Papers

We study the large-population limit of a discrete-time Moran process featuring multiple strategies, drawn from a possibly infinite strategy space $\mathcal{V}$ (a metric space), under both weak and strong selection. In the Eulerian density formulation, the limiting dynamics depend critically on the relative scaling of population size, mutation rate, and selection intensity. Depending on these scalings, the limit behavior is governed either by a purely deterministic replicator-type continuity equation or by a diffusion-enhanced PDE. In the latter case, the diffusion operator recovers the classical Fleming–Viot operator and the Kimura equation as special instances. Methodologically, we derive the limit by reinterpreting the discrete process in Eulerian coordinates, constructing interpolating curves that satisfy an approximate PDE, and establishing convergence via a compactness argument in a suitable topology on the space of probability measures over probabilities over $\mathcal{V}$.

A spacetime seal, a cryptographic guarantee that a digital event has occurred at an approved location and time, can augment a digital signature with location attestation for legal, financial, and regulatory use cases. In adversarial settings any purely classical realization of such a seal can be spoofed. Quantum position verification (QPV) offers a physics-based solution, exploiting the no-cloning theorem and the no-signaling principle to certify a party's spacetime coordinates. While the feasibility of QPV has been recently shown via entanglement- and coherent light-based protocols, achieving loss tolerance for these schemes substantially increases implementation complexity at metropolitan scales. Here, we introduce and experimentally demonstrate a loss-tolerant QPV (LT-QPV) protocol whose security is independent of channel loss. We prove finite-size security against quantum polynomial-time entangled adversaries in the quantum random oracle model instantiated with cryptographically secure hash functions. Implemented entirely with commercial off-the-shelf components, our system certifies position within 22 minutes of net data collection time against a restricted adversary, with a clear path to real-time certification (<1s) with upgraded hardware. Our architecture, requiring only a single quantum verifier node alongside classical infrastructure, is naturally compatible with metropolitan-area quantum networks, establishing the foundations for scalable, physics-backed spacetime certification as a deployable service.

We establish a comprehensive algebraic, geometric, and physical classification of the wave closure space generated from primitive plane wave sequences on the discrete spatial lattice $\mathbb{Z}$. Resolving lattice degeneracies via unwrapped phase spaces, we prove that the linear wave group $\mathcal{G}$ under pointwise product, inversion, and root extraction is an infinite divisible abelian torsion group isomorphic to $(\mathbb{Q}/\mathbb{Z}) \times (\mathbb{Q}/\mathbb{Z})$, with canonical decomposition into Prüfer $p$-groups and maximal cyclotomic value field $\mathbb{Q}^{\mathrm{ab}}$. Extending to coordinate powers yields the polynomial phase group $\mathcal{G}_{\mathrm{poly}}$, classified by integer-valued polynomials $\operatorname{Int}(\mathbb{Z})$. Adjoining addition yields the group algebra $\mathbb{C}[\mathcal{G}_{\mathrm{poly}}]$, for which we establish the Permutation-Symmetric Phasor Superposition Theorem, factoring superpositions into collective barycentric carriers and closed Born probability envelopes. Using algebraic sieves over roots of unity, we construct idempotent projectors and complementary Not-sieve notch filters, achieving exact algebraic synthesis of both momentum and localized Kronecker position bases. Generalizing to non-abelian quaternions $\mathcal{V}_\mathbb{H}$, we prove a polar decomposition into scalar envelopes and $\mathrm{SU}(2)$ spinor rotors. Identifying coordinate advance with time, biquaternion determinants intrinsically yield Minkowski spacetime $s^2 = c^2 t^2 - \|\mathbf{x}\|^2$ and the Lorentz group $\mathrm{SO}^+(1,3)$. We demonstrate emergent vacuum zero-point jitter, topological selection of rational frequencies, and correspondences with discrete qudits.

Bayesian graph alignment estimates correspondence probabilities, but convergence of an alignment-score trace need not imply accurate correspondence marginals. We audit this gap on 240 new exact graph pairs from four source families, 240 larger pairs with 20-100 vertices, and a separate 60-case exact implementation check. Under an explicit edge-flip likelihood, we compare three samplers and score, marginal, indicator, categorical, and classifier-based diagnostics. Marginal disagreement improves error discrimination over score R-hat for the exact informed sampler, but its improvement for vanilla local sampling is uncertain. Assignment-based R* and short indicator panels are competitive; no diagnostic dominates across samplers and endpoints. At larger sizes, diagnostics predict subsequent marginal changes, not posterior error, and classification performance depends on the drift threshold. Disjoint-window and held-out-chain checks attenuate but preserve positive associations. Only 22 of 240 original reference sets pass an agreement screen. On forty failure-selected cases, eightfold SMC particle escalation does not resolve disagreement, whereas additional rejuvenation helps. Longer informed runs remain unstable. An elementary feasible-alignment bound demonstrates severely unrepresentative SMC and informed-chain scores in concentrated 100-vertex cases, independently of approximate reference consensus. We also exhibit common-start chains with near-zero disagreement despite exact marginal error near .967. These results support assignment-sensitive auditing while identifying limits of finite budgets, diagnostic rankings, and reference agreement as evidence of accuracy.

Cross-lingual information retrieval (CLIR) is increasingly important in multi-national industries, where critical technical evidence may exist in a different language than the query. However, existing benchmarks do not adequately capture domain-specific cross-lingual retrieval or the retrieval-depth and recoverability failures that aggregate recall hides. In this work, we benchmark CLIR in the chemical domain, with a focus on patent data. We construct a multilingual dataset from Google Patents and the European Patent Office (EPO) data, spanning five languages (covering major Eastern and Western languages) and reflecting the diversity and complexity of real-world industrial documentation. Using this dataset, we systematically evaluate eight state-of-the-art embedding models for cross-lingual retrieval. Our results show a substantial performance gap between monolingual and cross-lingual settings: for the best-performing model, Recall@10 drops from 0.72 to 0.53 in cross-lingual setting. Retrieval depth also degrades significantly, with relevant documents ranked lower across languages in cross-lingual scenarios. Furthermore, some multilingual embedding models that perform strongly in monolingual settings exhibit sharp declines when queries and documents are in different languages, providing practical insights for model selection in cross-lingual use cases. These findings highlight critical limitations of current approaches and emphasize the need for more robust cross-lingual retrieval methods in domain-specific settings. Our benchmark provides actionable insights for model selection and establishes a controlled diagnostic evaluation framework for CLIR over industrial technical text. Data and code are publicly available at https://github.com/MohammadKhodadad/Multi-Lingual-QAC.

We study the large-time behavior of the displacement for wave equations with non-integrable speeds and zero initial displacement. For initial velocities in $L^2\cap L^1$, we derive an $L^2$ upper bound under a weighted Poincaré inequality and symbol estimates up to order two on the propagation speed. The rate is determined by the speed and its time integral. For these speeds, an additional symbol estimate for the third derivative and the nonzero mean condition on the initial velocity give a lower bound of the same order, obtained without using an explicit formula for the Fourier multiplier. At the endpoint $a(t)=(1+t)^{-1}$, the sharp rate is $t(\ln t)^{-\frac{n}{4}}$. The estimates apply to polynomial, logarithmically modified, oscillating, and exponential-type speeds.

Obesity is a major risk factor for breast cancer, yet how obesity-associated changes in the physical properties of white adipose tissue (WAT) influence tumor invasion remains poorly understood. Here, we combine experimental and computational approaches to investigate this question. Using mouse models and human mastectomy samples, we show that obesity not only increases adipocyte size, but also asphericity. We observe that tumor-conditioned media drives lipid loss and dedifferentiation of adipocytes into myofibroblast-like cells in vitro and that adipocytes in high-fat diet-fed mice lose lipid at a faster rate than those in normal diet-fed mice in response to mammary tumors in vivo. We then develop discrete element method (DEM) simulations to understand how these obesity-induced changes in cell and tissue properties alter breast cancer invasion under high-fat and normal diet conditions. In DEM simulations, adipocytes are modeled as deformable polygons and polyhedra, and cancer cells are modeled as soft, adhesive disks and spheres in two and three dimensions, respectively. Invasion is driven by cancer cell proliferation and tumor-induced lipid loss is modeled as corresponding decreases in adipocyte size. DEM simulations indicate that obesity-associated increases in tissue pressure induce adipocyte deformation, consistent with elevated asphericity in experimental data. At small lipid loss rates, the degree of cancer invasion is only weakly affected by tissue pressure. However, at sufficiently large lipid loss rates, we find that tissue pressure accelerates cancer invasion. Together, these results suggest that lipid loss physically remodels WAT to facilitate breast cancer invasion, and that obesity may exacerbate this effect.

For a propositional proof system $\mathcal{P}$ and a polynomial-time function $G: \{0, 1\}^n \to \{0, 1\}^N$ ($N > 10n$), we say that $G$ is a *proof complexity generator* against $\mathcal{P}$ if $\mathcal{P}$ cannot efficiently prove the (suitably encoded) statement "$y\not\in\mathrm{Range}(G)$" for every $y \in \{0, 1\}^N$, and $G$ is a *demi-bits generator* against $\mathcal{P}$ if $\mathcal{P}$ cannot efficiently prove the statement "$y \not\in\mathrm{Range}(G)$" for a noticeable fraction of $y \in \{0, 1\}^N$. As can be seen from the definitions, proof complexity generators are qualitatively stronger objects than demi-bits generators. Our main result is that, perhaps counter-intuitively, every demi-bits generator "contains" exponentially many proof complexity generators. In fact, a random subset of output bits of a demi-bits generator forms a proof complexity generator with constant probability. This result is an extremely simple corollary of Pajor's Lemma (a strengthening of the well-known Sauer–Shelah Lemma). This result allows us to exhibit the hardness of the Range Avoidance problem ($\text{Avoid}$) in several new, restricted settings of interest. Along the way, we introduce the notion of *zero-error disperser families* that can transform demi-bits generators into proof complexity generators, and show that this family can be computed by projections (i.e., without any circuit complexity overhead). Using a different instantiation of this family, we construct proof complexity generators from *barely non-trivial* demi-bits generators $G: \{0, 1\}^n \to \{0, 1\}^N$, where the number of hard-to-prove statements of the form "$y \not \in \mathrm{Range}(G)$" just slightly exceeds $2^n$ (which is the number of *false* statements of this form).

For a smooth, connected, oriented surface $\Sigma$, topologically a sphere, immersed in the homogeneous space $E(\kappa,\tau)$ with $\tau\ne0$, we show that a one-sided Cauchy–Riemann-type inequality on the original Abresch–Rosenberg differential $\mathcal{Q}_{AR}$ – much weaker than requiring $\mathcal{Q}_{AR}$ to be holomorphic – already forces the mean curvature $H$ to be constant. The proof is uniform across every $E(\kappa,\tau)$, including the round sphere, and combines the Bers–Vekua similarity principle with the Poincaré-Hopf index formula for line fields; it is logically independent of the algebraic argument used, in the companion paper~\cite{AlencarRosenberg2026}, to characterize CMC immersions by holomorphy of $\mathcal{Q}_{AR}$ alone. We also show that the topological hypothesis is sharp: on every Berger sphere, including the round one, there exist compact non-CMC tori – the preimages under the Hopf-type submersion $\pi:E(\kappa,\tau)\to M^2(\kappa)$ of simple closed curves in the base with nonconstant geodesic curvature – satisfying the same inequality with a constant bound. As an elementary consequence of the classical Poincaré-Hopf theorem for line fields, we record that every closed surface on which $\det S<0$ everywhere – a condition satisfied, in particular, by every Hopf tube, where $\det S\equiv-\tau^2$ – is a torus.

Agent-Integrated Software (AIS) embeds an intelligent agent in a conventional application, supporting both human operations and agent actions. Human operations let users make precise changes and inspect results, while agent actions carry out routine or multi-step tasks. These complementary roles make coexistence a likely long-term feature of many software systems. Human operations and agent actions affect the same software state and can use one another's results. Therefore, security policies must remain effective across both paths. We argue that AIS security must be assessed at the level of the whole software system. Protecting the agent and the conventional software core separately does not establish that they are secure together. To guide security analysis of AIS as a whole, we organize the problems arising from this coexistence into four categories: context misuse, authorization violation, execution control, and effect integrity. Using these categories, we examine how current practices address the security problems in AIS and where their protection remains limited. Building on this analysis, we identify research opportunities in preserving information provenance, enforcing policy across operation paths, maintaining valid authorization over time, and managing persistent effects and recovery. This resulting perspective provides a conceptual framework for understanding and improving the security of AIS.

The ideas behind the screened hybrid functional HSE took shape in the early 2000s, about a quarter century ago. HSE was not explicitly designed to predict band gaps, although at the time we did recognize that hybrids could help address the band-gap problem. It was designed to make exact exchange affordable and physically defensible in extended systems. Keeping exact exchange at short range and screening it away at long range inverted the prescription developed for molecules. When HSE band gaps for ordinary semiconductors and selected metal oxides came out substantially closer to experiment than those obtained with conventional semilocal approximations, we were pleasantly surprised and encouraged by the path we had taken. The revision from HSE03 to HSE06 improved molecular thermochemistry while preserving the good accuracy for semiconductor gaps. The one-quarter exchange fraction was inherited unchanged from the successful unscreened PBE hybrid for molecules, rather than fitted to solids. HSE established a practical screened-hybrid approach for accurate band-gap calculations, expanding first-principles work in defect physics, photovoltaics, power electronics, and quantum information, among other areas. The first part of this Perspective recounts the decisions made at HSE's birth and their influence on subsequent developments. The second part assesses recent advances in semilocal and hybrid functionals against benchmarks on sets of solids. Recent meta-GGAs achieve HSE06-level accuracy for selected semiconductors, while dielectric-dependent and tuned hybrids address the limitations of fixed screening. These hybrid developments share with our early work on local hybrids and local range separation the aim of adapting exact exchange to the electronic environment. The comparisons assess band gaps alongside structures, energetics, computational cost, and transferability.

This paper studies symmetry-induced reductions of scalar recurrence complexity for balanced banded Toeplitz determinants and permanents. Three mechanisms emerge: determinant state and spectral compression, exceptional permanent–determinant conversion, and symmetries of permanent state spaces. For symmetric determinants, straightening reduces the row-column states from $\binom{2m}{m}$ to the Catalan number $C_{m+1}$; primitive symplectic weight compression then leaves $(3^m+1)/2$ distinct autonomous modes. Skew symmetry has a parallel compound/Hodge explanation: the middle exterior representation splits into two Hodge halves of dimension $\binom{2m}{m}/2$, the one-step compound transfer exchanges the halves, and its two-step restriction has $3^{m-1}$ generic ternary modes. Thus the full skew bound is $2\cdot3^{m-1}$. Widom–Hankel arguments prove generic scalar minimality in both symmetry classes, and every even skew Toeplitz determinant admits an explicit half-size square factorization. For the zero-diagonal pentadiagonal support, a Pólya–Kasteleyn signing converts every permanent to a determinant. Among consecutive zero-diagonal two-sided bands, universal entrywise conversion—and separately Toeplitz diagonal-wise conversion—occurs only in the Hessenberg families and this pentadiagonal case. Paired renewal identities recover all restored-diagonal determinant and permanent layers. For permanents, transposition gives the open symmetric bound $(\binom{2m}{m}+2^m)/2$, while cyclic defect-sector pairing gives $(4^m+\binom{2m}{m})/2$. Skew-symmetry forces odd-size vanishing and corresponding even-subsequence bounds. In semibandwidth two the open symmetric bound is generically sharp; higher-semibandwidth permanent minimality is left separate from the symmetry reductions proved here.

Day-ahead electricity price forecasts support trading and storage decisions, but for battery arbitrage predicting intraday price spreads is more relevant than predicting individual hourly prices. Here we show that a temporal hierarchy forecasting (THieF) framework that jointly reconciles forecasts of hourly electricity prices and all intraday price spreads consistently improves performance across two major European electricity markets and three different forecasting architectures. Using five years of out-of-sample data from Germany and Spain, we obtain accuracy improvements of up to 19.7% and profit gains of up to 10.4% relative to unreconciled hourly price forecasts. The gains persist even for a highly accurate pretrained TabPFN foundation model. Our results demonstrate that exploiting coherent relationships between economically relevant forecasting targets can improve both predictive accuracy and decision value, and that better statistical forecasts do not necessarily imply better economic decisions.

We study the problem of finding tight explicit bounds on sums over zeros of $L$-functions near the $1$-line with a variety of zero-free regions and zero-density estimates. As an application to demonstrate the techniques, we apply them to the case of the Riemann zeta function and the prime number theorem to obtain new best estimates on the error terms in the prime number theorem. We show that for $x\geqslant21$ we have \begin{multline*} |\psi(x)-x| < 0.239\, x \exp\bigl(-0.1982767\,(1+\log\log\log(x)/(15\log\log(x)))\log(x)^{3/5}\log\log(x)^{-1/5}\bigr). \end{multline*} Combined with the zero-free half-plane $\Re(s)>7/8$ announced by OpenAI in October 2026, the same methods give $|\psi(x)-x| < 0.04934\,x^{3/4}\log(x)^2 + 0.02636425\,x^{7/8}$ for $x\geqslant 21$.

In the spirit of the Brown-Susskind conjecture, proven in \cite{haf} and \cite{li}, we study the growth of the dimension of the set of $n$-qubit unitaries which can be obtained as a product of a fixed number of $2$-qubit gates from a fixed set of pairs of qubits. We show that this dimension increases strictly when one more pair is added, for at least some choice of the $2$-qubit pairs at each step.

We provide sufficient conditions for the consistency of penalized least squares procedures that select the order (dimension) of a regression model from a sequence of nested classes, allowing for dependent, martingale-difference errors. The main contribution is to relax the classical identifiability requirement: parameters indexing classes larger than the true order need not be identified, provided the additional, excess directions admit a linear approximation to the truth in a neighbourhood of the true parameter. This relaxation lets the number of candidate models grow with the sample size, removing the usual need for a fixed upper bound. We verify the resulting high-level conditions for two classes of nonlinear regression models used in applied work: multiple-regime smooth transition regression and mixture-of-experts models with generic generalized linear model experts. Under a BIC-type penalty, the resulting order selection rule is consistent in either model class whenever the number of candidate models grows slower than the logarithm of the sample size.

Unobserved confounding is a fundamental challenge in causal inference from observational data. This article develops a mixture-learning perspective, viewing latent confounders as sources of heterogeneity that induce mixture structure in observed data. Under suitable structural and identifiability assumptions, recovering the mixing distribution and component mechanisms enables estimation of interventional distributions and causal estimands. Using variants of Bernoulli mixtures as a running example, we contextualize mixture-learning techniques and their structural assumptions, and connect them to causal inference in panel-data settings, including latent factor models and synthetic interventions.We then consider high-dimensional exponential-family mixtures with dependent outcome trajectories, moving beyond counterfactual means to model counterfactual distributions. We situate this perspective relative to complementary approaches for unobserved confounding. Together, these ideas provide a bridge between mixture learning and causal inference, connecting recent advances in high-dimensional mixture learning to scalable identification and estimation of causal effects while raising new challenges for mixture learning.

Automated kernel bug discovery has advanced rapidly. Continuous fuzzing and static analysis systems, such as syzbot, now expose Linux kernel bugs at a scale that downstream processes struggle to absorb. Yet a crash report is only the beginning. Before a bug is eliminated, it must be triaged, understood, patched, validated, reviewed, integrated, and often backported. These later stages remain far less automated, creating a persistent gap between bug discovery and patch deployment. This SoK systematizes the Linux kernel bug lifecycle from discovery to deployment. We organize prior work and production systems into five stages: discovery, triage, patch generation, patch validation, and integration. We explain the resulting automation gradient through kernel-specific challenges such as concurrency, implicit invariants, cross-syscall state, hardware dependence, lack of fault isolation, and architecture/configuration multiplicity. We further ground the analysis in a measurement of real syzbot-fixed bugs. The data shows that the crash-to-patch gap is not merely a backlog of unfixed reports but a structural failure mode of the repair pipeline: even after being fixed, bugs often remain open for weeks, require review-driven patch revisions, or lack reproducers that current repair and validation systems assume. This exposes a mismatch between where kernel-security automation is mature and where bug closure actually breaks down. These findings expose a deeper mismatch: today's repair and validation techniques often assume reliable reproducers, localized root causes, and checkable correctness oracles, yet these are precisely the artifacts missing from many real kernel bug reports. Closing the crash-to-patch gap, therefore, requires treating such artifacts as outputs to be produced, not prerequisites to be assumed.

This paper provides a rigorous computer-assisted proof of the existence of generalized transmission eigenvalues arising in nonlinear optics in the context of high-order harmonic generation, a result conjectured in Cakoni, Hovsepyan, Lassas, and Vogelius, SIAM J. Math. Anal. 57 (2025), 1370-1405. The analysis is carried out for a one-dimensional nonlinear medium, where the problem reduces to a coupled system of nonlinear homogeneous ordinary differential equations subject to nonstandard boundary conditions. These eigenvalues correspond to probing frequencies $\omega$ for which there exists a nontrivial incident $\omega$-wave such that the second-harmonic field generated does not persist outside the compact support of the nonlinear medium, thereby rendering its nonlinear properties undetectable to an external observer. Building on earlier numerical evidence, we establish that, in the low-frequency regime, there exist generalized transmission eigenvalues whose associated eigenfunctions exhibit blow-up behavior as the frequency tends to zero. The proof combines analytical arguments with validated numerics, employing a Newton-Kantorovich framework together with interval arithmetic to rigorously control approximation errors. The algorithmic implementation underlying the computer-assisted proof is made available in the GitHub repository TransmissionEigenvalues.jl by Dominic Blanco (2026).

In this manuscript, we analyze a two-phase free boundary problem of Alt-Phillips-type driven by fully nonlinear elliptic equations with lower-order ingredients. More precisely, we consider equations of the type \[ F(x,D^{2}u)+\mathscr{H}(x,Du)=\mathscr{F}(x,u^{+},u^{-}) \quad \text{in } B_1 \subset \mathbb{R}^{n}, \] where the Hamiltonian term exhibits mixed linear and nonlinear gradient dependence, namely \[ \mathscr{H}(x,\xi):= \langle \mathcal{B}(x),\xi\rangle+\varrho(x)|\xi|^{\sigma}, \quad 0< \sigma\leq2, \,\,\, \sigma \neq 1, \] and the reaction is governed by a two-phase power-type nonlinearity of semilinear-type \[ \mathscr{F}(x,u^{+},u^{-}):=\mathfrak{g}(x)\big[(u^{+})^{m}-(u^{-})^{m}\big] \quad \text{for} \quad 0<m<1. \] Under suitable structural assumptions on the coefficients and the operator $F$, we develop a robust analytical framework to capture the fine properties of solutions near the free boundary. In particular, we establish improved growth estimates at higher-order singular nodal points, points where both phases meet and the solution exhibits critical degeneracy. The results are based upon a fine blow-up argument, stability and a Liouville-type theorem. We further prove quantitative non-degeneracy results, ensuring that solutions detach from zero at a controlled rate on each phase. As a consequence of our analysis, we derive a Liouville-type theorem for global solutions within this class. Our results extend and unify several previously known scenarios, and remain new even in the presence of gradient-dependent nonlinearities of order ($\sigma \neq 1$), thereby encompassing models with nonlinear drift and absorption effects arising in phase transition and reaction-diffusion phenomena.

LLM explainers are increasingly attached to autonomous agents as runtime oversight, with operators reading a generated account of the agent's beliefs and actions rather than its internal state. We audit the account itself, pairing an Active Inference (AIF) agent that tracks German grid demand and adjusts generation with an LLM explainer on three backends (GPT-4o, Claude-3-Opus, Gemini), and probing the pair with three black-box triggers. Corrupting the observation stream by 600 MW per step moves the agent's posterior by 490 MW, roughly 0.9% of grid capacity. None of the 30 explanations produced during the injection flag anything under a stated rubric, and each narrates the corrupted belief fluently. On timesteps where the agent takes an objectively wrong action, all three explainers produce a sycophantic rationalization 80-95% of the time (n = 20 per backend). Attacker-controlled text in the observation metadata field steers the explainer, with susceptibility differing by provider and data exfiltration succeeding on all three. We propose mitigations for each failure but do not evaluate them. In every failure we observed, the explanation was fluent and wrong. Moreover, nothing in the explainer architecture checks whether an explanation is true before an operator acts on it. Testing the explainer therefore belongs in any audit of an agentic deployment.

We study the variance of the number of lattice points in smoothed narrow sectors, where the average is taken over the position of the sectors. There are three regimes corresponding to specific ranges for the open angle of the sectors and in each we obtain an asymptotic formula for the variance.

We report the synthesis and characterization of lanthanum nitride by direct reaction of lanthanum metal with high pressure nitrogen gas in a laser floating-zone furnace. Combined synchrotron X-ray and neutron diffraction measurement verifies a rocksalt structure with stoichiometry LaN$_{0.94}$, consistent with first principles prediction that LaN is dynamically unstable in the stoichiometric limit. Electrical transport, heat capacity, and magnetization measurements show that LaN$_{0.94}$ is metallic with a superconducting transition at $T_c=5.95$ K. Our data establish LaN as the first bulk superconducting member of the rare earth mononitride family and motivates future exploration of the series using the unique processing space afforded by high-pressure optical reactors.

The Deck-of-cards-based Ordinal Regression (DOR) infers a value function from a ranking of reference alternatives in which the Decision Maker (DM) inserts blank cards between consecutive levels to express preference intensity. DOR, and its stochastic extension (SMAA-DOR), treat these answers as hard constraints defining a set of compatible value functions. We propose B-DOR, a probabilistic reformulation of DOR in which each pair of adjacent levels yields an ordinal observation, the declared direction and the number of cards, modelled through a cumulative-link likelihood that relates the number of blank cards to the latent value difference between alternatives. Two Bayesian inference algorithms are proposed: BAYES-DOR samples the whole posterior distribution by Hamiltonian Monte Carlo; FTRL-DOR tracks the maximum a posteriori estimate by constrained convex optimization. Moreover, through a multi-step elicitation process, elicitation can be spread over several short sessions reducing the cognitive burden on the DM. Both algorithms enjoy logarithmic regret bounds for prediction that hold for any sequence of DM responses and that guide the choice of the prior hyperparameters. A Monte Carlo study over 768 configurations shows that accuracy grows with the number of sessions, that blank cards add significant information over preference directions alone, that both algorithms maintain good performance under inconsistent answers, and that both outperform DOR and SMAA-DOR. An illustrative application to Italian regional healthcare performance demonstrates the practical applicability of the approach for building composite indicators.

Clinical trials in critical care often evaluate outcomes that are truncated by death, such as time to discharge alive, for which treatment effects are not well defined among patients who would not survive under one or both treatment strategies. Principal stratification provides a natural framework for defining survivor causal effects, but existing approaches often rely on monotonicity assumptions that may be implausible in settings where treatment can affect survival in competing directions. This concern is illustrated by the ARDS Network trial of lower versus higher positive end-expiratory pressure (PEEP), in which higher PEEP may benefit some patients by improving alveolar recruitment while harming others through over-distention or hemodynamic compromise. Moreover, the largely null average findings of the original trial do not rule out the possibility of clinically meaningful subgroups that may benefit from or be harmed by higher PEEP. We propose a monotonicity-free framework for identifying conditional survivor average causal effects (CSACE) using an interpretable sensitivity parameter that characterizes latent principal-stratum membership. We then develop a flexible Bayesian estimation strategy based on BART, a posterior-mean-based variable importance measure, and a distribution-aware conditional inference tree that uses the full posterior draws of individualized CSACEs. In simulation studies, the proposed BART-based approach improves individualized CSACE estimation and variable-importance recovery compared with linear modeling, especially under nonlinear treatment-effect heterogeneity. Applied to the ARDS PEEP trial, our method reveals clinically interpretable heterogeneity among estimated always-survivors, identifying subgroups with posterior evidence of benefit or harm that are obscured by the overall null trial results.

We prove that the lattice covering density of the regular octahedron is $9/8$, establishing the optimality of the classical body-centered cubic covering. A hypothetical lattice covering of smaller density would yield a lattice basis whose six Kuhn tetrahedra each fit in a translate of the octahedron. A sharp determinant bound then gives a contradiction. As a consequence, the maximum order of an undirected Abelian Cayley graph with three generators and their inverses and diameter at most $k$ is $\frac{32}{27}k^3+O(k^2)$ as $k\to\infty$.

Large Language Model (LLM)-based applications are increasingly deployed across domains including customer service, education, and mobility. These systems are prone to inaccurate, fictitious, or harmful responses, and their vast, high-dimensional input space makes systematic testing particularly challenging. In this paper, we present a search-based testing framework for LLM-based applications that incorporates failure diversity as an explicit optimization objective. Building on a discretization along stylistic, content-related, and perturbation dimensions, our framework maintains an archive of generated tests and rewards distance from that archive, while a repopulation operator periodically replaces non-failing tests in the population to sustain exploration. The repopulation operator is parameterized by its sampling strategy: replacement candidates are drawn either uniformly or by greedy distance maximization, two settings we compare empirically. We evaluate both against several baselines across three case studies—LLM safety, in-car navigation, and in-vehicle function control—covering five systems under test, eight LLMs, and 18 distinct test configurations, with over one million executed tests. Our results show that all guided variants detect substantially more failures than random and combinatorial search in nearly all configurations. Among them, diversified search detects fewer failures, but covers a broader range of failure types in all case studies, with greedy repopulation offering the best tradeoff.

Hybrid physics-machine-learning solvers improve under-resolved simulations by embedding trainable corrections into the time integration. During autoregressive inference, repeated solver-model interactions can amplify small errors, motivating multi-timestep solver-in-the-loop training. However, production solvers rarely expose the derivatives needed to backpropagate through such rollouts. We introduce Diff-NekRS, a scalable differentiable framework that embeds neural corrections directly in the GPU-accelerated NekRS incompressible-flow solver. NekRS computes the authoritative forward trajectory, a manually implemented exact discrete adjoint differentiates the supported fully discrete timestep, and LibTorch supplies neural vector-Jacobian products and parameter gradients. End-to-end Taylor and centered finite-difference tests verify the assembled gradient for two-dimensional cylinder flow (2Dcyl) and the three-dimensional Taylor-Green vortex (3DTGV) across five horizons and 12-1,020 MPI ranks. At 1,020 ranks, optimizer-enabled post-setup training updates retain 54.5%-78.0% and 80.7%-81.9% weak-scaling efficiency for 2Dcyl and 3DTGV, respectively. In 200-step autoregressive inference, the M = 50 model reduces the three-seed median terminal relative L2 velocity error by 59.2% for 2Dcyl and 12.1% for 3DTGV relative to the uncorrected coarse-grid P = 2 baseline, and retains wall-clock speedups of 5.38x and 2.49x, respectively, relative to the corresponding P = 7 configurations for equal simulated-time intervals. These results establish a verified and scalable path for multi-timestep solver-in-the-loop training that improves coarse-grid trajectory accuracy while retaining a speed advantage over the high-order reference

The classical theory of the wind-driven surface layer of a rotating ocean closes the momentum balance with a local flux-gradient law, in which the turbulent stress at a given depth is proportional to the shear at that depth. It predicts a surface current deflected forty-five degrees from the wind, which exceeds most direct measurements. This study asks what follows when locality is relaxed. Beginning from the exact integral relation between turbulent stress and mean shear, and requiring the memory kernel to carry no preferred vertical scale, we obtain a power-law kernel and therefore a stress law of fractional order. The resulting equation cannot be posed on the velocity, because the fractional derivative of a bounded profile vanishes at the surface, so the wind stress cannot be applied, while the alternative definition of the derivative leaves the surface current unbounded. Posed on the stress instead, the problem is solvable in closed form in Mittag-Leffler functions at every order between zero and one. The surface deflection then depends on the closure order alone and is smaller than forty-five degrees throughout, whereas the depth-integrated transport stays exactly normal to the wind, because that constraint follows from the momentum balance and not from the closure. The far field decays as a power law rather than exponentially, and its amplitude vanishes in the local limit, so that limit is singular. Under a suddenly applied stress the surface transient decays algebraically. Three independent algorithms agree closely, and no observational or model data are used.

Green's function based operator approximation offers an efficient route for solving linear partial differential equations under varying boundary conditions and source terms. Once the Green's function is learned, solutions for new configurations are obtained through integration rather than by solving the differential equation again. Existing Green's function learning methods face two structural challenges. The first is the singular behavior of the Green's function near the source point, which places a difficult approximation burden on neural networks. The second is the absence of explicit consistency between the learned Green's function and its gradient, although both quantities enter the integral solution representation directly. This work proposes SDC-GON, a Singular Decomposition and Consistency-Regularized Green's Operator Network that addresses both challenges within a unified framework. The Green's function is decomposed into an analytically known singular component and a smooth correction learned by the network, so that the neural approximation targets only the regular part of the response kernel. A self-consistency loss enforces agreement between the gradient and the autodifferentiation gradient of the smooth correction. The method is evaluated on two dimensional Poisson, three dimensional heat conduction, heterogeneous reaction diffusion, and Stokes benchmarks, consistently outperforming the compared baselines across all cases. On the heterogeneous pipe benchmark, SDC-GON achieves a testing error of $3.70\times10^{-4}$ with a smaller network architecture, compared with $9.60\times10^{-4}$ for the same-width baseline and $4.63\times10^{-4}$ for a larger configuration, demonstrating that structural improvements are more effective than increasing model size.