the field read almost perfectly has the highest entropy in the corpus. A linear probe on frozen SigLIP features gives per-glyph recoverability p~0.61, predicting 1.9% against an observed 0.00; tiling nearly doubles recoverability on two fields, yet the end-to-end model gets worse. Transplanting a natively-trained visual tower onto the same frozen decoder and recipe takes that address field from 0.00 to 0.81 exact, on a coarser token budget than the tiling condition that recovered nothing: pretraining regime, not resolution, sets how far the losses reach. A later, separately trained checkpoint adds one positive result: on B8 (34 fields, 16 templates, 2,040 items, dual shuffled-image/best-constant control), 9 fields pass, confirming genuine grounding within trained template-field combinations only. Sharpest new finding: inside a well-trained template, an untrained field returns a near-constant wrong answer independent of the image -- landmark-keyed lookup, not free-text reading. B6/B7 tool identification stays at 0.0 on every checkpoint including this one; we retract an earlier 0.08 tool-id score after finding three harness defects a stronger model would conceal. We release code, all three benchmarks, configs, and all training checkpoints, including the B8 corpus.

cs.CL↗
arXiv2026

Multi-tracer mass bias in matched cosmic voids from SDSS DR7 and the ELUCID constrained simulation

Cosmic voids provide a unique environment for studying the relationship between galaxies, subhaloes, and dark matter in the underdense Universe. Using the SDSS galaxy catalogue and the ELUCID constrained simulation, we establish an observationally anchored framework for measuring multi-tracer mass bias within matched cosmic voids. A sample of 102 matched void pairs is constructed to directly compare galaxy, subhalo, and dark matter mass distributions within an observationally constrained realisation of the local Universe. We find that both the galaxy-to-dark matter and subhalo-to-dark matter mass ratios decrease toward void centres, indicating that luminous and halo tracers become increasingly depleted relative to the underlying matter distribution in the deepest underdensities. In contrast, the galaxy-to-subhalo mass ratio exhibits substantially larger statistical uncertainties within the inner void regions ($r/R_{\rm v}\lesssim0.5$). By comparing measurements obtained using independent and common coordinate frameworks, we show that coordinate offsets contribute to the observed scatter but cannot fully account for the large uncertainties. The remaining uncertainty primarily arises from the severe scarcity of massive subhaloes ($\log_{10}(M_{\rm sub}/h^{-1}M_\odot)\ge11.8$) within void interiors, which greatly reduces the number of statistically valid measurements near void centres. Our results provide an empirical characterization of multi-tracer mass bias in observationally constrained cosmic environments and demonstrate the importance of accounting for coordinate consistency and tracer scarcity when interpreting such measurements in extreme underdense regions.

astro-ph.CO↗