Generalized Reed-Solomon (GRS) and Gabidulin codes have been proposed for various code-based cryptosystems, though most such schemes without elaborate disguising techniques have been successfully attacked. Both code classes are prominent examples of the isometric families of (generalized) skew and linearized Reed-Solomon ((G)SRS and (G)LRS) codes which are obtained as evaluation codes from skew polynomials. Both GSRS and GLRS codes share the advantage of achieving the maximum possible error-decoding radius and thus promise smaller key sizes than e.g. Classic McEliece. We investigate whether these generalizations can avoid the known structural attacks on GRS and Gabidulin codes. In particular, we prove that both GSRS and GLRS codes decompose into GRS subcodes and are thus efficiently distinguishable from random codes with a square code method. This applies to all parameters for which the code length $n$ and its dimension $k$ over the field $\mathbb{F}_{q^m}$ satisfy $m + 1 < k < n - \tfrac{1}{2} (m^2 + 3m)$. The distinguishability extends to GSRS and GLRS codes with Hamming-isometric disguising. We further relate these findings to existing distinguishers for GRS, Gabidulin, and LRS codes, and extend known results on duals of SRS and LRS codes to the generalized setting allowing nonzero column multipliers. Finally, we provide explicit transformations between GSRS and GLRS codes, clarifying the algebraic relationship between the skew and linearized frameworks.

We present GlintMarkers, the first system to perform gaze-driven spatial perception using the inward-facing cameras on XR eyewear. Our key observation is that the cornea acts as a mirror that encodes both gaze direction and visual information about the environment in a small, low-contrast reflection. To extract spatial and semantic information from this reflection despite the camera's limited pixel budget, we present a passive retroreflective marker design that concentrates reflected near-infrared light onto the cornea, producing bright glint patterns. We develop a custom Perspective-n-Point (PnP) estimation framework adapted to corneal imaging and perform orientation and distance estimation of tagged objects, as well as unique object identification.

The tunable interaction between stationary quantum bits and propagating modes of light allows for the encoding of quantum information in the state of itinerant photons. This ability fulfills a central requirement for quantum networking, enabling quantum state transfer between distant quantum devices. Conventionally, a symmetric envelope of the photon wavepacket is used for such purposes. Yet, the use of alternative \textit{temporal modes} enables multiple applications in waveguide quantum electrodynamics that remain unexplored experimentally. Here, we use superconducting quantum circuits to generate individual itinerant microwave photons shaped in three mutually orthogonal temporal modes. We transfer the created photons across a 30-m cryogenic link, showing that the orthogonality allows us to decide at the receiver which mode to absorb, reflecting the other two with a selectivity ratio of 40. This experimental capability extends the microwave-frequency quantum communication toolbox, enabling a new photonic degree of freedom.

The Wiedemann--Franz law, which determines the universality of the ratio of thermal conductivity to electrical conductivity, is studied in the hydrodynamic electron transport regime, where electron--electron scattering predominates over scattering by disorder. In this case, the different relaxation of electric and thermal currents can lead to a violation of the Wiedemann--Franz law, which is expected to be even more pronounced in mesoscopic electron systems. This paper reports the propagation of hot electrons in a GaAs hydrodynamic narrow channel, studied using micrometer-resolution photoluminescence thermometry. A temperature dependence of the Lorenz number was obtained, indicating a violation of the Wiedemann--Franz law. The important role of narrow constrictions in this violation was also demonstrated, and theoretical arguments are presented.

Recently, the statistical properties of empirical Entropic Optimal Transport (EOT) have attracted great interest, as this quantity has been shown to be useful for complex data analysis, among other reasons due to its computational efficiency. In several applications, it has been observed that the EOT plan provides valuable information beyond just the optimal value. For example, in cell biology, colocalization analysis based on the EOT plan has been introduced as a measure for quantification of spatial proximity of different protein assemblies. Despite recent progress in the analysis of its risk properties, a precise understanding of its statistical fluctuations to make it accessible for inference remains elusive to a large extent. In this paper, we derive asymptotic weak convergence result for a large class of functionals of the EOT plan, in which the colocalization process is included. The proof is based on Hadamard differentiability and the extended delta method. As an application, we obtain uniform confidence bands for colocalization curves and bootstrap consistency. Our theory is supported by simulation studies and is illustrated by real world data analysis from mitochondrial protein colocalization.

In Hermitian impurity scattering, each isolated late-time exponential is the fingerprint of a bound state. We show that this correspondence breaks down in non-Hermitian bands. For a single impurity in a non-Hermitian lattice, the late-time signal is controlled by isolated complex frequencies selected by the analytic continuation of the Green's function relevant to real-time dynamics, which we term dynamical poles (DPs). DPs need not coincide with static bound states: one may appear without any bound-state counterpart, while a static bound state may be dynamically invisible. The remainder of the signal is an incoherent background set by complex continuum edges. Our results establish that the real-time analytic structure of the Green's function, not the static eigenvalue problem alone, organizes non-Hermitian impurity scattering.

Slender magnetic elements provide a versatile platform for programmable shape-morphing under remote magnetic actuation. However, a general and physically interpretable framework for the inverse design of a `magneto-elastica' under prescribed boundary conditions remains lacking. In this work, we develop an explicit analytical formulation for the inverse design of a magneto-elastica based on the integral form of the moment equilibrium equations. This approach yields direct constraints on the admissible curvature and rotation fields, enabling a systematic characterization of the feasible design space. We identify the key dimensionless parameters that govern the competition between magnetic torques and elastic restoring moments and show that the applied boundary conditions are an essential ingredient. We obtain closed-form solutions for the beam tapering profiles required to generate desired actuated shapes in the cases of clamped--free and clamped--clamped configurations; in the latter case, this includes analytical expressions for the boundary reactions. The formulation recovers the classical inverse elastica in the absence of magnetic fields and reveals a linear scaling between curvature deviation and magnetic mismatch. A tessellation strategy based on stiffness tailoring is further proposed for the design of discretized morphing surfaces. The theoretical predictions are validated against discrete elastic rod simulations and experiments across representative geometries. This work establishes a consistent analytical framework for the inverse design of a magneto-elastica and provides new insight into magnetically-induced shape programming in slender structures.

Let $V$ be a grading-restricted vertex algebra and let $A^\infty(V)=U^\infty(V)/Q^\infty(V)$ be the associative algebra constructed by Huang, where $U^\infty(V)$ is the space of column-finite infinite matrices with entries in V and $Q^\infty(V)$ is an ideal of a (nonassociative) algebra structure on $U^\infty(V)$ defined by Huang. Huang introduced families of elements in $Q^\infty(V)$ and conjectured that these elements generate $Q^\infty(V)$. We discover and prove that Huang's elements all satisfy what we call ``the diagonal shift property". On the other hand, in the case that $V$ is the rank one Heisenberg vertex operator algebra, we construct infinitely many linearly independent elements in $Q^\infty(V)$ that do not satisfy the diagonal shift property. As a corollary, we disprove Huang's conjecture.

Hypertrophic Cardiomyopathy (HCM) is a genetic heart disease affecting approximately 1 in 500 people and is the leading cause of sudden cardiac death in young athletes. Current diagnostic methods -- cardiovascular magnetic resonance (CMR), echocardiography, and genetic testing -- are limited by high costs, operator dependency, or insufficient accuracy, while standard electrocardiogram (ECG) analysis cannot reliably distinguish HCM from acquired left ventricular hypertrophy (LVH). This paper presents a wearable ECG device paired with a classification algorithm that differentiates HCM from acquired LVH using ECG signals alone. The portable device integrates a 3-lead electrode system, an AD8232 signal conditioning module, an Arduino Nano 33 BLE microcontroller, and a lithium polymer battery. The algorithm extracts two quantitative indices -- HCM Index~1 and HCM Index~2 -- from each heartbeat and classifies patients via dual statistical thresholds. Validation on 483 LVH patients (PhysioNet) and 29 HCM patients (digitized clinical records) yields 75.86\% sensitivity, 99.17\% specificity, and an F1-score of 80.00\%. Leave-one-out cross-validation confirms generalizability, with cross-validated sensitivity of 72.41\%, specificity of 98.96\%, and F1-score of 76.36\% (95\% confidence intervals reported). A digitization confound analysis demonstrates that the classification is driven by physiological cardiac features rather than data source artifacts. A simulated device acquisition chain analysis confirms that the wearable hardware's signal characteristics are compatible with the classification algorithm. The system offers a promising tool for affordable HCM screening in resource-limited settings.

Recent advances in data-driven modeling have shown that diffusion models can successfully generate synthetic Lagrangian trajectories in turbulent flows. Building on this progress, we extend the method to the joint generation of pairs of Lagrangian velocity trajectories, enabling a fully data-driven representation of turbulent pair dispersion, a long-standing fundamental problem with broad relevance in fluid dynamics. We demonstrate that diffusion models accurately reproduce the evolution of particle-pair separation, including deviations from Richardson's classical scaling law, while simultaneously preserving all key single-particle statistical properties reported in previous studies. These findings underscore the potential of diffusion-based generative models to emulate high-dimensional, multi-scale turbulent dynamics, further establishing them as a powerful tool for scientific modeling and for future geophysical and astrophysical applications.

We develop a Quantile Bayesian Vector Autoregression (QBVAR) to forecast real oil prices across different quantiles of the conditional distribution. The model allows predictor effects to vary across quantiles, capturing asymmetries that standard mean-focused approaches miss. Using monthly data from 1975 to 2025, we document three findings. First, the QBVAR improves median forecasts by 2-5\% relative to Bayesian VARs, demonstrating that quantile-specific dynamics matter even for point prediction. Second, uncertainty and financial condition variables strongly predict downside risk, with left-tail forecast improvements of 10-25\% that intensify during crisis episodes. Third, right-tail forecasting remains difficult; stochastic volatility models dominate for upside risk, though forecast combinations that include the QBVAR recover these losses. The results show that modeling the conditional distribution yields substantial gains for tail risk assessment, particularly during major oil market disruptions.

The LogQ algorithm encodes Quadratic Unconstrained Binary Optimization (QUBO) problems, which are often encountered in the industry (portfolio optimization, fleet optimization, charging stations, etc.). It was developed within the framework of quantum computing, designed as a pragmatic approach to quantum combinatorial optimization that drastically reduces the number of required qubits and quantum circuit depth. While LogQ has recently been made compliant with gradient-inspired methods, greatly improving parameter optimization efficiency, it still faced hurdles regarding Pauli decomposition and measurement overhead. We here demonstrate that LogQ can be fully reformulated within a classical framework, which effectively eliminates the need for Pauli decomposition and bypasses the measurement challenges altogether. This finally leads to a classical heuristic based on a non-linear continuous relaxation of variables and is, to the best of our knowledge, novel. The LogQ story illustrates how quantum computing can inspire classical algorithms, leading to so-called "quantum-inspired" methods.

We prove that a Le Page-type inequality is also valid for metrically characterizing those JB$^*$-triples that are commutative. More precisely, we establish that the following statements are equivalent for any JB$^*$-triple $E$: $(a)$ $E$ is commutative. $(b)$ There exists $γ>0$ satisfying $$\big\|\{a,b,\{x,y,z\}\}\big\|\leq γ\ \! \big\|\{x,y,\{a,b,z\}\}\big\|, \hbox{ for all } a,b,x,y,z\in E.$$

We consider nonlinear GMRES (NGMRES) as an acceleration technique for the Navier-Stokes Picard iteration, a direction that has not previously been explored. We identify the optimal norm for the least squares optimization problem arising in the NGMRES algorithm, and establish a convergence analysis for NGMRES with general depth that proves NGMRES scales the Picard Lipschitz constant by the gain of the optimization problem. To our knowledge, this is the first convergence proof for NGMRES that identifies the mechanism responsible for convergence acceleration. Numerical experiments demonstrate that the convergence estimates are remarkably sharp. In addition, NGMRES greatly improves the performance of the Picard iteration, even in cases where the unaccelerated iteration diverges.

We propose a variant of the model-independent method for determining meson charge radii from spatial moments of correlation functions on the lattice. Traditional determinations based on fits to the momentum transfer squared dependence of form factors are subject to systematic uncertainties arising from the choice of fit ansatz. By contrast, model-independent methods based on spatial moments provide a useful framework for determining the slope of the form factor without assuming its functional form. Recently, Feng et al. proposed a model-independent method, which drastically suppresses the finite-volume effect in the charge radius coming from higher-order contributions of the expansion of the form factor with respect to the momentum transfer squared. In this work, we introduce an auxiliary function of the momentum transfer squared and reformulate the method in terms of its product with the form factor, rather than the form factor itself, thereby further suppressing higher-order contributions, notably in cases of small volume and large radius. In particular, we investigate quadratic and logarithmic forms as practical choices for this auxiliary function. Applying this method to mock data based on a monopole form factor, as well as to actual lattice QCD data using $N_f=2+1$ gauge ensembles at $m_π\simeq 0.5$ and $0.3$ GeV, we find that it reduces residual finite-volume effects and provides an effective framework for meson charge radius determinations.

In 2005, Parreau proved that if a measure preserving system is not strongly mixing then it contains a non-trivial factor that is disjoint from every strongly mixing system. Taking this construction as the starting point, we develop the complementary notions of $\mathcal U$-generated and $\mathcal U$-mixing systems, for a set $\mathcal U$ of ultrafilters, and use them to recover several classical results in ergodic theory as special cases of a unified framework. We prove that a system is $\mathcal U$-mixing if and only if it is disjoint from all $\mathcal U$-generated systems. In fact, we show that if $\mathcal Y$ is a $\mathcal U$-generated system and $\mathcal Z$ is disjoint from every $\mathcal U$-mixing system, then any joining of $\mathcal Y$ and $\mathcal Z$ remains disjoint from all $\mathcal U$-mixing systems. We also show that every partially rigid system is a finite extension of some $\mathcal{U}$-generated system.

We present an efficient implementation of the low-cost linear-response coupled-cluster singles and doubles (LR-CCSD) method for computing static and frequency-dependent polarizabilities in systems with significant relativistic and electron-correlation effects. The approach employs X2C-based Hamiltonians (X2CAMF and X2CMP) and incorporates Cholesky decomposition to reduce memory requirements. In the current implementation, costly three- and four-external index integrals are generated on the fly, eliminating the need for their storage. Benchmark results indicate that the X2CMP Hamiltonian provides more consistent performance than X2CAMF, particularly for large and highly augmented basis sets. The proposed FNS++CD-X2CMP-LR-CCSD method shows excellent agreement with four-component reference values across a wide range of systems. Additionally, different strategies for constructing the FNS++ basis were assessed, and an averaged density approach was found to offer a favorable balance between accuracy and computational cost. On average, about 73% of the virtual spinor space is removed, demonstrating the efficiency and consistency of the FNS++ density-based truncation approach. The present implementation enables accurate and scalable relativistic response calculations for large molecular systems, as demonstrated by the calculation of the static polarizability of the Uranium Hexafluoride complex with a triple-zeta basis set more than 1400 basis functions.

This work addresses the challenge of ignition timing and load control in homogeneous charge compression ignition engines operating subject to uncertainty from complex combustion dynamics and external disturbances. To handle this issue, we propose a nonlinear stochastic model predictive control approach explicitly incorporating distributional information of uncertainties. Specifically, we integrate an uncertainty model learned from empirical residual data to capture realistic probabilistic characteristics and handle the nonlinear additive uncertainty propagation within the prediction horizon based on polynomial chaos expansion. Additionally, we introduce a novel cost function based on maximum mean discrepancy, enabling direct penalization of the discrepancy between predicted and desired distributions of combustion indicators. The simulation results demonstrate that our proposed method achieves over a 28 \% reduction on combustion phasing variation and more than a 26 \% improvement in load tracking accuracy compared to traditional nonlinear and Gaussian-based predictive control strategies. These findings indicate the effectiveness of explicitly modeling uncertainty distributions and highlight the advantages of distribution-level performance index in robust combustion control.

Hilbert space fragmentation refers to exponential growth in the number of dynamically disconnected Krylov sectors with system size. It is taken as evidence of ergodicity breaking, since conventional symmetries generate at most a polynomial number of sectors. However, we demonstrate that generalized symmetries can fragment the Hilbert space. Models with higher-form, subsystem, and gauge symmetries can have exponentially many symmetry sectors. We further prove that non-invertible symmetries can induce additional fragmentation within individual symmetry sectors. Fragmentation in several known models arises from generalized symmetries, and the presence of exponentially many Krylov sectors therefore does not by itself imply ergodicity breaking. Finally, we show that disorder free localization arises naturally from Krylov-restricted thermalization when sectors lack translation invariance, requiring neither ergodicity breaking nor gauge symmetry.

Stimulated Brillouin scattering (SBS), an optical nonlinearity arising from photon-phonon interactions, has formed the basis for a large class of optical signal processing devices, including Brillouin amplifiers. A limiting factor of such amplifiers is the noise due to thermal-mechanical fluctuations that the phonons imprint on the optical signal. Prior work has either inferred or experimentally observed a noise factor ($F$) that depends only on the thermal occupation of the phonons ($F\approx 1+n_{th}$). We show that this noise factor results naturally from a Hamiltonian-based spatio-temporal coupled mode treatment in the limit of large Brillouin amplification and when phonon propagation is neglected. Moreover, this theoretical framework allows us to extend our treatment to a much larger and more representative parameter space for emerging SBS systems; specifically, this analysis accounts for the forward or backward nature of the scattering process and the effects of phonon propagation, optical loss, and small Brillouin gains. Our results demonstrate that the noise factor can deviate radically from $F\approx 1+n_{th}$ for a host of modern SBS devices, especially those in which phonon propagation significantly changes the coupled mode dynamics.

Machine learning models are increasingly deployed in wireless networks with stringent performance requirements. However, dynamic propagation environments and fluctuating traffic densities introduce concept drift, which complicates the ability to maintain accurate predictive machine learning models. We propose a distributed optimization framework that jointly clusters cells and trains cluster-level predictive models, enabling nodes to cooperatively predict quality of service (QoS) distributions under communication constraints. The proposed method models QoS as a multivariate Gaussian/lognormal distribution and uses a novel clustering mechanism that groups cells with similar network conditions, allowing each cell to select the most appropriate predictor without retraining new models for each cell. By leveraging block coordinate descent, our solution efficiently clusters the cells and updates the predictive models to mitigate concept drift, while maintaining a compact model set to minimize computation overhead. Evaluation using data from realistic simulations with the Sionna ray-tracer and the ns-3 simulator shows that the method converges and yields cluster constellations that adapt to changes in the network that cause concept drift. The experimental evaluation focuses on providing a prediction of the distribution latency, jitter, and RSRP over a one-hour prediction horizon. The proposed method significantly outperforms the traditional single global predictive model approach and reduces the mean absolute error by 9-27% compared to local cell-level predictors. This demonstrates that the proposed method effectively captures local variability using far fewer models through scalable distributed clustering.

Many techniques in program synthesis, superoptimization, and array programming require parallel rollouts of general-purpose programs. GPUs, while capable targets for domain-specific parallelism, are traditionally underutilized by such workloads. Motivated by this opportunity, we introduce a pleasingly parallel virtual machine and benchmark its performance by evaluating millions of concurrent array programs, observing speedups up to $147\times$ relative to serial evaluation.

The doubling conjecture predicts that a manifold admits positive scalar curvature with mean convex boundary if and only if its double admits positive scalar curvature. We show that it holds true for manifolds where the inclusion of the boundary satisfies a certain split-condition on fundamental groups. Our proof is based on surgery-techniques for positive scalar and mean curvature. If the boundary is non-connected, we use existence of area-minimizing hypersurfaces and the monotonicity-formula. Furthermore, we investigate if a psc-metric on a closed manifold can be adjusted so that a given embedded hypersurface is minimal, stable minimal or totally geodesic. While not true in general, such an adjustment is possible in many cases.

Both cluster randomized trials and quasi-experimental designs are used to evaluate the impact of health and social policies and interventions. Stepped-wedge cluster randomized trials randomize a staggered adoption approach, while recent difference-in-differences methods allow analysis of non-randomized settings where similar policies are adopted at different time points. These approaches have become common, but the sheer variety of methods for analyzing observational studies with staggered adoption makes it challenging to clearly design and report such studies. We propose that observational and quasi-experimental study investigators can address these challenges by emulating stepped-wedge cluster randomized trials in the target trial emulation framework. The conceptual framework and reporting standards of trial emulation will encourage consideration of key features of these designs, such as policy heterogeneity and time-varying effects, and clear reporting of the estimand and assumptions. It also highlights areas where those interested in randomized trials and quasi-experimental designs can benefit from one another's experience by bringing insights across disciplines. Questions of treatment effect heterogeneity, power, spillovers, and anticipation effects, among others, are common to both fields and can benefit from cross-pollination. This article also demonstrates how trial emulation can identify settings that are not well-served by either approach, thereby avoiding studies unlikely to generate high-quality causal evidence. Finally, it informs the bias-variance-generalizability trade-off that arises with design and analysis choices made in these settings, supporting better evidence generation and interpretation in settings where important questions can be answered.

Wireless communications in intelligent rail transit face harsh propagation conditions, including severe penetration loss, frequent blockages, and amplified large-scale fading. Existing leaky coaxial cables (LCX) provide wired-to-wireless conversion and stable coverage, but can be energy- and spectrum-inefficient, particularly at high carrier frequencies. Motivated by the growing demand for high-capacity and high-reliability rail services, this article introduces pinching-antenna systems (PASS), which are flexible waveguide-based architectures that enable reconfigurable radiation points with low deployment overhead and a natural fit to predominantly straight track geometries. We discuss the key benefits and deployment flexibility of PASS, evaluate their performance relative to LCX via representative simulations, and present a deep learning (DL)-enabled channel-estimation framework to cope with mobility-induced channel dynamics. Finally, we summarize the major open challenges for practical deployment and outline promising research directions.

The term 'neutrinoless' is a cornerstone of modern particle physics, yet it defines a fundamental process by what is missing rather than what is created. We trace the origins of this privative neologism to a 1953 experimental claim and show how a 'sociology of suspicion' transformed Ettore Majorana's affirmative ontology into an agnostic shorthand. By examining this linguistic shift, we argue that our current terminology may obscure the profound physical meaning of the search. Reclaiming the language of 'matter creation' is not merely a semantic choice, but a timely conceptual shift to bridge the gap between experimental caution and the radical character of the laws of nature we aim to uncover.

Glueballs represent a fascinating aspect of the strong interaction in nature. Gluons that serve as the mediators of the strong interaction are massless particles, but they possess a property unique to the strong interaction called color charge, which is analogous to electric charge in the electromagnetic interaction. Glueballs are composed of multiple gluons and would be massless without color charges. The interaction of the color charges, however, makes glueballs becoming massive objects. Glueballs thus offer a unique way to study the mass creation of strongly interacting particles.

Neuromorphic photonics is emerging as a powerful platform for fast and efficient optical information processing and sensing. However, future brain-inspired photonic systems require compact and scalable light sources, capable of generating the neuro-mimetic optical signals needed for their operation. This work demonstrates a single-stack laser that delivers optical and electrical neural-like spiking emission under solitary operation. Termed the Neuron Surface-Emitting Laser (NeuronSEL), this compact, multi-junction Vertical-Cavity Surface Emitting Laser (VCSEL) exhibits non-linear Negative Differential Resistance (NDR), similar to that observed in memristive devices. Leveraging this NDR behaviour enables the novel demonstration of multiple neuronal features in the NeuronSEL including refractoriness and threshold-/integrate-and-fire dynamics. We demonstrate the NeuronSEL's behaviour as an optical spiking neuron and its ability to perform processing functions, such as coincidence detection and exclusive OR operations. Its scalability is illustrated by proposing a network based on an array of NeuronSELs, able to perform classification tasks. The NeuronSEL emerges as a strong candidate for practical and scalable neuromorphic photonic hardware, with potential impact across a range of applications in optical sensing, communications and computing technologies, whilst benefitting from the inherent advantages of VCSEL technology -low manufacturing cost, compactness, efficiency, vertical emission, and straightforward integration into large arrayed-structures and networks.

The actinides, including thorium (Th), are the heaviest observable elements synthesized in the universe, holding clues to the extremes of the astrophysical and nuclear conditions of $r$-process sites. We present Th abundances based on high-resolution spectroscopy for 47 metal-poor stars, the largest homogeneously analyzed sample to date. The chemical evolution of Th exhibits a decrease in dispersion in [Th/H] and [Th/Fe] from $\sim$0.6 dex at the lowest metallicities to $\sim$0.2 dex at higher metallicities. We also find that Th and the lanthanides Eu and Dy are co-produced remarkably well, with average [Th/Eu]$\sim0.0$ across $-3.0 \lesssim$ [Fe/H] $\lesssim -1.5$, as well as across stars with $0.0\lesssim$ [Eu/Fe] $\lesssim2.5$. Even so, the absolute range of $\logε$(Th/Eu) is 1.02 dex, with an observed standard deviation of $\pm0.20$ dex and an intrinsic standard deviation of $\pm0.11$ dex at the lowest metallicities. We infer that $68\%$ of $r$-process events have $\logε$(Th/Eu) yields that only vary within a factor of $\pm1.3$ or $\pm30\%$, while $5\%$ of $r$-process events have $\logε$(Th/Eu) yields that vary by factors $>3.3$ approaching $\sim$10. This serves as a strong constraint for the nuclear and astrophysical models of $r$-process sites, and suggests that achieving an $r$-process site that is both prompt and produces a robust $\logε$(Th/Eu) ratio is a challenge for current models.

Three-dimensional (3D) photonic integrated circuits (PIC) are emerging as an indispensable scheme for high density and multifunctional photonic systems. However, the wafer-scale scaling of PICs towards a 3D configuration is constrained by two key factors: (i) the trade-off between inter-layer taper efficiency and footprint, and (ii) wafer-scale uniformity of inter-layer transition loss. In this work, we introduce etch-back assisted chemical mechanical polishing (E-CMP) to achieve high wafer-scale uniformity of the spacer layer. Moreover, we break the efficiency-footprint trade-off by demonstrating a novel $κ$-engineered taper, achieving a reliability metric that is 75\% higher than the traditional linearly tapered structure. Building on these design and fabrication developments, we enable reliable 3D PICs with typical loss of 0.077 and 0.068 dB/cm on two silicon nitride (SiN) waveguide layers and typical 3D transition loss as low as 6 mdB. Furthermore, the low 3D transition loss enables the first class of 3D high-Q optical cavities occupying two distinct device layers, providing new design space for high-Q optical cavities. The scalable fabrication process and design methodology provide routes for wafer-scale reliable 3D PICs that are promising in a series of applications ranging from photonic interconnects and computing networks to high-density photonic sensors and nonlinear photonics.