This article proposes an online bootstrap scheme for nonparametric level estimation in nonstationary time series. Our approach applies to a broad class of level estimators expressible as weighted sample averages over time windows, including exponential smoothing methods and moving averages. The bootstrap procedure is motivated by asymptotic arguments and provides well-calibrated uniform-in-time coverage, enabling scalable uncertainty quantification in streaming or large-scale time-series settings. This makes the method suitable for tasks such as adaptive anomaly detection, online monitoring, or streaming A/B testing. Simulation studies demonstrate good finite-sample performance of our method across a range of nonstationary scenarios. In summary, this offers a practical resampling framework that complements online trend estimation with reliable statistical inference.

In generalized extreme value model for the r largest order statistics, denoted by rGEV, the selection of r is critical. The existing entropy difference test for selecting r is applicable to large sample. Another existing method (the score test with parametric bootstrap) is applicable to small sample, but computationally demanding. To address this problem for small sample, we propose a new method using a sequence of the goodness-of-fit tests based on the conditional cumulative distribution function (CCDF). The proposed CCDF test is easy to implement and computationally fast. The Cram{é}r-von Mises test was employed for the goodness-of-fit purpose. The proposed method is compared via Monte Carlo simulations with existing methods including the spacings, the score, and the entropy difference tests. The proposed CCDF test turned out to perform well for both small and large samples, comparable to the spacings and entropy difference tests. The utility of the proposed method is illustrated by an application to the r largest daily rainfall data in Korea. Additionally, we extended the existing methods and the CCDF test to a nonstationary rGEV model. Wide applicability of the proposed method are discussed.

We have studied the effect of zwitterionic form of the amino acid on the electron attachment to DNA using thymine glycine as a model system. The electron attachment to thymine in the presence of glycine takes place through a "doorway mechanism", where the electron density in the initial anionic state remains away from the nuclear framework of thymine. The electron gets transferred to nucleobase through mixing of electronic and nuclear degrees of freedom and glycine plays an important role in modulating the rate of electron transfer. The charge separation in the zwitterionic glycine make them a better trap for the electron than the native glycine and it can act as a better shield to the incoming electron. However, in the bulk solvated thymine anion the presence of zwitterionic glycine does not lead to any proton transfer to nucleobase, which makes it more resilient to base damage.

Let $S$ be a finite set, $s=|S|\ge6$. Given a non-negative integer $t$, there exists an inclusion-minimal non-Bondy system $\mathscr{A}$ of size $t$ on $S$ if and only if $s+1\le t\le2s$.

AI coding agents allow software developers to generate code quickly, which raises a practical question for project managers and open source maintainers: can vibe coders with less development experience substitute for expert developers? To explore whether developer experience still matters in AI-assisted development, we study $22,953$ Pull Requests (PRs) from $1,719$ vibe coders in the GitHub repositories of the AIDev dataset. We split vibe coders into lower experience vibe coders ($\mathit{Exp}_{Low}$) and higher experience vibe coders ($\mathit{Exp}_{High}$) and compare contribution magnitude and PR acceptance rates across PR categories. We find that $\mathit{Exp}_{Low}$ submits PRs with larger volume ($2.15\times$ more commits and $1.47\times$ more files changed) than $\mathit{Exp}_{High}$. Moreover, $\mathit{Exp}_{Low}$ PRs, when compared to $\mathit{Exp}_{High}$, receive $4.52\times$ more review comments, and have $31\%$ lower acceptance rates, and remain open $5.16\times$ longer before resolution. Our results indicate that low-experienced vibe coders focus on generating more code while shifting verification burden onto reviewers. For practice, project managers may not be able to safely replace experienced developers with low-experience vibe coders without increasing review capacity. Development teams should therefore combine targeted training for novices with adaptive PR review cycles.

We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +C(t)x. Specifically, we find bounds on the number of its rational solutions when A(t), B(t) and C(t) are polynomials with real or complex coefficients; and on the number of rational limit cycles when A(t), B(t) and C(t) are trigonometric polynomials with real coefficients.

If $S$ is a non-empty finite set, $|S|=s$, then a system $\mathscr{A}$ of subsets of $S$ is a size-minimal hypercompletely separable system (i.e., for every $a\in S$ there are $A,B\in\mathscr{A}$ such that $A\cap B=\{a\}$) if and only if $|\mathscr{A}|=\left\lceil\frac{1+\sqrt{8s+1}}2\right\rceil$.

The Hubble tension and the recently reported anomaly in data from the Dark Energy Spectroscopic Instrument (DESI) are considered to pose serious challenges to the standard $Λ$CDM model of cosmology. In this work, we show that resolving the Hubble tension with a scenario featuring dark radiation-matter decoupling (DRMD) predicts the presence of dark acoustic oscillations (DAO) similar in scale to baryon acoustic oscillations (BAO). Using an inference independent of large-scale structure data, relying only on Planck measurements of the cosmic microwave background and SH$0$ES-calibrated supernova data, we find evidence for a DAO signal with drag-horizon scale $r_{d,\mathrm{DAO}} \in[54,65]\,\mathrm{Mpc}/h$ ($68\%\,\mathrm{C.I.}$) and amplitude $A_\mathrm{DAO} \in [0.02,0.05]$ ($68\%\,\mathrm{C.I.}$). These predictions provide a concrete target for current and upcoming large-scale structure surveys, including DESI, Euclid, and the Roman Space Telescope. Remarkably, the predicted DAO properties are consistent with those required to explain the DESI anomaly, offering both an alternative to evolving dark energy and a preliminary validation of the relevance of a dark radiation-matter decoupling scenario for addressing the Hubble tension.

The transformation of the energy system has raised concerns about the reliability of fully renewable energy systems. We address this question for a 2050 European energy system using an economically optimal adequacy assessment. Our results show that a cost-optimal, fully renewable European system can be as reliable as a fossil-based one, with an average loss of load of only 0.03% due to variability in renewable generation. Outages primarily affect industrial and service sectors, while household supply remains largely uninterrupted. Regional differences in supply security emerge, with outages concentrated in countries with a low Value of Lost Load (VoLL). We demonstrate that system reliability can be fully ensured at negligible additional cost (+0.17%) by modestly increasing hydrogen turbine (+10%) and battery capacities (+15%) beyond the cost-optimal levels. We conclude that well-designed renewable energy systems are stable, with hydrogen-based backup being a key enabler of reliability.

This paper presents an optimization-based behavioral model for mixers driven by multi-tone local oscillator (LO) signals, considered specifically for frequency comb orthogonal frequency-division multiplexing radar applications. Unlike traditional models, the proposed approach is designed and tested for multi-tone LO excitations. The model uses polynomial nonlinearities for both intermediate frequency and LO ports, supported by spectrum-domain fitting that selectively emphasizes strong intermodulation products. In addition, a polynomial block is introduced to capture input power-dependent phase nonlinearity. The approach is validated using circuit-level simulations and supported by measurements. Radar processing results show the model replicates distortive effects in simulations. The proposed model enables rapid system-level performance estimations and waveform optimization, replacing computationally expensive circuit-level simulations.

Understanding how individuals navigate mental health challenges over time is critical yet methodologically challenging. Traditional approaches analyze community-level snapshots, failing to capture dynamic individual recovery trajectories. We introduce a novel framework applying Topological Data Analysis (TDA) specifically persistent homology to model users' longitudinal posting histories as trajectories in semantic embedding space. Our approach reveals topological signatures of trajectory patterns: loops indicate cycling back to similar states (stagnation), while flares suggest exploring new coping strategies (growth). We propose Semantic Recovery Velocity (SRV), a novel metric quantifying the rate users move away from initial distress-focused posts in embedding space. Analyzing 15,847 r/depression trajectories and validating against multiple proxies, we demonstrate topological features predict self-reported improvement with 78.3% accuracy, outperforming sentiment baselines. This work contributes: (1) a TDA methodology for HCI mental health research, (2) interpretable topological signatures, and (3) design implications for adaptive mental health platforms with ethical guardrails.

A subset $S$ of vertices, in a connected graph $G$, is called a distance-equalizer set if for every pair of distinct vertices outside $S$, there exists a vertex in $S$ equidistant to both. The equidistant dimension, denoted by $ξ(G)$, is defined as the minimum cardinality of such sets. While several distance-based parameters have been studied for different graph products, the equidistant dimension of corona product graphs has remained unexplored. In this paper, we investigate the equidistant dimension of the corona product $G \odot H$ of two graphs $G$ and $H$. We introduce the empty bisector graph $\widehat{G}$, an auxiliary construction that relates pairs of vertices in $G$ that cannot be equidistant from any third vertex. Using this framework, we establish tight bounds on the equidistant dimension of $G \odot H$ and derive exact values for several classical families of graphs. Moreover, we show that for any fixed base graph $G$, the equidistant dimension of $G \odot H$ depends on $H$ only through its order and eventually becomes linear in $\n(H)$.

The Kochen Specker theorem revealed contextuality as a fundamental nonclassical feature of nature. Nonlocality arises as a special case of contextuality, where entangled states shared by space like separated parties exhibit nonlocal correlations. The notion of maximality in correlations, analogous to maximal entanglement, is less explored in multipartite systems. In our work, we have defined maximal correlations in terms of contextual models, which are analogous to absolutely maximally entangled (AME) states. Employing the sheaf theoretic framework, we introduce maximal contextual correlations associated with the corresponding maximal contextual model. The formalism introduces the contextual fraction CF as a measure of contextuality, taking values from 0 (noncontextual) to 1 (fully contextual). This enables the formulation of a new class of correlations termed absolutely maximal contextual correlations (AMCC), which are both maximally contextual and maximal marginals. In the bipartite setting, the canonical example is the Popescu Rohrlich (PR) box, while in the tripartite case, it includes Greenberger Horne Zeilinger (GHZ) correlations and three way nonlocal correlations. In this work, we extend these findings to four party correlations. Notably, no AME state exists for four qubits, which introduces a subtle difference between AMCC and AME. The construction follows the constraint satisfaction problem (CSP) and parity check methods. In particular, the explicit realization of a non AMCC correlation that is maximally contextual yet not maximal marginal is obtained within the CSP framework.

The scalar-tensor representation of $f(R,T)$ gravity is extended to incorporate the Herglotz variational principle. The field equations are derived in both the geometric and scalar-tensor frameworks. Although the divergence of the energy-momentum tensor in matter-geometry coupling theories is generally nonvanishing, conservation can be achieved through the introduction of the Herglotz vector. The generalized Friedmann equations in scalar-tensor Herglotz $f(R,T)$ theory are obtained, and a conservative cosmological model is shown to be consistent with late-time observational data. Comparisons with analogous nonconservative models and with the standard $Λ$CDM model are also provided.

In the present paper, we are concerned with the study of matrix-sequences arising from the discretization of PDEs and FDEs on domains $Ω\subset \mathbb{R}^d$ with finite measure. When $Ω$ is either a hypercube or a bounded domain, the theory of Generalized Locally Toeplitz (GLT) sequences and of reduced GLT sequences cover the spectral analysis of the matrix-sequences derived from the approximation of the continuous problem. This work aims to extend the machinery and tools of the GLT apparatus to the case of unbounded domains with finite measure. For any unbounded domain $Ω\subset \mathbb{R}^d$ with finite measure, we define a new class of sequences, which we call unbounded GLT, and study their spectral properties.

Differentiable logics are a family of quantitative logics originated in the machine learning literature. Because of their origin, differentiable logics often come equipped with analytic properties that guarantee that they are differentiable. However, they usually lack an accompanying theory that describes their algebraic and proof-theoretic properties. Meanwhile, fuzzy logics, seen as substructural logics, have been studied algebraically and proof-theoretically, and some fuzzy logics with desirable analytic properties have also been used in machine learning. Our aim is to systematically compare analytic, algebraic and proof-theoretical properties of both fuzzy and differentiable logics. To this end, we formalize differentiable and fuzzy logics in a unified framework, encoded using the Mathcomp library in the Rocq proof assistant. We propose a single language specification to encompass multiple logics, using intrinsic typing to only allow valid and well-typed formulas for each of the logics that we encode: Yager, Łukasiewicz, Gödel and product fuzzy logics, as well as the differentiable logics DL2 and STL. Algebraically, we show how these logics can be interpreted using residuated lattices, which are prevalent in the theory of substructural logics. Analytically, we formalise the existence of a positive derivative for certain logical connectives, and to this end we formalise L'Hôpital's, contributing it to the Mathcomp library. Proof-theoretically, we formalise established sequent calculi for fuzzy logics, and we propose new sequent calculi for DL2 and STL$_{\infty}$, and formalise their soundness in our framework.

We propose a difference-in-differences (DiD) framework with mediation for possibly multivalued discrete or continuous treatments and mediators, aimed at identifying the direct effect of the treatment on the outcome (net of effects operating through the mediator), the indirect effect via the mediator, and the joint effects of treatment and mediator, consistent with the framework of dynamic treatment effects. Identification relies on a conditional parallel trends assumption imposed on the mean potential outcome across treatment and mediator states, or (depending on the causal parameter) additionally on the mean potential outcomes and potential mediator distributions across treatment states. We propose ATET estimators for repeated cross sections and panel data within the double/debiased machine learning framework, which allows for data-driven control of covariates, and we establish their asymptotic normality under standard regularity conditions. We investigate the finite-sample performance of the proposed methods in a simulation study and illustrate our approach in an empirical application to the US National Longitudinal Survey of Youth, estimating the direct effect of health care coverage on general health as well as the indirect effect operating through routine checkups.

Merge-width is a recently introduced family of graph parameters that unifies treewidth, clique-width, twin-width, and generalised colouring numbers. We prove the equivalence of several alternative definitions of merge-width, thus demonstrating the robustness of the notion. Our characterisation via definable merge-width uses vertex orderings inspired by generalised colouring numbers from sparsity theory, and enables us to obtain the first non-trivial approximation algorithm for merge-width, running in time $n^{O(1)} \cdot 2^n$. We also obtain a new characterisation of bounded clique-width in terms of vertex orderings, and establish that graphs of bounded merge-width admit sparse quotients with bounded strong colouring numbers, are quasi-isometric to graphs of bounded expansion, and admit neighbourhood covers with constant overlap. We also discuss several other variants of merge-width and connections to adjacency labelling schemes.

The automated detection of sexism in memes is a challenging task due to multimodal ambiguity, cultural nuance, and the use of humor to provide plausible deniability. Content-only models often fail to capture the complexity of human perception. To address this limitation, we introduce and validate a human-centered paradigm that augments standard content features with physiological data. We created a novel resource by recording Eye-Tracking (ET), Heart Rate (HR), and Electroencephalography (EEG) from 16 subjects (8 per experiment) while they viewed 3984 memes from the EXIST 2025 dataset. Our statistical analysis reveals significant physiological differences in how subjects process sexist versus non-sexist content. Sexist memes were associated with higher cognitive load, reflected in increased fixation counts and longer reaction times, as well as differences in EEG spectral power across the Alpha, Beta, and Gamma bands, suggesting more demanding neural processing. Building on these findings, we propose a multimodal fusion model that integrates physiological signals with enriched textual-visual features derived from a Vision-Language Model (VLM). Our final model achieves an AUC of 0.794 in binary sexism detection, a statistically significant 3.4% improvement over a strong VLM-based baseline. The fusion is particularly effective for nuanced cases, boosting the F1-score for the most challenging fine-grained category, Misogyny and Non-Sexual Violence, by 26.3%. These results show that physiological responses provide an objective signal of perception that enhances the accuracy and human-awareness of automated systems for countering online sexism.

This paper proposes a novel low probability of intercept (LPI) waveform design approach for orthogonal frequency-division multiplexing (OFDM)-based integrated sensing and communication systems by introducing artificial phase and Doppler shifts. These controlled impairments, unknown to eavesdroppers, effectively disrupt passive radar processing and intercept attempts. At legitimate receivers, they can be fully compensated, so that standard OFDM communication and sensing performance are preserved. To support the effectiveness of the proposed LPI waveform design for OFDM-based ISAC, measurement results with 1 GHz bandwidth at 27 GHz are presented considering different impairment introduction approaches, all with no impact on cooperative system performance, and compensation capabilities at the eavesdropper.

Communication networks often rely on some form of local failover rules for fast forwarding decisions upon link failures. While on undirected networks, up to two failures can be tolerated, when just matching packet origin and destination, on directed networks tolerance to even a single failure cannot be guaranteed. Previous results have shown a lower bound of at least $\lceil\log(k+1)\rceil$ rewritable bits to tolerate $k$ failures. We improve on this lower bound for cases of $k\geq 2$, by constructing a network, in which successful routing is linked to the \textit{Covering Array Problem} on a binary alphabet, leading to a lower bound of $Ω(k + \lceil\log\log(\lceil\frac{n}{4}\rceil-k)\rceil)$ for $k$ failures in an $n$ node network.

Network design under uncertainty arises in countless real-world settings and can be captured by the Stochastic Steiner Tree Problem (SSTP). Although there are a few approaches specifically tailored to this stochastic optimization problem, there are considerably more state-of-the-art heuristics for its deterministic variant, the Steiner Tree Problem (STP). In this work, we show how to leverage an existing STP heuristic in building a novel method for solving its stochastic variant, the SSTP. This approach is a powerful, yet simple and easy-to-implement way of solving this complex problem. We test our method using benchmark instances from the literature. Numerical results show considerably faster computation times compared to the state-of-the-art, with a gap of approximately 5%.

Lead-free iodide double perovskites are an interesting class of materials since they combine a relatively low toxicity (compared to the lead counterpart) with the small bandgap typical of iodide-based perovskite structures. Their reported number is small due to their lower structural stability compared to the chloride and bromide analogues, hence their difficult synthesis. The structural constraints that limit stability, on the other hand, can be much relieved in layered, organic-inorganic perovskites. Following this line of thought, we report here a successful fast precipitation route to iodide layered (CnH(2n+1)NH3)4AgBiI8 (n = 10, 12, and 14) double perovskites that borrow concepts from the synthesis of colloidal nanocrystals. X-ray diffraction studies revealed for these compounds a monoclinic crystal structure containing edge-sharing alternated [AgI6] and [BiI6] octahedra. These materials have experimental band gaps of 2.1 eV, as also corroborated by theoretical calculations. We have also investigated their phase transitions by thermal analysis and temperature-dependent diffraction and found them to be similar to their lead-based layered perovskite counterparts.

This paper studies a downlink multi-user multiple-input multiple-output (MU-MIMO) system, where the precoding matrix is computed at a baseband unit (BBU) and then transmitted to the remote antenna array over a limited-capacity digital fronthaul. The limited bit resolution of the fronthaul introduces quantization effects that are explicitly modeled. We propose a novel sum rate maximization framework that directly incorporates the quantizer's constraints into the precoding design. The resulting maximization problem, a non-convex mixed-integer program, is addressed using a new iterative algorithm inspired by the weighted minimum mean square error (WMMSE) methodology. The precoding optimization subproblem is reformulated as an integer least-squares problem and solved using a novel sphere decoding (SD) algorithm. Additionally, a low-complexity expectation propagation (EP)-based method is introduced to enable the practical implementation of quantized precoding in MU-massive MIMO (MU-mMIMO) systems. Furthermore, numerical evaluations demonstrate that the proposed precoding schemes outperform conventional approaches that optimize infinite-resolution precoding followed by element-wise quantization. We also propose a heuristic quantization-aware precoding method with comparable complexity to the baseline but superior performance. In particular, the EP-based approach offers near-optimal performance with substantial complexity reduction, making it well-suited for real-time MU-mMIMO applications.

We present a timing and scintillation study of the young Galactic halo pulsar PSR J1740+1000 using observations from the Nanshan, FAST, and Parkes radio telescopes. From timing analysis, we measure the pulsar's proper motion for the first time, indicating motion away from the Galactic plane at a position angle of 16.7 +/- 4.8 degrees (Galactic coordinates), with a total proper motion of 56.9 +/- 8.0 mas/yr and a corresponding transverse velocity of 329 +/- 80 km/s. This velocity suggests that PSR J1740+1000 is a typical-velocity young pulsar born within the Galactic halo. In scintillation studies, we detect scintillation arcs, arclets, and double-layered adjacent arcs in the secondary spectra. Under isotropic and anisotropic scattering assumptions, the screen-to-pulsar distance is 370 +/- 72 pc and 1 +/- 12 pc, respectively. The latter closely matches the scale of the pulsar wind nebula associated with PSR J1740+1000 and provides a better fit, suggesting that scattering is likely dominated by the PWN. The double-layered adjacent arcs observed on MJD 60180 imply that the pulsar's scattered image consists of two dominant components (A and B) and multiple weaker components. Component A is located at the pulsar's geometric position (0 uas), while Component B is located 112 +/- 16 uas and 23 +/- 17 uas from the central component under isotropic and anisotropic scattering, respectively. The frequency-independent angular position of Component B hints at refraction by an AU-scale structure within the scattering region, possibly originating from the PWN.

We show that a universal propagation scale can emerge from purely relational, scale-invariant $N$-body dynamics formulated on shape space, i.e. the space of configurations modulo translations, rotations, and dilatations. Although pure shape dynamics treats only unparametrized curves as fundamental, we adopt an affine parametrization as a gauge choice to perform a perturbative analysis, with all physically meaningful results expressed in parametrization-independent terms. Linear perturbations around central configurations satisfy second-order equations on shape space whose high-frequency spectrum defines a dimensionless constant $c_{\mathrm{rel}}$. Under general conditions of reparametrization invariance, spectral universality, and strict hyperbolicity, $c_{\mathrm{rel}}$ functions as an emergent light-cone velocity, endowing the product space $\mathbb{R} \times \mathcal{S}$ with an effective Lorentzian structure. These results suggest that causal structure and a maximal signal speed may arise dynamically from the geometry and spectral properties of relational configuration space, providing a novel perspective on the origin of relativistic kinematics.

Observations of linear structure connecting pulsars to gamma-ray halos reveal injection of TeV electrons into the interstellar medium (ISM). In some cases, this could be attributed to nearly scattering-free electron transport along large-scale magnetic fields connected to pulsar winds with very slow diffusion across the field lines. In this work we model this process with a magnetic flux tube emerging from the pulsar and attached to the ISM magnetic field. We show that particles in this case have an anisotropic distribution of magnetic pitch angle, such that the overall emission is highly beamed. We apply this model to pulsar tails and filaments, including the extended X-ray and TeV emission associated with PSR J1740+1000 and the misaligned X-ray jet in the Guitar Nebula, to constrain their particle population and magnetic fields.

The early evolution of unstable hydrodynamic and magnetohydrodynamic equilibria is often governed by a few dominant linear eigenmodes. We investigate whether initialising a simulation with a superposition of linear eigenmodes that contains the most unstable mode saves computation time, and how the selection of the included modes affects the non-linear evolution. Using the non-linear (magneto)hydrodynamic simulation code MPI-AMRVAC, the evolutions of a flow-sheared fluid interface, a Harris current sheet, and a flow-sheared plasma interface were simulated for various initial perturbations. The novel initial perturbations were linear eigenmodes of the equilibrium, or superpositions thereof, and calculated with the Legolas code. We benchmarked to initialisation with velocity noise and, in the case of the Harris sheet, initialisation with an analytic magnetic field perturbation. By perturbing an unstable equilibrium with a superposition of linear eigenmodes that contains the most unstable mode, significantly less computation time is spent on the linear stage of the evolution compared to traditional perturbation methods. In the best case, the simulation time needed to reach the non-linear stage is reduced by an order of magnitude. The inclusion or omission of certain modes in the initial perturbation is observed to affect the non-linear evolution to various degrees. The perturbation of equilibria with a superposition of linear eigenmodes that contains the most unstable mode allows simulations to reach a late-evolution stage faster, thus saving computation time. Additional eigenmodes can be included alongside the fastest growing mode to obtain further benefits, for example, to accelerate symmetry breaking in the non-linear stage, or to delay their effect on the non-linear evolution. Coupling spectroscopic codes with (magneto)hydrodynamic codes therefore offers significant advantages.

Multiple sequence alignment (MSA) is a core operation for comparing genome sequences and is widely used in bio-informatics. MAFFT, a practical MSA tool, repeatedly shifts a pair of sequences and computes a distance. Because the number of sequence pairs grows quadratically with the number of sequences, this procedure can become a bottleneck. We propose Quantum Shift-based Sequence Alignment (QShift-SA), which implements this ``shift-wise score computation'' as a gate-based quantum circuit and searches over shift amounts and sequence pairs using Grover algorithm. QShift-SA constructs an oracle circuit that compute the Hamming distance (the number of mismatches) between two sequences with data encoding, controlled shift, comparison, and addition. This oracle can search for candidates with small distances. QShift-SA does not aim to replace the full MSA workflow; instead, it targets the screening steps that often dominate the runtime in classical MAFFT as stated above. We evaluate circuit resources (number of qubits, gate count, and depth) and benchmark simulation time across multiple quantum circuit simulators. We find that a decision diagram (DD)-based quantum circuit simulator runs more than 1,000$\times$ faster than state-vector and MPS simulators and can handle larger circuits.

We respond to the critique by Aleksandr V. Korolev and Evgeny F. Talantsev on the superconducting phase fraction ($f$) calculations in Li et al. Nature 649, 871-878 (2026). First, the weak upturn in the low-temperature tail of our data has been confirmed to originate from the background, and the paramagnetic Meissner effect is absent in our case; thus, field-cooled (FC) data can be used for superconducting phase fraction calculations. Second, demagnetization effect must be calculated based on the actual measured moment as a function of $f$, which has been well-established and routinely employed in the superconductivity community. In contrast, Korolev and Talantsev treated the demagnetization field as a constant; thus, their calculation underestimates $f$ by a factor of $(1-Nχ_{meas})(1-N)$. This factor is close to 1/3, given $N$ = 0.849, $χ_{meas}$ = -1.313 in our study, which explains the origin of their deviated result (nearly three times smaller than our results). Third, our sample is a homogeneous high-quality bulk single crystal, evidenced by various techniques, making the existence of multiple discrete superconducting regions highly unlikely. We conclude that the superconducting phase fraction calculations reported in Li et al. Nature 649, 871-878 (2026) are not invalidated by the analyses presented in Korolev et al. arXiv: 2602.23240 (2026).