Quantum-chaotic systems exhibit several universal properties, ranging from level repulsion in the energy spectrum to wavefunction delocalization. On the other hand, if wavefunctions are localized, the levels exhibit no level repulsion and their statistics is Poisson. At the boundary between quantum chaos and localization, however, one observes critical behavior, not complying with any of those characteristics. An outstanding open question is whether there exist yet another type of universality, which is genuine for the critical point. Previous work suggested that there may exist a relation between the global characteristics of energy spectrum, such as spectral compressibility $χ$, and the degree of wavefunction delocalization, expressed via the fractal dimension $D_1$ of the Shannon--von Neumann entropy in a preferred (e.g., real-space) basis. Here we study physical systems subject to local and non-local hopping, both with and without time-reversal symmetry, with the Anderson models in dimensions three to five being representatives of the first class, and the banded random matrices as representatives of the second class. Our thorough numerical analysis supports validity of the simple relation $χ+ D_1 = 1$ in all systems under investigation. Hence we conjecture that it represents a universal property of a broad class of critical models. Moreover, we test and confirm the accuracy of our surmise for a closed-form expression of the spectral compressibility in the one-parameter critical manifold of random banded matrices. Based on these findings we derive a universal function $D_{1}(r)$, where $r$ is the averaged level spacing ratio, which is valid for a broad class of critical systems.

The rapid and unprecedented dominance of Artificial Intelligence (AI), particularly through Large Language Models (LLMs), has raised critical trust challenges in high-stakes domains like politics. Biased LLMs' decisions and misinformation undermine democratic processes, and existing trust models fail to address the intricacies of trust in LLMs. Currently, oversimplified, one-directional approaches have largely overlooked the many relationships between trustor (user) contextual factors (e.g. ideology, perceptions) and trustee (LLMs) systemic elements (e.g. scientists, tool's features). In this work, we introduce a bowtie model for holistically conceptualizing and formulating trust in LLMs, with a core component comprehensively exploring trust by tying its two sides, namely the trustor and the trustee, as well as their intricate relationships. We uncover these relationships within the proposed bowtie model and beyond to its sociotechnical ecosystem, through a mixed-methods explanatory study, that exploits a political discourse analysis tool (integrating ChatGPT), by exploring and responding to the next critical questions: 1) How do trustor's contextual factors influence trust-related actions? 2) How do these factors influence and interact with trustee systemic elements? 3) How does trust itself vary across trustee systemic elements? Our bowtie-based explanatory analysis reveals that past experiences and familiarity significantly shape trustor's trust-related actions; not all trustor contextual factors equally influence trustee systemic elements; and trustee's human-in-the-loop features enhance trust, while lack of transparency decreases it. Finally, this solid evidence is exploited to deliver recommendations, insights and pathways towards building robust trusting ecosystems in LLM-based solutions.

Elucidating dielectric losses, structural heterogeneity, and interface imperfections is critical for improving coherence in superconducting qubits. However, most diagnostics rely on destructive electron microscopy or low-throughput millikelvin quantum measurements. Here, we demonstrate noninvasive terahertz (THz) nano-imaging/-spectroscopy of encapsulated niobium transmon qubits, revealing sidewall near-field scattering that correlates with qubit coherence. We further employ a THz hyperspectral line scan to probe dielectric responses and field participation at Al junction interfaces. These findings highlight the promise of THz near-field methods as a high-throughput proxy characterization tool for guiding material selection and optimizing processing protocols to improve qubit and quantum circuit performance.

This article deals with the following fractional $(p,q)$-Choquard equation with exponential growth of the form: $$\varepsilon^{ps}(-Δ)_{p}^{s}u+\varepsilon^{qs}(-Δ)_q^su+ Z(x)(|u|^{p-2}u+|u|^{q-2}u)=\varepsilon^{μ-N}[|x|^{-μ}*F(u)]f(u) \ \ \mbox{in} \ \ \mathbb{R}^N,$$ where $s\in (0,1),$ $\varepsilon>0$ is a parameter, $2\leq p=\frac{N}{s}<q,$ and $0<μ<N.$ The nonlinear function $f$ has an exponential growth at infinity and the continuous potential function $Z$ satisfies suitable natural conditions. With the help of the Ljusternik-Schnirelmann category theory and variational methods, the multiplicity and concentration of positive solutions are obtained for $\varepsilon>0$ small enough. In a certain sense, we generalize some previously known results.

Despite significant advancements in deep learning based CSI compression, some key limitations remain unaddressed. Current approaches predominantly treat CSI compression as a source-coding problem, thereby neglecting transmission errors. Conventional separate source and channel coding suffers from the cliff effect, leading to significant deterioration in reconstruction performance under challenging channel conditions. While existing autoencoder-based compression schemes can be readily extended to support joint source-channel coding, they struggle to capture complex channel distributions and exhibit poor scalability with increasing parameter count. To overcome these inherent limitations of autoencoder-based approaches, we propose Residual-Diffusion Joint Source-Channel Coding (RD- JSCC), a novel framework that integrates a lightweight autoencoder with a residual diffusion module to iteratively refine CSI reconstruction. Our flexible decoding strategy balances computational efficiency and performance by dynamically switching between low-complexity autoencoder decoding and sophisticated diffusion-based refinement based on channel conditions. Comprehensive simulations demonstrate that RD-JSCC significantly outperforms existing autoencoder-based approaches in challenging wireless environments. Furthermore, RD-JSCC offers several practical features, including a low-latency 2-step diffusion during inference, support for multiple compression rates with a single model, robustness to fixed-bit quantization, and adaptability to imperfect channel estimation.

We give a complete classification of free arrangement of three smooth conics on complex projective plane admitting only ${\rm ADE}$ singularities and $J_{2,0}$ singularities.

Protocol art has recently proliferated through blockchain-based smart contracts, building on a century-long lineage of conceptual, participatory, interactive, systematic, algorithmic, and generative art practices. Few studies have examined the characteristics and appreciation of this emerging art form. To address this gap, this paper presents an annotated portfolio analysis of protocol artworks by Pak, a pioneering and influential pseudonymous artist who treats smart contracts as medium and collective participation through protocol as message. Tracing the evolution from early open-edition releases of The Fungible (2021) and the dynamic mechanics of Merge (2021) to the soul-bound messaging of Censored (2022) and the reflective absence of Not Found (2023), we examine how Pak choreographs distributed agency across collectors and autonomous code, demonstrating how programmable protocols become a social fabric in artistic meaning-making. Through thematic analysis of Pak's works, we identify seven core characteristics distinguishing protocol art from other art forms: (1) system-centric rather than object-centric composition, (2) autonomous governance enabling open-ended control, (3) distributed agency and communal authorship, (4) temporal dynamism and lifecycle aesthetics, (5) economy-driven engagement, (6) poetic message embedded in interaction rituals, and (7) interoperability enabling composability for emergent complexity. We then discuss how these features set protocol art apart from adjacent movements such as conceptual, generative, participatory, interactive, and performance art. By analyzing principles grounded in Pak's practice, we contribute to the emerging literature on protocol art (or "protocolism") and offer design implications for future artists exploring this evolving form.

Primer vector theory using averaged dynamics is well suited for optimizing low-thrust, many-revolution spacecraft trajectories, but is difficult to implement in a way that maintains both optimality and computational efficiency. An improved model is presented that combines advances from several past works into a general and practical formulation for minimum-fuel, perturbed Keplerian dynamics. The model maintains computational efficiency of dynamics averaging with optimal handling of the eclipsing constraint and bang-bang control through the use of the Leibniz integral rule for multi-arc averaging. A subtle, but important singularity arising from the averaged eclipsing constraint is identified and fixed. A maximum number of six switching function roots per revolution is established within the averaged dynamics. This new theoretical insight provides a practical upper-bound on the number of thrusting arcs required for any low-thrust optimization problem. Variational equations are provided for fast and accurate calculation of the state transition matrix for use in targeting and optimization. The dynamics include generic two-body perturbations and an expanded state to allow for sensitivity calculations with respect to launch date and flight time. The new model is illustrated on a GTO to GEO transfer, including up to 486 revolutions.

In this article, we contribute to the study of the dynamics induced by the conjugation action on the space of subgroups of Baumslag-Solitar groups BS(m,n), via the mixing properties of elements asymptotically produced by suitable random walks on the group. In an acylindrically hyperbolic context, the authors of [HMO] demonstrated strong mixing situations, namely topological mu-mixing, a strengthening of high topological transitivity. Regarding non-metabelian and non-unimodular BS(m,n), we exhibit here a radically different situation on each of the pieces except one of the partition introduced in [CGLMS22] (although it is highly topologically transitive on each piece). On the other hand, when BS(m,n) is unimodular, we demonstrate the topological mu-mixing character on each of the pieces.

We investigate the coherence of mixed Greenberger-Horne-Zeilinger (GHZ) and W states for bosonic and fermionic fields when a subset of $n$ ($n<N$) qubits experiences Hawking radiation near a Schwarzschild black hole. Analytical expressions are derived for the coherence of mixed N-qubit systems, including both the physically accessible and inaccessible parts in curved spacetime. The results show that the mixed W state maintains its coherence more effectively than the GHZ state as the Hawking temperature increases, even though its entanglement is weaker. As the number of qubits grows, W-state coherence becomes increasingly resistant to gravitational decoherence. Furthermore, fermionic fields preserve stronger entanglement, while bosonic fields retain higher coherence, highlighting a clear contrast between different particle statistics. These findings demonstrate how the Schwarzschild spacetime reshapes the balance between quantum coherence and entanglement, offering guidance for future relativistic quantum information applications.

Polarized particle sources have a plethora of applications, ranging from deep-inelastic scattering to nuclear fusion. One crucial challenge in laser-plasma interaction is maintaining the initial polarization of the target. Here, we propose the acceleration of spin-polarized Helium-3 from near-critical density targets using high-intensity Laguerre-Gaussian laser pulses. Three-dimensional particle-in-cell simulations show that Magnetic Vortex Acceleration with these modes yields higher polarization on the 90%-level compared to conventional Gaussian laser pulses, while also providing low-divergence beams.

Let $V$ be one of the following unitary strongly-rational VOAs: unitary WZW models, discrete series W-algebras of type ADE, even lattice VOAs, parafermion VOAs, their tensor products, and their strongly-rational cosets. Let $U$ be a (unitary) VOA extension of $V$, described by a Q-system $Q$. We prove that $U$ is strongly local. Let $\mathcal A_V,\mathcal A_U$ be the conformal nets associated to $V,U$ in the sense of Carpi-Kawahigashi-Longo-Weiner (CKLW). We prove that $\mathcal A_U$ is canonically isomorphic to the conformal net extension of $\mathcal A_V$ defined by the Q-system $Q$. We prove that all unitary $U$-modules are strongly integrable in the sense of Carpi-Weiner-Xu (CWX). We show that the CWX $*$-functor from the $C^*$-category of unitary $U$-modules to the $C^*$-category of finite-index $\mathcal A_U$-modules is naturally isomorphic to $*$-functor defined by $Q$.

We first prove homogenization for the nonlinear Schrödinger equation with sprinkled nonlinearity introduced in [19]. We then investigate how solutions fluctuate about the homogenized solution.

Using Swedish register data spanning 250 years, we estimate multigenerational effects of smallpox vaccination on longevity and occupational achievements. Employing mother fixed-effects, difference-in-differences, and shift-share instrumental-variables designs, we find vaccination improves outcomes for three generations. We explore mechanisms through which benefits transmit across generations, finding evidence consistent with both improved health behaviors and epigenetic inheritance. Effects persist even in milder disease environments, demonstrating vaccination lasting benefits beyond epidemic contexts. These findings underscore the importance of accounting for multigenerational returns when evaluating early-life health interventions.

Silicon strip detectors developed for the Inner Tracker (ITk) of the ATLAS experiment will operate in a harsh radiation environment of the HL-LHC accelerator. The ITk is thus designed to endure a total fluence of 1.6E15 1MeV n_eq/cm2 and a total ionizing dose (TID) of 66 Mrad in the strip detector region. A radiation-hard n^+-in-p technology is implemented in the ITk strip sensors. To achieve the required radiation hardness, extensive irradiation studies were conducted during sensor development, primarily performed up to the maximal expected total fluence and TID to ensure a full functionality of the detector at its end-of-life. These studies included irradiations of sensors with various particle types and energies, including the Co60 gamma-rays. Our previous results obtained for gamma-irradiated diodes and strip sensors indicate a linear increase of bulk current with TID, while the surface current saturates at the lowest TID levels checked (66 Mrad), preventing a determination of the exact TID for which the observed saturation occurs. This work presents the results coming from irradiations by Co60 gamma-rays to multiple low TIDs, ranging from 0.5 to 100 krad. The detailed study of total, bulk, and surface currents of diodes explores an unknown dependence of surface current on the TID, annealing, and temperature. Additionally, the effect of the p-stop implant between the bias and the guard ring of measured samples is shown. The observations are relevant for the initial operations of the new ATLAS tracker.

We build the canonical ensemble of a hot self-gravitating matter thin shell in anti-de Sitter (AdS) space by finding its partition function through the Euclidean path integral approach with fixed temperature at the conformal boundary. We obtain the reduced action of the system by restricting the path integral to spherically symmetric metrics with given boundary conditions and with the Hamiltonian constraint satisfied. The stationary conditions, i.e., the mechanical equilibrium and the thermodynamic equilibrium, are obtained from minimizing the reduced action. Evaluating the perturbed reduced action at the stationary points yields the mechanical stability condition and the thermodynamic stability condition. The reduced action calculated at the stationary points gives the partition function in the zero-loop approximation and from it the thermodynamic properties of the system are acquired. Within thermodynamics alone, the only stability condition that one can establish is thermodynamic stability, which follows from the computation of the heat capacity. For given specific pressure and temperature equations of state for the shell, we obtain the solutions of the ensemble. There are four different thin shell solutions, one of them is fully stable, i.e., is stable mechanically and thermodynamically. For the equations of state given, we find a first order phase transition from the matter thermodynamic phase to the Hawking-Page black hole phase. Moreover, there is a maximum temperature above which the shell ceases to exist, presumably at these high temperatures the shell inevitably collapses to a black hole.

Reachability in pushdown vector addition systems with states (PVASS) is among the longest standing open problems in Theoretical Computer Science. We show that the problem is decidable in full generality. Our decision procedure is similar in spirit to the KLMST algorithm for VASS reachability, but works over objects that support an elaborate form of procedure summarization as known from pushdown reachability.

Gene expression in response to stimuli is regulated by transcription factors (TFs) through feedback loop motifs, aimed at maintaining the desired TF concentration despite uncertainties and perturbations. In this work, we consider a stochastic model of the positive gene autoregulating feedback loop and we probabilistically quantify its resilience, \textit{i.e.}, its ability to preserve the equilibrium associated with a prescribed concentration of TFs, and the corresponding basin of attraction, in the presence of noise. We show that the formation of larger oligomers, corresponding to larger Hill coefficients of the regulation function, and thus to sharper non-linearities, improves the system resilience, even close to critical concentrations of TFs. We also explore a complementary definition of resilience that can be assessed within a stochastic formulation relying on the Fokker-Planck equation. Our formal results are accompanied by numerical simulations.

The article examines a boundary-value problem in a domain consisting of perforated and imperforate regions, with Neumann conditions prescribed at the boundaries of the perforations. Assuming the porous medium has symmetric, periodic structure with a small period $\varepsilon,$ we analyse the limit behavior of the problem as $\varepsilon \to 0.$ A crucial aspect of this study is deriving correct coupling conditions at the common interface, which is achieved using inner-layer asymptotics. For the flat interface, we construct and justify a complete asymptotic expansion of the solution in the $H^1$-Sobolev space. Furthermore, for the $\varepsilon$-periodically oscillating interface of amplitude $\mathcal{O}(\varepsilon),$ we provide an approximation to the solution and establish the corresponding asymptotic estimates in $H^1$-Sobolev spaces.

In this paper we study the Chow quotient ${\mathcal C}X$ of a convex variety $X$ of Picard number one by the action of a one dimensional torus having no non-trivial finite isotropy. Examples of these actions can be found in the rational homogeneous framework. We prove that the subvariety of ${\mathcal C}X$ parametrizing reducible torus-invariant cycles is a simple normal crossing divisor, we compute the Nef and Mori cones of ${\mathcal C}X$, and its anticanonical divisor.

We study a centralized discrete-time dynamic two-way matching model with finitely many agent types. Agents arrive stochastically over time and join their type-dedicated queues waiting to be matched. We focus on availability-based policies that make matching decisions based solely on agent availability across types (i.e., whether queues are empty or not), rather than relying on complete queue-length information (e.g., the longest-queue policy). We aim to achieve constant regret at all times with optimal scaling in terms of the general position gap, $ε$, which measures the distance of the fluid relaxation from degeneracy. We classify availability-based policies into global and local policies based on the scope of information they utilize. First, for general networks (possibly cyclic), we propose a global availability-based policy, probabilistic matching, and prove that it achieves the optimal all-time regret scaling of $O(ε^{-1})$, matching the known lower bound established by [KAG24]. Second, for acyclic networks, we focus on the class of local availability-based policies, specifically static priority policies that prioritize matches based on a fixed order. Within this class, we derive the first explicit regret bound for the previously proposed tree priority policy, showing all-time regret scaling of $O(ε^{-(d+1)/2})$, where $d$ is the network depth. Next, we introduce a new truncated tree priority policy and prove that it is the first static priority policy to achieve the optimal all-time regret scaling of $O(ε^{-1})$. These policies are appealing for matching systems such as queueing and load balancing; they reduce operational costs by using minimal information while effectively balancing the trade-off between immediate and future rewards.

Restarted GMRES is a robust and widely used iterative solver for linear systems. The control of the restart parameter is a key task to accelerate convergence and to prevent the well-known stagnation phenomenon. We focus on the Proportional-Derivative GMRES (PD-GMRES), which has been derived using control-theoretic ideas in [Cuevas Núñez, Schaerer, and Bhaya (2018)] as a versatile method for modifying the restart parameter. Several variants of a quadtree-based geometric optimization approach are proposed to find a best choice of PD-GMRES parameters. We show that the optimized PD-GMRES performs well across a large number of matrix types and we observe superior performance as compared to major other GMRES-based iterative solvers. Moreover, we propose an extension of the PD-GMRES algorithm to further improve performance by controlling the range of values for the restart parameter.

Despite algorithms' potential to improve public services, adoption has been limited by concerns about effectiveness and equity. We conduct a randomized controlled trial ($N=4,681$) providing real-time algorithm support to Child Protective Services (CPS) workers allocating investigations. Algorithm access reduced maltreatment-related hospitalizations, especially among disadvantaged groups, while reducing CPS surveillance of Black children. Notably, child injury admissions decreased by 21 percent. Workers reallocated investigations toward children at greater likelihood of harm, without mechanically following algorithmic predictions. Discussion notes suggest the algorithm shifted worker attention to complementary information. Counterfactual exercises show that human-algorithm complementarity would outperform algorithmic automation in efficiency and equity.

The interaction of gravitational waves (GWs) with matter is thought to be typically negligible in the Universe. We identify an exception in the case of resonant interactions, where GWs emitted by a background binary system, such as an inspiraling supermassive black hole (SMBH) binary, cause a resonant response in a stellar-mass foreground binary and the frequencies of the two systems become, and remain, synchronized. We point out that this previously unexplored dynamical phenomenon is not only possible, but can lead to $\mathcal{O}(30)$ binary systems becoming resonantly locked in the host galaxy of merging SMBHs of mass $10^{8.5-11}\Msun$, each of which has a significantly reduced merger time. We predict $\mathcal{O}(10^{10})$ binary systems have been locked in the Universe's history. Resonant locking could be detected through anomalous inspiral of binary systems.

We examine the quantum field description of massive Carrollian field theories, emphasizing the critical role of gauge fixing within the Carrollian sector. We illustrate this importance using scalar Carrollian Electrodynamics (sCED) as a primary example. We also present the quantum field description for complex Carrollian scalar fields, Carrollian fermions, and Carrollian Electrodynamics. We highlight the challenges in scalar Carrollian electrodynamics (sCED), where the renormalized mass appears gauge-dependent, and clarify this discrepancy by carefully constructing completely gauge-fixed propagators. We discuss how certain abelian Carrollian field theories do not admit any loop corrections and are trivial in that sense.

We introduce a new bilevel version of the classic shortest path problem and completely characterize its computational complexity with respect to several problem variants. In our problem, the leader and the follower each control a subset of the edges of a graph and together aim at building a path between two given vertices, while each of the two players minimizes the cost of the resulting path according to their own cost function. We investigate both directed and undirected graphs, as well as the special case of directed acyclic graphs. Moreover, we distinguish two versions of the follower's problem: Either they have to complete the edge set selected by the leader such that the joint solution is exactly a path, or they have to complete the edge set selected by the leader such that the joint solution is a superset of a path. In general, the bilevel problem turns out to be much harder in the former case: We show that the follower's problem is already NP-hard here and that the leader's problem is even hard for the second level of the polynomial hierarchy, while both problems are one level easier in the latter case. Interestingly, for directed acyclic graphs, this difference turns around, as we give a polynomial-time algorithm for the first version of the bilevel problem, but it stays NP-hard in the second case. Finally, we consider restrictions that render the problem tractable. We prove that, for a constant number of leader's edges, one of our problem variants is actually equivalent to the shortest-$k$-cycle problem, which is a known combinatorial problem with partially unresolved complexity status. In particular, our problem admits a polynomial-time randomized algorithm that can be derandomized if and only if the shortest-$k$-cycle problem admits a deterministic polynomial-time algorithm.

We measure the entropy change of charge transitions in an electrostatically defined quantum dot in bilayer graphene. Entropy provides insights into the equilibrium thermodynamic properties of both ground and excited states beyond transport measurements. For the one-carrier regime, the obtained entropy shows that the ground state has a two-fold degeneracy lifted by an out-of-plane magnetic field. This observation is in agreement with previous direct transport measurements and confirms the applicability of this novel method. For the two-carrier regime, the extracted entropy indicates a non-degenerate ground state at zero magnetic field, contrary to previous studies suggesting a three-fold degeneracy. We attribute the degeneracy lifting to the effect of Kane-Mele type spin--orbit interaction on the two-carrier ground state, which has not been observed before. Our work demonstrates the validity and efficacy of entropy measurements as a unique, supplementary experimental tool to investigate the degeneracy of the ground state in quantum devices build in materials such as graphene. This technique, applied to exotic systems with fractional ground state entropies, will be a powerful tool in the study of quantum matter.

The goal of this paper is to give a new construction of the free monodromic categories defined by Yun. We then use this formalism to give simpler constructions of the free monodromic Hecke categories and then compute the trace of Frobenius and of the identity on them. As a first application of the formalism, we produce new proofs of key theorems in Deligne--Lusztig theory.

The coalescent is a foundational model of latent genealogical trees under neutral evolution, but suffers from intractable sampling probabilities. Methods for approximating these sampling probabilities either introduce bias or fail to scale to large sample sizes. We show that a class of cost functionals of the coalescent with recurrent mutation and a finite number of alleles converge to tractable processes in the infinite-sample limit. A particular choice of costs yields insight about importance sampling methods, which are a classical tool for coalescent sampling probability approximation. These insights reveal that the behaviour of coalescent importance sampling algorithms differs markedly from standard sequential importance samplers, with or without resampling. We conduct a simulation study to verify that our asymptotics are accurate for algorithms with finite (and moderate) sample sizes. Our results constitute the first theoretical description of large-sample importance sampling algorithms for the coalescent, provide heuristics for the a priori optimisation of computational effort, and identify settings where resampling is harmful for algorithm performance. We observe strikingly different behaviour for importance sampling methods under the infinite sites model of mutation, which is regarded as a good and more tractable approximation of finite alleles mutation in most respects.

Recent James Webb Space Telescope observations of cool, rocky exoplanets reveal a probable lack of thick atmospheres, suggesting prevalent escape of the secondary atmospheres formed after losing primordial hydrogen. Yet, simulations indicate that hydrodynamic escape of secondary atmospheres, composed of nitrogen and carbon dioxide, requires intense fluxes of ionizing radiation (XUV) to overcome the effects of high molecular weight and efficient line cooling. This transonic outflow of hot, ionized metals (not hydrogen) presents a novel astrophysical regime ripe for exploration. We introduce an analytic framework to determine which planets retain or lose their atmospheres, positioning them on either side of the cosmic shoreline. We model the radial structure of escaping atmospheres as polytropic expansions - power-law relationships between density and temperature driven by local XUV heating. Our approach diagnoses line cooling with a three-level atom model and incorporates how ion-electron interactions reduce mean molecular weight. Crucially, hydrodynamic escape onsets for a threshold XUV flux dependent upon the atmosphere's gravitational binding. Ensuing escape rates either scale linearly with XUV flux when weakly ionized (energy-limited) or are controlled by a collisional-radiative thermostat when strongly ionized. Thus, airlessness is determined by whether the XUV flux surpasses the critical threshold during the star's active periods, accounting for expendable primordial hydrogen and revival by volcanism. We explore atmospheric escape from Young-Sun Mars and Earth, LHS 1140 b and c, and TRAPPIST-1 b. Our modeling characterizes the bottleneck of atmospheric loss on the occurrence of observable Earth-like habitats and offers analytic tools for future studies.