Daily Papers

The quantum Hall effect, topological insulators and quantum capacitance are usually treated as distinct topics, yet they share a common thread. Bulk topology determines a protected boundary state, and quantum capacitance is one of the few electrostatic methods able to detect the density of states. This paper examines that connection. The integer quantum Hall effect showed that quantised boundary transport has a topological origin given by the Chern number, later extended to zero field in Z2 topological insulators with helical edge and Dirac surface states. Quantum capacitance, developed for two-dimensional electron gases, has since become a key technique for mapping the Landau level density of states underlying the quantum Hall effect and the linear Dirac density of states of topological surface states. This picture is complicated by self-consistent screening theory, which shows that the quantum Hall bulk splits into compressible and incompressible regions that shift with field. Local quantum capacitance measurements confirm that the bulk is not uniformly insulating across a quantised plateau, even though the plateau stays quantised. The paper argues that quantum capacitance is not a secondary feature of these phenomena but one of the clearest windows into the boundary density of states that topology imposes, and that it reveals the local, field-dependent nature of the insulating bulk assumption underlying the analogy between the quantum Hall effect and topological insulators.

Self-consistency (SC) is a widely used test-time inference technique for improving performance in chain-of-thought reasoning. It involves generating multiple responses, or samples from a large language model (LLM) and selecting the most frequent answer. This procedure can naturally be viewed as a majority vote or empirical mode estimation. Despite its effectiveness, SC is prohibitively expensive at scale when naively applied to datasets, and it lacks a unified theoretical treatment of sample efficiency and scaling behavior. In this paper, we provide the first comprehensive analysis of SC's scaling behavior and its variants, drawing on mode estimation and voting theory. We derive and empirically validate power law scaling for self-consistency across datasets, and analyze the sample efficiency for fixed-allocation and dynamic-allocation sampling schemes. From these insights, we introduce Blend-ASC, a novel variant of self-consistency that dynamically allocates samples to questions during inference, achieving state-of-the-art sample efficiency. Our approach uses 6.8x fewer samples than vanilla SC on average, outperforming both fixed- and dynamic-allocation SC baselines, thereby demonstrating the superiority of our approach in terms of efficiency. In contrast to existing variants, Blend-ASC is hyperparameter-free and can fit an arbitrary sample budget, ensuring it can be easily applied to any self-consistency application.

In this note, we show that the Local Molecular Field theory of Weeks et. al. can be re-derived as an extremum problem for an approximate Helmholtz free energy. Using the resulting free energy as a classical, fluid density functional yields an implicit solvent method identical in form to the Molecular Density Functional theory of Borgis et. al., but with an explicit formula for the 'ideal' free energy term. This new expression for the ideal free energy term can be computed from all-atom molecular dynamics of a solvent with only short-range interactions. The key hypothesis required to make the theory valid is that all smooth (and hence long-range) energy functions obey Gaussian statistics. This is essentially a random phase approximation for perturbations from a short-range only, 'reference,' fluid. This single hypothesis is enough to prove that the self-consistent LMF procedure minimizes a novel density functional whose 'ideal' free energy is the molecular system under a specific, reference Hamiltonian, as opposed to the non-interacting gas of conventional density functionals. Implementation of this new functional into existing software should be straightforward and robust.

For the large and chemically diverse GMTKN55 benchmark suite, we have studied the performance of density-corrected density functional theory (HF-DFT), compared to self-consistent DFT, for several pure and hybrid GGA and meta-GGA exchange-correlation (XC) functionals (PBE, BLYP, TPSS, SCAN) as a function of the percentage of HF exchange in the hybrid. The D4 empirical dispersion correction has been added throughout. For subsets dominated by dynamical correlation -- particularly noncovalent interaction subsets -- HF-DFT is highly beneficial, particularly at low HF exchange percentages. For subsets with significant static correlation (i.e., where a Hartree-Fock determinant is not a good zero-order wavefunction), HF-DFT may do more harm than good. While the self-consistent series show optima at or near 37.5% (i.e., 3/8) for all four XC functionals -- consistent with Grimme's proposal of the PBE38 functional -- HF-BnLYP-D4, HF-PBEn-D4, and HF-TPSSn-D4 all exhibit minima nearer 25% (i.e., 1/4). Intriguingly, for HF-SCANn-D4, the minimum is near 10%, but the weighted mean absolute error (WTMAD2) for GMTKN55 is only barely lower than that of HF-SCAN-D4 (i.e., where the post-HF step is a pure meta-GGA). The latter becomes an attractive option, only slightly more costly than pure Hartree-Fock, and devoid of adjustable parameters other than the three in the dispersion correction. Moreover, its WTMAD2 is only surpassed by the highly empirical M06-2X and by the combinatorically optimized empirical range-separated hybrids wB97X-V and wB97M-V.

Experiments in recent years on high T_c superconductors find a puzzling nodal scale-covariant self-energy with an exponent varying continuously with doping. We propose a mechanism: nonlocality induced by poorly screened effective repulsions V_α(r) sim 1/r^α, where a continuously doping-dependent exponent 1 le αle 3 naturally interpolates between the Mott insulating and Fermi liquid limits. We develop a phenomenology of hydrodynamic screening, finding a scale-covariant quasiparticle decay rate Γ(ω,T) propto T^γ Φ(ω/T) in energy ω and temperature T, with γ= 2-1α for nonlocal 1 < α< 2. Our results naturally capture the optimally doped to overdoped regimes, whereas the underdoped regime is qualitatively distinct. In our theory, spectroscopy-fitted exponents directly probe the charged fluid's effective spatial nonlocality. Nonlocality shows that quantum criticality is not necessary to explain scale-covariant phenomena.

Iterative self-training (self-distillation) repeatedly refits a model on pseudo-labels generated by its own predictions. We study this procedure in overparameterized linear regression: an initial estimator is trained on noisy labels, and each subsequent iterate is trained on fresh covariates with noiseless pseudo-labels from the previous model. In the high-dimensional regime, we derive deterministic-equivalent recursions for the prediction risk and effective noise across iterations, and prove that the empirical quantities concentrate sharply around these limits. The recursion separates two competing forces: a systematic component that grows with iteration due to progressive signal forgetting, and a stochastic component that decays due to denoising via repeated data-dependent projections. Their interaction yields a U-shaped test-risk curve and an optimal early-stopping time. In spiked covariance models, iteration further acts as an iteration-dependent spectral filter that preserves strong eigendirections while suppressing weaker ones, inducing an implicit form of soft feature selection distinct from ridge regression. Finally, we propose an iterated generalized cross-validation criterion and prove its uniform consistency for estimating the risk along the self-training trajectory, enabling fully data-driven selection of the stopping time and regularization. Experiments on synthetic covariances validate the theory and illustrate the predicted denoising-forgetting trade-off.

Recursive synthetic-data training can degrade a fitted model, but its effects on the variable-screening stage that precedes model fitting are less understood. When each generation labels data for the next, a marginal screener ranks variables against the current pseudo-label target rather than the original response. Across generations, this target drift can erode or reverse the active--inactive score margin and cause screening to discard variables that clean-label screening would retain. We call this process screening collapse. In a Gaussian linear model with minimum-norm recursive generators, we establish finite-sample screening bounds, characterize a late-generation information-loss limit, and derive a minimax impossibility result. Screening collapse can precede and compound ordinary model collapse: omitted variables incur a covariance-weighted loss that no refit restricted to the selected variables can eliminate, while recursive labels further degrade estimation on the retained variables. We also examine two approaches to mitigating screening collapse. Fixed-fraction real-label retention preserves a clean-label component but leaves resolution fixed, whereas data accumulation can avoid screening collapse if the pooled target preserves a positive margin and the growing sample resolves it. Numerical experiments on gene-expression data support the theoretical findings.

Predicting the relationship between surface charge density and electrostatic potential for spherical particles remains a fundamental challenge in colloid science. Because the governing non-linear Poisson--Boltzmann equation lacks a general exact analytical solution, researchers typically rely on numerical calculations or various semi-empirical approximations. In this paper, we overcome these limitations by developing a dual-asymptotic framework that provides explicit, closed-form expressions for this surface charge--potential relationship across the entire spectrum of particle curvature. For weakly curved systems, where the Debye length λ_D is much smaller than the particle radius R (λ_D/R ll 1), a formal mathematical derivation rigorously establishes the Ohshima--Healy--White formula as an exact regular perturbation expansion. Conversely, for highly curved spheres (R/λ_D ll 1), we employ singular perturbation analysis using the scaled particle radius as the small parameter. This approach provides a first-principles mathematical justification for the spherical Debye--Hückel theory, proving that its leading-order expansion remains asymptotically exact within the full non-linear Poisson--Boltzmann framework due to the geometric deactivation of non-linearity. Crucially, we map the exact limits of this geometric regulation, demonstrating how non-linear screening re-emerges as the particle radius increases, with the breakdown threshold governed by the interplay between curvature and surface charge density.

I investigate the spontaneous-fission (SF) half-lives of ^{254,256}Rf within the multidimensionally constrained relativistic Hartree-Bogoliubov (MDC-RHB) model combined with the local quasiparticle random-phase approximation (LQRPA) method. The collective inertia is evaluated using the finite-amplitude method (FAM), which enables a fully self-consistent treatment of dynamical residual effects beyond the conventional cranking approximation. The effective collective potential is constructed by subtracting both the vibrational zero-point energy and rotational energy correction from the deformation energy surface. The SF half-lives are evaluated within the Wentzel-Kramers-Brillouin (WKB) approximation. Compared with the nonperturbative and perturbative cranking approximation, the FAM-LQRPA collective inertia is significantly enhanced, especially around the ground-state minimum, due to the dynamical residual interaction included in the QRPA. As a consequence, the predicted SF half-lives increase by several orders of magnitude and become consistent with experimental data. The calculated SF half-lives are 5.65 μs for ^{254}Rf and 0.95 ms for ^{256}Rf, both within approximately two orders of magnitude of the corresponding experimental values. These results indicate that a self-consistent treatment of dynamical residual effects and beyond mean-field correlations is crucial for quantitative microscopic descriptions of SF within covariant density functional theory.

We present dla-ideal-solver, a high-performance framework for simulating two-dimensional Diffusion-Limited Aggregation (DLA) using Numba-accelerated Python. By leveraging just-in-time (JIT) compilation, we achieve computational throughput comparable to legacy static implementations while retaining high-level flexibility. We investigate the Laplacian growth instability across varying injection geometries and walker concentrations. Our analysis confirms the robustness of the standard fractal dimension D_f approx 1.71 for dilute regimes, consistent with the Witten-Sander universality class. However, we report a distinct crossover to Eden-like compact growth (D_f approx 1.87) in high-density environments, attributed to the saturation of the screening length. Beyond standard mass-radius scaling, we employ generalized Rényi dimensions and lacunarity metrics to quantify the monofractal character and spatial heterogeneity of the aggregates. This work establishes a reproducible, open-source testbed for exploring phase transitions in non-equilibrium statistical mechanics.

Self-distillation (SD) is the process of retraining a student on a mixture of ground-truth labels and the teacher's own predictions using the same architecture and training data. Although SD has been empirically shown to often improve generalization, its formal guarantees remain limited. We study SD for ridge regression in unconstrained setting in which the mixing weight ξ may be outside the unit interval. Conditioned on the training data and without any distributional assumptions, we prove that for any squared prediction risk (including out-of-distribution), the optimally mixed student strictly improves upon the ridge teacher for every regularization level λ> 0 at which the teacher ridge risk R(λ) is nonstationary (i.e., R'(λ) neq 0). We obtain a closed-form expression for the optimal mixing weight ξ^star(λ) for any value of λ and show that it obeys the sign rule: sign(ξ^star(λ))=-sign(R'(λ)). In particular, ξ^star(λ) can be negative, which is the case in over-regularized regimes. To quantify the risk improvement due to SD, we derive exact deterministic equivalents for the optimal SD risk in the proportional asymptotics regime (where the sample and feature sizes n and p both diverge but their aspect ratio p/n converges) under general anisotropic covariance and deterministic signals. Our asymptotic analysis extends standard second-order ridge deterministic equivalents to their fourth-order analogs using block linearization, which may be of independent interest. From a practical standpoint, we propose a consistent one-shot tuning method to estimate ξ^star without grid search, sample splitting, or refitting. Experiments on real-world datasets and pretrained neural network features support our theory and the one-shot tuning method.

Self-consistency (SC), a widely used decoding strategy for chain-of-thought reasoning, shows significant gains across various multi-step reasoning tasks but comes with a high cost due to multiple sampling with the preset size. Its variants, Adaptive self-consistency (ASC) and Early-stopping self-consistency (ESC), dynamically adjust the number of samples based on the posterior distribution of a set of pre-samples, reducing the cost of SC with minimal impact on performance. Both methods, however, do not exploit the prior information about question difficulty. It often results in unnecessary repeated sampling for easy questions that could be accurately answered with just one attempt, wasting resources. To tackle this problem, we propose Difficulty-Adaptive Self-Consistency (DSC), which leverages the difficulty information from both prior and posterior perspectives to adaptively allocate inference resources, further reducing the cost of SC. To demonstrate the effectiveness of DSC, we conduct extensive experiments on three popular categories of reasoning tasks: arithmetic, commonsense and symbolic reasoning on six benchmarks. The empirical results show that DSC consistently surpasses the strong baseline ASC and ESC in terms of costs by a significant margin, while attaining comparable performances.

Self-Consistency improves reasoning reliability through multi-sample aggregation, but incurs substantial inference cost. Adaptive self-consistency methods mitigate this issue by adjusting the sampling budget; however, they rely on count-based stopping rules that treat all responses equally, often leading to unnecessary sampling. We propose Reliability-Aware Adaptive Self-Consistency (ReASC), which addresses this limitation by reframing adaptive sampling from response counting to evidence sufficiency, leveraging response-level confidence for principled information aggregation. ReASC operates in two stages: a single-sample decision stage that resolves instances confidently answerable from a single response, and a reliability-aware accumulation stage that aggregates responses by jointly leveraging their frequency and confidence. Across five models and four datasets, ReASC consistently achieves the best accuracy-cost trade-off compared to existing baselines, yielding improved inference efficiency across model scales from 3B to 27B parameters. As a concrete example, ReASC reduces inference cost by up to 70\% relative to self-consistency while preserving accuracy on GSM8K using Gemma-3-4B-it.

Many-body perturbation theory is a powerful framework to calculate spectroscopic quantities, as well as the total energy. However, the field suffers from the fact that different flavors of approximations lead to a wide range of results. We present several in principle exact expansions and discuss how to use them consistently. We concentrate on the first two orders in the respective screened Coulomb interaction. Of special interest is a consistent expansion around the Kohn-Sham system. We illustrate the performance of the various choices using the symmetric Hubbard dimer over a wide range of correlation strength. In particular, the expansion based on exact Kohn-Sham ingredients stands out as a promising route towards good energies and good spectra.

Self-alignment, whereby models learn to improve themselves without human annotation, is a rapidly growing research area. However, existing techniques often fail to improve complex reasoning tasks due to the difficulty of assigning correct rewards. An orthogonal approach that is known to improve correctness is self-consistency, a method applied at inference time based on multiple sampling in order to find the most consistent answer. In this work, we extend the self-consistency concept to help train models. We thus introduce self-consistency preference optimization (ScPO), which iteratively trains consistent answers to be preferred over inconsistent ones on unsupervised new problems. We show ScPO leads to large improvements over conventional reward model training on reasoning tasks such as GSM8K and MATH, closing the gap with supervised training with gold answers or preferences, and that combining ScPO with standard supervised learning improves results even further. On ZebraLogic, ScPO finetunes Llama-3 8B to be superior to Llama-3 70B, Gemma-2 27B, and Claude-3 Haiku.

Test-time scaling can improve model performance by aggregating stochastic reasoning trajectories. However, achieving sample-efficient test-time self-consistency under a limited budget remains an open challenge. We introduce PETS (Principled and Efficient Test-TimeSelf-Consistency), which initiates a principled study of trajectory allocation through an optimization framework. Central to our approach is the self-consistency rate, a new measure defined as agreement with the infinite-budget majority vote. This formulation makes sample-efficient test-time allocation theoretically grounded and amenable to rigorous analysis. We study both offline and online settings. In the offline regime, where all questions are known in advance, we connect trajectory allocation to crowdsourcing, a classic and well-developed area, by modeling reasoning traces as workers. This perspective allows us to leverage rich existing theory, yielding theoretical guarantees and an efficient majority-voting-based allocation algorithm. In the online streaming regime, where questions arrive sequentially and allocations must be made on the fly, we propose a novel method inspired by the offline framework. Our approach adapts budgets to question difficulty while preserving strong theoretical guarantees and computational efficiency. Experiments show that PETS consistently outperforms uniform allocation. On GPQA, PETS achieves perfect self-consistency in both settings while reducing the sampling budget by up to 75% (offline) and 55% (online) relative to uniform allocation. Code is available at https://github.com/ZDCSlab/PETS.

Self-correction is an effective technique for maintaining parallel sampling in discrete diffusion models with minimal performance degradation. Prior work has explored self-correction at inference time or during post-training; however, such approaches often suffer from limited generalization and may impair reasoning performance. GIDD pioneers pretraining-based self-correction via a multi-step BERT-style uniform-absorbing objective. However, GIDD relies on a continuous interpolation-based pipeline with opaque interactions between uniform transitions and absorbing masks, which complicates hyperparameter tuning and hinders practical performance. In this work, we propose a Self-Correcting Discrete Diffusion (SCDD) model to reformulate pretrained self-correction with explicit state transitions and learn directly in discrete time. Our framework also simplifies the training noise schedule, eliminates a redundant remasking step, and relies exclusively on uniform transitions to learn self-correction. Experiments at the GPT-2 scale demonstrate that our method enables more efficient parallel decoding while preserving generation quality.

Large language models (LLMs) are expected to respond accurately but often exhibit deficient reasoning or generate hallucinatory content. To address these, studies prefixed with ``Self-'' such as Self-Consistency, Self-Improve, and Self-Refine have been initiated. They share a commonality: involving LLMs evaluating and updating itself to mitigate the issues. Nonetheless, these efforts lack a unified perspective on summarization, as existing surveys predominantly focus on categorization without examining the motivations behind these works. In this paper, we summarize a theoretical framework, termed Internal Consistency, which offers unified explanations for phenomena such as the lack of reasoning and the presence of hallucinations. Internal Consistency assesses the coherence among LLMs' latent layer, decoding layer, and response layer based on sampling methodologies. Expanding upon the Internal Consistency framework, we introduce a streamlined yet effective theoretical framework capable of mining Internal Consistency, named Self-Feedback. The Self-Feedback framework consists of two modules: Self-Evaluation and Self-Update. This framework has been employed in numerous studies. We systematically classify these studies by tasks and lines of work; summarize relevant evaluation methods and benchmarks; and delve into the concern, ``Does Self-Feedback Really Work?'' We propose several critical viewpoints, including the ``Hourglass Evolution of Internal Consistency'', ``Consistency Is (Almost) Correctness'' hypothesis, and ``The Paradox of Latent and Explicit Reasoning''. Furthermore, we outline promising directions for future research. We have open-sourced the experimental code, reference list, and statistical data, available at https://github.com/IAAR-Shanghai/ICSFSurvey.

Electron-electron (e-e) and electron-phonon (e-ph) interactions are challenging to describe in correlated materials, where their joint effects govern unconventional transport, phase transitions, and superconductivity. Here we combine first-principles e-ph calculations with dynamical mean field theory (DMFT) as a step toward a unified description of e-e and e-ph interactions in correlated materials. We compute the e-ph self-energy using the DMFT electron Green's function, and combine it with the e-e self-energy from DMFT to obtain a Green's function including both interactions. This approach captures the renormalization of quasiparticle dispersion and spectral weight on equal footing. Using our method, we study the e-ph and e-e contributions to the resistivity and spectral functions in the correlated metal Sr_2RuO_4. In this material, our results show that e-e interactions dominate transport and spectral broadening in the temperature range we study (50-310~K), while e-ph interactions are relatively weak and account for only sim10\% of the experimental resistivity. We also compute effective scattering rates, and find that the e-e interactions result in scattering several times greater than the Planckian value k_BT, whereas e-ph interactions are associated with scattering rates lower than k_BT. Our work demonstrates a first-principles approach to combine electron dynamical correlations from DMFT with e-ph interactions in a consistent way, advancing quantitative studies of correlated materials.

We propose modifications to the functional form of the SCAN density functional to eliminate numerical instabilities. This is necessary to allow reliable, automatic generation of pseudopotentials (including PAW potentials). The regularized SCAN is designed to match the original form very closely, and we show that its performance remains comparable.

Self-training is a well-known approach for semi-supervised learning. It consists of iteratively assigning pseudo-labels to unlabeled data for which the model is confident and treating them as labeled examples. For neural networks, softmax prediction probabilities are often used as a confidence measure, although they are known to be overconfident, even for wrong predictions. This phenomenon is particularly intensified in the presence of sample selection bias, i.e., when data labeling is subject to some constraint. To address this issue, we propose a novel confidence measure, called T-similarity, built upon the prediction diversity of an ensemble of linear classifiers. We provide the theoretical analysis of our approach by studying stationary points and describing the relationship between the diversity of the individual members and their performance. We empirically demonstrate the benefit of our confidence measure for three different pseudo-labeling policies on classification datasets of various data modalities. The code is available at https://github.com/ambroiseodt/tsim.

Computational catalysis is playing an increasingly significant role in the design of catalysts across a wide range of applications. A common task for many computational methods is the need to accurately compute the minimum binding energy - the adsorption energy - for an adsorbate and a catalyst surface of interest. Traditionally, the identification of low energy adsorbate-surface configurations relies on heuristic methods and researcher intuition. As the desire to perform high-throughput screening increases, it becomes challenging to use heuristics and intuition alone. In this paper, we demonstrate machine learning potentials can be leveraged to identify low energy adsorbate-surface configurations more accurately and efficiently. Our algorithm provides a spectrum of trade-offs between accuracy and efficiency, with one balanced option finding the lowest energy configuration, within a 0.1 eV threshold, 86.33% of the time, while achieving a 1331x speedup in computation. To standardize benchmarking, we introduce the Open Catalyst Dense dataset containing nearly 1,000 diverse surfaces and 85,658 unique configurations.

Detecting LLM reasoning failures at inference time without ground-truth labels has motivated a wide range of confidence baselines, including self-consistency, semantic entropy, and P(True), built on within-question sampling and self-evaluation. Operad theory, the formalism for systems built by iterated substitution, suggests a complementary diagnostic: a model's direct answer to a compositional query should agree with the answer it produces by composing a stated decomposition of the same query. We instantiate this idea as operadic consistency (OC), a per-question signal. Across twelve instruction-tuned LLMs (4B to 671B parameters, open-weights and closed-source) on four multi-hop QA datasets, OC is strongly correlated with accuracy on every dataset (Pearson r in [0.86, 0.94], all p leq 0.0004), and is the only signal we evaluate with r geq 0.85 uniformly across all four datasets. Chain-of-thought self-consistency (CoT-SC; Wang et al., 2023) matches OC on HotpotQA and DROP (r = 0.93, 0.87) but drops to r approx 0.45 on MuSiQue and StrategyQA. At the per-question level, OC contributes information beyond CoT-SC and semantic entropy on every dataset (cluster-robust p leq 10^{-16} for the OC coefficient), and the conclusion is robust to additionally controlling for constructed decomposition-aware baselines (p leq 10^{-13}). The same signal yields selective-prediction improvements (accuracy at fixed coverage) over a tuned CoT-SC baseline at the equal-cost K = 3 budget (AUARC lifts of +0.086 to +0.096 and AUROC lifts of +0.092 to +0.164; 95% CIs exclude zero on every cell). On five frontier thinking models, where the decomposition is extracted from the model's own chain of thought, the same equal-cost comparison gives positive selective-prediction point-estimate lift on all 16 (dataset, budget, metric) cells tested, with 95% CIs excluding zero on 12 of the 16.

Tuning the regularization parameter in penalized regression models is an expensive task, requiring multiple models to be fit along a path of parameters. Strong screening rules drastically reduce computational costs by lowering the dimensionality of the input prior to fitting. We develop strong screening rules for group-based Sorted L-One Penalized Estimation (SLOPE) models: Group SLOPE and Sparse-group SLOPE. The developed rules are applicable to the wider family of group-based OWL models, including OSCAR. Our experiments on both synthetic and real data show that the screening rules significantly accelerate the fitting process. The screening rules make it accessible for group SLOPE and sparse-group SLOPE to be applied to high-dimensional datasets, particularly those encountered in genetics.

Serial verification gates are a core reliability primitive in LLM harnesses: a candidate answer is returned only if k verifier calls all accept it. Under conditionally independent gates, the recent Odds Law (arXiv:2606.15712) shows that posterior log-odds grow linearly in k, so failure decays exponentially, and states that "a tight theory of partially correlated verifier cascades remains open." This note gives a minimal such theory. Modeling the per-instance false-accept rate on the generator's own errors as a latent variable αsim G (de Finetti), the exact cascade posterior is ell_k = ell_0 - ln m_k, with m_k the k-th moment of G. Then: (i) ell_k is concave in k for every non-degenerate G -- the Odds Law is its tangent at the first gate and an upper bound; (ii) for Beta(a,b) latents, failure decays polynomially, 1-r_k asymp k^{-b}, with correlation parameter ρ_v = 1/(a+b+1); (iii) a blind-spot atom of mass 1-π at α=1 caps the evidence extractable from any number of gates at -ln(1-π) nats, so reliability saturates below 1; (iv) letting the true-accept rate also vary (βsim H) yields a trichotomy -- gates eventually always help, plateau, or actively harm -- decided by the upper-tail exponents of G and H, with closed-form crossover k^dagger. The mechanism is survivorship: errors surviving gates are the high-α ones. The theory is measurable: R repeated verdicts per instance identify the first R moments of G, so two verdicts identify ρ_v; beta-binomial likelihood and NPMLE recover the reliability curve and the ill-posed ceiling. In synthetic tests, independence-based extrapolation underestimates failure by 20x at k=5 and ~3000x at k=10; the correlated fit at R=8 tracks held-out depths. The practical lever is decorrelation -- changing model family, modality, or evidence source -- not adding gates.

Machine-learned exchange--correlation (XC) functionals offer a route to improve Kohn--Sham density-functional theory without incurring the cost of explicitly correlated electronic-structure methods. Their use in production simulation codes, however, requires a well-defined mapping between the learned model and the host-code density representation. We formulate and implement a Skala-1.1 interface in CP2K through the external GauXC library. CP2K supplies the geometry, Gaussian basis, spin-resolved atomic-orbital density matrix, and communicator, while GauXC evaluates the XC energy, atomic-orbital potential matrix, and available nuclear derivatives. The interface accepts both all-electron and valence-only density matrices. The latter may arise from separable dual-space pseudopotentials or molecular effective-core potentials. Implementation errors are isolated from functional differences by comparing the Perdew--Burke--Ernzerhof (PBE) functional evaluated through GauXC with native CP2K PBE. The resulting interface gives consistent energies, forces validated against finite-difference total-energy checks, and force-based molecular-virial diagnostics for representative molecular cases. The dietGMTKN55 benchmark suite is evaluated with an all-electron Gaussian augmented plane-wave treatment for elements up to bromine and def2 effective-core potentials for the heavier elements. The resulting aggregate mean absolute deviation of 1.255 kcal/mol is within 0.020 kcal/mol of the corresponding Skala reference value of 1.235 kcal/mol. This work establishes a validated molecular implementation of Skala in CP2K through GauXC.

Consistency models have been proposed for fast generative modeling, achieving results competitive with diffusion and flow models. However, these methods exhibit inherent instability and limited reproducibility when training from scratch, motivating subsequent work to explain and stabilize these issues. While these efforts have provided valuable insights, the explanations remain fragmented, and the theoretical relationships remain unclear. In this work, we provide a theoretical examination of consistency models by analyzing them from a flow map-based perspective. This joint analysis clarifies how training stability and convergence behavior can give rise to degenerate solutions. Building on these insights, we revisit self-distillation as a practical remedy for certain forms of suboptimal convergence and reformulate it to avoid excessive gradient norms for stable optimization. We further demonstrate that our strategy extends beyond image generation to diffusion-based policy learning, without reliance on a pretrained diffusion model for initialization, thereby illustrating its broader applicability.

Multireference density functional theory (MR-DFT) provides a pivotal microscopic framework for the description of the ground state properties, low-lying nuclear spectra and transition properties of atomic nuclei. Conventionally, practical implementations of MR-DFT rely on empirically chosen generator coordinates, which may omit relevant collective degrees of freedom and thus fail to capture sufficient collective correlations. Here we introduce the stochastic-basis multireference density functional theory (MR-SDFT). This is an extended scheme that augments the MR-DFT toolkit by (i) generating a diverse ensemble of mean-field reference configurations via a stochastic external field and (ii) selecting a compact subspace with Projection-Selection method. The chosen reference configurations are then linearly superposed within the MR-DFT framework to yield spectroscopic observables. Applying this framework to \nuclide[20]{Ne}, \nuclide[24]{Mg} and \nuclide[28]{Si} with the covariant density functional theory (CDFT), it is demonstrated that the MR-SCDFT leads to lower ground-state energies, smaller point-proton rms radius, and a softer ground-state band compared to the conventional MR-CDFT.

In this work we present a new method for the estimation of Mutual Information (MI) between random variables. Our approach is based on an original interpretation of the Girsanov theorem, which allows us to use score-based diffusion models to estimate the Kullback Leibler divergence between two densities as a difference between their score functions. As a by-product, our method also enables the estimation of the entropy of random variables. Armed with such building blocks, we present a general recipe to measure MI, which unfolds in two directions: one uses conditional diffusion process, whereas the other uses joint diffusion processes that allow simultaneous modelling of two random variables. Our results, which derive from a thorough experimental protocol over all the variants of our approach, indicate that our method is more accurate than the main alternatives from the literature, especially for challenging distributions. Furthermore, our methods pass MI self-consistency tests, including data processing and additivity under independence, which instead are a pain-point of existing methods.

Hydrogen-rich materials under pressure are candidates for conventional phonon-mediated superconductivity, but first-principles electron--phonon-coupling (EPC) calculations are computationally demanding. We develop an interpretable multistage workflow for prioritizing binary hydride superconductors before full EPC calculations. Using controlled cubic X_mH_n prototypes at 200~GPa, we introduce the spectral moment A_1 as an intermediate measure of electron--phonon perturbation strength. In its linewidth-based construction, A_1 remains finite as harmonic phonons soften, allowing related structures with imaginary harmonic modes to be screened; it does not establish superconductivity or physical realizability for unstable structures. For dynamically stable cubic systems, A_1 correlates strongly with the H-1s density-of-states fraction at the Fermi level, α, a projected Fermi-surface descriptor, β, and atomic number density, ρ. These trends are summarized heuristically by A_1approx hαβρ, where h is framework- and pressure-dependent; the descriptor is used for ranking, not as a universal T_c predictor. We incorporate αβρ and enthalpic competitiveness into a three-stage evolutionary search comprising multiobjective structural screening, harmonic-phonon stability filtering, and coarse density-functional perturbation-theory (DFPT) EPC calculations. Application to more than 100,000 binary hydride structures at 50 and 200~GPa identifies candidates and prioritizes NaH_6 structural families. Higher-accuracy calculations show that Pm3m NaH_6 is dynamically stable down to 50~GPa and has a harmonic Allen--Dynes estimate of T_capprox230~K. The framework combines electronic descriptors with progressively more expensive stability and EPC calculations for scalable hydride exploration.

Given a black-box AI system and a task, at what confidence level can a practitioner trust the system's output? We answer with a reliability level -- a single number per system-task pair, derived from self-consistency sampling and conformal calibration, that serves as a black-box deployment gate with exact, finite-sample, distribution-free guarantees. Self-consistency sampling reduces uncertainty exponentially; conformal calibration guarantees correctness within 1/(n+1) of the target level, regardless of the system's errors -- made transparently visible through larger answer sets for harder questions. Weaker models earn lower reliability levels (not accuracy -- see Definition 2.4): GPT-4.1 earns 94.6% on GSM8K and 96.8% on TruthfulQA, while GPT-4.1-nano earns 89.8% on GSM8K and 66.5% on MMLU. We validate across five benchmarks, five models from three families, and both synthetic and real data. Conditional coverage on solvable items exceeds 0.93 across all configurations; sequential stopping reduces API costs by around 50%.

Semi-supervised semantic segmentation has long turned on one question, which pseudo-labels to trust, and a generation of selection rules, dynamic thresholds, per-class curricula, soft confidence weights, answered it for the noisy, under-confident ResNet teachers of their day. Self-supervised foundation encoders change the regime: with a DINOv2 teacher, confidence saturates, so the filtering that helped a weak teacher can hurt a strong one. We propose CW-BASS v2, a saturation-aware pseudo-label selection method that reads the teacher's confidence regime rather than committing to one rule. It pairs held-out calibration, an unbiased per-class noise estimate, with a self-adaptive confidence floor that provably bounds retention away from 1, and combines them in a one-pass gate: measure the reliability of the teacher's confident set, pi_kept = Pr[correct | c >= tau], on a held-out slice, and filter strictly when it meets the confidence demanded (pi_kept >= tau), falling back to the adaptive floor otherwise. The boundary is the pre-existing operating threshold, not a value tuned to mIoU, and across six DINOv2 teachers it makes the correct strict-vs-floor call blind. CW-BASS v2 thus recovers the UniMatch V2 operating point on the saturated benchmarks by selecting strict (Pascal VOC 1/8 87.4 against its reported 87.9; Cityscapes within 0.5), and improves on it where the confident set is unreliable (pi_kept ~ 89%, ADE20K), where the floor edges ahead (+1.5 mIoU, single seed). The gate is principled because the failure it avoids is measured, not assumed: on a reliable, saturated teacher the confidence distribution's dynamic range collapses (98% of Pascal pixels >= 0.95), so an adaptive cutoff floods the retention mask and self-training decays into confirmation bias.

Two critical limitations of organic-inorganic lead halide perovskite materials for solar cells are their poor stability in humid environments and inclusion of toxic lead. In this study, high-throughput density functional theory (DFT) methods are used to computationally model and screen 1845 halide perovskites in search of new materials without these limitations that are promising for solar cell applications. This study focuses on finding materials that are comprised of nontoxic elements, stable in a humid operating environment, and have an optimal bandgap for one of single junction, tandem Si-perovskite, or quantum dot-based solar cells. Single junction materials are also screened on predicted single junction photovoltaic (PV) efficiencies exceeding 22.7%, which is the current highest reported PV efficiency for halide perovskites. Generally, these methods qualitatively reproduce the properties of known promising nontoxic halide perovskites that have either been experimentally evaluated or predicted from theory. From a set of 1845 materials, 15 materials pass all screening criteria for single junction cell applications, 13 of which have not been previously investigated, such as (CH3NH3)0.75Cs0.25SnI3, ((NH2)2CH)Ag0.5Sb0.5Br3, CsMn0.875Fe0.125I3, ((CH3)2NH2)Ag0.5Bi0.5I3, and ((NH2)2CH)0.5Rb0.5SnI3. These materials, together with others predicted in this study, may be promising candidate materials for stable, highly efficient, and non-toxic perovskite-based solar cells.

Despite its simplicity and efficacy, the high token expenditure of self-consistency can limit its practical utility. Here we investigate if self-consistency can be made more token-efficient for long chain-of-thought reasoning tasks, while preserving its parallelism, through early hypothesis pruning. Concretely, we generate all solutions in parallel, but periodically prune intermediate hypotheses that are deemed unnecessary based on two lightweight indicators: (a) the model's own confidence in individual hypotheses, and (b) lexical coverage of all current hypotheses by candidate subsets that are under consideration for continued retention. We design a fast weighted set cover algorithm that utilizes the two indicators; our evaluation of five LLMs on three math benchmarks shows that this method can improve token efficiency for all models, by 10-35% in many cases.

We present a SemiEmpirical Theory of Learning (SETOL) that explains the remarkable performance of State-Of-The-Art (SOTA) Neural Networks (NNs). We provide a formal explanation of the origin of the fundamental quantities in the phenomenological theory of Heavy-Tailed Self-Regularization (HTSR): the heavy-tailed power-law layer quality metrics, alpha and alpha-hat. In prior work, these metrics have been shown to predict trends in the test accuracies of pretrained SOTA NN models, importantly, without needing access to either testing or training data. Our SETOL uses techniques from statistical mechanics as well as advanced methods from random matrix theory and quantum chemistry. The derivation suggests new mathematical preconditions for ideal learning, including a new metric, ERG, which is equivalent to applying a single step of the Wilson Exact Renormalization Group. We test the assumptions and predictions of SETOL on a simple 3-layer multilayer perceptron (MLP), demonstrating excellent agreement with the key theoretical assumptions. For SOTA NN models, we show how to estimate the individual layer qualities of a trained NN by simply computing the empirical spectral density (ESD) of the layer weight matrices and plugging this ESD into our SETOL formulas. Notably, we examine the performance of the HTSR alpha and the SETOL ERG layer quality metrics, and find that they align remarkably well, both on our MLP and on SOTA NNs.

Self-consistency (SC), leveraging multiple samples from LLMs, shows significant gains on various reasoning tasks but struggles with free-form generation due to the difficulty of aggregating answers. Its variants, UCS and USC, rely on sample selection or voting mechanisms to improve output quality. These methods, however, face limitations due to their inability to fully utilize the nuanced consensus knowledge present within multiple candidate samples, often resulting in suboptimal outputs. We propose Fine-Grained Self-Consistency (FSC) to addresses these limitations by extracting and integrating segment-level commonalities from candidate samples, enhancing the performance of LLMs both in open-ended and reasoning tasks. Based on this, we present two additional strategies: candidate filtering, which enhances overall quality by identifying highly similar candidate sets, and merging, which reduces input token requirements by combining similar samples. The effectiveness of FSC is demonstrated through extensive experiments on various tasks, including summarization, code generation, and mathematical reasoning, using GPT-3.5-turbo and GPT-4. The results indicate significant improvements over baseline methods, showcasing the potential of FSC to optimize output quality by effectively synthesizing fine-grained consensus knowledge from multiple samples.

The ground-state energy, electron density, and related properties of ordinary matter can be computed efficiently when the exchange-correlation energy as a functional of the density is approximated semilocally. We propose the first meta-GGA (meta-generalized gradient approximation) that is fully constrained, obeying all 17 known exact constraints that a meta-GGA can. It is also exact or nearly exact for a set of appropriate norms, including rare-gas atoms and nonbonded interactions. This SCAN (strongly constrained and appropriately normed) meta-GGA achieves remarkable accuracy for systems where the exact exchange-correlation hole is localized near its electron, and especially for lattice constants and weak interactions.

The search for low-cost, durable, and effective catalysts is essential for green hydrogen production and carbon dioxide upcycling to help in the mitigation of climate change. Discovery of new catalysts is currently limited by the gap between what AI-accelerated computational models predict and what experimental studies produce. To make progress, large and diverse experimental datasets are needed that are reproducible and tested at industrially-relevant conditions. We address these needs by utilizing a comprehensive high-throughput characterization and experimental pipeline to create the Open Catalyst Experiments 2024 (OCX24) dataset. The dataset contains 572 samples synthesized using both wet and dry methods with X-ray fluorescence and X-ray diffraction characterization. We prepared 441 gas diffusion electrodes, including replicates, and evaluated them using zero-gap electrolysis for carbon dioxide reduction (CO_2RR) and hydrogen evolution reactions (HER) at current densities up to 300 mA/cm^2. To find correlations with experimental outcomes and to perform computational screens, DFT-verified adsorption energies for six adsorbates were calculated on sim20,000 inorganic materials requiring 685 million AI-accelerated relaxations. Remarkably from this large set of materials, a data driven Sabatier volcano independently identified Pt as being a top candidate for HER without having any experimental measurements on Pt or Pt-alloy samples. We anticipate the availability of experimental data generated specifically for AI training, such as OCX24, will significantly improve the utility of computational models in selecting materials for experimental screening.

We consider beta-Dyson Brownian motion, with β>= 1, started from a deterministic configuration with uniformly bounded support. Let μ_t be the semicircular free-convolution flow issued from the initial empirical measure, and set S_t = supp(μ_t). For every fixed T, ε> 0, with probability at least 1 - C exp(-(log n)^2), every particle remains within an epsilon-neighborhood of S_t for all 0 <= t <= T. The result holds from time zero, requires no regularity assumption at the initial spectral edges, and applies to multi-cut supports with macroscopic interior gaps. A key ingredient is a deterministic local resolvent exclusion principle: an o((n η)^(-1)) comparison of Stieltjes transforms on a complex disc above a real point separated from the reference support excludes eigenvalues from the corresponding real interval. This gives a model-independent mechanism for converting local resolvent estimates into spectral confinement.

Data contamination poses a significant threat to the reliable evaluation of Large Language Models (LLMs). This issue arises when benchmark samples may inadvertently appear in training sets, compromising the validity of reported performance. While detection methods have been developed for the pre-training and Supervised Fine-Tuning stages, a critical research gap exists for the increasingly significant phase of Reinforcement Learning (RL) post-training. As RL post-training becomes pivotal for advancing LLM reasoning, the absence of specialized contamination detection methods in this paradigm presents a critical vulnerability. To address this, we conduct the first systematic study of data detection within RL post-training scenario and propose Self-Critique. Our method is motivated by a key observation: after RL phase, the output entropy distribution of LLMs tends to collapse into highly specific and sparse modes. Self-Critique probes for the underlying policy collapse, i.e., the model's convergence to a narrow reasoning path, which causes this entropy reduction. To facilitate this research, we also introduce RL-MIA, a benchmark constructed to simulate this specific contamination scenario. Extensive experiments show that Self-Critique significantly outperforms baseline methods across multiple models and contamination tasks, achieving an AUC improvement of up to 30%. Whereas existing methods are close to a random guess for RL-phase contamination, our method makes detection possible.

The independent atom ansatz of density functional theory yields an accurate analytical expression for dynamic correlation energy in the H_{2} molecule: E_{c} = 0.5(1 - 2)(ab|ba) for the atom-additive self-consistent density rho = |a|^{2} + |b|^{2}. Combined with exact atomic self-exchange, it recovers more than 99.5 % of nearly exact SCAN exchange-correlation energy at R > 0.5 A, differing by less than 0.12 eV. The total energy functional correctly dissociates the H-H bond and yields absolute errors of 0.002 A, 0.19 eV, and 13 cm^{-1} relative to experiment at the tight binding computational cost. The chemical bond formation is attributed to the asymptotic Heitler-London resonance of quasi-orthogonal atomic states (- (ab|ba)) with no contributions from kinetic energy or charge accumulation in the bond.

We present TraDE, a self-attention-based architecture for auto-regressive density estimation with continuous and discrete valued data. Our model is trained using a penalized maximum likelihood objective, which ensures that samples from the density estimate resemble the training data distribution. The use of self-attention means that the model need not retain conditional sufficient statistics during the auto-regressive process beyond what is needed for each covariate. On standard tabular and image data benchmarks, TraDE produces significantly better density estimates than existing approaches such as normalizing flow estimators and recurrent auto-regressive models. However log-likelihood on held-out data only partially reflects how useful these estimates are in real-world applications. In order to systematically evaluate density estimators, we present a suite of tasks such as regression using generated samples, out-of-distribution detection, and robustness to noise in the training data and demonstrate that TraDE works well in these scenarios.

Motivated by the recent observation of spatially modulated screening in different paramagnetic heavy-fermion materials, we propose a theoretical mechanism for the emergence of such hybridization waves in frustrated Kondo lattices with no magnetic order. By applying a parton mean-field theory and a perturbative calculation to a minimal Kondo-Heisenberg lattice model, we demonstrate that a heavy Fermi liquid can naturally develop hybridization waves when it approaches a quantum phase transition where Kondo screening breaks down and the local magnetic moments form a Dirac spin liquid. We relate the strength of this mechanism to a kinematic condition on the electronic and spinon band structures and show that the spatial periodicity of the hybridization wave is set by a momentum Q connecting the nodes in the dispersion of the Dirac spinons to low-energy portions of the electronic band structure.

Test-time scaling seeks to improve the reasoning performance of large language models (LLMs) by adding computational resources. A prevalent approach within the field is sampling-based test-time scaling methods, which enhance reasoning by generating multiple reasoning paths for a given input during inference. However, despite its practical success, the theoretical foundations remain underexplored. In this paper, we provide the first theoretical framework for analyzing sampling-based test-time scaling methods, grounded in the perspective of confidence estimation. Based on the framework, we analyze two dominant paradigms: self-consistency and perplexity, and reveal key limitations: self-consistency suffers from high estimation error while perplexity exhibits substantial modeling error and possible degradation of the estimation error convergence. To address these limitations, we introduce RPC, a hybrid method that leverages our theoretical insights through two key components: Perplexity Consistency and Reasoning Pruning. Perplexity Consistency combines the strengths of self-consistency and perplexity, boosting the convergence rate of estimation error from linear to exponential while preserving model error. Reasoning Pruning prevents degradation by eliminating low-probability reasoning paths. Both theoretical analysis and empirical results across seven benchmark datasets demonstrate that RPC has a strong potential for reducing reasoning error. Notably, RPC achieves reasoning performance comparable to self-consistency while not only enhancing confidence reliability but also reducing sampling costs by 50%. The code and resources are available at https://wnjxyk.github.io/RPC.

We study the Cooper instability in jellium model in the controlled regime of small to intermediate values of the Coulomb parameter r_s leq 2. We confirm that superconductivity naturally emerges from purely repulsive interactions described by the Kukkonen-Overhauser vertex function. By employing the implicit renormalization approach we reveal that even in the small-r_s limit, the dominant mechanism behind Cooper instability is based on dynamic screening of the Coulomb interaction--accurately captured by the random phase approximation, whereas the Kohn-Luttinger contribution is negligibly small and, thus, not relevant.

The advent of Scientific Machine Learning has heralded a transformative era in scientific discovery, driving progress across diverse domains. Central to this progress is uncovering scientific laws from experimental data through symbolic regression. However, existing approaches are dominated by heuristic algorithms or data-hungry black-box methods, which often demand low-noise settings and lack principled uncertainty quantification. Motivated by interpretable Statistical Artificial Intelligence, we develop a hierarchical Bayesian framework for symbolic regression that represents scientific laws as ensembles of tree-structured symbolic expressions endowed with a regularized tree prior. This coherent probabilistic formulation enables full posterior inference via an efficient Markov chain Monte Carlo algorithm, yielding a balance between predictive accuracy and structural parsimony. To guide symbolic model selection, we develop a marginal posterior-based criterion adhering to the Occam's window principle and further quantify structural fidelity to ground truth through a tailored expression-distance metric. On the theoretical front, we establish near-minimax rate of Bayesian posterior concentration, providing the first rigorous guarantee in context of symbolic regression. Empirical evaluation demonstrates robust performance of our proposed methodology against state-of-the-art competing modules on a simulated example, a suite of canonical Feynman equations, and single-atom catalysis dataset.

Self-improving agents are state-of-the-art (SOTA) on agentic coding benchmarks and have recently been extended to general domains. However, their search methods generally assume a stationary evaluation criterion: a fixed verifier, benchmark, or labeled dataset that remains valid as the agent improves. This ignores a central feature of evolution: species adapt as their environments change with them. We aim to bring the same principle to recursive self-improvement, making evaluation part of the improvement loop and opening search to evolving evaluators, adversarial objectives, and dynamic utilities that may surpass static benchmarks. We introduce the Red Queen Godel Machine (RQGM), an evolutionary framework for recursive self-improvement under non-stationary utilities. The RQGM makes this possible through controlled utility evolution: search is organized into epochs with a fixed within-epoch evaluation criterion, while the utility can be updated at epoch boundaries, so self-improvement guarantees hold per epoch as the objective evolves across them. We begin by showing that even on verifiable coding tasks, the RQGM improves test pass rate over the prior SOTA by adding a complementary agent-as-a-judge code-review signal. This signal is cheaper and the RQGM uses 1.35x-1.72x fewer tokens. We then turn to scientific paper writing and reviewing, and Olympiad-level proof writing and grading, where the RQGM improves performance over prior self-improving agents: co-evolved writers reach 1.78x-1.86x higher acceptance rates under a diverse agent-as-a-judge panel, while co-evolved graders reach 9% higher ground-truth accuracy. In paper reviewing, the strongest baseline reviewer over-accepts AI-generated papers at up to 1.91x the human rate. The RQGM corrects this by introducing an adversarial objective that discovers reviewers equally stringent on AI and human work.

Fermi operator expansion (FOE) methods are powerful alternatives to diagonalization type methods for solving Kohn-Sham density functional theory (KSDFT). One example is the pole expansion and selected inversion (PEXSI) method, which approximates the Fermi operator by rational matrix functions and reduces the computational complexity to at most quadratic scaling for solving KSDFT. Unlike diagonalization type methods, the chemical potential often cannot be directly read off from the result of a single step of evaluation of the Fermi operator. Hence multiple evaluations are needed to be sequentially performed to compute the chemical potential to ensure the correct number of electrons within a given tolerance. This hinders the performance of FOE methods in practice. In this paper we develop an efficient and robust strategy to determine the chemical potential in the context of the PEXSI method. The main idea of the new method is not to find the exact chemical potential at each self-consistent-field (SCF) iteration iteration, but to dynamically and rigorously update the upper and lower bounds for the true chemical potential, so that the chemical potential reaches its convergence along the SCF iteration. Instead of evaluating the Fermi operator for multiple times sequentially, our method uses a two-level strategy that evaluates the Fermi operators in parallel. In the regime of full parallelization, the wall clock time of each SCF iteration is always close to the time for one single evaluation of the Fermi operator, even when the initial guess is far away from the converged solution. We demonstrate the effectiveness of the new method using examples with metallic and insulating characters, as well as results from ab initio molecular dynamics.

The Large Language Model from Power Law Decoder Representations (PLDR-LLM) and its attention, Power Law Graph Attention (PLGA), replace the fixed bilinear form of scaled dot-product attention (SDPA) with a learned, input-generated bilinear operator G_{LM}, built from a positive tensor A_{LM} by elementwise power laws. The architecture is fully specified, verified against pinned reference releases; claims are labeled theorem, conditional theorem, measurement, or conjecture. Unconditionally: PLGA contains SDPA exactly at G_{LM}=I; A_{LM} and A_P are strictly entrywise positive, with Perron-Frobenius structure on A_{LM}; the DAG regularizer has the NOTEARS walk-counting form and positivity obstructs exact acyclicity; and, under nonresonance (satisfied by standard rotary frequencies), a commutant criterion identifies which operators preserve relative-position dependence. An inference-collapse theorem: exact input invariance of deductive outputs collapses inference to generalized SDPA with a constant operator. Measured invariance: relative fluctuations of 10^{-6} and below; perturbation bounds quantify but do not certify cached inference; the assembled proxy misses the decoding margin. A conditional three-stage mechanism (rotary twirl, concentration, row-map contraction) is measured on a released checkpoint. Blockwise training and scoring under the global Gram are stated with explicit target exposure; on tested samples, block and sequential scoring select identical answers and agree on the published TruthfulQA probability-mass metric within 5times 10^{-5} per item. Self-organized criticality enters as a phenomenological framework with an intrinsic order parameter; open claims become falsifiable conjectures. Selected proof cores are machine-checked in Lean 4.

Multi-path reasoning methods such as self-consistency (SC) sample K reasoning paths and choose the most frequent answer. However, their gains quickly plateau as K increases, and existing methods do not predict when this saturation will occur. We formalize multi-path LLM reasoning as a diversity combining problem from wireless communications: each path is a noisy channel observation, and the pairwise correlation of path correctness caps the design-effect effective sample size of the vote at a finite ceiling. Generalized least squares (GLS) analysis shows that, under exchangeability, the optimal symmetric linear combiner of latent embeddings is uniform, supporting majority vote as the natural default in standard SC while leaving room for weighting or pruning under heterogeneous prompt-template branches. Across 5 models and 12 benchmarks, prompt-template diversity reduces path correlation in 55 of 57 valid cells, with the strongest effect on open-ended QA. We derive an Adaptive-K rule that uses a four-path pilot to select K^*, retaining 96--103% of MV@K{=}32 accuracy across Math, QA, and NLU.

This work presents a non-perturbative quantum field theoretic computation of spinor tunnelling through a potential barrier. Within position space potential formalism, we identify coupled recursion relations forming a self-consistent integral equation. Our method, enabling both analytical and numerical approaches, transforms into a formalism admitting both numerical framework for arbitrary potentials and analytical exact results for select barriers. For the rectangular barrier, the QFT-derived amplitudes reproduce the relativistic quantum mechanical (RQM) results, thus establishing a foundation for future calculation of beyond-RQM field-theoretic corrections.

Describing fermionic systems with attractive interactions, such as superconductors and atomic nuclei, beyond the mean-field level remains a central challenge. In this work, we investigate the performance of Bogoliubov fully self-consistent GW (Bog.~scGW) and second-order Green's function (Bog.~scGF2) methods, implemented within a unified imaginary-time and imaginary-frequency framework. Using the attractive Hubbard model as a benchmark against full configuration interaction (FCI), we find that Bog. scGF2 provides a more accurate description than Bog.~scGW. In particular, Bog.~scGW systematically overestimates the correlation energy, with errors increasing rapidly as the attractive interaction strength grows. These results highlight the importance of exchange contributions in strongly correlated fermionic systems, even when exchange effects are included only at low order in perturbation theory.

A continuous-time projection quantum Monte Carlo algorithm is employed to simulate the ground state of a short-range quantum spin-glass model, namely, the two-dimensional Edwards-Anderson Hamiltonian with transverse field, featuring Gaussian nearest-neighbor couplings. We numerically demonstrate that guiding wave functions based on self-learned neural networks suppress the population control bias below modest statistical uncertainties, at least up to a hundred spins. By projecting a two-fold replicated Hamiltonian, the spin overlap is determined. A finite-size scaling analysis is performed to estimate the critical transverse field where the spin-glass transition occurs, as well as the critical exponents of the correlation length and the spin-glass susceptibility. For the latter two, good agreement is found with recent estimates from the literature for different random couplings. We also address the spin-overlap distribution within the spin-glass phase, finding that, for the workable system sizes, it displays a non-trivial double-peak shape with large weight at zero overlap.

In high energy physics, self-supervised learning (SSL) methods have the potential to aid in the creation of machine learning models without the need for labeled datasets for a variety of tasks, including those related to jets -- narrow sprays of particles produced by quarks and gluons in high energy particle collisions. This study introduces an approach to learning jet representations without hand-crafted augmentations using a jet-based joint embedding predictive architecture (J-JEPA), which aims to predict various physical targets from an informative context. As our method does not require hand-crafted augmentation like other common SSL techniques, J-JEPA avoids introducing biases that could harm downstream tasks. Since different tasks generally require invariance under different augmentations, this training without hand-crafted augmentation enables versatile applications, offering a pathway toward a cross-task foundation model. We finetune the representations learned by J-JEPA for jet tagging and benchmark them against task-specific representations.

The stability of the Berezinskii--Kosterlitz--Thouless phase against long-range interactions remains an open problem. We address this question in the two-dimensional XY model with couplings decaying as r^{-2-σ}, by developing a real-space renormalization-group treatment near the Gaussian manifold of quasi-long-range ordered (QLRO) states. We retain the full many-body vortex interaction and show that, at leading order in the fugacity, screening by small vortex--antivortex pairs preserves its Gaussian-average representation without an expansion in the long-range coupling. Including spin waves perturbatively then yields a local stability criterion: a stable QLRO interval exists only for σ>7/4. We also investigate a Wilson--Fisher-like fixed point at finite coupling for σ<7/4, for which the retained flow predicts ν^{-1}sim7/4-σ as it merges with the Gaussian line.

Many chemical concepts can be well defined in the context of quantum chemical theories. Examples are the electronegativity scale of Mulliken and Jaffe and the hard and soft acids and bases concept of Pearson. The sound theoretical basis allows for a systematic definition of such concepts. However, while they are often used to describe and compare chemical processes in terms of reactivity, their predictive power remains unclear. In this work, we elaborate on the predictive potential of chemical reactivity concepts, which can be crucial for autonomous reaction exploration protocols to guide them by first-principles heuristics that expoit these concepts.

In this paper, we present a novel approach for improving the quality and consistency of generated outputs from large-scale pre-trained language models (LLMs). Self-consistency has emerged as an effective approach for prompts with fixed answers, selecting the answer with the highest number of votes. In this paper, we introduce a generalized framework for self-consistency that extends its applicability beyond problems that have fixed-answer answers. Through extensive simulations, we demonstrate that our approach consistently recovers the optimal or near-optimal generation from a set of candidates. We also propose lightweight parameter-free similarity functions that show significant and consistent improvements across code generation, autoformalization, and summarization tasks, even without access to token log probabilities. Our method incurs minimal computational overhead, requiring no auxiliary reranker models or modifications to the existing model.

We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as ``the simple method'') optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi--Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. To outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems.

Building local surrogates to accelerate stationary point searches on potential energy surfaces spans decades of effort. Done correctly, surrogates can reduce the number of expensive electronic structure evaluations by roughly an order of magnitude while preserving the accuracy of the underlying theory, with the gain depending on oracle cost, search distance, and the availability of analytical forces. We present a unified Bayesian optimization view of minimization, single-point saddle searches, and double-ended path searches: all three share one six-step surrogate loop and differ only in the inner optimization target and the acquisition criterion. The framework uses Gaussian process regression with derivative observations, inverse-distance kernels, and active learning, and we develop optional extensions for production use, including farthest-point sampling with the Earth Mover's Distance, MAP regularization, an adaptive trust radius, and random Fourier features for scaling. Accompanying pedagogical Rust code demonstrates that all three applications use the same Bayesian optimization loop, bridging the gap between theoretical formulation and practical execution.

Semi-supervised algorithms aim to learn prediction functions from a small set of labeled observations and a large set of unlabeled observations. Because this framework is relevant in many applications, they have received a lot of interest in both academia and industry. Among the existing techniques, self-training methods have undoubtedly attracted greater attention in recent years. These models are designed to find the decision boundary on low density regions without making additional assumptions about the data distribution, and use the unsigned output score of a learned classifier, or its margin, as an indicator of confidence. The working principle of self-training algorithms is to learn a classifier iteratively by assigning pseudo-labels to the set of unlabeled training samples with a margin greater than a certain threshold. The pseudo-labeled examples are then used to enrich the labeled training data and to train a new classifier in conjunction with the labeled training set. In this paper, we present self-training methods for binary and multi-class classification; as well as their variants and two related approaches, namely consistency-based approaches and transductive learning. We examine the impact of significant self-training features on various methods, using different general and image classification benchmarks, and we discuss our ideas for future research in self-training. To the best of our knowledge, this is the first thorough and complete survey on this subject.

Self-Distillation Fine-Tuning (SDFT) enables a language model to act as its own teacher: by conditioning on a demonstration, the model produces an implicit reward via pointwise mutual information, which guides on-policy learning without external supervision. However, SDFT operates at training time: it requires gradient updates and access to expert demonstrations, making it inapplicable at inference. We propose test-time self-distillation, a decoding-time method that extracts a steering signal from the self-distillation framework without any parameter updates, reward models, or training data. Our key insight is that counterfactual contexts, i.e. fixed textual templates that hypothetically prime the model for excellent versus poor reasoning, can substitute for the demonstration. The log-odds ratio of a candidate answer under these two counterfactual conditions defines a new reward signal. We derive the optimal KL-regularized policy under this reward, which takes the form of a Gibbs reweighting of the base distribution. Crucially, this reweighting is global: it cannot be decomposed into independent per-token operations without ignoring future trajectory quality. We therefore approximate the target distribution via beam search. Experiments on mathematical reasoning (MATH500), code generation (HumanEval), and graduate-level science QA (GPQA) across multiple model scales show that test-time self-distillation improves over standard sampling, low temperature, beam search and power sampling baselines on average, demonstrating that the self-distillation principle can be operationalized at inference time.

We study linear perturbations of the Maldacena-Milekhin-Popov (MMP) wormhole in the parity sector that pairs axial metric perturbations with polar gauge perturbations. It contains the axial gravitational mode and the electric field along the magnetic field lines. The lowest Landau level (LLL) of the charged fermions responds through its charge. The gauge components a_τ, a_ρ form a two-dimensional gauge field on the worldsheet of each field line, and the fermions respond with the exact Schwinger current. Lifted over the sphere, it gives the electric field the MMP screening mass m_S^2=αq^2, which is not small and survives in the mouth. In the throat the perturbation tilts the worldsheets, and the Casimir stress tensor acquires mixed components. Force balance then requires a current across the field lines, and charge conservation fixes an accompanying polarisation current, the counterpart of the Hall current of the polar sector. These currents close the Bianchi identities at first order in the fermionic backreaction α. The system reduces to three coupled master equations, given in closed form and matched to the fermion-free mouth. Consistency alone does not select the fermionic constitutive relation, but locality, covariance and the index theorem do. At α=0 the throat reduces to Poschl-Teller problems with masses l(l-1) and (l+1)(l+2) at m_S^2=0. At finite m_S^2 one mode is screened, one tends to the mass l^2+l+2, and one neutral charge-density wave stays exactly massless. The O(α) shifts of the massive normal frequencies are real and negative. The two parity sectors are isospectral only without fermions. The neutral channel sees an attractive region in each mouth, and a numerical solution coupled to the gravitational channel in the exact mouth potential finds no bound state. The sector has no growing mode for lge2 at this order.

In recent years, diffusion models trained on equilibrium molecular distributions have proven effective for sampling biomolecules. Beyond direct sampling, the score of such a model can also be used to derive the forces that act on molecular systems. However, while classical diffusion sampling usually recovers the training distribution, the corresponding energy-based interpretation of the learned score is often inconsistent with this distribution, even for low-dimensional toy systems. We trace this inconsistency to inaccuracies of the learned score at very small diffusion timesteps, where the model must capture the correct evolution of the data distribution. In this regime, diffusion models fail to satisfy the Fokker--Planck equation, which governs the evolution of the score. We interpret this deviation as one source of the observed inconsistencies and propose an energy-based diffusion model with a Fokker--Planck-derived regularization term to enforce consistency. We demonstrate our approach by sampling and simulating multiple biomolecular systems, including fast-folding proteins, and by introducing a state-of-the-art transferable Boltzmann emulator for dipeptides that supports simulation and achieves improved consistency and efficient sampling. Our code, model weights, and self-contained JAX and PyTorch notebooks are available at https://github.com/noegroup/ScoreMD.

In-context learning is commonly interpreted as a form of conditional inference, in which the prompt specifies a context and the model's output is treated as an estimate of the corresponding conditional distribution. If this interpretation holds, then LLM estimates should satisfy basic probabilistic identities. In particular, the law of total probability asserts that prior-weighted conditional distributions aggregate into population-level marginals over any valid partition of the population. In this work, we investigate to what extent LLM estimates adhere to this self-consistency principle. We use binary trees as an evaluation scaffold to recursively partition a population into increasingly fine-grained subpopulations. We then prompt LLMs with verbalized subpopulation descriptions in context, aggregate the resulting estimates back into population-level estimates, and compare them across partitions of varying granularity. Applying this protocol across problem domains and state-of-the-art frontier models, we show widespread violations of basic consistency properties. An in-depth study of persona prompting reveals a pattern we call the macro fallacy: estimates reconstructed from more fine-grained subpopulation responses are often better aligned with human reference data than direct population-level estimates. This effect persists across variations in tree structure and estimation task, and can be partially recovered through implicit prompting. Together, these findings suggest that models possess relevant subpopulation knowledge but do not reliably propagate it into aggregate estimates. This gap establishes statistical self-consistency as an unsaturated, reference-free criterion for evaluating LLMs.

The emergence of multi-agent systems built from large language models (LLMs) offers a promising paradigm for scalable collective intelligence and self-evolution. Ideally, such systems would achieve continuous self-improvement in a fully closed loop while maintaining robust safety alignment--a combination we term the self-evolution trilemma. However, we demonstrate both theoretically and empirically that an agent society satisfying continuous self-evolution, complete isolation, and safety invariance is impossible. Drawing on an information-theoretic framework, we formalize safety as the divergence degree from anthropic value distributions. We theoretically demonstrate that isolated self-evolution induces statistical blind spots, leading to the irreversible degradation of the system's safety alignment. Empirical and qualitative results from an open-ended agent community (Moltbook) and two closed self-evolving systems reveal phenomena that align with our theoretical prediction of inevitable safety erosion. We further propose several solution directions to alleviate the identified safety concern. Our work establishes a fundamental limit on the self-evolving AI societies and shifts the discourse from symptom-driven safety patches to a principled understanding of intrinsic dynamical risks, highlighting the need for external oversight or novel safety-preserving mechanisms.

This paper studies a machine learning regression problem as a multivariate approximation problem using the framework of the theory of random functions. An ab initio derivation of a regression method is proposed, starting from postulates of indifference. It is shown that if a probability measure on an infinite-dimensional function space possesses natural symmetries (invariance under translation, rotation, scaling, and Gaussianity), then the entire solution scheme, including the kernel form, the type of regularization, and the noise parameterization, follows analytically from these postulates. The resulting kernel coincides with a generalized polyharmonic spline; however, unlike existing approaches, it is not chosen empirically but arises as a consequence of the indifference principle. This result provides a theoretical foundation for a broad class of smoothing and interpolation methods, demonstrating their optimality in the absence of a priori information.

We introduce marginalization models (MaMs), a new family of generative models for high-dimensional discrete data. They offer scalable and flexible generative modeling with tractable likelihoods by explicitly modeling all induced marginal distributions. Marginalization models enable fast evaluation of arbitrary marginal probabilities with a single forward pass of the neural network, which overcomes a major limitation of methods with exact marginal inference, such as autoregressive models (ARMs). We propose scalable methods for learning the marginals, grounded in the concept of "marginalization self-consistency". Unlike previous methods, MaMs support scalable training of any-order generative models for high-dimensional problems under the setting of energy-based training, where the goal is to match the learned distribution to a given desired probability (specified by an unnormalized (log) probability function such as energy function or reward function). We demonstrate the effectiveness of the proposed model on a variety of discrete data distributions, including binary images, language, physical systems, and molecules, for maximum likelihood and energy-based training settings. MaMs achieve orders of magnitude speedup in evaluating the marginal probabilities on both settings. For energy-based training tasks, MaMs enable any-order generative modeling of high-dimensional problems beyond the capability of previous methods. Code is at https://github.com/PrincetonLIPS/MaM.

Self-supervised learning excels at learning representations from large amounts of data. At the same time, generative models offer the complementary property of learning information about the underlying data generation process. In this study, we aim at establishing a principled connection between these two paradigms and highlight the benefits of their complementarity. In particular, we perform an analysis of self-supervised learning objectives, elucidating the underlying probabilistic graphical models and presenting a standardized methodology for their derivation from first principles. The analysis suggests a natural means of integrating self-supervised learning with likelihood-based generative models. We instantiate this concept within the realm of cluster-based self-supervised learning and energy models, introducing a lower bound proven to reliably penalize the most important failure modes and unlocking full unification. Our theoretical findings are substantiated through experiments on synthetic and real-world data, including SVHN, CIFAR10, and CIFAR100, demonstrating that our objective function allows to jointly train a backbone network in a discriminative and generative fashion, consequently outperforming existing self-supervised learning strategies in terms of clustering, generation and out-of-distribution detection performance by a wide margin. We also demonstrate that the solution can be integrated into a neuro-symbolic framework to tackle a simple yet non-trivial instantiation of the symbol grounding problem. The code is publicly available at https://github.com/emsansone/GEDI.

We combine a regularized variant of the strongly constrained and appropriately normed semilocal density functional [J. Sun, A. Ruzsinszky, and J. P. Perdew, Phys. Rev. Lett. 115, 036402 (2015)] with the latest generation semi-classical London dispersion correction. The resulting density functional approximation r2SCAN-D4 has the speed of generalized gradient approximations while approaching the accuracy of hybrid functionals for general chemical applications. We demonstrate its numerical robustness in real-life settings and benchmark molecular geometries, general main group and organo-metallic thermochemistry, as well as non-covalent interactions in supramolecular complexes and molecular crystals. Main group and transition metal bond lengths have errors of just 0.8%, which is competitive with hybrid functionals for main group molecules and outperforms them for transition metal complexes. The weighted mean absolute deviation (WTMAD2) on the large GMTKN55 database of chemical properties is exceptionally small at 7.5 kcal/mol. This also holds for metal organic reactions with an MAD of 3.3 kcal/mol. The versatile applicability to organic and metal-organic systems transfers to condensed systems, where lattice energies of molecular crystals are within chemical accuracy (errors <1 kcal/mol).

The Sherrington-Kirkpatrick spin-glass model used the replica symmetry method to find the phase transition of the system. In 1979-1980, Parisi proposed a solution based on replica symmetry breaking (RSB), which allowed him to identify the underlying phases of complex systems such as spin-glasses. Regardless of the method used for detection, the intrinsic phase of a system exists whether or not replicas are considered. We introduce a single replica method of spin-glass phase detection using the field's variation experienced by each spin in a system configuration. This method focuses on a single replica with quenched random couplings. Each spin inevitably observes a different field from the others. Our results show that the mean and variance of fields named "Spontaneous Configurational Field" experienced by spins are suitable indicators to explore different ferromagnetic, paramagnetic, and mixed phases. To classify different phases of the system with defined indicators we have developed an algorithm based on machine learning to analyze the desired samples.

Large Language Models often improve accuracy on reasoning tasks by sampling multiple Chain-of-Thought (CoT) traces and aggregating them with majority voting (MV), a test-time technique called self-consistency. When we truncate a CoT partway through and regenerate the remainder, we observe that traces with correct answers reproduce their original answer more often than traces with wrong answers. We use this difference as a reliability signal, prefix consistency, that weights each candidate answer by how often it reappears under regeneration. It requires no access to token log-probabilities or self-rating prompts. Across five reasoning models and four math and science benchmarks, prefix consistency is the best correctness predictor in most settings, and reweighting votes by it reaches Standard MV plateau accuracy at up to 21x fewer tokens (median 4.6x). Our code is available at https://github.com/naoto-iwase/prefix-consistency.

Density functional theory (DFT) stands as a cornerstone method in computational quantum chemistry and materials science due to its remarkable versatility and scalability. Yet, it suffers from limitations in accuracy, particularly when dealing with strongly correlated systems. To address these shortcomings, recent work has begun to explore how machine learning can expand the capabilities of DFT; an endeavor with many open questions and technical challenges. In this work, we present Grad DFT: a fully differentiable JAX-based DFT library, enabling quick prototyping and experimentation with machine learning-enhanced exchange-correlation energy functionals. Grad DFT employs a pioneering parametrization of exchange-correlation functionals constructed using a weighted sum of energy densities, where the weights are determined using neural networks. Moreover, Grad DFT encompasses a comprehensive suite of auxiliary functions, notably featuring a just-in-time compilable and fully differentiable self-consistent iterative procedure. To support training and benchmarking efforts, we additionally compile a curated dataset of experimental dissociation energies of dimers, half of which contain transition metal atoms characterized by strong electronic correlations. The software library is tested against experimental results to study the generalization capabilities of a neural functional across potential energy surfaces and atomic species, as well as the effect of training data noise on the resulting model accuracy.

Humans need to study only a handful of well-written textbooks to master a discipline and attempt its hardest problems. We argue that an ideal self-evolution method should share the same property, that is autonomously learning from raw training material for transferable problem-solving capability. However, we still lack a direct measurement for it. We introduce StudyBench, a controlled physics benchmark that directly measures how efficiently a self-evolution method converts training material into capability. We organise the test set into an Application Set, consisting of difficult textbook problems and evaluating absorption ability, and a Transfer Set, consisting of olympiad-level problems and evaluating transfer ability. Benchmarking representative self-evolution methods across three base models, we find that improvements on the Application Set rarely translate to the harder Transfer Set. A guidance ablation exposes a Guidance Gap: even the strongest method closes only a small fraction of what the same material unlocks when supplied as in-context guidance. Besides, every method hits a Compute Plateau, saturating well before exhausting its compute budget. The remaining gap is therefore a method problem rather than a data or compute problem. By offering a clean and controlled benchmark, StudyBench turns self-evolution progress from an open-ended pursuit into a measurable target for future research. Our code is released at https://github.com/thunlp/StudyBench.

A number of recent self-supervised learning methods have shown impressive performance on image classification and other tasks. A somewhat bewildering variety of techniques have been used, not always with a clear understanding of the reasons for their benefits, especially when used in combination. Here we treat the embeddings of images as point particles and consider model optimization as a dynamic process on this system of particles. Our dynamic model combines an attractive force for similar images, a locally dispersive force to avoid local collapse, and a global dispersive force to achieve a globally-homogeneous distribution of particles. The dynamic perspective highlights the advantage of using a delayed-parameter image embedding (a la BYOL) together with multiple views of the same image. It also uses a purely-dynamic local dispersive force (Brownian motion) that shows improved performance over other methods and does not require knowledge of other particle coordinates. The method is called MSBReg which stands for (i) a Multiview centroid loss, which applies an attractive force to pull different image view embeddings toward their centroid, (ii) a Singular value loss, which pushes the particle system toward spatially homogeneous density, (iii) a Brownian diffusive loss. We evaluate downstream classification performance of MSBReg on ImageNet as well as transfer learning tasks including fine-grained classification, multi-class object classification, object detection, and instance segmentation. In addition, we also show that applying our regularization term to other methods further improves their performance and stabilize the training by preventing a mode collapse.

Self-consistency (SC) has been a widely used decoding strategy for chain-of-thought reasoning. Despite bringing significant performance improvements across a variety of multi-step reasoning tasks, it is a high-cost method that requires multiple sampling with the preset size. In this paper, we propose a simple and scalable sampling process, Early-Stopping Self-Consistency (ESC), to greatly reduce the cost of SC without sacrificing performance. On this basis, one control scheme for ESC is further derivated to dynamically choose the performance-cost balance for different tasks and models. To demonstrate ESC's effectiveness, we conducted extensive experiments on three popular categories of reasoning tasks: arithmetic, commonsense and symbolic reasoning over language models with varying scales. The empirical results show that ESC reduces the average number of sampling of chain-of-thought reasoning by a significant margin on six benchmarks, including MATH (-33.8%), GSM8K (-80.1%), StrategyQA (-76.8%), CommonsenseQA (-78.5%), Coin Flip (-84.2%) and Last Letters (-67.4%), while attaining comparable performances.

Diffusion models trained on different, non-overlapping subsets of a dataset often produce strikingly similar outputs when given the same noise seed. We trace this consistency to a simple linear effect: the shared Gaussian statistics across splits already predict much of the generated images. To formalize this, we develop a random matrix theory (RMT) framework that quantifies how finite datasets shape the expectation and variance of the learned denoiser and sampling map in the linear setting. For expectations, sampling variability acts as a renormalization of the noise level through a self-consistent relation σ^2 mapsto κ(σ^2), explaining why limited data overshrink low-variance directions and pull samples toward the dataset mean. For fluctuations, our variance formulas reveal three key factors behind cross-split disagreement: anisotropy across eigenmodes, inhomogeneity across inputs, and overall scaling with dataset size. Extending deterministic-equivalence tools to fractional matrix powers further allows us to analyze entire sampling trajectories. The theory sharply predicts the behavior of linear diffusion models, and we validate its predictions on UNet and DiT architectures in their non-memorization regime, identifying where and how samples deviates across training data split. This provides a principled baseline for reproducibility in diffusion training, linking spectral properties of data to the stability of generative outputs.

A ubiquitous feature of data of our era is their extra-large sizes and dimensions. Analyzing such high-dimensional data poses significant challenges, since the feature dimension is often much larger than the sample size. This thesis introduces robust and computationally efficient methods to address several common challenges associated with high-dimensional data. In my first manuscript, I propose a coherent approach to variable screening that accommodates nonlinear associations. I develop a novel variable screening method that transcends traditional linear assumptions by leveraging mutual information, with an intended application in neuroimaging data. This approach allows for accurate identification of important variables by capturing nonlinear as well as linear relationships between the outcome and covariates. Building on this foundation, I develop new optimization methods for sparse estimation using nonconvex penalties in my second manuscript. These methods address notable challenges in current statistical computing practices, facilitating computationally efficient and robust analyses of complex datasets. The proposed method can be applied to a general class of optimization problems. In my third manuscript, I contribute to robust modeling of high-dimensional correlated observations by developing a mixed-effects model based on Tsallis power-law entropy maximization and discussed the theoretical properties of such distribution. This model surpasses the constraints of conventional Gaussian models by accommodating a broader class of distributions with enhanced robustness to outliers. Additionally, I develop a proximal nonlinear conjugate gradient algorithm that accelerates convergence while maintaining numerical stability, along with rigorous statistical properties for the proposed framework.

Large language models (LLMs) have demonstrated strong capabilities in translating natural language questions about relational databases into SQL queries. In particular, test-time scaling techniques such as Self-Consistency and Self-Correction can enhance SQL generation accuracy by increasing computational effort during inference. However, these methods have notable limitations: Self-Consistency may select suboptimal outputs despite majority votes, while Self-Correction typically addresses only syntactic errors. To leverage the strengths of both approaches, we propose CSC-SQL, a novel method that integrates Self-Consistency and Self-Correction. CSC-SQL selects the two most frequently occurring outputs from parallel sampling and feeds them into a merge revision model for correction. Additionally, we employ the Group Relative Policy Optimization (GRPO) algorithm to fine-tune both the SQL generation and revision models via reinforcement learning, significantly enhancing output quality. Experimental results confirm the effectiveness and generalizability of CSC-SQL. On the BIRD development set, our 3B model achieves 65.28% execution accuracy, while the 7B model achieves 69.19%. The code will be open sourced at https://github.com/CycloneBoy/csc_sql.

Chain-of-thought prompting combined with pre-trained large language models has achieved encouraging results on complex reasoning tasks. In this paper, we propose a new decoding strategy, self-consistency, to replace the naive greedy decoding used in chain-of-thought prompting. It first samples a diverse set of reasoning paths instead of only taking the greedy one, and then selects the most consistent answer by marginalizing out the sampled reasoning paths. Self-consistency leverages the intuition that a complex reasoning problem typically admits multiple different ways of thinking leading to its unique correct answer. Our extensive empirical evaluation shows that self-consistency boosts the performance of chain-of-thought prompting with a striking margin on a range of popular arithmetic and commonsense reasoning benchmarks, including GSM8K (+17.9%), SVAMP (+11.0%), AQuA (+12.2%), StrategyQA (+6.4%) and ARC-challenge (+3.9%).

We present the first complete machine learning method for accelerating plane-wave density functional theory (DFT) in materials under the projector augmented wave (PAW) formalism. We formalize seven criteria that a Complete Neural Electronic Initializer must satisfy for practical end-to-end PAW DFT acceleration. Applying these criteria to prior work reveals two missing structure-dependent components, augmentation occupancies and spin initialization, that prevent existing methods from providing complete reference-free initialization. Controlled ablations show that omitting these components can eliminate or reverse the acceleration obtained via models that only predict the smooth valence density. We satisfy these missing requirements by introducing AugNet, the first general equivariant model for PAW augmentation occupancies, and the first general spin density model for materials, which predicts the smooth spin-difference density and spin-difference PAW augmentation occupancies using predicted magnetic moments to constrain the global magnetic state. Combined with existing valence density models, these components satisfy all seven criteria and form a fully reference-free electronic initializer for materials DFT, requiring no electronic quantities from a converged target calculation. Our method reduces end-to-end DFT wall time by up to ~25% on unseen structures while preserving converged energies.

This paper introduces a continuous-time stochastic dynamical framework for understanding how large language models (LLMs) may self-amplify latent biases or toxicity through their own chain-of-thought reasoning. The model posits an instantaneous "severity" variable x(t) in [0,1] evolving under a stochastic differential equation (SDE) with a drift term μ(x) and diffusion σ(x). Crucially, such a process can be consistently analyzed via the Fokker--Planck approach if each incremental step behaves nearly Markovian in severity space. The analysis investigates critical phenomena, showing that certain parameter regimes create phase transitions from subcritical (self-correcting) to supercritical (runaway severity). The paper derives stationary distributions, first-passage times to harmful thresholds, and scaling laws near critical points. Finally, it highlights implications for agents and extended LLM reasoning models: in principle, these equations might serve as a basis for formal verification of whether a model remains stable or propagates bias over repeated inferences.

Self-Consistency mitigates hallucinations in Large Language Models (LLMs) by sampling multiple reasoning paths,but it lacks a systematic approach to determine the optimal number of samples or select the most faithful rationale. To address this limitation, we introduce Reasoning-Aware Self-Consistency (RASC), a novel framework that enhances sampling efficiency and reasoning faithfulness by dynamically evaluating both outputs and rationales. RASC assesses the quality of reasoning and the consistency of answers for each generated sample, using these assessments to guide early stopping decisions and rationale selection. The framework employs criteria-based stopping and weighted majority voting, enabling more informed choices on when to halt sampling and which rationale to select. Our comprehensive experiments across diverse question-answering datasets demonstrate that RASC outperforms existing methods, reducing sample usage by approximately 70% while maintaining accuracy. Moreover, RASC facilitates the selection of high-fidelity rationales, thereby improving the faithfulness of LLM outputs. Our approach effectively addresses the efficiency-accuracy trade-off in LLM reasoning tasks, offering a new perspective for more nuanced, faithful, and effective utilization of LLMs in resource-constrained environments.

Growing biochemical systems are intrinsically noisy, with one important source of stochasticity arising from the discrete reaction events of the underlying biochemical network. Because growing systems do not generally admit stationary abundance distributions, it is unclear how to separate this intrinsic chemical noise from other sources of variability in population and single-cell data. Here we develop a large-deviation theory for exponentially growing stochastic chemical reaction networks by decomposing abundance into volume and composition. In these coordinates, balanced growth corresponds to a stable composition together with exponentially increasing volume. We show that composition fluctuations satisfy a large-deviation principle with speed equal to the volume of the growing system, and the corresponding quasipotential is selected by a solution of the contact Hamilton--Jacobi equation. We also derive a fluctuation theory for accumulated observables and show that their covariances decay with accumulated-volume, rather than with physical time. Applications to a minimal autocatalytic network and a coarse-grained cellular growth model demonstrate how the theory predicts composition quasipotentials, growth-rate fluctuations, and correlations between observables. The framework therefore provides a stochastic null model for single-cell heterogeneity generated by the intrinsic stochasticity of the underlying chemical reaction network.

Many environmental remediation and energy applications (conversion and storage) for sustainability need design and development of green novel materials. Discovery processes of such novel materials are time taking and cumbersome due to large number of possible combinations and permutations of materials structures. Often theoretical studies based on Density Functional Theory (DFT) and other theories, coupled with Simulations are conducted to narrow down sample space of candidate materials, before conducting laboratory-based synthesis and analytical process. With the emergence of artificial intelligence (AI), AI techniques are being tried in this process too to ease out simulation time and cost. However tremendous values of previously published research from various parts of the world are still left as labor-intensive manual effort and discretion of individual researcher and prone to human omissions. AIMS-EREA is our novel framework to blend best of breed of Material Science theory with power of Generative AI to give best impact and smooth and quickest discovery of material for sustainability. This also helps to eliminate the possibility of production of hazardous residues and bye-products of the reactions. AIMS-EREA uses all available resources -- Predictive and Analytical AI on large collection of chemical databases along with automated intelligent assimilation of deep materials knowledge from previously published research works through Generative AI. We demonstrate use of our own novel framework with an example, how this framework can be successfully applied to achieve desired success in development of thermoelectric material for waste heat conversion.

Probabilistic decoding in Large Language Models (LLMs) often yields inconsistent outputs, particularly on complex or long-form questions. Self-Consistency (SC) mitigates this for short-form QA by majority voting over exact strings, whereas Universal Self-Consistency (USC) and Weighted Unigram Consistency Score (WUCS) extend to long-form responses but lose accuracy on short-form benchmarks. We introduce Latent Self-Consistency (LSC), which selects the most semantically consistent response using learnable token embeddings. LSC's lightweight forward processing of summary tokens only introduces negligible runtime overhead (at most 0.9%) on top of standard decoding of the base LLM, and requires no changes to the model architecture. Across 6 short-form and 5 long-form reasoning benchmarks (e.g., MATH, MMLU, TruthfulQA), LSC surpasses SC, USC, and WUCS on both short-form and long-form on average performance, while adding negligible computational overhead on vanilla inference. These results position LSC as a reliable consistency-selection method that works effectively across various answer formats. Additionally, LSC provides well-calibrated confidence estimates, maintaining low expected calibration error across both answer formats.

Quantum many-body methods provide a systematic route to computing electronic properties of molecules and materials, but high computational costs restrict their use in large-scale applications. Due to the complexity in many-electron wavefunctions, machine learning models capable of capturing fundamental many-body physics remain limited. Here, we present a deep learning framework targeting the many-body Green's function, which unifies predictions of electronic properties in ground and excited states, while offering physical insights into many-electron correlation effects. By learning the GW or coupled-cluster self-energy from mean-field features, our graph neural network achieves competitive performance in predicting one- and two-particle excitations and quantities derivable from one-particle density matrix. We demonstrate its high data efficiency and good transferability across chemical species, system sizes, molecular conformations, and correlation strengths in bond breaking, through multiple molecular and nanomaterial benchmarks. This work opens up opportunities for utilizing machine learning to solve many-electron problems.

Evaluating the step-by-step reliability of large language model (LLM) reasoning, such as Chain-of-Thought, remains challenging due to the difficulty and cost of obtaining high-quality step-level supervision. In this paper, we introduce Self-Play Critic (SPC), a novel approach where a critic model evolves its ability to assess reasoning steps through adversarial self-play games, eliminating the need for manual step-level annotation. SPC involves fine-tuning two copies of a base model to play two roles, namely a "sneaky generator" that deliberately produces erroneous steps designed to be difficult to detect, and a "critic" that analyzes the correctness of reasoning steps. These two models engage in an adversarial game in which the generator aims to fool the critic, while the critic model seeks to identify the generator's errors. Using reinforcement learning based on the game outcomes, the models iteratively improve; the winner of each confrontation receives a positive reward and the loser receives a negative reward, driving continuous self-evolution. Experiments on three reasoning process benchmarks (ProcessBench, PRM800K, DeltaBench) demonstrate that our SPC progressively enhances its error detection capabilities (e.g., accuracy increases from 70.8% to 77.7% on ProcessBench) and surpasses strong baselines, including distilled R1 model. Furthermore, applying SPC to guide the test-time search of diverse LLMs significantly improves their mathematical reasoning performance on MATH500 and AIME2024, outperforming state-of-the-art process reward models.

The recently proposed rSCAN functional [J. Chem. Phys. 150, 161101 (2019)] is a regularized form of the SCAN functional [Phys. Rev. Lett. 115, 036402 (2015)] that improves SCAN's numerical performance at the expense of breaking constraints known from the exact exchange-correlation functional. We construct a new meta-generalized gradient approximation by restoring exact constraint adherence to rSCAN. The resulting functional maintains rSCAN's numerical performance while restoring the transferable accuracy of SCAN.

We present a state-specific second-order perturbation theory based on a configuration interaction singles (CIS) reference and its generalized Fock operator. Starting from a fully internally contracted construction, we retain the complete doubles space and internally contracted triples in the first-order wave function. We consider two partitions, Canonical and Block-diagonal, with reference excitation energies given by the Fock excitation gap ω^{(0)} and the CIS excitation energy ω_{rm CIS}, respectively. Our analysis establishes that strict size-intensivity is preserved when the contracted-triples denominators use bare Fock gaps and the occupied--virtual Fock block is projected to remove residual spectator coupling. Based on this observation, combining Canonical triples with Block-diagonal doubles yields a size-intensive Hybrid partition. These properties are verified for water with non-interacting helium atoms. Our formulation enables O(o^3v^2) cost per matrix--vector product in a semicanonical basis. On the QUEST\#1 benchmark, Canonical and Block-diagonal show opposite systematic biases, whereas Hybrid gives the best overall accuracy among the three partitions. Its mean absolute errors for singlets and triplets are 0.24 and 0.15~eV, compared with 0.28 and 0.22~eV for CIS(D). The improvement is largest for Rydberg excitations, although CIS(D) remains more accurate for valence singlets.

In the search for novel intermetallic ternary alloys, much of the effort goes into performing a large number of ab-initio calculations covering a wide range of compositions and structures. These are essential to build a reliable convex hull diagram. While density functional theory (DFT) provides accurate predictions for many systems, its computational overheads set a throughput limit on the number of hypothetical phases that can be probed. Here, we demonstrate how an ensemble of machine-learning spectral neighbor-analysis potentials (SNAPs) can be integrated into a workflow for the construction of accurate ternary convex hull diagrams, highlighting regions fertile for materials discovery. Our workflow relies on using available binary-alloy data both to train the SNAP models and to create prototypes for ternary phases. From the prototype structures, all unique ternary decorations are created and used to form a pool of candidate compounds. The SNAPs are then used to pre-relax the structures and screen the most favourable prototypes, before using DFT to build the final phase diagram. As constructed, the proposed workflow relies on no extra first-principles data to train the machine-learning surrogate model and yields a DFT-level accurate convex hull. We demonstrate its efficacy by investigating the Cu-Ag-Au and Mo-Ta-W ternary systems.

Kohn--Sham density functional theory (DFT) underpins electronic-structure simulations, but repeated orbital diagonalizations lead to cubic scaling, restricting quantum calculations to modest scales only. Eliminating these auxiliary orbitals while retaining Kohn--Sham accuracy is the central goal of orbital-free DFT, but both analytical and machine-learning methods have so far fallen short. Prior learning approaches either try to learn the variational kinetic-energy functionals, which are ill-conditioned, or directly predict the ground state, which extrapolate poorly to larger systems. Instead, we identify the Kohn--Sham map as the right learning target for orbital-free DFT. It maps a Kohn--Sham potential directly to the corresponding density and noninteracting kinetic energy, quantities otherwise obtained through an orbital diagonalization. Focusing on the density component in this work, a domain-invariant SE(3)-equivariant Fourier neural operator learns to predict it from the potential as input on real-space grids, enabling stable quasi-linear scaling SCFs. Trained jointly on 8,504 molecules and solids, a single model generalizes to out-of-distribution organic molecules, insulators, and metals. For the first time, the same method converges SCFs across these systems without explicitly constructing Kohn--Sham orbitals, while reproducing densities, electronic spectra, and structural observables at Kohn--Sham DFT accuracy. Linear-scaling SCFs additionally allow converging magnesium dislocation densities containing up to 82,500 valence electrons on a single GPU.

Machine Learning Force Fields (MLFFs) are a promising alternative to expensive ab initio quantum mechanical molecular simulations. Given the diversity of chemical spaces that are of interest and the cost of generating new data, it is important to understand how MLFFs generalize beyond their training distributions. In order to characterize and better understand distribution shifts in MLFFs, we conduct diagnostic experiments on chemical datasets, revealing common shifts that pose significant challenges, even for large foundation models trained on extensive data. Based on these observations, we hypothesize that current supervised training methods inadequately regularize MLFFs, resulting in overfitting and learning poor representations of out-of-distribution systems. We then propose two new methods as initial steps for mitigating distribution shifts for MLFFs. Our methods focus on test-time refinement strategies that incur minimal computational cost and do not use expensive ab initio reference labels. The first strategy, based on spectral graph theory, modifies the edges of test graphs to align with graph structures seen during training. Our second strategy improves representations for out-of-distribution systems at test-time by taking gradient steps using an auxiliary objective, such as a cheap physical prior. Our test-time refinement strategies significantly reduce errors on out-of-distribution systems, suggesting that MLFFs are capable of and can move towards modeling diverse chemical spaces, but are not being effectively trained to do so. Our experiments establish clear benchmarks for evaluating the generalization capabilities of the next generation of MLFFs. Our code is available at https://tkreiman.github.io/projects/mlff_distribution_shifts/.

Self-consistency improves large language models by sampling multiple reasoning paths and selecting the most frequent answer, but majority voting often fails to recover correct answers that are already present among the samples. We address this limitation with Ranking-Improved Self-Consistency (RISC), which reformulates answer selection in self-consistency as a ranking problem. Instead of relying on a single uncertainty or confidence signal, RISC uses a lightweight LambdaRank model to score candidate answers with five carefully designed features that capture answer frequency, semantic centrality, and reasoning-trace consistency. We evaluate RISC on three datasets under a range of test-time budgets. Across datasets, RISC consistently achieves a better accuracy-efficiency trade-off than standard self-consistency and strong baselines, with particularly large gains on question answering benchmarks. Further analysis shows that the proposed features are individually useful and, more importantly, complementary, highlighting the value of learning to combine multiple informative signals for test-time answer selection.

We introduce Self Forcing, a novel training paradigm for autoregressive video diffusion models. It addresses the longstanding issue of exposure bias, where models trained on ground-truth context must generate sequences conditioned on their own imperfect outputs during inference. Unlike prior methods that denoise future frames based on ground-truth context frames, Self Forcing conditions each frame's generation on previously self-generated outputs by performing autoregressive rollout with key-value (KV) caching during training. This strategy enables supervision through a holistic loss at the video level that directly evaluates the quality of the entire generated sequence, rather than relying solely on traditional frame-wise objectives. To ensure training efficiency, we employ a few-step diffusion model along with a stochastic gradient truncation strategy, effectively balancing computational cost and performance. We further introduce a rolling KV cache mechanism that enables efficient autoregressive video extrapolation. Extensive experiments demonstrate that our approach achieves real-time streaming video generation with sub-second latency on a single GPU, while matching or even surpassing the generation quality of significantly slower and non-causal diffusion models. Project website: http://self-forcing.github.io/

Conditional independence testing (CIT) is essential for reliable scientific discovery. It prevents spurious findings and enables controlled feature selection. Recent CIT methods have used machine learning (ML) models as surrogates of the underlying distribution. However, model-agnostic approaches require a train-test split, which reduces statistical power. We introduce Semi-knockoffs, a CIT method that can accommodate any pre-trained model, avoids this split, and provides valid p-values and false discovery rate (FDR) control for high-dimensional settings. Unlike methods that rely on the model-X assumption (known input distribution), Semi-knockoffs only require conditional expectations for continuous variables. This makes the procedure less restrictive and more practical for machine learning integration. To ensure validity when estimating these expectations, we present two new theoretical results of independent interest: (i) stability for regularized models trained with a null feature and (ii) the double-robustness property.

Reinforcement learning from human feedback (RLHF) has been a central technique for recent large language model (LLM) alignment. However, its heavy dependence on costly human or LLM-as-Judge preference feedback could stymie its wider applications. In this work, we introduce Self-Contrast, a feedback-free large language model alignment method via exploiting extensive self-generated negatives. With only supervised fine-tuning (SFT) targets, Self-Contrast leverages the LLM itself to generate massive diverse candidates, and harnesses a pre-trained embedding model to filter multiple negatives according to text similarity. Theoretically, we illustrate that in this setting, merely scaling negative responses can still effectively approximate situations with more balanced positive and negative preference annotations. Our experiments with direct preference optimization (DPO) on three datasets show that, Self-Contrast could consistently outperform SFT and standard DPO training by large margins. And as the number of self-generated negatives increases, the performance of Self-Contrast continues to grow. Code and data are available at https://github.com/THUDM/Self-Contrast.

Recently, Test-Time Scaling (TTS) has gained increasing attention for improving LLM reasoning performance at test time without retraining the model. A notable TTS technique is Self-Consistency (SC), which generates multiple reasoning chains in parallel and selects the final answer via majority voting. While effective, the order-of-magnitude computational overhead limits its broad deployment. Prior attempts to accelerate SC mainly rely on model-based confidence scores or heuristics with limited empirical support. For the first time, we theoretically and empirically analyze the inefficiencies of SC and reveal actionable opportunities for improvement. Building on these insights, we propose Slim-SC, a step-wise pruning strategy that identifies and removes redundant chains using inter-chain similarity at the thought level. Experiments on three STEM reasoning datasets and two recent LLM architectures show that Slim-SC reduces inference latency and KVC usage by up to 45% and 26%, respectively, with R1-Distill, while maintaining or improving accuracy, thus offering a simple yet efficient TTS alternative for SC.

We develop a Gaussian comparison theory for posterior inference in non-Gaussian high-dimensional models. The framework allows for model misspecification and does not require the posterior distribution itself to be approximately Gaussian. Working directly with likelihood processes on separable function spaces, we establish explicit nonasymptotic free energy bounds controlled by first two moment discrepancies and local third moments. A perturbation-and-convexity argument transfers these comparisons to sandwich bounds for posterior mean squared error and posterior variance, including one-sided conclusions at nondifferentiable phase transitions. In high-dimensional product models, these conditions reduce to local moment controls and a canonical feature-radius scaling. Under weak regularity requirements, the resulting universality theory applies to a broad class of likelihood models, ranging from exponential families to more general regular local log-likelihoods. The framework is illustrated on sparse Bernoulli hypergraph inference.

We study the learning dynamics of self-predictive learning for reinforcement learning, a family of algorithms that learn representations by minimizing the prediction error of their own future latent representations. Despite its recent empirical success, such algorithms have an apparent defect: trivial representations (such as constants) minimize the prediction error, yet it is obviously undesirable to converge to such solutions. Our central insight is that careful designs of the optimization dynamics are critical to learning meaningful representations. We identify that a faster paced optimization of the predictor and semi-gradient updates on the representation, are crucial to preventing the representation collapse. Then in an idealized setup, we show self-predictive learning dynamics carries out spectral decomposition on the state transition matrix, effectively capturing information of the transition dynamics. Building on the theoretical insights, we propose bidirectional self-predictive learning, a novel self-predictive algorithm that learns two representations simultaneously. We examine the robustness of our theoretical insights with a number of small-scale experiments and showcase the promise of the novel representation learning algorithm with large-scale experiments.

Accurate free-energy estimation is essential in molecular simulation, yet the periodic boundary conditions (PBC) commonly used in computer simulations have rarely been explicitly exploited. Equilibrium methods such as umbrella sampling, metadynamics, and adaptive biasing force require extensive sampling, while non-equilibrium pulling with Jarzynski's equality suffers from poor convergence due to exponential averaging. Here, we introduce a physics-informed, score-based diffusion framework: by mapping PBC simulations onto a Brownian particle in a periodic potential, we derive the Fokker-Planck steady-state score that directly encodes free-energy gradients. A neural network is trained on non-equilibrium trajectories to learn this score, providing a principled scheme to efficiently reconstruct the potential of mean force (PMF). On benchmark periodic potentials and small-molecule membrane permeation, our method is up to one order of magnitude more efficient than umbrella sampling.

The high cost of fine-tuning LLMs poses a significant economic barrier; pre-hoc performance prediction offers a critical solution to substantially reduce this expense. However, the theoretical limits of pre-hoc performance prediction remain unexplored. We formulate it as a stochastic estimation problem under information constraints, decomposing prediction risk into two components: an intrinsic limit (static data-model compatibility) and a reducible optimization variance. We prove that optimization variance admits a necessary lower bound on its decay rate, implying fundamental constraints on how quickly uncertainty dissipates, regardless of the predictor used. Based on these dynamics, we derive a budget-optimal probing principle and introduce a predictability phase diagram that organizes tasks into three distinct regimes: Static-Sufficient, Dynamic-Critical, and Noise-Dominant. Extensive experiments on synthetic and real-world benchmarks validate these theoretical regimes and demonstrate the efficiency of our probing strategy.

We endow Large Language Models (LLMs) with fine-grained self-evaluation to refine multi-step reasoning inference. We propose an effective prompting approach that integrates self-evaluation guidance through stochastic beam search. Our approach explores the reasoning search space using a well-calibrated automatic criterion. This enables an efficient search to produce higher-quality final predictions. With the self-evaluation guided stochastic beam search, we also balance the quality-diversity trade-off in the generation of reasoning chains. This allows our approach to adapt well with majority voting and surpass the corresponding Codex-backboned baselines by 6.34%, 9.56%, and 5.46% on the GSM8K, AQuA, and StrategyQA benchmarks, respectively, in few-shot accuracy. Analysis of our decompositional reasoning finds it pinpoints logic failures and leads to higher consistency and robustness. Our code is publicly available at https://github.com/YuxiXie/SelfEval-Guided-Decoding.

Understanding electron irradiation effects is vital not only for reliable transmission electron microscopy characterization, but increasingly also for the controlled manipulation of two-dimensional materials. The displacement cross sections of monolayer hBN are measured using aberration-corrected scanning transmission electron microscopy in near ultra-high vacuum at primary beam energies between 50 and 90 keV. Damage rates below 80 keV are up to three orders of magnitude lower than previously measured at edges under poorer residual vacuum conditions where chemical etching appears to have been dominant. Notably, is possible to create single vacancies in hBN using electron irradiation, with boron almost twice as likely as nitrogen to be ejected below 80 keV. Moreover, any damage at such low energies cannot be explained by elastic knock-on, even when accounting for vibrations of the atoms. A theoretical description is developed to account for lowering of the displacement threshold due to valence ionization resulting from inelastic scattering of probe electrons, modelled using charge-constrained density functional theory molecular dynamics. Although significant reductions are found depending on the constrained charge, quantitative predictions for realistic ionization states are currently not possible. Nonetheless, there is potential for defect-engineering of hBN at the level of single vacancies using electron irradiation.

We study the hardmax limit of self-attention dynamics for token embeddings obtained in the zero-temperature (βto+infty) regime, and relate it to the finite-β setting. In this limit, the update rule can be viewed as a Frank-Wolfe step for a quadratic objective over the convex hull of the current token embeddings. When the key-query matrix is negative semidefinite, the method linearly contracts all tokens to a single cluster at the origin. When it is positive semidefinite, extending the hardmax rule to the entire convex hull induces a Voronoi diagram: vertices are stationary, interior points remain in their initial cells, and each token moves along a straight line toward its cell's vertex, yielding (super-)exponential convergence. As a byproduct, we also establish well-posedness of the associated ODE limit in this regime. Returning to the finite-β regime, we model self-attention dynamics as a Markov chain and prove dynamic metastability: with high probability, interior tokens reach near-vertex configurations in a constant number of steps and remain within a small neighborhood for times that grow exponentially in the inverse temperature β, before ultimately collapsing to the origin. Thus, the hardmax dynamics accurately approximate the finite-β process over exponentially long time horizons.

Conceptual density functional theory (CDFT) defines chemical reactivity descriptors as derivatives of the electronic energy with respect to the number of electrons and the external potential, or combinations thereof; in practice, however, researchers almost always replace these derivatives with finite-difference and frontier-orbital approximations, and no common software exists for their analytical evaluation. We present PyCDFT, to our knowledge, the first standardized, open-source, and scriptable code that analytically computes the descriptors of conceptual density (matrix) functional theory for ground and excited states up to second order, including the orbital hardness, the Fukui function and Fukui matrix, and the linear response function, from a single converged mean-field wavefunction imported from virtually any electronic-structure package. The descriptors are obtained from matrix-free, preconditioned Krylov-subspace solutions of the coupled-perturbed self-consistent-field equations for both spin-unpolarized and spin-polarized references, with the work distributed over MPI ranks and all data flowing through a single HDF5 checkpoint file that supports restart, post-processing, and the export of real-space descriptors as cube files for visualization. The design delegates integrals, grids, and exchange-correlation kernels to PySCF and Libxc, keeping the library compact, user-friendly, and interoperable. Worked examples on the ground state of H_2O and a broken-symmetry ΔSCF excited state of NH_3 show that a complete second-order CDFT analysis requires only a few lines of user code.

Accurate potential energy surface (PES) descriptions are essential for atomistic simulations of materials. Universal machine learning interatomic potentials (UMLIPs)^{1-3} offer a computationally efficient alternative to density functional theory (DFT)^4 for PES modeling across the periodic table. However, their accuracy today is fundamentally constrained due to a reliance on DFT relaxation data.^{5,6} Here, we introduce MatPES, a foundational PES dataset comprising sim 400,000 structures carefully sampled from 281 million molecular dynamics snapshots that span 16 billion atomic environments. We demonstrate that UMLIPs trained on the modestly sized MatPES dataset can rival, or even outperform, prior models trained on much larger datasets across a broad range of equilibrium, near-equilibrium, and molecular dynamics property benchmarks. We also introduce the first high-fidelity PES dataset based on the revised regularized strongly constrained and appropriately normed (r^2SCAN) functional^7 with greatly improved descriptions of interatomic bonding. The open source MatPES initiative emphasizes the importance of data quality over quantity in materials science and enables broad community-driven advancements toward more reliable, generalizable, and efficient UMLIPs for large-scale materials discovery and design.

Generations from large language models (LLMs) can be improved by sampling and scoring multiple solutions to select a final answer. Current "sample and select" methods such as self-consistency (SC) rely on majority voting to score answers. However, when tasks have many distinct and valid answers, selection by voting requires a large number of samples. This makes SC prohibitively expensive for interactive tasks that involve generating multiple actions (answers) sequentially. After establishing that majority voting fails to provide consistent gains on such tasks, we demonstrate how to increase success rates by softening the scoring criterion. We introduce Soft Self-Consistency (SOFT-SC), which replaces SC's discontinuous scoring with a continuous score computed from model likelihoods, allowing for selection even when actions are sparsely distributed. SOFT-SC improves both performance and efficiency on long-horizon interactive tasks, requiring half as many samples as SC for comparable or better performance. For a fixed number of samples, SOFT-SC leads to a 1.3% increase over SC in absolute success rate on writing bash programs, a 6.6% increase on online shopping (WebShop), and a 4.7% increase for an interactive household game (ALFWorld). Finally, we show that SOFT-SC can be applied to both open-source and black-box models.

Self-consistency has emerged as a popular technique for improving large language model accuracy on reasoning tasks. The approach is straightforward: generate multiple reasoning paths and select the most common answer through majority voting. While this reliably boosts accuracy, it remains unclear whether these gains reflect genuine improvements in reasoning quality. We investigate a fundamental question that has not been studied before: does inference scaling improve reasoning faithfulness? We conduct a comprehensive empirical study across four frontier models (GPT-5.2, Claude Opus 4.5, Gemini-3-flash-preview, and DeepSeek-v3.2) on 100 GSM8K mathematical reasoning problems. Our analysis employs bootstrap confidence intervals, McNemar's tests for paired comparisons, and Cohen's d effect sizes to quantify the effects rigorously. The results reveal striking differences across models that challenge common assumptions about self-consistency. GPT-5.2 shows the expected pattern: accuracy improves from 78% to 90% at N=5, with faithfulness remaining relatively stable (0.540 to 0.510). Claude Opus 4.5 tells a completely different story. Its accuracy actually drops from 78% to 74.3% while faithfulness jumps dramatically from 0.270 to 0.891 at N=5. DeepSeek-v3.2, already at 98% accuracy, shows ceiling effects with modest faithfulness gains (0.440 to 0.541). Gemini-3-flash improves from 81% to 86% accuracy with a slight faithfulness decrease (0.260 to 0.212). Problem difficulty analysis reveals that GPT-5.2 solves 82% of hard problems while breaking only 13% of easy ones. Claude, in contrast, breaks 23% of easy problems, explaining its accuracy decrease. These findings matter for practitioners: self-consistency is not universally beneficial, and teams should test their specific models before deployment. We release our code and provide practical recommendations for navigating these tradeoffs.

The initial state of neural networks plays a central role in conditioning the subsequent training dynamics. In the context of classification problems, we provide a theoretical analysis demonstrating that the structure of a neural network can condition the model to assign all predictions to the same class, even before the beginning of training, and in the absence of explicit biases. We show that the presence of this phenomenon, which we call "Initial Guessing Bias" (IGB), depends on architectural choices such as activation functions, max-pooling layers, and network depth. Our analysis of IGB has practical consequences, in that it guides architecture selection and initialization. We also highlight theoretical consequences, such as the breakdown of node-permutation symmetry, the violation of self-averaging, the validity of some mean-field approximations, and the non-trivial differences arising with depth.

Parameterized tight-binding models fit to first principles calculations can provide an efficient and accurate quantum mechanical method for predicting properties of molecules and solids. However, well-tested parameter sets are generally only available for a limited number of atom combinations, making routine use of this method difficult. Furthermore, most previous models consider only simple two-body interactions, which limits accuracy. To tackle these challenges, we develop a density functional theory database of nearly one million materials, which we use to fit a universal set of tight-binding parameters for 65 elements and their binary combinations. We include both two-body and three-body effective interaction terms in our model, plus self-consistent charge transfer, enabling our model to work for metallic, covalent, and ionic bonds with the same parameter set. To ensure predictive power, we adopt a learning framework where we repeatedly test the model on new low energy crystal structures and then add them to the fitting dataset, iterating until predictions improve. We distribute the materials database and tools developed in this work publicly.

Signal decay and regime shifts pose recurring challenges for data-driven investment strategies in non-stationary markets. Conventional time-series and machine learning approaches, which rely primarily on historical correlations, often struggle to generalize when the economic environment changes. While large language models (LLMs) offer strong capabilities for processing unstructured information, their potential to support quantitative factor screening through explicit economic reasoning remains underexplored. Existing factor-based methods typically reduce alphas to numerical time series, overlooking the semantic rationale that determines when a factor is economically relevant. We propose Alpha-R1, an 8B-parameter reasoning model trained via reinforcement learning for context-aware alpha screening. Alpha-R1 reasons over factor logic and real-time news to evaluate alpha relevance under changing market conditions, selectively activating or deactivating factors based on contextual consistency. Empirical results across multiple asset pools show that Alpha-R1 consistently outperforms benchmark strategies and exhibits improved robustness to alpha decay. The full implementation and resources are available at https://github.com/FinStep-AI/Alpha-R1.

LLM-as-judge is essential for evaluating open-ended text and steering post-training, yet improving the judge itself typically relies on expensive annotations, reward models, or distillation from stronger teachers. In this work, we eliminate external gold supervision from the RL training reward: the model's own evaluative capability generates learning signals for its optimization -- a closed-loop setting of bounded recursive self-improvement (RSI) termed Recursive Self-Evaluation (RecurSE). We study two central questions: when can self-improvement occur, and when must it stop? First, RecurSE pairs a trainable judge evaluating candidate responses under per-rule rubrics (Pass 1) with a synchronized policy-copy checker that audits the judge's reasoning against meta-rubrics to supply a scalar process reward (Pass 2). To enable learning, interface decoupling structurally isolates the checker's scalar score from the judge's verdict tokens, eliminating a degenerative token-copying shortcut that inflates self-assigned rewards. Second, because unanchored recursive learning is inherently bounded, Pairwise Advantage Validity (PAV) serves as an unbiased validation monitor that jointly tracks judge accuracy and checker fidelity to reliably identify the optimal early-stopping window. Across Qwen3.5-9B, Gemma-4-E4B-it, and Qwen3.6-27B, RecurSE achieves consistent generalization gains across held-out medical, pairwise, summarization, and professional benchmarks. Ablations demonstrate that synchronized judge-checker co-evolution outperforms frozen checkers, external meta-judges, self-consistency, and scaled teacher distillation. Furthermore, preference pairs curated by our judge effectively enhance downstream policy alignment. Bounded RSI for LLM-as-judge is thus viable when self-produced reward validity is explicitly decoupled and monitored.

In geometrically frustrated assemblies local inter-subunit misfits propagate to intra-assembly strain gradients, giving rise to anomalous self-limiting assembly thermodynamics. Here, we use theory and coarse-grained simulation to study a recently developed class of ``curvamer'' particles, flexible shell-like particles that exhibit self-limiting assembly due to the build up of curvature deformation in cohesive stacks. To address a generic, yet poorly understood aspect of frustrated assembly, we introduce a model of curvamer assembly that incorporates both {\it intra-particle} shape deformation as well as compliance of {\it inter-particle} cohesive gaps, an effect we can attribute to a {\it finite range of attraction} between particles. We show that the ratio of intra-particle (bending elasticity) to inter-particle stiffness not only controls the regimes of self-limitation but also the nature of frustration propagation through curvamer stacks. We find a transition from uniformly-bound, curvature-focusing stacks at small size to gap-opened, uniformly curved stacks at large size is controlled by a dimensionless measure of inter- versus intra-curvamer stiffness. The finite range of inter-particle attraction determines range of cohesion in stacks are self-limiting, a prediction which is in strong agreement with numerical studies of our coarse-grained colloidal model. These predictions provide critical guidance for experimental realizations of frustrated particle systems designed to exhibit self-limitation at especially large multi-particle scales.

Disorder and defective coordination in the catalytic plane are crucial for enhancing the Hydrogen Evolution Reaction (HER) on two-dimensional catalysts. Amorphous materials are disordered, making them catalytically adaptive for many reactions. In this work, the HER capabilities of Monolayer Amorphous Carbon (MAC) were studied in comparison with crystalline carbon derivatives, such as pristine graphene (GE) and graphyne derivatives. MAC generated from melt-quench simulations revealed a diverse framework of predominantly sp2 and sp3 carbons with numerous 5-, 6-, and 7-membered rings. Density Functional Theory (DFT) calculations investigated free-energy variations in hydrogen adsorption for each material. According to Sabatier's principle, optimum activity is achieved when the Gibbs free energy (Delta GH) change approaches zero. Crystalline carbon materials possess limited active sites, with beta-graphyne showing the best Delta GH value of +0.34 eV. The adsorption study for MAC was conducted in 30 distinct local environments, where core structural properties were analyzed against varying radii. Calculations showed a Delta GH distribution for MAC ranging from -0.02 eV to +1.35 eV. To evaluate activity across the entire MAC surface, a MACE MLIP foundation model was finetuned, achieving optimal energy and force fitting of 1.67 meV/atom and 29.15 meV/A, respectively. The MLIP predicted Delta GH values from -0.91 eV to +1.70 eV, with approximately 15% of sites exhibiting values below +0.25 eV. Feature analysis revealed that 7-membered rings, curvature, and ripple height enhance HER activity. Our findings suggest that, with careful optimization of local features, MAC can be tuned to compete with noble metal catalysts.

Algorithms based on normalizing flows are emerging as promising machine learning approaches to sampling complicated probability distributions in a way that can be made asymptotically exact. In the context of lattice field theory, proof-of-principle studies have demonstrated the effectiveness of this approach for scalar theories, gauge theories, and statistical systems. This work develops approaches that enable flow-based sampling of theories with dynamical fermions, which is necessary for the technique to be applied to lattice field theory studies of the Standard Model of particle physics and many condensed matter systems. As a practical demonstration, these methods are applied to the sampling of field configurations for a two-dimensional theory of massless staggered fermions coupled to a scalar field via a Yukawa interaction.

P. Kabamba developed generation theory as a tool for studying self-reproducing systems. We provide an alternative definition of a generation system and give a complete solution to the problem of finding optimal seeds for a finite self-replicating system. We also exhibit examples illustrating a connection between self-replication and fixed-point theory.

We study the fine-grained uniform convergence behavior of halfspaces beyond worst-case VC bounds. For inhomogeneous halfspaces in R^d with dge 2, we show that standard first-order VC bounds are essentially tight: even consistent hypotheses can incur population error Θ(dln(n/d)/n), and in the agnostic setting the deviation scales as τln(1/τ) at true error τ. In contrast, homogeneous halfspaces in R^2 exhibit a markedly different behavior. In the realizable case, every hypothesis consistent with the sample has error O(1/n). In the agnostic case, we prove a bandwise, log-free deviation bound on each dyadic risk band via a critical-wedge localization argument. Unioning over bands incurs only a lnln n overhead, and we establish a matching lower bound showing this overhead is unavoidable. Together, these results give a fine-grained and nearly complete picture of uniform convergence for halfspaces, revealing sharp dimensional and structural thresholds.

One of the main goals and challenges of materials discovery is to find the best candidates for each interest property or application. Machine learning rises in this context to efficiently optimize this search, exploring the immense materials space, consisting of simultaneously the atomic, compositional, and structural spaces. Topological insulators, presenting symmetry-protected metallic edge states, are a promising class of materials for different applications. However, further, development is limited by the scarcity of viable candidates. Here we present and discuss machine learning-accelerated strategies for searching the materials space for two-dimensional topological materials. We show the importance of detailed investigations of each machine learning component, leading to different results. Using recently created databases containing thousands of ab initio calculations of 2D materials, we train machine learning models capable of determining the electronic topology of materials, with an accuracy of over 90%. We can then generate and screen thousands of novel materials, efficiently predicting their topological character without the need for a priori structural knowledge. We discover 56 non-trivial materials, of which 17 novel insulating candidates for further investigation, for which we corroborate their topological properties with density functional theory calculations. This strategy is 10times more efficient than the trial-and-error approach while few orders of magnitude faster and is a proof of concept for guiding improved materials discovery search strategies.

The prediction of phase diagrams in the search for new phases is a complex and computationally intensive task. Density functional theory provides, in many situations, the desired accuracy, but its throughput becomes prohibitively limited as the number of species involved grows, even when used with local and semi-local functionals. Here, we explore the possibility of integrating machine-learning models in the workflow for the construction of ternary convex hull diagrams. In particular, we train a set of spectral neighbour-analysis potentials (SNAPs) over readily available binary phases and we establish whether this is good enough to predict the energies of novel ternaries. Such a strategy does not require any new calculations specific for the construction of the model, but just avails of data stored in binary-phase-diagram repositories. We find that a so-constructed SNAP is capable of accurate total-energy estimates for ternary phases close to the equilibrium geometry but, in general, is not able to perform atomic relaxation. This is because during a typical relaxation path a given phase traverses regions in the parameter space poorly represented by the training set. Different metrics are then investigated to assess how an unknown structure is well described by a given SNAP model, and we find that the standard deviation of an ensemble of SNAPs provides a fast and non-specie-specific metric.

Single-atom catalysts (SACs) have emerged as frontiers for catalyzing chemical reactions, yet the diverse combinations of active elements and support materials, the nature of coordination environments, elude traditional methodologies in searching optimal SAC systems with superior catalytic performance. Herein, by integrating multi-branch Convolutional Neural Network (CNN) analysis models to hybrid descriptor based activity volcano plot, 2D SAC system composed of diverse metallic single atoms anchored on six type of 2D supports, including graphitic carbon nitride, nitrogen-doped graphene, graphene with dual-vacancy, black phosphorous, boron nitride, and C2N, are screened for efficient CO2RR. Starting from establishing a correlation map between the adsorption energies of intermediates and diverse electronic and elementary descriptors, sole singular descriptor lost magic to predict catalytic activity. Deep learning method utilizing multi-branch CNN model therefore was employed, using 2D electronic density of states as input to predict adsorption energies. Hybrid-descriptor enveloping both C- and O-types of CO2RR intermediates was introduced to construct volcano plots and limiting potential periodic table, aiming for intuitive screening of catalyst candidates for efficient CO2 reduction to CH4. The eDOS occlusion experiments were performed to unravel individual orbital contribution to adsorption energy. To explore the electronic scale principle governing practical engineering catalytic CO2RR activity, orbitalwise eDOS shifting experiments based on CNN model were employed. The study involves examining the adsorption energy and, consequently, catalytic activities while varying supported single atoms. This work offers a tangible framework to inform both theoretical screening and experimental synthesis, thereby paving the way for systematically designing efficient SACs.