While massive black hole binaries (MBHBs) merge at gravitational-wave frequencies above the pulsar timing array (PTA) sensitivity band, we show that they leave orphaned low-frequency contributions in the PTA pulsar term. Due to the light-propagation time between each pulsar in the array and Earth, the pulsar term acts as a time-delayed probe of a chirping merger with a specific frequency response determined by the direction of origin and intrinsic properties of the MBHB. We provide a detailed consideration of how such a multiband signal would manifest in a full PTA, demonstrate an approach to stack these orphaned pulsar terms across the array, and discuss prospects for an archival, multiband search in conjunction with MBHB mergers observed in astrometric data or spaceborne interferometers like LISA.

We investigate how laser-driven, cooperative dipole-dipole interactions in weakly trapped atomic arrays give rise to self-organized configurations. Starting from an analytically tractable two-emitter system, we identify the possible steady-state spatial arrangements accessible to the atoms. We then extend this analysis to larger ensembles in both linear and ring geometries. In linear chains, we demonstrate the emergence of topologically nontrivial dimerized configurations across a range of initial interatomic spacings. In ring geometries, we find that the system undergoes self-organized contraction and expansion, enabling access to length scales below those set by the trapping lattice. Our results demonstrate that collective light-matter interactions in free space can spontaneously generate modified ordered geometries, even when the emitters are initially separated by distances larger than their transition wavelength.

Pulsar timing arrays record gravitational waves from supermassive black hole binaries at two spacetime points: an Earth term, measured when the wave passes the Earth, and a pulsar term, measured when the wave passed each pulsar at an earlier epoch. We show that a future $μ$Hz-band detection of a nearby massive binary by a mission such as $μ$Ares would turn PTA pulsar terms into targeted probes of binary evolution. In analogy with supernova light echoes, each pulsar term acts as a gravity echo: a dated snapshot of the binary at an earlier stage of its inspiral. Together, the $μ$Hz Earth-term measurement and the nHz pulsar-term echoes provide a temporal baseline that neither detector could access alone. For a fiducial equal-mass binary with total mass $10^9\,M_\odot$ at 80~Mpc, we find a combined pulsar timing array echo signal-to-noise ratio of 33, with up to 24 pulsars individually resolving the signal among pulsars with 50-year baselines. The angular dependence of the single-pulsar echo sensitivity alone enables independent sky localization of the source to $\sim$10--100~deg$^2$, and the resolved pulsar-term frequencies directly measure the binary inspiral rate hundreds to thousands of years ago. With sufficient pulsar distance precision, a small set of anchor pulsars could additionally phase-connect the array and trace the post-Newtonian evolution coherently over kpc baselines. The source population required for gravity echoes is drawn from the same massive-end census responsible for the observed nanoHertz stochastic background.

Variable selection in linear regression has been a central topic in statistical research for decades. Bayesian variable selection methods, which account for uncertainty in both the regression coefficients and the noise variance, have achieved broad success through the use of discrete or continuous shrinkage priors and efficient collapsed Gibbs samplers. Despite their popularity and strong empirical performance, an enigma remains: the marginal likelihood, obtained by integrating out the regression coefficients and noise variance, is not log-concave; therefore, there is no guarantee of reliably finding its global optimum. In this article, we study this problem from an optimization perspective. Taking the negative log-marginal likelihood as a loss function of the latent precision parameters, we can rewrite it as a difference of convex functions (DC), and then optimize it via a simple iterative algorithm. Under mild compact set conditions, the DC algorithm converges to the global optimum at a linear rate. The positive finding applies to type-II maximum likelihood and extends to maximum marginal posterior under suitable priors, indicating that the problem of mode finding in Bayesian variable selection is much more benign than the lack of log-concavity might suggest. Besides the theoretical insight, the proposed algorithm is easy to implement, free of tuning, and extensible to structured sparsity, and thus can serve as an efficient alternative or warm-start for traditional Markov chain Monte Carlo solutions. The method is illustrated through numerical studies and a spatial data application for quantifying the aftershock risk following the 2019 Ridgecrest earthquakes. The source code for the algorithm is publicly available at https://github.com/leoduan/dca_optimization_variable_selection.

We report on the spectroscopic confirmation of overdense regions of massive quiescent galaxies (QGs) in the early Universe with JWST/NIRSpec. Based on data from the DeepDive NIRSpec program and archival data from the Dawn JWST Archive, we confirm three QGs in the vicinity of Jekyll & Hyde, a pair of massive QG and a dusty star-forming galaxy, at $z=3.71$ and two QGs around SXDS-27434 at $z=4.01$. According to the analysis of galaxy number density with photometric redshifts, Jekyll & Hyde (SXDS-27434) are in an overdense region, where the number density of galaxies is three times higher than the average in the COSMOS (SXDS) field. SED fitting suggests that most of the QGs follow similar star formation histories and have consistent formation and quenching epochs. The same trend is observed in other proto-clusters hosting QGs that were already identified by ground-based telescopes, indicating that the large-scale environment plays an important role in the formation of QGs. In addition, JWST spectra reveal a broad H$α$ emission line from SXDS-27434 and faint emission lines from other three QGs, which are identified as AGN-driven based on their emission line ratios. The overdensity is also reproduced by the Illustris TNG300 simulation at $z=3.71$, in which the member QGs also have similar quenching epochs. These results suggest that large-scale structure may enhance merger activity and/or gas accretion and trigger AGN feedback, which simultaneously drives galaxy quenching in the overdensity.

Accurate modeling of ion-molecule reaction networks is essential for understanding the chemical evolution of planetary ionospheres, particularly for giant planets where proton-transfer chains drive atmospheric composition. However, predicting reaction rates in these ultracold environments remains a challenge due to the non-trivial interplay between vibrational dynamics and quantum tunneling. In this work we present a chaos-diagnostic framework that integrates multireference electronic structure theory, Adiabatic Gauge Potentials (AGP), and Random Matrix Theory (RMT) to characterize the microscopic dynamics of proton transport. Using the formation of H+3 and the proton-bound cluster H+5 as representative model systems relevant to Jovian atmospheres, we demonstrate that the transition state acts as a dynamical bottleneck where quantum chaos is notably suppressed, effectively enhancing tunneling probabilities. We introduce a fragility index based on the AGP slope to quantify how specific vibrational modes reintroduce chaos and suppress reactivity. This diagnostic approach offers a generalizable, data-driven metric for identifying vibrationally gated pathways in complex astrochemical networks, providing a theoretical basis for refining kinetic models of planetary and interstellar plasmas

We present an invariant relational path-integral quantization framework for general-relativistic gauge field theories based on the Dressing Field Method. The construction implements an automatic anomaly-cancellation mechanism that encompasses Bardeen-Wess-Zumino counterterms. The resulting framework unifies invariant schemes across contexts ranging from electroweak theory to cosmology, and is amenable to lattice implementations, key to high-precision tests in both domains.

Password-based authentication is one of the most commonly used methods for verifying user identities, and its widespread usage continues in virtual reality (VR) applications. As a result, various forms of attacks on password-based authentication in traditional environments such as keystroke inference and shoulder surfing, are still effective in VR applications. While keystroke inference attacks on virtual keyboards have been studied extensively, few efforts have developed an effective and cost-efficient defense strategy to mitigate keystroke inferences in VR. To address this gap, this paper presents a novel QWERTY keyboard called \textit{VRSafe} that is resilient to keystroke inference attacks. The proposed keyboard carefully introduces false positive keystrokes into the information collected by attackers during the typing process, making the inference of the original password difficult. \textit{VRSafe} also incorporates a novel malicious login detector that can effectively identify unauthorized login attempts using credentials inferred from keystroke inference attacks with high detection rate and minimal time and memory cost. The proposed design is evaluated through both simulation experiments and a real-world user study, and the results show that \textit{VRSafe} can significantly reduce the accuracy of keystroke inference attacks while incurring a modest overhead from a usability standpoint.

Bubbles entrained by breaking waves rise to the ocean surface where they cluster and burst, emitting sea spray aerosols into the atmosphere. Bubble bursting thereby links seawater biogeochemistry and aerosol chemistry, influencing the ability of emitted aerosols to serve as cloud condensation nuclei or ice nucleating particles. The mechanisms of film drop and jet drop production are modulated by organic material present in seawater, which may affect the size, number, and composition of resulting aerosols. We disentangle the effect of surfactant on collective bursting processes using laboratory experiments with detailed bubble and aerosol measurements down to small sizes, multiple bubble size configurations, and measurements of bubble lifetime. Submicron aerosol emission, linked to film drop production, increased with surfactant up to an optimal concentration, while production of supermicron aerosols emitted through jet drop production was shut down. Our work paves the way to integrate organic composition into sea spray emission functions.

A search for dark matter produced in association with a dark Higgs boson decaying into a bottom quark-antiquark pair has been performed using proton-proton collision data at a center-of-mass energy of 13 TeV. The search uses data collected with the CMS detector at the CERN LHC during the 2016$-$2018 data-taking period, corresponding to an integrated luminosity of 138 fb$^{-1}$. The results are interpreted in terms of a theoretical model of dark matter production that, together with a spin-1 gauge boson mediator, predicts the existence of a Higgs-boson-like particle in the dark sector (i.e., a dark Higgs boson). This search focuses on an experimental signature with large missing transverse momentum from dark matter production and a resonant structure in the invariant mass of the bottom quark-antiquark pair from the dark Higgs boson decay. Upper limits at 95% confidence level on the signal strength for dark Higgs boson mass hypotheses below 160 GeV are set. Values of the mediator mass up to 4.5 (2.5) TeV are excluded at 95% confidence level for a dark Higgs boson mass of 50 (150) GeV. This represents the most stringent limits set to date for the dark Higgs boson masses considered in this study.

We establish new strong factorization properties for the smooth vectors of representations of exponential solvable Lie groups on Fréchet spaces. In particular, our results improve upon the Dixmier-Malliavin factorization theorem for simply connected nilpotent Lie groups.

The localization of slow and fast slip in fault gouges may play a crucial role in understanding the mechanics of earthquakes and slow slip events. Here, we investigate the fracture energy accompanying this localization and the subsequent thermal weakening. We develop an analytical framework, complemented by numerical simulations, for a gouge governed by rate-and-state-dependent friction with flash-heating at high strain rate and thermal pressurization of pore fluids. The model captures the transition from initially distributed shearing to a co-seismic principal slip ``surface'' at slip $δ_{\mathrm{loc}} \approx γ_c h$, and yields a decomposition of the fracture energy, $G = G_\mathrm{loc}(h) + ΔG(δ)$. The minimum, localization-related component $G_\mathrm{loc}$ scales with gouge thickness $h$, which in turn scales linearly with fault size. Flash heating is activated only upon localization for fast earthquake slip, producing an abrupt strength drop, and contributing to the magnitude of $G_\mathrm{loc}$. The post-localization term $ΔG$ increases with co-seismic slip due to efficient thermal pressurization and is insensitive to $h$. Localization is predicted to occur for both rate-weakening and rate-strengthening gouges because transient state evolution drives apparent weakening after a slip-rate increase. These results unify field, laboratory, and seismological observations of shear band thickness, critical slip, and fracture-energy scaling, and they clarify why small events can be governed by scale-dependent $G_\mathrm{loc}$ whereas large ruptures become increasingly fault-invariant as $ΔG$ dominates. Our framework provides testable predictions for the relation of gouge thickness to lower bounds of co-seismic fracture energy, and the mechanics of slow-slip transients and fast earthquakes.

In this workshop, we present a compact but rigorous introduction to second-order optimality conditions for mathematical programs with equilibrium constraints (MPECs). We start from the classical nonlinear programming template, then explain why it fails in the equilibrium-constrained setting, and develop the three main viewpoints used in the literature: (i) multiplier-based conditions, (ii) implicit-programming conditions based on the solution map of the lower-level equilibrium system, and (iii) piecewise-programming conditions obtained by decomposing complementarity structure into smooth pieces. The emphasis is on conceptual structure, critical cones, strong regularity, and the exact role of curvature terms.

This paper explores the use of artificial neural networks for the stable and data-driven selection of the frequency parameter in hyperbolic polynomial penalized splines (HP-splines). This parameter defines the underlying spline space and is essential for adapting the model to exponential patterns in the data, such as those encountered in signal processing. The theoretical approximation properties of deep neural network architectures are investigated to establish a connection between classical spline-based regression and modern data-driven learning methods. Based on this analysis, a neural network is designed to predict optimal HP-spline parameters by balancing approximation accuracy, stability analysis, and complexity control, thereby producing neural architectures that are both expressive and stable. Numerical experiments confirm that the proposed approach achieves both high accuracy and stable performance, validating the theoretical findings.

Relativistic effects are sensitive to subtle changes in dark energy. These effects grow on very large scales and at high redshifts, which will be the reach of upcoming surveys. We investigate these effects in both the linear and the angular galaxy power spectra in a late-time universe dominated by cold dark matter and k-essence, focusing on three core models (dilaton, tachyon, and DBI scalar fields) and contrasting their predictions with those of the concordance model. By enforcing identical present-day cosmological parameters, we isolate the imprints of k-essence dynamics and perturbations on very large scales. We found that relativistic corrections dominate on very large scales and grow with redshift, but are largely insensitive to k-essence microphysics in Fourier space, leading to strong degeneracies among the models. However, in the angular power spectrum, where line-of-sight integrals are naturally included, relativistic effects are significantly amplified, yielding better sensitivity to clustering k-essence. In particular, the tachyon exhibits clear deviations across multipoles and redshifts, with distinct imprints in the Doppler and the combined (velocity and gravitational) potentials contributions. Furthermore, our results show that neglecting relativistic corrections can lead to systematic misestimation of deviations of k-essence from the cosmological constant. Our results show the relativistic angular galaxy power spectrum as a more consistent and robust probe of ultra-large-scale physics. These findings underscore the need for full relativistic modelling in next-generation surveys that are targeting horizon-scale modes, where the imprint of non-standard dark energy is most pronounced.

In this workshop, we present a compact but rigorous introduction to the basic language of nonlinear programming, variational inequalities, and complementarity systems. The goal is twofold. First, we explain the mathematical logic of hypotheses under which first-order optimality conditions for MPECs become valid. Second, we explain how to use that theory in research practice: how to classify a model, choose the appropriate verification route, prove the right hypotheses, and derive a correct first-order analysis.

Optical vortex knots have been realized in monochromatic laser beams, but monochromatic fields are stationary and their topology is frozen. Here we show that knotted spatiotemporal vortices, whose phase singularities form closed loops in spacetime, undergo topology changing reconnections with free space propagation. When null lines of different vector components unlink, the electric spin, magnetic spin, linear momentum, and electromagnetic helicity densities, each built from a specific pair of field components, twist to exactly compensate the change in linking number. This compensation is enforced by the argument principle where the total for each component pair, combining mutual phase twist, geometric linking, and open-line threading, vanishes identically and remains exactly zero through all reconnection events.

A graph reaction--diffusion (RD) equation is a system of differential equations that is defined on the nodes of a graph. Consider a sequence of growing graphs that converges in cut norm to a limiting graphon. We show that the solutions of the sequence of graph RD equations converge in $L^p$ norm, for $p \in [1,\infty]$, to the solution of a limiting nonlocal RD equation, which we call a graphon RD equation. Furthermore, we show a large numbers result for a stochastic particle process that consists of a random walk and a birth-death process on graphs. For a sequence of graphs that converge in cut norm to a limiting graphon, the sequence of stochastic processes converges in probability to the solution of the graphon RD equation.

We present MediaGraph, a network-theoretic framework for analyzing reporting preferences in news media through entity co-occurrence networks. Using articles from four Indian news-sources, two mainstream (The Times of India and The Indian Express) and two fringe outlets (dna and firstpost), we construct source-specific co-occurrence networks around the 2020-21 and 2024 Farmers Protests. We analyze these networks along three network theoretic axes of centrality, community structure, and co-occurrence link predictability. The link predictability metric is a novel metric proposed that quantifies the consistency of entity associations over time using a GraphSAGE-based model. Our results reveal significant differences in reporting preferences across sources for the same event, and a consistent under-representation of farmer leaders across sources. By shifting the focus from textual signals to relational structures, our approach offers a scalable, label-independent perspective on media analysis and introduces link predictability as a complementary measure of reporting behavior.

The Riccati differential equation is examined in light of its connection to second order linear time varying systems. In that light it becomes the clear generalization for the characteristic equation of linear time invariant systems, and is called the Riccati Characteristic Equation (RCE). Consequently, the RCE becomes the unifying centerpiece for the study of linear systems. Its solutions are considered in complementary pairs that form a continuum based on a primitive pair. Pairs may always be found as purely real solutions, despite the fact that complex conjugate primitive solutions are shown to exist in many cases. Not only is the pairing unique, but the general form of solutions, shown here for the first time, is uniquely compact and encompasses all known solutions, while allowing for all initial conditions. Classical engineering mathematics examples are shown to conform to this approach, which provides new insights to all, especially Floquet theory.

This paper presents a unifying theory of Linear second order systems that allows time-varying and time invariant systems to be treated in the same way for the first time. In the process, a transformation is given that diagonalizes an arbitrary time varying state matrix in a spectrum invariant way. A canonical form for the fundamental matrix is given that depends on dynamic eigenvalues and related eigenvectors dependent upon the Riccati Characteristic Equation for the system, which intuitively generalizes the standard characteristic equation for time invariant systems. The technique is shown by examples to give a unified approach to the solutions of time invariant, time-varying, and periodic systems.

In many spatial and spatial-temporal models, and more generally in models with complex dependencies, it may be too difficult to carry out full maximum likelihood (ML) analysis. Remedies include the use of pseudo-likelihood (PL) and quasi-likelihood (QL) (also called the composite likelihood). The present article studies the ML, the PL and the QL methods for general Markov chain models, partly motivated by the desire to understand the precise behaviour of PL and QL methods in settings where this can be analysed. We present limiting normality results and compare performances in different settings. The PL and QL methods can be seen as maximum penalised likelihood methods. We find that the QL strategy is typically preferable to the PL, and that it loses very little to the ML, while earning in model robustness. It has also appeal and potential as a modelling tool. Our methods are illustrated for analysis of DNA sequence evolution type models.

We show that finite-size violations of local electroneutrality in confined electrolytes are governed by the topology of the confining domain, yielding a universal hierarchy of deviations across spherical, cylindrical, and planar geometries. Within Poisson-Boltzmann theory, we introduce an electroneutrality deviation ratio that quantifies how global electrostatic constrains associated with compacness and boundary multiplicity modify charge balance inside confined domains. Although electroneutrality is asymptotically restored in all geometries, finite-size deviations are strongest in compact spherical cavities, weaker in cylindrical confinement, and weakest in planar slits. These results identify topology as the structural origin of confinement-induced charge redistribution and stablish the violation of local electroneutrality as global constraint underlying phenomena such as overcharging anf charge reversal, demostrating that confinement-not local-not local geometric details-controls the emergence of these effects.

By monitoring the times of arrival of radio pulses from millisecond pulsars, Pulsar Timing Arrays (PTAs) serve as unique gravitational wave (GW) laboratories in the nanohertz band. To date, the primary astrophysical sources of GWs targeted in this frequency range have been inspiraling supermassive black hole binaries (SMBHBs) on circular and eccentric orbits. In this work, we demonstrate that, thanks to the so-called pulsar term in the timing residual waveform of GW signals, PTAs can probe individual SMBHBs that merged before timing observations began. We refer to the latter as \emph{zombie binaries}. Using SMBHB population models consistent with current PTA constraints, we find that while the probability of detecting such systems in existing PTA datasets remains low, the Square Kilometer Array observatory is expected to achieve sufficient sensitivity to have a few zombie binaries with a signal-to-noise ratio exceeding 3 in its data. Although their confident identification might be challenging, this new class of PTA sources opens a novel window for studying the most massive SMBHBs in our local universe.

Architected materials can exhibit remarkable combinations of stiffness, strength, and toughness, yet their application is currently limited by an incomplete understanding of how cracks initiate and propagate through their discrete architecture. Elucidating the mechanisms that underpin these processes is challenging because lattice failure is governed by highly localized deformations of slender beams, which fall outside the resolution and assumptions of optical methods developed for continuum solids, such as digital image correlation (DIC). Thus, characterizing crack propagation within lattice materials requires measurement strategies capable of resolving lattice-scale deformations while accounting for both the intrinsic discreteness of lattice architectures and the progressive formation of material discontinuities during failure. This work introduces a global DIC framework tailored to architected materials, in which the correlation problem is solved directly on the lattice mesh and damaged elements are automatically removed during the analyses. Damage detection, which relies on a data-driven residual criterion, enables the robust tracking of localized deformation and crack-tip motion under different testing conditions. The method provides physically consistent displacement field measurements on the evolving intact lattice topology and resolves the crack path over time. Validations on 3D-printed regular and imperfect triangular lattices under mode-I loading demonstrate that the approach accurately captures both damage initiation and crack propagation. Furthermore, we demonstrate that identifying damaged elements provides an estimate of the critical failure strain, which can be used directly in numerical models or adopted as an alternative element-deletion threshold in DIC analyses.

Effective hyperlocal communication is critical in the Philippines, where delayed or algorithm-filtered updates can leave residents uninformed about emergency advisories and community events. We conducted a user-centered study consisting of contextual inquiry and semi-structured interviews to identify four key barriers: delayed alerts, algorithm-driven noise, language gaps, and digital divides. Guided by these insights, we designed KUBO (Kumunidad at Balitang Opisyal), a prototype that integrates a home module for verified local government unit advisories and curated headlines, and a community module for resident-powered neighborhood reports and discussions. Using a within-subjects evaluation design, KUBO significantly reduced task completion times (p-value < 0.001), improved information recall on post-task quizzes (p-value = 0.010), and yielded higher user satisfaction ratings for ease of use, overall satisfaction, and perceived effectiveness compared to Facebook, the commonly used communication platform in the Philippines. These results demonstrate that a dual-channel, inclusive platform can substantially enhance real-time information access, comprehension, and civic engagement in hyperlocal settings.

Gravitational wave (GW) detector LISA will observe near-coalescence extreme mass ratio inspirals (EMRIs), which typically form in galactic central accretion disks. Gas torques on EMRI will alter its GW-driven inspiral trajectory from the vacuum expectation, leading to potentially LISA-observable GW dephasing ($Δψ_{\rm gas}$). Most studies compute $Δψ_{\rm gas}$ for a thin, laminar disk, with negligible flow turbulence, where the disk exerts a fairly well-understood linear torque ($T_{\rm lin}$). However, these disks must be turbulent due to magneto-rotational instability in the inner regions. Hence, we present a proof-of-concept general, agnostic prescription for the turbulent torque ($T_{\rm turb}$) acting on an EMRI by modeling it as a Gaussian distribution around $T_{\rm lin}$, based on recent advances from a global hydrodynamical (HD) study. We compute $Δψ_{\rm gas}$ for the ``golden'' circular EMRI with total source mass $M=10^6~{\rm M}_\odot$ and mass ratio $q=5\times10^{-5}$ in its final four-year evolution at redshift $z=0.276$ and signal-to-noise ratio (SNR) $=50$ by varying Eddington ratio ${\rm f}_{\rm Edd}$, turbulence normalization $C$ ($=~360$ in the aforementioned HD study), disk aspect ratio $h_0$, and turbo-viscous coefficient $α$ in a reasonable parameters space. We find that for ${\rm f}_{\rm Edd}\gtrsim0.3$, $C\gtrsim300$, $h_0\gtrsim0.03$, and $α\gtrsim0.1$, gas-induced dephasings are unobservable if only considering $T_{\rm lin}$ but could become detectable ($Δψ_{\rm gas}>8/$SNR) if EMRIs exhibit chaotic migration due to turbulent gas flow. Hence, this work motivates running MHD simulations of accretion disks with embedded LISA EMRIs in the early in-spiral phase over long enough timescales to understand the evolution of their orbital elements and the imprint of the turbulent environment on their gravitational waveforms.

We produce infinitely many local systems on (level covers of) the moduli space of smooth cubic threefolds, with algebraic monodromy group equal to the exceptional group $E_6$. These local systems arise in the middle cohomology of abelian étale covers of the Fano scheme parametrizing lines in the universal cubic threefold.

We investigate the effects of coupling a quantum-magnetic cavity field to molecules. Our high-precision auxiliary-field quantum Monte Carlo calculations capture the effect of the cavity field in the presence of electron correlations, and their interplay and competition. In H$_2$, we find that a strong enough cavity coupling makes the original bound ground state metastable, along with inverting the singlet-triplet gap. In ring molecules (e.g., H$_n$), the magnetic cavity coupling stabilizes symmetric geometries. As a consequence, open-shell rings such as H$_4$, H$_8$, or C$_4$H$_4$, which would undergo Jahn-Teller distortions outside of the cavity, obtain exotic spin or ring-current polarized, antiaromatic ground states. These effects are enhanced by increasing the molecule concentration inside the cavity. Our results suggest cavity quantum electrodynamics beyond the long-wavelength approximation as a promising avenue for cavity-altered chemistry.

We study the long-time dynamics of small-amplitude solutions to the three-dimensional gravity-capillary water waves equations for an inviscid and irrotational fluid with periodic boundary conditions. We prove that, for almost all values of the surface tension parameter, solutions with initial size $\varepsilon$ exist and remain small over time intervals of order $\varepsilon^{-2}$. A major difficulty arises from the loss of derivatives caused by the quasilinear nature of the equations combined with severe quadratic and cubic small-divisor interactions in high space dimensions. Classical normal form methods applied to 3D water waves system typically fail to prevent derivative loss due to the accumulation of near-resonances. To overcome this obstruction, we develop a new analytical strategy that combines a sharp frequency partition with a quasi-resonant normal form transformation acting only on selected interaction scales. Our microlocal analysis reveals that the potentially dangerous interactions terms exhibit a block-diagonal structure, which stems from both the geometric properties of the quasi-resonant frequency sets and the Hamiltonian structure of the water waves system. As a consequence, these operators preserve Sobolev norms and do not produce energy growth. This structural insight, together with the quasi-resonant normal-form transformation, allows us to prevent derivative-loss mechanisms while avoiding the accumulation of harmful small denominators.