AI for Science

Large language models (LLMs) are increasingly embedded in AI for Science (AI4Science) workflows, from scientific question answering and literature analysis to laboratory planning and autonomous discovery. This progress creates an urgent need for safety benchmarks that evaluate not only scientific competence, but also whether models recognize and avoid risks in high-stakes scientific contexts. Existing AI4Science safety datasets cover several disciplines and task formats, leaving the underlying risk dimensions underspecified. We introduce \textbf{SciRisk-Bench}, a benchmark designed to evaluate AI4Science safety from two complementary perspectives: explicit risk dimensions and scientific disciplines. SciRisk-Bench covers 7 disciplines, 31 subdisciplines and 10 risk dimensions. In the experimental section, we evaluate both mainstream LLMs and science-oriented LLMs across risk dimensions, disciplines, and sub-disciplines, enabling fine-grained diagnosis of where scientific models remain unsafe.

Decision making in lithium production is challenging, whether from an investor's perspective or a strategic production standpoint. Determining which mines to open and when to open them involves not only geological and price uncertainties, but also complexities around the choice of extraction method, from direct lithium extraction to hard rock mining. Prior work explored models of this problem and different methods to optimize mining decisions; these models did not account for uncertainty in pricing, uncertainty in demand, or different mining technologies to extract lithium. Incorporating different pricing models and extraction technology into these models enables more robust strategies for determining not only when and where to open a mine, but also which method of production to pursue. We frame the problem as a partially observable Markov decision process (POMDP) and solve using belief state planning methods to get optimal decision making. In our study, we show that POMDP solvers outperform human inspired heuristics by dynamically adapting to shifting lithium price regimes (static, linear, exponential, and stochastic) through belief state planning and explicit uncertainty management. By optimally sequencing exploration, production, and technology choice, the framework achieves higher demand fulfillment and more balanced economic environmental outcomes over the projects lifetime in all different pricing and deposit scenarios.

Deep research agents are Large Language Model (LLM)-based systems designed for autonomous, multi-step scientific reasoning, and they hold immense potential for accelerating research in the physical sciences. However, comprehensive and in-depth evaluations of their capabilities within this domain remain lacking. To address this gap, we introduce PhySciBench, a benchmark highly relevant to physical science research, comprising 200 expert-curated questions, balanced between physics and chemistry, across six task categories that reflect real-world scientific workflows. Evaluations of state-of-the-art models and agent systems on PhySciBench reveal limited performance; even the strongest baseline, Gemini Deep Research, achieves an accuracy of only 33.5%. Analysis of failure cases identifies three recurrent deficiencies: fragility in extended reasoning chains, limited knowledge transfer across steps, and a lack of physics-grounded self-verification. Motivated by these findings, we develop DelveAgent, a modular multi-agent framework equipped with an adaptive planning loop, dual-granularity memory, and a hierarchical physics-grounded reflection mechanism. Across four scientific benchmarks, DelveAgent improves accuracy by up to 7.5 percentage points while reducing inference costs to approximately one-third of the strongest baseline. These results establish the significance of PhySciBench as a critical benchmark for evaluating AI systems in the physical sciences and demonstrate that architectural specialization can effectively enhance the reliability of autonomous scientific research.

A predictive physics-based model of human respiratory, phonatory, and articulatory subsystems is developed to simulate voice production. Representing lungs, compressible airways, and vocal folds as spring-damper-mass controlled piston-cylinder systems, our mathematical model robustly captures the intricate dynamics of airways during phonation. The nonlinear viscoelastic properties of lung tissues and compressible airways were investigated, yielding a responsive and expressive baseline respiratory model with the capability to further extend into a patient-specific model for both respiration and phonation. The resulting framework was subsequently integrated with a mechanical representation of the vocal tract, governed by the glottal area waveform (GAW) capturing the motion of vocal folds during sustained phonation. The GAW is extracted from laryngeal high-speed videoendoscopy data of a normophonic participant using deep learning. Our novel paradigm transcends beyond modeling the respiratory system, enabling AI-driven modeling of vocalization, including vocal fold dynamics, interactions with flow aerodynamics, and flow resistances, induced by the oscillatory behavior of vocal folds. Our investigation leads to the first-ever simulation of respiratory dynamics for vocalization, directly advancing the prediction of subglottal pressure distributions, impossible to measure directly and noninvasively in humans, dynamic resistances, and energy transfer mechanisms during phonation in voice mechanics.

The Remeika series superconductor, $(\text{Ca}_x\text{Sr}_{1-x})_3\text{Rh}_4\text{Sn}_{13}$, shows a rare nonmagnetic quantum critical point (QCP) associated with the continuous charge-density wave (CDW) and structural transition under the ``dome'' of superconductivity achieved by tuning composition and applying pressure. Here we use a nonmagnetic point-like disorder induced by 2.5 MeV electron irradiation to suppress the CDW and drive the system to and even beyond the QCP. This conclusion is based on a clear evolution of temperature-dependent resistivity, $ρ\left(T\right)$, from the Fermi liquid to the non-Fermi liquid regime with increasing amount of disorder. Starting on the CDW side, below the suggested QCP concentration of $x_c=0.9$, added disorder resulted in a progressively larger linear term and a reduced quadratic term in $ρ\left(T\right)$. Nearly perfect $T-$linear dependence is observed at the dose at which long-range CDW order is suppressed to $T=$0, consistent with the expectations. We refine the QCP location in this system and place it in the interval between $x=$0.75 and 0.85. Our results strongly support the concept that the disorder can tune the system to the quantum critical regime and even beyond. It follows from the argument by Imry and Ma that any ordered phase is unstable toward quenched disorder. Introduced in a controlled way, this disorder becomes a novel non-thermal tuning parameter likely applicable to a variety of different systems.

Disorder in magnetic systems typically suppresses long-range order, promoting short-range states such as spin glasses and magnetic clusters. This is particularly prominent in high-entropy materials, characterized by the random distributions of local magnetic entities and exchange interactions. However, in rare exceptions, long-range magnetic order can persist in high-entropy systems, while the microscopic characters and underlying mechanisms remain elusive, especially the magnetic behaviors of individual elements. Here, combining neutron diffraction and resonant soft x-ray scattering, we have conducted an element-specific investigation into the magnetic order of a high-entropy honeycomb-lattice van der Waals material (Mn1/4Fe1/4Co1/4Ni1/4)PS3. Despite significant atomic disorder, long-range zigzag antiferromagnetic order is observed below 72 K, with all four transition-metal elements participating in a unified phase transition. However, the spin orientations of various elements are distinct, attributed to the competition between single-ion anisotropies and exchange interactions. Our findings showcase a novel form of long-range magnetic order with disordered spin orientations, which is synergically stabilized by distinct magnetic elements in a high entropy magnet, offering a new paradigm for understanding complex magnetic systems.

Generating quantum data by learning the underlying quantum distribution poses challenges in both theoretical and practical scenarios, yet it is a critical task for understanding quantum systems. A fundamental question in quantum machine learning (QML) is the universality of approximation: whether a parameterized QML model can approximate any quantum distribution. We address this question by proving a universality theorem for the Many-body Projected Ensemble (MPE) framework, a method for quantum state design that uses a single many-body wave function to prepare random states. This demonstrates that MPE can approximate any distribution of pure states within a 1-Wasserstein distance error. This theorem provides a rigorous guarantee of universal expressivity, addressing key theoretical gaps in QML. For practicality, we propose an Incremental MPE variant with layer-wise training to improve the trainability. Numerical experiments on clustered quantum states and quantum chemistry datasets validate MPE's efficacy in learning complex quantum data distributions.

In this paper we prove a lower bound result for extremely large values of $L(\frac{1}{2},χ_p)$ with prime numbers $p\equiv 1\pmod 8$.

We study almost complete intersection ideals in a polynomial ring, generated by powers of all the variables together with a power of their sum. Our main result is an explicit description of the reduced Gröbner bases for these ideals under any term order. Our approach is primarily combinatorial, focusing on the structure of the initial ideal. We associate a lattice path to each monomial in the vector space basis of an Artinian monomial complete intersection and introduce a reflection operation on these paths, which enables a key counting argument. As a consequence, we provide a new proof that Artinian monomial complete intersections possess the strong Lefschetz property over fields of characteristic zero. Our results also offer new insights into the longstanding problem of classifying the weak Lefschetz property for such intersections in characteristic $p$. Furthermore, we show that the number of Gröbner basis elements in each degree is connected to several well-known sequences, including the (generalized) Catalan, Motzkin, and Riordan numbers, and connect these numbers to the study of entanglement detection in spin systems within quantum physics.

We consider an inverse boundary value problem for the biharmonic operator with the first order perturbation in a bounded domain of dimension three or higher. Assuming that the first and the zeroth order perturbations are known in a neighborhood of the boundary, we establish log-type stability estimates for these perturbations from a partial Dirichlet-to-Neumann map. Specifically, measurements are taken only on an arbitrarily small open subsets of the boundary.

Transversally Kähler foliations are a generalisation of Kähler manifolds, appearing naturally in the complex non-Kähler setting. We give a self-contained proof of how the classical methods used in the proof of the Aubin-Yau theorem adapt to the transversally Kähler case under the homological orientability condition. We apply this result to obtain a new, simpler proof of the already known Vaisman Aubin-Yau theorem.

On biomedical tabular data, flexible models such as deep networks, gradient-boosted trees, and kernel methods are repeatedly matched or beaten by linear and logistic regression given the same features. The usual reaction is to treat this as a model-side shortfall, to be fixed with more data, a better architecture, or tuning, on the assumption that the nonlinear structure is there and the model has failed to capture it. We argue that these fixes cannot help when the binding limit is the measurement rather than the model, as it frequently is in biomedicine. Additive noise blurs the population-optimal predictor, and because blurring removes a function's fine, rapidly varying detail before its broad shape, it erases nonlinear structure faster than linear structure. A degree-$k$ interaction is attenuated by the $k$-th power of feature reliability, while the linear part is attenuated only once. At the reliabilities typical of biomedical measurement, the nonlinear advantage can vanish even when the underlying biology is strongly nonlinear, and what the noise removes cannot be recovered by a larger cohort or a more flexible model, only by better measurement. The nonlinearity is hidden, not absent, and a tie between linear and flexible models is not by itself a verdict on the biology. These pieces are classical, drawn from measurement-error statistics, psychometrics, and Gaussian analysis, and we assemble them into an exact excess-risk identity. Measurement reliability is one of three conditions, alongside sample size and feature representation, that must align for a flexible model to help, and together they leave only a narrow window that most biomedical tasks fall outside. Across 140 UK Biobank tasks, the gap between flexible and linear models, where it exists, carries the predicted noise signature, and the three conditions can be separated by intervention but not by a benchmark alone.

Nanoscale three-dimensional localization across large biological volumes remains an outstanding challenge in optical microscopy, with existing approaches typically limited by imaging speed, volumetric field of view and localization precision when required simultaneously. Here, we overcome these limitations by combining a twin Airy engineered point spread function with two-photon light sheet fluorescence microscopy, enabling nanoscale localization throughout large volumetric fields of view. Our framework explicitly incorporates the broadband fluorescence emission characteristic of biological fluorophores, ensuring accurate localization under realistic imaging conditions. We achieve localization precisions of $<20$~nm laterally and 42~nm axially over volumes measuring 295~$μ$m x 330~$μ$m x 100~$μ$m , with a projected path to sub-10-nm localization in millimeter-scale specimens. Experiments in fluorescent bead phantoms and live mammalian oocytes confirm robust performance in both controlled and biologically complex environments. These results establish a scalable strategy for localization-based super-resolution imaging across biologically relevant volumes, bridging the gap between nanoscale precision and large-scale volumetric microscopy.

Tin vacancy (SnV) centers in diamond are promising solid state qubits for integrated quantum photonics. Here, we fabricate and characterize a diamond on Al2O3 dual taper waveguide structure containing SnV centers, demonstrating optical coupling between the diamond nanobeam and the underlying Al2O3 waveguide. The devices are realized using a bilayer fabrication approach compatible with wafer scale lithography. Clear guided SnV- emission is observed in all optically active devices, indicating effective optical coupling in the integrated structure. These results demonstrate a scalable fabrication approach toward integrating diamond color centers with photonic waveguides.

To simulate the relaxation of an entire 2D cellular network, this study proposes a symmetric relaxation method for both inner and marginal vertices. The relaxations of these two types of vertices are determined by the central angle symmetry of associated cells and the angle symmetry at each vertex, but with different major considerations. Trimmed Voronoi networks with varying irregularity are used as initial networks for the relaxation simulation. In particular, we propose a regular hexagon disordering method to generate Voronoi networks and find that the inner cells of networks with an irregularity value of one exhibit a conserved edge number distribution, as found in other 2D cellular networks. Simulation results agree with the von Neumann-Mullins law for both inner and marginal cells, and a modified equation including a geometric correction term significantly improves prediction quality. The Aboav-Weaire law and Lewis law are also reproduced, with the latter showing that relaxed cells tend to approach the ellipses' maximum inscribed polygons. Analysis of edge length, interior angle, and shape index reveals that symmetric relaxation inhibits T1 (neighbour exchange) topological transitions by reducing short edges while increasing area disparity among neighbouring cells. The findings suggest that T1 events may be triggered when force disequilibrium overcomes the stabilising effect of symmetric relaxation, providing a possible mechanistic explanation for T1 in 2D foams.

Differential Interference Contrast (DIC) microscopy and chiral analysis are two imaging techniques that measure the birefringence, i.e., the phase difference introduced by a sample on two orthogonal polarizations. Conventional approaches employ Gaussian beams and infer birefringence from polarization changes, resulting in phase-estimation sensitivities that depend on the unknown phase. We demonstrate here a new type of birefringence detector. It makes use of a vector vortex beam, a type of structured light endowed with optical modes that carry opposite orbital angular momentum (OAM). Using quantum estimation theory tools, we demonstrate that the sensitivity of phase estimation is independent of the value of the unknown phase, and can be even better, in principle, than the conventional approach. We experimentally validate the proposed scheme, demonstrating the potential of structured light for robust and uniform birefringence sensing.

We realize an injector device that supplies warm gas directly into a cryogenic environment. This injector has several advantageous features, including robustness, rigidity, simple installation, and excellent thermal isolation between a hot ($\sim$300 K) copper fill line and a cold ($<$3 K) cryogenic buffer gas cell. Less than 200 mW heat load on the cell is observed in realistic conditions of a molecular precision measurement experiment. A polyamide-imide (PAI) tube is the essential design feature. The fill line is epoxied to one end of the tube while the other end of the tube is connected to the cell via a slip-fit onto a brass nipple, realizing a complete vacuum-tight seal. PAI contracts on the brass nipple when cooled, forming a cryogenic leak-tight seal. The injector is easily (de-)mountable and rigid, with no significant displacement of the fill line relative to the cell observed during cooldown to 4 K. We characterize injector performance by flowing into the cell $\text{SF}_6$ through the hot fill line and cold $\text{He}$ buffer gas through a separate cryogenic fill line while laser ablating a barium-containing target. This produces cold BaF free radicals, detected using absorption spectroscopy. This injector design will be employed to laser cool radium-containing molecules, such as $\text{RaF}$ and $\text{RaOH}$, where leak-tight delivery of $\text{SF}_6$ and $\text{H}_2\text{O}$ reagents into a cryogenic buffer gas cell is required for scientific and safety reasons. These molecules are of particular interest for the study of symmetry-violating nuclear properties and searches for physics beyond the Standard Model.

Understanding how swine farms are interconnected, directly and indirectly, is essential to characterizing infectious disease transmission. This study aimed to describe the connectivity of swine farms across 11 network types, including vehicle movements (i.e., trucks and trailers), animal movements, and distance-based farm-to-farm contacts, to identify links among production types and farms likely to be consistently characterized as super-spreaders. Truck and trailer movement networks were the most densely connected, particularly for feed transport, showing connectivity levels between 98.7% and 99.7% higher than those of pig movement and distance-based networks. These networks also exhibited the highest degree and frequency of connections between farms, while the aggregated truck network, which included all truck types, showed the greatest potential to act as a bridge connecting farms. Finisher farms were highly interconnected with other farm types across all networks. Sow farms were frequently reached by other farm types, especially through feed truck movements, representing up to 8.7% of these links. We demonstrated that in vehicle movements and proximity networks, finisher farms played a major role as super-spreaders. When comparing the top 50 farms ranked by super-spreader score in each network, vehicle-based networks showed the highest similarity, with up to 89% of top-ranked farms shared between vehicle networks. In contrast, pig movement and distance-based networks identified largely distinct sets of top-ranked farms, sharing at most 4% and 8%, respectively, with other contact networks. Overall, each network exhibited a distinct connectivity structure, resulting in different sets of high-risk farms, particularly regarding potential transmission to breeding farms. These findings support the integration of multiple transmission pathways into disease surveillance.

In this work, we describe the development of track- and energy-reconstruc-tion algorithms for atypical Time Projection Chambers (TPCs) that will be used at the Institute of Experimental and Applied Physics, Czech Technical University in Prague, to search for the anomalous internal pair creation reported by the ATOMKI group. These chambers operate with an inhomogeneous toroidal magnetic field oriented orthogonally to the electric field; we therefore refer to them as Orthogonal-Field TPCs (OFTPCs). Although this configuration distorts the drift of ionization electrons and complicates the resulting electron and positron trajectories, it also offers several practical advantages. We present the most effective of several tested approaches, which employs a simulated ionization-electron drift map for track reconstruction and a Runge--Kutta-based fit for energy reconstruction. Using simulations, we demonstrate that -- under idealized conditions, namely an ideal charge readout with no amplification and no noise and with known initial track positions and directions -- it is possible to achieve a fitted Gaussian width (sigma) better than 1\% in relative energy for both electrons and positrons, after applying corrections for systematic effects that depend on the track parameters.

AI systems can increasingly automate scientific workflows, but the reasoning that links prior evidence, generated ideas, experiments and final claims often remains implicit inside model inference. Here we introduce Xcientist, a research harness that externalizes research synthesis and experimental validation into inspectable, contract-governed processes. Xcientist organizes literature evidence, idea states, implementation plans, ablation records and repair traces as persistent research artifacts, so that generated mechanisms can be grounded, executed, tested and revised without losing their evidential basis. We identify claim drift as a failure mode of automated research, where runnable artifacts no longer support the mechanism originally claimed. Across training-free memory systems, graph-structured traffic forecasting and multi-scale physics-informed neural networks, Xcientist preserves traceable trajectories from problem formulation to mechanism design, validation and bounded revision. These results suggest that AI scientists should be evaluated not only by their final artifacts, but by whether their synthesis and validation processes remain attributable, inspectable and scientifically accountable.

Accurately estimating treatment effects in time series is essential for evaluating interventions in real-world applications, especially when treatment assignment is biased by unobserved factors. In many practical settings, interventions are adopted at different times across individuals, leading to staggered treatment exposure and heterogeneous pre-treatment histories. In such cases, aggregating outcome trajectories across treated units is ill-defined, making individual treatment effect (ITE) estimation a prerequisite for reliable causal inference. We therefore study the problem of estimating the average treatment effect for the treated (ATT) by first recovering individual-level counterfactuals. We introduce a neural framework that learns simultaneously low-dimensional latent representations of individual time series and propensity scores. These estimates are then used to approximate the individual treatment effects through a flexible matching procedure that avoids classical convexity constraints commonly used in synthetic control methods. By operating at the individual level, our approach naturally accommodates staggered interventions and improves counterfactual estimation under latent bias, without relying on explicit temporal modeling assumptions. We illustrate our approach on both real-world energy consumption data and clinical time series, including high-frequency electricity demand-response programs and semi-synthetic data for individuals in intensive care unit (ICU), where hidden confounding, staggered treatment adoption, and non-stationary dynamics are prevalent.

This work presents an extension to Pareto Front Guided Sampling (PFGS), a Human-in-the-Loop (HitL) Bayesian Optimization (BO) framework in which Gaussian process (GP) surrogate-derived quantities are reformulated as objectives of a multi-objective optimization problem, and the resulting Pareto front is exposed to a domain expert for interactive candidate selection rather than returning a single automated recommendation. The framework is extended in two directions: constrained optimization is addressed by incorporating the posterior probability of satisfying output specification limits as an explicit Pareto objective, computed analytically from the GP posterior distribution; robust optimization is addressed by a Monte Carlo sampling strategy that estimates expected lower-confidence performance over a user-defined variability of input perturbations, capturing performance degradation under likely implementation deviations. The resulting multi-dimensional Pareto representation renders trade-offs between predicted performance, model uncertainty, probabilistic constraint satisfaction, and input robustness simultaneously visible through pairwise two-dimensional projections on an interactive dashboard, enabling selection criteria to be iteratively refined as the surrogate model improves and development objectives evolve. The framework is showcased on an eight-dimensional fed-batch Chinese Hamster Ovary (CHO) cell culture simulator demonstrating systematic identification of high-performing, feasibility-compliant, and perturbation-resilient operating conditions, and illustrating how expert-defined requirements provide a principled stopping criterion and support informed allocation of experimental resources.

Multivariate time series anomaly detection (MTSAD) is critical for a wide range of application areas, such as industrial monitoring, cybersecurity, or healthcare. Real-world data is often sparse, irregularly sampled or partially observed, yet existing methods assume uniformly sampled time series. We propose a generative approach based on Latent SDEs that projects the observed time series on a continuous-time stochastic dynamical system, directly being able to handle missing observations and irregular sampling, while also naturally capturing possible cyclic behavior that many real-world use cases inherently possess. Experiments on six anomaly benchmark datasets show that our proposed method ranks first among state-of-the-art baselines. We further demonstrate that our method remains robust under severe data sparsity, while performance significantly degrades for the tested baseline methods. These results highlight latent SDEs as a natural inductive bias for anomaly detection in multivariate time series, especially in presence of real-world irregularities.

Sparse matrix kernels form the computational backbone of scientific computing, traditionally relying on C/C++ and Fortran implementations that prioritize performance over memory safety. This work evaluates Rust as a systems-level alternative for sparse linear algebra by implementing and benchmarking three core workloads: sparse matrix-vector multiplication (SpMV), Lanczos-based Krylov methods, and matrix-exponential evaluation. We compare native Rust code against established baselines (Intel oneMKL, Eigen, PETSc, and PSBLAS) across a suite of representative matrices. Our results show that Rust's sparse kernels achieve performance comparable to Eigen and PSBLAS, tracking the state-of-the-art for CSC formats, while trailing PETSc's advanced blocked CSR optimizations. By analyzing compile-time monomorphization, SIMD vectorization, and FFI boundaries, we assess the practical impact of Rust's safety model and ecosystem readiness. The study provides concrete, evidence-based guidance for modernizing high-performance numerical software stacks.

We study the $I^G(J^{PC})=0^+(1^{++})$ $K\bar Kπ$ system in an infinite-volume three-body unitary framework, focusing on the pole content of the region of the $f_1(1285)$ and $f_1(1420)$ resonances. The coupled $πa_0$-$K\bar K^*$ amplitude is constructed in the spectator-isobar representation, where the one-particle-exchange interaction required by three-body unitarity automatically incorporates the triangle-singularity mechanism. The short-range three-body interaction is constrained by fitting the $0^+(1^{++})$ component of the BESIII $K^0_SK^0_Sπ^0$ invariant-mass distribution in the $J/ψ\toγ(K^0_SK^0_Sπ^0)$ decay. Analytically continuing the fitted amplitude to the relevant unphysical Riemann sheets, we find two robust poles: \begin{align} \sqrt{s_{f_1(1285)}}&= \left(1277\pm2\pm1\right) -i\left(12\pm1\pm0\right)\text{MeV}\,,\notag\\ \sqrt{s_{f_1(1420)}}&= \left(1435\pm2\pm7\right) -i\left(40\pm2\pm1\right)\text{MeV}\,.\notag \end{align} The pole trajectories indicate that the $f_1(1285)$ originates from dressing a bare state introduced in the potential. In contrast, the $f_1(1420)$ is predominantly dynamically generated, and a single-channel analysis traces it to an $S$-wave $K\bar K^*$ quasi-bound state mixed with the nearby bare state, supporting its hadronic-molecule interpretation. We also find an additional pole deeper in the complex plane in the best-fit amplitude on the same Riemann sheet as the $f_1(1285)$. This additional pole is generated by the $P$-wave $πa_0$ contact interaction alone. It has a sizable cutoff and two-body-input dependence, and leaves little visible imprint on the physical lineshape. Finally, we provide a detailed and pedagogical appendix on how three-body cuts affect the solution of the integral equation.

We prove that every natural number is a finite sum of distinct unit fractions whose denominators are semiprimes (products of two distinct primes). This is the $ω=2$ integer case of a problem of Erdős and Graham, stated only as a conjecture by Butler, Erdős and Graham (Integers 15 (2015), A51), who proved the $ω=3$ analogue. Counterintuitively the problem hardens as $ω$ decreases -- the induction's feed thins -- so $ω=2$ is the hard case; our proof adapts the Butler-Erdős-Graham induction to this thin-feed regime, where the entire content of the induction step reduces to an explicit onset inequality $Y_0(N)\le\min\{β(N),β'(N)\}$, proved for all $N\ge10$ by Olson's addition theorem and elementary Chebyshev bounds above a finite, machine-checked base range. The same engine extends to the rationals: for every squarefree $b$, every $a/b$ above an explicit threshold $\min\{B_{N_b}/6,\,1/5\}$ is $ω=2$ representable, unconditionally. As an application we give the first complete proof of the rational $ω=3$ statement -- every $a/b$ with squarefree $b$ is a sum of distinct sphenic unit fractions -- that Butler, Erdős and Graham conjectured but left unpublished; a descent settles every $ω\ge3$. What remains open is the $ω=2$ regime below this threshold, which we reduce to a single explicit conjecture -- that the gap-free floor of a semiprime subset-sum set tends to zero. This work is a human-AI collaboration: AI tools (notably Anthropic's Claude, used through Claude Code) contributed substantially to the Lean formalisation, the experiments, and the writing; correspondingly, every result is machine-checked in Lean 4 / Mathlib (no sorry; two cited classical axioms, plus the native_decide compiler-trust base for the finite computations), so its correctness is independent of the tools used.

We prove a growth dichotomy for infinite approximate groups, and more generally approximate semigroups, in hyperbolic groups. If \(G\) is a finitely generated hyperbolic group and \(A\subseteq G\) is infinite with \[ A^2\subseteq AX \] for some finite \(X\subseteq G\), then either \(\langle A\rangle\) is virtually cyclic, or \(A\) has positive exponential growth in the ambient word metric. We also introduce a product-growth criterion for the existence of growth rates of approximate semigroups. The criterion applies to hyperbolic groups: if \(G\) is hyperbolic with finite generating set \(S\), then there is a constant \(c_{G,S}>0\) such that \[ |UV| \geq c_{G,S}\,\frac{|U||V|}{n+k+1}, \qquad U\subseteq B_n,\; V\subseteq B_k. \] The linear loss is optimal in order whenever \(G\) contains an element of infinite order. In the free group with its standard generating set one may take \(c_{G,S}=1/4\). We also prove that, in a free group, if \(U\subseteq S_n\) and \(V\subseteq S_k\), then \[ |UV|\geq \left(\frac{2}{3}+\frac{1}{3\cdot 4^{\min\{n,k\}}}\right)|U||V|, \] and this constant is sharp for all \(n,k\).

We extend the application of the Guillemin--Abreu formalism to holomorphic sectional and bisectional curvature on $\mathbb C^n$, $\mathcal O(-\ell)$, and Hirzebruch manifolds $M_{n,\ell}$, and further apply it to the total spaces of certain higher-rank vector bundles. The resulting formulas recover known positivity criteria and we show that, when the slope is sufficiently close to $1$, the extremal metrics on $M_{n,\ell}$ have positive holomorphic sectional curvature. We construct complete scalar-flat Kähler metrics on $ \operatorname{Tot}\bigl(\mathcal O(-k)\oplus\mathcal O(-k)\to\mathbb{CP}^n\bigr), $ and identify the Ricci-flat case, which occurs precisely when $2k=n+1$.

We develop a framework for conformal prediction in dyadic regression problems under complex missingness mechanisms. At the theoretical level, we develop general technical tools for establishing finite-sample validity of conformal prediction under distributional invariance conditions weaker than exchangeability. A key result handles the case where the sample itself is a random subset of the index set, a setting not covered by existing theory, via a novel bijection argument that constructs an explicit measure-preserving correspondence between events. In addition, we propose conformal prediction procedures for jointly exchangeable arrays, including full conformal, split conformal, a row-column approach exploiting similarities within rows and columns, and a selective conformal procedure achieving mask-conditional validity. For missing elements, we establish asymptotic validity of a weighted conformal procedure under a nonparametric graphon model for the missingness mechanism. We further establish conditional validity results for both continuous and discrete responses; to the best of our knowledge, this is the first formal proof of asymptotic conditional validity for weighted conformal prediction under a missing-not-at-random assumption. The proposed methods are illustrated on synthetic and real network data.

The 2024-2025 Nobel and Turing awards recognised AI and quantum science simultaneously. Yet no deployed system has brought these streams together for the public. This paper presents QSignAI, a production-deployed platform demonstrating a bidirectional AI-quantum relationship in a real-time event participation system. We address three questions: can quantum-randomness generation via a two-source extractor be embedded in an AI-driven social platform with acceptable latency; can an AI bot make quantum phenomena perceptually legible to general audiences; and does the combined system work in practice? A conversational bot routes each participant's first message through a quantum pipeline comprising a Toeplitz two-source extractor over independent single-qubit Hadamard measurements on SV1 and DM1 simulators, plus a 2-qubit Bell state, producing a unique quantum-randomness-seeded identity signature per participant. The first two questions are answered through system architecture and qualitative deployment evidence from live events; the third through successful production deployment. The current deployment uses cloud quantum simulators; physical QPU randomness is the near-term extension. Measurable benchmarks are identified as priority future work.