Papers

Real-world document processing systems rely on rigid, predefined schemas, yet critical target fields often lack direct visual counterparts on the page. Extracting these implicit values requires multi-hop derivation, such as aggregating sub-categories or reasoning over visual marks. While existing methods handle explicit text spans or simple implicit queries, they fail at multi-hop visual reasoning even after standard fine-tuning: models retrieve incorrect visual evidence, or retrieve it correctly and then skip the intermediate steps of the derivation. To address this, we introduce DocMIDE, a fine-tuning framework that trains compact vision-language models to retrieve visual evidence explicitly before deriving an answer. DocMIDE constrains generation to a plan-retrieve-derive structure and optimizes it with Group Relative Policy Optimization under a four-component, rule-based reward that scores output format, the retrieved evidence block, every intermediate derivation step, and the final value against a verified reference trace. On a 4,151-pair implicit extraction benchmark, DocMIDE raises accuracy from 70.8% to 95.9% on Qwen3.5-4B from only a small set of annotated examples, and transfers to a second backbone architecture. Supervised demonstrations alone do not close this gap at any budget we tested; rewarding the intermediate steps is what does.

Koornwinder moments generalize the Askey–Wilson moments arising in the asymmetric simple exclusion process. Rains conjectured that, under the specialization $t=q$, the minimal numerators of Koornwinder moments have nonnegative integer coefficients. While the one-row case of this conjecture was previously proved by Corteel, Mandelshtam, and Williams using rhombic staircase tableaux, its dual counterpart, the one-column case, has remained open. In this paper, we derive a closed, manifestly positive combinatorial formula for the normalized numerators of the coefficients of the rescaled Askey–Wilson polynomials. This proves Rains' conjecture for one-column partitions, thereby providing the exact dual counterpart to the previous result.

Long-horizon autonomous intelligent systems rely on heterogeneous components such as large language models, databases, external APIs, and rule engines, while their external states continuously change during execution. Re-executing the entire workflow after every change introduces substantial redundant computation. This paper proposes an incremental consistency execution method based on task fact contracts, field-level dependency masks, and state perturbation result invariant domains. After an initial verified execution, the system constructs conservative invariant domains for critical inputs and uses them to determine whether downstream results can be safely renewed without re-invoking expensive components. When re-execution is required, only the smallest affected output fields are recomputed, and an equivalence barrier prevents unnecessary downstream propagation. A submission-time version consistency gate further ensures the safety of side-effecting actions. Experiments on industrial fault diagnosis, enterprise analytics, and LLM-based multi-tool assistants show that the proposed method significantly reduces expensive component calls and end-to-end latency while maintaining high consistency and low incorrect-reuse rates.

Transformer models built on the attention mechanism have become a central building block in modern deep learning, yet softmax attention remains a major bottleneck for long-context workloads. While FlashAttention makes the forward and first backward passes I/O-efficient, it does not support backward-over-backward (BoB), which enables exact differentiation through the backward pass for applications such as second-order optimization, test-time training, gradient-based memory, and meta-learning. Existing BoB implementations either materialize large intermediate tensors or exhaust GPU memory at long sequence lengths. We present FlashBoB, an exact, I/O-efficient algorithm for BoB in softmax attention that keeps computation within on-chip tiles and avoids all $N \times N$ intermediate tensors, where $N$ is the sequence length. The key insight is a hierarchical affine structure in the softmax double backward: two row-wise scalars determine all outputs through affine transformations. This yields a two-pass schedule with bounded on-chip static random-access memory (SRAM) usage and minimal off-chip high-bandwidth memory (HBM) traffic. FlashBoB achieves $\Theta(N^2 d^2/M)$ HBM traffic ($d$ is the head dimension and $M$ is the memory size) and, within the standard FlashAttention-style score-recomputation model, matches the inherited large-cache lower bound for exact forward attention. Empirically, it scales exact attention BoB to $N=262\text{K}$ on a single A100 80GB GPU, where prior PyTorch exact baselines fail by $N=16\text{K}$, and is up to $6.3\times$ faster than FlashBack. These results make exact second-order attention practical at long-context sequence lengths where prior implementations cannot run efficiently.

In latent generative models, reconstruction quality is often assumed to correlate with generative performance. However, reconstruction FID (rFID) can exhibit weak or even negative correlation with generation FID (gFID). We attribute this misalignment to a latent distribution mismatch: reconstruction evaluates the decoder on encoder-induced latents, whereas generation uses the same decoder on latents produced by the generative model. To characterize this shift, we introduce generation-aware reconstruction (GAR), which constructs a continuous trajectory from standard reconstruction toward generation by perturbing encoder latents with noise and denoising them through the generative model before decoding. GAR probes the decoder behavior along this trajectory, making the transition from encoder to generation-time latent distributions observable and diagnosable. The resulting trajectory-based diagnostic, GAR-FID, exhibits strong empirical correlation with gFID across diverse tokenizers and scales. Importantly, intermediate GAR latents become more generation-aware while preserving correspondence with their source images, thereby retaining paired supervision that is absent for fully generated latents. This correspondence enables decoder adaptation on intermediate GAR latents, consistently improving generative quality across model scales. Overall, latent distribution mismatch provides a useful perspective for evaluating and improving latent generative models.

Given two weighted directed graphs of order \(n\), we study how much their overlap can deviate from its random average under relabelling, and how concentrated the distribution of their intersection is when they are placed at random. Bollobás and Scott studied these problems on weighted $k$-uniform hypergraphs and raised the corresponding questions for directed graphs. In this paper, we study these problems for weighted directed graphs and obtain results analogous to those in the weighted hypergraph setting.

Large-scale multipurpose cohort studies and biobanks often omit covariates needed for specific downstream analyses. We study target-population inference for generalized linear regression when key covariates are completely absent from the target data but observed in a related source population. Standard missing covariate methods are not directly applicable because they require at least partial observation of the covariates in the target population. We develop a doubly robust transfer learning framework under a sub-population shift assumption, which allows the distribution of the observed outcome and covariates to differ between populations while requiring the conditional distribution of the missing covariates given the observed variables to be shared. Unlike linear regression, nonlinear target estimating equations require parameter-indexed conditional functionals of the missing covariates rather than a fixed collection of low-order conditional moments. Our estimator combines importance-weighted source estimating equations with imputation of these conditional functionals. Under the identifying assumption, the estimator remains consistent when either the density-ratio model or the conditional-covariate model is correctly specified. Under regularity conditions, it is root-$n$ consistent and asymptotically normal, and attains the semiparametric efficiency bound when both nuisance models are correctly specified.

Axion-like dark matter (ALDM) is a well-motivated dark matter candidate whose Chern-Simons coupling to photons can induce cosmic birefringence, producing temporal oscillations and spatial correlation in polarization angle (PA) signals from different sources. Hyperactive repeating fast radio bursts (FRBs) with high linear polarization are a particularly promising probe of such signatures. We develop a Bayesian framework to search for PA oscillations caused by ALDM in individual repeating FRBs, and combine multiple sources in an FRB polarization array (FRBPA) by exploiting correlations arising from the common Earth term. We apply this framework to PA datasets from three well-studied hyperactive repeating FRBs, and constrain the Chern-Simons coupling over a broad range of ALDM masses. For each FRB, we estimate the dark matter density at its host location using a simulation-calibrated halo profile. The inferred dark matter densities are comparable to the fiducial local density $\rho_0 = 0.4~\mathrm{GeV/cm^3}$. The comparable source and Earth dark matter densities, together with the redshift-induced difference between the frequencies of the source and Earth terms, motivate our adoption of the full two-component birefringence model. We find no compelling evidence for ALDM-induced PA oscillations in either the individual source analyses or the joint FRBPA analysis. We place 95% credible upper limits on the Chern-Simons coupling strength as a function of ALDM mass. Across the axion mass from $10^{-23}$ to $10^{-18}~\mathrm{eV}$, the resulting upper limits of the coupling strength of FRBPA lie between $10^{-12.6}$ and $10^{-7.9}~\mathrm{GeV}^{-1}$. Our results establish long-term polarimetric monitoring of hyperactive repeating FRBs as a promising and independent probe of ALDM, with sensitivity expected to improve with longer observational baselines and larger source samples.

Open-weight large language models (LLMs) can be copied, modified, and redeployed behind black-box APIs, making post-release ownership verification difficult. Existing black-box fingerprints often rely on secret query-key pairs that reproduce predefined responses, and can therefore be easily disrupted by fine-tuning, pruning, quantization, model merging, and serving-time prompt changes. We propose SimPrint, a recoverable semantic fingerprinting framework for black-box LLM ownership verification. Rather than relying on isolated exact matches, SimPrint encodes a private owner signature into a coded semantic fingerprint domain, distributing ownership evidence across natural binary question-answering probes. It implants only base-deviating probes through a low-interference batch update that preserves the original model behavior, and later recovers the signature by parsing suspect-model responses into reliable bits or erasures with an error-correcting recovery mechanism. Because verification only uses input-output queries, SimPrint remains applicable when model weights or activations are inaccessible. Experiments on three open-weight LLMs show that SimPrint reliably recovers the owner signature in both clean and modified settings, remains robust under fine-tuning, pruning, quantization, model merging, and serving-time perturbations, and maintains comparable downstream utility.

Generative AI enables scalable production of educational videos, but current systems largely focus on producing visually coherent content rather than supporting learning. As a result, generated videos often lack explicit pedagogical structure, reliable quality control, and mechanisms for assessing learner understanding or addressing misconceptions. In this work, we introduce PIVOT (Pedagogy-guided Instructional VideO Tutoring), a generative video tutoring framework for STEM learning via learning-centered instructional support.1 Inspired by conventional teaching workflows, our framework integrates pedagogy into the full generation pipeline: it first uses instructional principles to guide storyboard generation, then produces verified multimodal videos through code-centric generation and a pedagogical verification harness, and finally connects videos with assessment and misconception-aware remediation. Experiments and expert evaluations across four STEM domains show that our framework produces educational videos with pedagogically aligned content, clear and engaging presentation, coherent instructional flow, and perceived effectiveness for learning. These findings suggest a human-centered perspective on educational content generation: generative systems should be evaluated and designed not only by what they produce, but also by how they support teaching practices, learner understanding, and corrective feedback.

In this article, we discuss a multivariable analogue of Lehmer's parity problem. We generalise the notions of Lehmer numbers as Lehmer $q$-tuples for a fixed odd prime $q$. We count the number of Lehmer $q$-tuples and derive an asymptotic estimate with explicit error terms. This deals with estimating various character sums and Deligne's bound for incomplete exponential sums.

(AlxGa1-x)2O3/beta-Ga2O3 transistors are an emerging candidate for high-power and high-frequency electronic devices. beta-Ga2O3 in particular is notably limited for anisotropic low thermal conductivity, which is further reduced in (AlxGa1-x)2O3 due to alloy and other defect-driven phonon scattering mechanisms. In this work we show that the thermal conductivity of (010) oriented (AlxGa1-x)2O3 thin films for 0.01 < x < 0.20, measured using time-domain thermoreflectance (TDTR), is limited by alloy scattering without observable adverse scattering by additional defects. This is enabled through the synthesis of the (AlxGa1-x)2O3 films using molecular beam epitaxy (MBE) on e̱ta}-Ga2O3 substrates, leveraging indium-catalyzed growth to suppress dislocation formation and phase separation to achieve single-phase pseudomorphic films up to x = 0.2. This growth process improved the thermal conductivity of (AlxGa1-x)2O3 by 2X as compared to previously reported values. For increasing Al composition (x), we observe a steady decline in (AlxGa1-x)2O3 thermal conductivity due to alloy scattering which is validated using virtual crystal approximation (VCA) model. We also show that the thermal boundary conductance across the Al/(AlxGa1-x)2O3 interface is reduced with increasing x, which we posit is due to the stiffening of the (AlxGa1-x)2O3 acoustic modes with increasing x by comparing experimental results with a diffuse mismatch model (DMM). Overall, these thermal characteristics provide valuable insights for designing heterostructures with optimized interfaces and composition.

A graph $G$ is $F$-free if it does not contain $F$ as a subgraph. Let $\mathrm{spex}(n,F)$ be the maximum spectral radius over all $n$-vertex $F$-free outerplanar graphs. For integers $t\geq1$ and $l\geq2$, let $P_{t\cdot l}$ be the starlike tree with $t$ branches of length $l-1$. For sufficiently large $n$, Yin, Li, and Meng [arXiv:2504.04364v1] characterized the unique extremal graph for $\mathrm{spex}(n,P_{t\cdot l})$ when $t=1$, $t=2$, or $t\geq4$. They left the case $t=3$ open and proposed a natural candidate for the extremal graph. We show that this candidate is not extremal and determine the unique extremal graph for $\mathrm{spex}(n,P_{3\cdot l})$. For every $l\geq3$ and all sufficiently large $n$, this unique extremal graph is $K_1\vee\bigl(2P_{2l-3}\cup qP_{l-2}\cup P_r\bigr),$ where $q$ and $r$ are integers satisfying $n=2(2l-3)+q(l-2)+r+1,$ $q\geq0,$ $0\leq r<l-2.$

This article describes our experiences in mathematical outreach over the past decade. We talk about specific activities, but also general principles that we've learned along the way.

We introduce determinantal ellipticity, or det-ellipticity, a quantitative structural condition for fully nonlinear elliptic operators on compact Kähler manifolds that provides the link between the $L^\infty$-estimates and algebraic/combinatorial properties of a large class of Hessian elliptic operators. On the analytic side, we introduce $\mathcal D$-subsolutions extending the determinant sublevel condition of Sui–Sun. Using the auxiliary comparison method of Guo–Phong–Tong and pluripotential theory for complex Monge–Ampère equations, we obtain relative $L^\infty$-estimates in big cohomology classes under a determinant-entropy bound for viscosity supersolutions and singular reference potentials. Det-ellipticity provides a systematic construction of such subsolutions. On the algebraic side, we study Gårding elliptic polynomial operators of degree $d$. We characterize the determinant increment bound with exponent $d/n$ by positivity of the determinantal capacity of the top homogeneous part. This is equivalent to balanced-point conditions for the Newton polytopes of rank-one scalarizations and to slope semistability of an associated subspace polymatroid. Together with the analytic hypotheses, these criteria yield relative $L^\infty$ estimates for the corresponding fully nonlinear equations.

Privileged language-model agents can satisfy a new system task by displacing a healthy incumbent that depends on the same file, process, socket, lock, or capacity allocation. This failure arises because execution privilege determines whether an operation can run, not whether the requester may preempt the current resource owner. We present LeaseGuard, a deterministic admission layer that represents preemption authority through canonical resource leases, incumbent-health checks, effect-aware admission, coexistence limits, safe alternatives, and resource-scoped overrides before adapter execution. We evaluate it on a frozen benchmark of 60 newly authored conflict scenarios with matched controls across two local model families. Relative to a preservation prompt, LeaseGuard reduces unauthorized preemption from 73.3% to 0.0% and increases safe completion by 70.0 percentage points (scenario-clustered 95% CI [60.8, 79.2]). Requested-task success changes by -3.3 points (95% CI [-9.2, 2.5]). The fully evaluated v0.2 broker also rejects a forged incumbent task identity in a hash-linked stress audit. Expiry-only reclamation can still expose a healthy incumbent after missed renewal. The evidence supports incumbent-preserving admission when effects are completely mediated, task ownership is authenticated, and lease expiry reflects incumbent liveness.

We apply our complex-scaled time-evolution operator to the two-proton decay of $^{6}$Be. The nucleus is described as an $\alpha+p+p$ three-body system with explicit Jacobi-coordinate rearrangement and the realistic Argonne $v8'$ NN interaction for the proton-proton subsystem. Starting from a confined initial wave packet, the decay dynamics are described by expansion over the complex-scaled eigenstates of the final Hamiltonian. The decay width extracted from the late-time survival probability agrees closely with that obtained from the CSM resonance pole. The time-dependent densities in different Jacobi coordinates reveal complementary aspects of the evolving three-body geometry, while the spin-singlet component remains dominant during the decay. In particular, the correlated two-proton configuration persists even in the presence of the strong short-range repulsion of the realistic NN interaction. These results demonstrate the applicability of the complex-scaled time-evolution framework to explicit three-body decay dynamics and provide a consistent description of the decay width, spatial evolution, and spin correlations of $^{6}$Be.

Computer-vision systems used for construction monitoring can degrade under adverse environmental and visual conditions, yet such conditions remain underrepresented in existing construction image datasets. We present ConSynth-X, a paired synthetic construction-site image dataset containing 34,199 images derived from 3,109 real-world source scenes. The dataset comprises 11 condition-specific subsets spanning precipitation, fog, nighttime illumination, adverse weather at night, and small-object or long-distance views. Each synthetic image is linked to its corresponding source scene, enabling controlled comparison across environmental and visual conditions. ConSynth-X includes source-derived annotations, generation metadata, provenance information, and image-quality indicators, supporting object detection, image captioning, visual grounding, and visual question answering. Technical validation evaluates source-synthetic fidelity and alignment with real adverse-condition imagery using embedding-based similarity and distributional analyses. The dataset provides a structured resource for evaluating and improving the robustness of construction vision and vision-language models under challenging field conditions.

The complementarity relations in an interferometer reflect an important phenomenon in quantum mechanics: wave-particle duality. Here, we develop a method to quantify both wave and particle behaviors in a multi-path interferometer. In particular, we find that the trace distance is a good candidate for wave and particle measures. As a result, some duality relations and triality ralations are established respectively. This work not only extends the application of the trace distance to the interferometer, but also opens up new perspectives on the quantification of waveness and particleness.

We present an optomechanical microphone based on a diaphragm-integrated photonic waveguide Mach-Zehnder interferometer. Acoustic pressure deforms the MEMS diaphragm, inducing strain in the sensing waveguide and changing its optical path length. We analytically evaluate the optical and mechanical transduction mechanisms and key figures of merit, including signal-to-noise ratio, dynamic range, acoustic overload pressure, and minimum detectable pressure. Two design cases are considered: a MEMS microphone and a measurement microphone. The results indicate competitive performance but no substantial overall advantage over state-of-the-art microphones in conventional applications. The architecture may nevertheless offer advantages for high-temperature and other harsh-environment sensing applications.

Service organizations use simulation, pilot studies, and historical data to select service policies that balance aggregate performance and fairness. Existing work primarily evaluates the operational cost of fairness, defined as the performance loss caused by restricting the feasible policy set. In this research, we identify and study implementation cost as a distinct and equally important dimension of fairness, defined as the sampling effort required to verify fairness and reliably select the best fair policy. Specifically, we consider fixed-budget policy selection under fairness requirements based on mean performance, under-service risk, quantiles, and upper tail outcomes, and show that different fairness metrics can induce the same fair policy set and the same operational cost while requiring substantially different amounts of evidence because their estimators have different local verification difficulty. We further show that, near a critical fairness tolerance, the required sampling budget for implementation scales inversely with the square of the distance to the boundary, creating a finite-budget implementation gap. These results imply that the fairness metric, the fairness tolerance, and the available sampling budget should be designed jointly, because a population-level fairness requirement may be operationally attractive but not statistically implementable at the available budget. To translate this perspective into practice, we develop a fairness-guided adaptive allocation algorithm that directs samples toward the ranking and fairness verification comparisons governing false selection. Experiments in synthetic, call center, and emergency department settings demonstrate the implementation cost mechanisms identified in this work and show that the proposed allocation uses the sampling budget much more effectively.

Chart reasoning agents are increasingly used to extract actionable insights in critical domains, achieving state-of-the-art performance on multiple benchmarks. Yet, high benchmark accuracy alone is insufficient for deployment, where stakeholders must be able to audit and verify how a model reaches its answer. Existing LVLM-based chart agents produce either answer-only predictions or free-form rationales that are hard to verify, obscuring whether an error arose from misreading the chart, extracting a wrong value, or miscomputing. We propose Chart-RVR, a reinforcement learning framework for training monitorable chart agents with verifiable process rewards. Chart-RVR decomposes chart reasoning into three auditable blocks: Structure, identifying the chart type; Evidence, reconstructing the underlying data table in JSON; and Derivation, exposing the stepwise trace that computes the answer. Across six in-domain and out-of-domain benchmarks, Chart-RVR attains state-of-the-art accuracy among comparable-sized LVLMs. Beyond accuracy, we assess monitorability using a triangulated protocol that combines ground-truth surrogate metrics, an oracle information-gain measure, and an LLM-as-auditor scoring Process Verifiability and Evidence Localization, showing that Chart-RVR yields rationales that are markedly more verifiable and evidence-grounded than those from CoT prompting, SFT, and existing chart-specific baselines.

We prove quantitative stability and instability estimates for reversible periodic solutions of strictly Legendre-convex Lagrangian systems. If the configuration dimension is $n$, the fixed-period Morse index is $k$, and $k_D,\nu_D$ are the Dirichlet index and nullity on a half-period, set \[ \widetilde k=k-2k_D-\nu_D. \] Then the algebraic number $\nu_+$ of positive real Floquet multipliers satisfies \[ \nu_+\ge 2n-2\widetilde k, \] and we obtain corresponding bounds for the positive real spectrum away from $1$, allowing arbitrary Jordan chains at the unit multiplier. Hence at most $2\widetilde k$ Floquet multipliers can lie outside the positive real axis. In particular, when $k=0$ we recover the positive-real-spectrum theorem for reversible minimizers, while for nonconstant autonomous brake orbits the bounds improve by two. If, moreover, $n\ge2$ and the algebraic multiplicity of $1$ is two, an index-one brake orbit is transversely hyperbolic, with $n-1$ stable and $n-1$ unstable directions. Decoupled oscillators show that the refined bounds can be sharp even when the estimate based on $k$ alone is vacuous, and the Hénon–Heiles straight-line orbit provides a nonlinear illustration of the half-period mechanism.

Point cloud-based CFD enables flow analysis directly on discrete points obtained from 3D scanning and medical imaging, bypassing surface reconstruction, geometry cleanup, and boundary-fitted mesh generation. Derived from immersogeometric analysis, the method immerses the point cloud in a background mesh and enforces no-slip conditions on discrete points through a Nitsche-based weak boundary condition (BC). The framework delivers accurate velocity fields, pressure distribution, and integrated loads; however, accurate prediction of the local wall shear stress (WSS) has remained a critical challenge. The geometry intersects the background mesh arbitrarily, producing cut elements that lack the regularity required for consistent gradient evaluation. The issue is compounded by the stabilization term of the weak BC, whose parameter estimation in the symmetric Nitsche formulation is dependent on the cut configuration and affects the variationally consistent definition of traction from which the WSS is computed. In this work, we propose a new method to obtain accurate wall shear stress in immersed flow analysis with application to point cloud-based CFD, using a non-symmetric Nitsche's formulation with near-wall modeling and a patch-based stress recovery approach with traction compatibility. The method is validated on canonical benchmarks and applied to turbulent flow past a sphere and to a patient-specific aorta, showcasing excellent agreement with reference results.

Dexterous grasping in clutter poses a basic sensing question: when do tactile measurements and external wrench estimates improve on visual geometry? Occlusion and contact can obscure grasp quality, motivating a controlled evaluation of these interaction signals. We present a controlled real-world study over five tabletop scene conditions on a dexterous hand (LEAP Hand) with vision, per-finger and wrist wrench estimates, and distributed fingertip taxels. With demonstrations, visual observations, action space, and compliant control fixed, we compare diffusion policies trained by imitation learning on vision only, wrench, taxels, or all combined, plus representation and fusion baselines. The combined policy succeeds in 24/25 trials versus 14/25 for vision only, and 15/15 versus 6/15 across the three confined conditions. Ablations show that wrench and taxel feedback are complementary. Behavioral comparisons show that interaction feedback enables earlier rejection of inadequate contacts, regrasping before lift, and more stable grasps. To our knowledge, this is the first real-world study to combine and separately evaluate these interaction modalities for target-oriented dexterous grasping in clutter. These results chart a widening vision-interaction gap and position cluttered dexterous grasping as a benchmark for determining when the learned policy needs interaction sensing. Project website: https://interaction-dex-grasp.github.io/

In order to explain an elastic collision, Johann Bernoulli considered two bodies connected by a spring. Motion is defined as a succession of states of rest. Then, considering the spring to be a lever with a body at each extremity, the laws of equilibrium imply that `motion' is described by the time variation of the (common) quantity of motion of the bodies; the dynamic law is thus deduced. This inspires a teaching sequence to introduce the dynamic law (in one dimension) in introductory physics course; we call it “bernoullian sequence”.

Best-of-$N$ is a widely used inference strategy for complex reasoning, whose effectiveness depends on whether sampled candidates can cover diverse and high-quality reasoning paths. However, post-trained reasoning models often suffer from exploration collapse, where independent rollouts repeatedly follow similar reasoning paths and limit the gains from increasing the rollout budget. Existing methods alleviate this issue by promoting broader exploration, but do not explicitly guide exploration toward continuations that make meaningful progress, resulting in limited exploration efficiency. To address this, we propose \emph{\underline{S}tate-conditioned \underline{P}rogress-guided \underline{S}teering} (SPS), a training-free latent steering framework. Specifically, SPS constructs a state-conditioned Direction Bank containing multiple progress-guided steering vectors for different prefix-state regions. During online inference, SPS retrieves a suitable steering vector based on the current prefix state and applies it at high-uncertainty transitions to guide the next reasoning step toward meaningful progress. Extensive experiments across multiple model scales and benchmarks demonstrate that SPS consistently outperforms strong baselines. Further analyses validate the effectiveness of its key designs and offer valuable insights for future research. The code is available at https://github.com/rattlesnakey/SPS.

A topological vector space E has the countable intersection property for subspaces if every family of closed subspaces intersecting to $0$ contains a countable subfamily that also intersects to $0$. We prove that every separable F-space satisfies this property, whereas a TVS with an uncountable Markushevich basis does not. Finally, we construct a separable TVS that fails to satisfy this property.

New Guinea is the world's richest island flora (~2,856 orchid species), yet most species are represented by only a handful of photographs, far fewer than direct species-level classification requires. Methods for fine-grained identification in such species-rich, data-poor floras are needed, and it remains unclear which backbone architecture and pretraining strategy best support them. We built a two-stage system that first predicts the genus of a query photograph, then retrieves visually similar reference images of candidate species using FAISS. We compared four pretrained backbones – two Vision Transformers (ViTs; DINOv2, BioCLIP 2) and two CNNs (ConvNeXt V2-L, EfficientNetV2-L) – fine-tuned under an identical protocol on a fixed, species-stratified partition of 16,701 photographs spanning 120 genera and 1,350 species, assessing accuracy, calibration, error structure, species retrieval, and open-set detection of novel genera. DINOv2 attained the best genus performance (macro top-1 66.9%, 95% CI 63.7-70.6; global top-1 88.9%); both ViTs outranked both CNNs, and general-purpose self-supervised pretraining (DINOv2) outperformed domain-matched biological pretraining (BioCLIP 2) by 7.1 points of macro top-1. Errors concentrated on two abundant genera acting as error attractors. DINOv2 embeddings achieved species Recall@5 of 86.6% and genus Recall@5 of 98.7%; temperature scaling reduced every backbone's Expected Calibration Error to about 0.03; and a distance-based open-set gate flagged unseen genera (mean AUROC 0.958). A self-supervised Vision-Transformer backbone combined with embedding retrieval is an effective, deployable strategy for fine-grained identification in species-rich, data-poor floras. The system is released as an open web application (the New Guinea Orchid Identifier), offering a practical template for other hyperdiverse, under-documented taxa.

We derive exact finite-size nonperturbative representations of equilibrium modal occupations and related statistics for three representative nonlinear systems: the Majda-McLaughlin-Tabak dispersive-wave model, the Fermi-Pasta-Ulam-Tsingou beta anharmonic chain, and the discrete nonlinear Schrodinger lattice field. Independent simulations confirm the predictions from weak to strong nonlinearity. For DNLS, the theory remains accurate across the weak-coupling quasicondensation crossover, where large low-mode occupations and long-range coherence amplify interaction effects even when the bare nonlinear coefficient is small. The finite-ring DNLS occupations are further resolved into a positive sum of Rayleigh-Jeans channels with distinct correlation lengths, explaining when a single Rayleigh-Jeans law applies and why it fails near quasicondensation. In MMT and DNLS, the exact occupations also determine the mean modal frequencies even when the dynamical spectra broaden or split. The nonperturbative results allow a direct assessment of two representative perturbative approaches. Treating the mean interaction appropriately yields accurate low-order approximations, including at strong nonlinearity. At higher orders, however, the corrections cease to decrease and successive approximations oscillate with increasing amplitude; both finite-order approaches also fail near weak-coupling quasicondensation. Thus neither low-order agreement nor a small bare coupling guarantees a reliable perturbative description. The results establish nonperturbative equilibrium theory for widely used nonlinear wave and lattice models and provide a quantitative basis for modal distributions of energy, particles, and optical power in nonlinear optics, dispersive waves, anharmonic lattices, and cold-atom systems.