Singular learning theory characterizes Bayesian models with non-identifiable parameterizations through two central quantities: the real log canonical threshold (RLCT), which governs marginal likelihood asymptotics, and the singular fluctuation, which determines second-order generalization behavior and the complexity term in WAIC. While the geometric meaning of the RLCT is well understood, the interpretation of singular fluctuation has remained comparatively opaque. We show that singular fluctuation admits a precise thermodynamic interpretation. Under a tempered (Gibbs) posterior, it is exactly the curvature of the Bayesian free energy with respect to inverse temperature; equivalently, the variance of the log-likelihood observable. In this sense, singular fluctuation is the statistical analogue of specific heat. This identity clarifies why singular fluctuation controls the equation of state relating training and generalization error and explains the success of WAIC in singular models: WAIC estimates a fluctuation coefficient rather than a parameter dimension. Across Gaussian mixture models and reduced-rank regression, we demonstrate that singular fluctuation behaves as a thermodynamic response coefficient. As temperature decreases, posterior reorganization suppresses fluctuation directions that affect predictive performance, and model-specific geometric observables track the decay of singular fluctuation. Rather than introducing new asymptotic expansions, this work unifies existing variance identities, equation-of-state results, and WAIC complexity corrections under a single free-energy curvature framework.

The dynamics of disordered nuclear spin ensembles are the subject of nuclear magnetic resonance studies. Due to the through-space long-range dipolar interaction generically many spins are involved in the time evolution, so that exact brute force calculations are impossible. The recently established spin dynamic mean-field theory (spinDMFT) represents an efficient and unbiased alternative to overcome this challenge. The approach only requires the dipolar couplings as input and the only prerequisite for its applicability is that each spin interacts with a large number of other spins. In this article, we show that spinDMFT can be used to describe spectral spin diffusion in static samples and to simulate zero-quantum line shapes which eluded an efficient quantitative simulation so far to the best of our knowledge. We perform benchmarks for two test substances that establish an excellent match with published experimental data. As spinDMFT combines low computational effort with high accuracy, we strongly suggest to use it for large-scale simulations of spin diffusion, which are important in various areas of magnetic resonance.

The Taishan Antineutrino Observatory (TAO or JUNO-TAO) is a satellite observatory for the Jiangmen Underground Neutrino Observatory (JUNO), located 44 meters away from the No.1 reactor of the Taishan Nuclear Power Plant. TAO can measure the reactor antineutrino energy spectrum with excellent energy resolution (better than 2% at 1 MeV) using state-of-the-art Silicon Photomultipliers (SiPMs) operated at low temperature. To achieve this goal, the SiPMs (together with their readout electronics) must be well calibrated. This paper presents the channel-level calibration methods for the dark count rate (DCR), relative photon detection efficiency (PDE), time offset, gain, and internal optical crosstalk (IOCT) of the SiPMs based on charge and time information of the collected events. For the tile-level calibration of the external optical crosstalk (EOCT), in terms of its rate and emission angle distribution, a novel method is proposed by switching on and off different groups of SiPMs with an LED placed in the detector. Using one million simulated events, the expected calibration biases are evaluated for all the aforementioned parameters: relative PDE (~3%), IOCT (1.4%), DCR (0.4%), EOCT Rate (<0.1%), gain (<0.1%), time offset (<0.2 ns). The emission angle distribution of the EOCT photons could be measured with a bias of less than 4% in the main angular range. With the this calibration accuracy, the overall impacts of SiPM parameter uncertainties and calibration biases on vertex precision and energy resolution are limited, with relative degradation below 2% and 3%, respectively. It verifies the validity of the calibration method for the JUNO-TAO detector.

Deviations from diffusive heat transport in high thermal conductivity crystalline insulators are generally understood within the framework of the phonon Boltzmann Transport Equation. However, for low thermal conductivity materials with large primitive cells or strong anharmonicity, the recently developed Wigner Transport Equation is more appropriate as it includes tunnelling between overlapping phonon bands. In this work, via solutions to the Wigner Transport Equation, we develop a scheme to obtain the dynamics of the phonon populations and coherences as a function of an arbitrary heat source. The approach is applied to predict size effects and dynamical thermal conductivities in CsPbBr$_\text{3}$ and La$_\text{2}$Zr$_\text{2}$O$_\text{7}$ using first-principles data as input. We predict significant deviations from the bulk thermal conductivity in these materials at length scales on the order of hundreds of nanometers to a few microns at room temperature, well within the reach of direct observation using current experimental techniques.

Many properties related to security or concurrency must be encoded as so-called hyperproperties, temporal properties that allow reasoning about multiple traces of a system. However, despite recent advances on model checking hyperproperties, there is still a lack of higher-level specification languages that can effectively support software engineering practitioners in verifying properties of this class at early stages of system design. Alloy is a lightweight formal method with a high-level specification language that is supported by automated analysis procedures, making it particularly well-suited for the verification of design models at early development stages. It does not natively support, however, the verification of hyperproperties. This work proposes HyperPardinus, a new model finding procedure that extends Pardinus -- the temporal logic backend of the Alloy language -- to automatically verify hyperproperties over relational models by relying on existing low-level model checkers for hyperproperties. It then conservatively extends Alloy to support the specification and automatic verification of hyperproperties over design models, as well as the visualization of (counter-)examples at a higher-level of abstraction. Evaluation shows that our approach enables modeling and finding (counter-)examples for complex hyperproperties with alternating quantifiers, making it feasible to address relevant scenarios from the state of the art.

We revisit the renormalization of the gauge coupling in massless QED coupled to a scaleless quadratic theory of gravity. We compare two alternative prescriptions for the running of the electric charge: (i) the conventional $μ$-running in minimal subtraction, and (ii) a ''physical'' running extracted from the logarithmic dependence of amplitudes on a hard scale $Q^{2}$ (e.g., $p^{2}$ or a Mandelstam invariant) after removing IR effects. At one loop, using dimensional regularization with an IR mass regulator $m$, we compute the photon vacuum polarization. We find a clean separation between UV and soft logarithms: the former is gauge and process independent and fixes the beta function, whereas the latter encodes nonlocal, IR-dominated contributions that may depend on gauge parameters and must not be interpreted as UV running. In the quadratic-gravity sector, the photon self-energy is UV finite--the $\lnμ^{2}$ pieces cancel--leaving only $\ln(Q^{2}/m^{2})$ soft logs. Consequently, quadratic gravity does not modify the one-loop UV coefficient and thus does not alter $β(e)$. Therefore, the physical running coincides with the $μ$-running in QED at one loop. Our analysis clarifies how to extract a gauge and process independent running in the presence of gravitational interactions and why soft logs from quadratic gravity should not contribute to $β(e)$.

Collective vertical swimming may generate aggregate-scale flows that contribute to mixing and transport in stratified environments. The strength of these flows depends not only on swimmer behavior but also on environmental properties. Here we examine how fluid density affects flow generated by vertically migrating swarms of brine shrimp $\textit{Artemia salina}$. Using simultaneous three-dimensional swimmer tracking and particle image velocimetry, we measured swimmer kinematics and the induced flow field during phototactically driven migrations under four controlled salinity conditions. Induced velocity increased with buoyancy forcing and scaled with the parameter $N(ρ_s - ρ)$, where $N$ is the number of swimmers and $ρ_s - ρ$ is the density difference between swimmers and the surrounding fluid ($R^2 = 0.70$, $p = 5.9 \times 10^{-5}$). A multiple regression including swimmer number, swimming speed, fluid density, and the swarm Gaussian width confirmed that density remained a significant predictor of induced velocity after controlling for the other variables ($p = 0.012$, $R^2 = 0.82$). A simplified actuator-disk model captures the first-order dependence of induced velocity on buoyancy forcing and swimmer momentum, suggesting that environmentally driven density variations can substantially modify the hydrodynamic impact of collective vertical migration.

Long-lived spin-helix states facilitate the study of non-equilibrium dynamics in quantum magnets. We consider the decay of transverse spin-helices in antiferromagnetic spin-$S$ XXZ chains with single-ion anisostropy. The spin-helix decay is observable in the time evolution of the local magnetization that we calculate numerically for the system in the thermodynamic limit using infinite time-evolving block decimation simulations. Although the single-ion anisotropy prevents helix states from being eigenstates of the Hamiltonian, they still can be long-lived for appropriately chosen wave numbers. In case of an easy-axis exchange anisotropy the single-ion anisotropy may even stabilize the helices. Within a spin-wave approximation, we obtain a condition giving an estimate for the most stable wave number $Q$ that agrees qualitatively with our numerical results.

Power systems face increasing weather-driven variability and, therefore, increasingly rely on flexible but energy-limited storage resources. Energy storage can buffer this variability, but its value depends on intertemporal decisions under uncertain prices. Without accounting for the future reliability value of stored energy, batteries may act myopically, discharging too early or failing to preserve reserves during critical hours. This paper introduces a stopping-time reward that, together with a state-of-charge (SoC) range target penalty, aligns arbitrage incentives with system reliability by rewarding storage that maintains sufficient SoC before critical hours. We formulate the problem as an online optimization with a chance-constrained terminal SoC and embed it in an end-to-end (E2E) learning framework, jointly training the price predictor and control policy. The proposed design enhances reachability of target SoC ranges, improves profit under volatile conditions, and reduces its standard deviation.

We study the emergent dynamics of the singular continuum Kuramoto model (in short, SCKM) and its graph limit. The SCKM takes the form of an integro-differential equation exhibiting two types of nonlocal singularities: a nonlocal singular interaction weight and a nonlocal singular alignment force. The natural frequency function determines the emergent dynamics of the SCKM, and we emphasize that singularity plays a crucial role in the occurrence of sticking phenomena. For the identical natural frequency function, we derive the complete phase synchronization in finite time under a suitable set of conditions for system parameters and initial data. In contrast, for a nonidentical natural frequency function, we show the emergence of practical synchronization, meaning that the phase diameter is proportional to the inverse of coupling strength asymptotically. Furthermore, we rigorously establish a graph limit from the singular Kuramoto model with a finite system size to the SCKM. We also provide several numerical simulations to illustrate our theoretical results.

We investigate the shadow of a rotating braneworld black hole in dispersive plasma environments and assess the potential of the Event Horizon Telescope (EHT) observations to constrain braneworld gravity. The spacetime around a rotating braneworld black hole is modelled by a Kerr-Newman-like metric determined by its mass $M$, spin $a$, and tidal charge $q$, which encodes the gravitational effects of the bulk spacetime. We consider both inhomogeneous and homogeneous plasma environments characterised by plasma parameters $α_i$ ($i=1,2\text{ and }3$) to study light propagation and the interplay of the background spacetime and the plasma environment in influencing the shadow size and shape. We find that as the plasma density increases, inhomogeneous plasma environments decrease the shadow size, however homogeneous plasma enlarges it. On studying the effect due to the background spacetime, we find that $q<0$ (negative tidal charge) increases the shadow diameter, while $q>0$ decreases it. Using the EHT measurements of M87* and Sgr A*, we constrain the $(q,α_i)$ parameter space. The EHT data constrains the tidal charge in the range $-1.15 \lesssim q \lesssim 0.45$ for M87* and $-0.65 \lesssim q \lesssim 0.8$ for Sgr A* in the low density plasma limit, which is indeed the case for M87* and Sgr A*. However, for black holes surrounded by high density plasma, the shadow size is governed both by the background geometry as well as by the plasma environment. In such cases, joint constraints from plasma density estimates and observed shadow angular diameters can provide valuable insights into the underlying spacetime geometry.

We aim to construct a formal theory of affordances seen as ternary relations. Beginning with a characterization of affordances proposed by James J. Gibson, and utilizing the tools provided by Zdzisław Pawlak's information systems and rough sets, we construct a mathematically precise definition of both crisp and rough affordances. Then, we analyze modal and approximation operators that enable reasoning about affordances in both scenarios.

Policy gradient algorithms are widely used in reinforcement learning and belong to the class of approximate dynamic programming methods. This paper studies two key policy gradient algorithms, the Natural Policy Gradient and the Gauss-Newton Method, for solving the Linear Quadratic Regulator (LQR) problem in unknown stochastic linear systems. The main challenge lies in obtaining an unbiased gradient estimate from noisy data due to errors-in-variables in linear regression. This issue is addressed by employing a primal-dual estimation procedure. Using this novel gradient estimation scheme, the paper establishes convergence guarantees with a sample complexity of order O(1/epsilon). Theoretical results are further supported by numerical experiments, which demonstrate the effectiveness of the proposed algorithms.

This paper deals with the problem of covariance stabilization for a class of linear stochastic discrete-time systems in the Stochastic Model Predictive Control (SMPC) framework. The considered systems are affected by independent and identically distributed (i.i.d.) additive and parametric stochastic uncertainties (potentially unbounded), in addition to polytopic deterministic uncertainties bounding the mean of the state and input parameters. The design conditions presented in this paper are formulated as Linear Matrix Inequalities (LMIs), using the S-variable approach in order to reduce the potential conservatism. These conditions are derived using a deterministic exact characterization of the covariance dynamics, the latter involves bilinear terms in the control gain. A technique to linearize such dynamics is presented, it results in a descriptor representation allowing to derive sufficient conditions for the design of a covariance-stabilizing controller. The derived condition is first compared with a known necessary and sufficient stability condition for systems without deterministic uncertainties and additive stochastic noise. Although more conservative, the proposed condition is more numerically tractable, with an LMI size scaling as O(n^2) instead of O(n^3). Then, the same condition is used to design controllers that are robust to both deterministic and stochastic uncertainties. Several numerical examples are presented for comparison and illustration.

It is believed that rays bent inside and outside photon spheres could affect partially the black hole shadow images by the Event Horizon Telescope and the rays near photon spheres would be detected by near-future space observations. The investigation of the rays near the photon spheres in not only black hole spacetimes but also exotic spacetimes would be important since one will need them to exclude black hole mimickers. The deflection angles of the rays deflected by the photon spheres diverge logarithmically and we can treat them by a strong-deflection-limit analysis. The error of the strong-deflection-limit analysis becomes large if antiphoton spheres exist in the spacetimes and the analysis breaks down when the photon spheres and the antiphoton spheres degenerate to form a marginally unstable photon sphere. This is because the deflection angles of the rays bent by the marginally unstable photon sphere diverge in powers. In this paper, we extend Eiroa, Romero, and Torres's method to gravitational lensing of rays inside and outside of the marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits and we apply it to a Reissner-Nordström spacetime and a Hayward spacetime with the marginally unstable photon sphere. We have also confirmed that the deflection angles in the strong deflection limits by the method converge correctly to the deflection angle without approximations, while there are the mismatches of the coefficient of the power-divergent term of the deflection angles of the rays deflected just outside of the marginally unstable photon sphere in a semianalytic calculation by the author previously.

Higgs pair production offers the opportunity to probe correlations among the couplings of one or two Higgs bosons to fermions and gauge bosons. In this context, it serves as a powerful test of the underlying Effective Field Theory (EFT) framework. In particular, while such couplings remain correlated in the Standard Model Effective Field Theory (SMEFT) at dimension six, they can become fully de-correlated in Higgs Effective Field Theory (HEFT) already at leading order in the EFT expansion. In this work, we study Higgs pair production via gluon fusion within the HEFT framework. We demonstrate that adopting a consistent power counting in combination with next-to-leading order (NLO) diagrams necessitates the inclusion of higher-dimensional operators beyond the leading ones. We analyze their phenomenological impact and re-assess critically the kinematic benchmark scenarios commonly used in experimental non-resonant di-Higgs searches in light of these additional contributions.

We investigate the non-equilibrium four-vector work in an expanding relativistic piston. We derive the exact work distribution in this pedagogical model and find that the joint distribution of four-vector work $(W^0, W^1)$ concentrates on the origin and some curves in the $(W^0, W^1)$ space, rather than being smoothly distributed. In the non-relativistic limit, our model consistently recovers the non-relativistic dynamics. We further demonstrate that the momentum component of four-vector work remains significant in both the Lorentz-relativistic and Galilean-relativistic frameworks. On top of the work distribution, we verify a family of covariant fluctuation theorems of work. In addition, we introduce a novel geometrical technique for analyzing the dynamics of relativistic collision processes, which can be straightforwardly extended to multi-dimensional piston models.

With the advancement of modern applications, an increasing number of composite optimization problems arise whose smooth component does not possess a globally Lipschitz continuous gradient. This setting prevents the direct use of the proximal gradient (PG) method and its variants, and has motivated a growing body of research on new PG-type methods and their convergence theory, in particular, global convergence analysis without imposing any explicit or implicit boundedness assumptions on the iterates. Until recently, the first complete analysis of this kind has been established for the PG method and its specific nonmonotone variants, which has since stimulated further exploration along this research direction. In this paper, we consider a general composite optimization model beyond the global Lipschitz gradient continuity setting. We propose a novel problem-parameter-free algorithm that incorporates a carefully designed nonmonotone line search to handle the non-global Lipschitz gradient continuity, together with an extrapolation step to achieve potential acceleration. Despite the added technical challenges introduced by combining extrapolation with nonmonotone line search, we establish a refined convergence analysis for the proposed algorithm under the Kurdyka-Ł ojasiewicz property, without requiring any boundedness assumptions on the iterates. This work thus further advances the theoretical understanding of PG-type methods in the non-global Lipschitz gradient continuity setting. Finally, we conduct numerical experiments to illustrate the effectiveness of our algorithm and highlight the advantages of integrating extrapolation with a nonmonotone line search.

This paper studies a dyadic version of fractional Sobolev spaces in $\mathbb{R}^n$ for $n\geq 1$. It provides new proofs of the corresponding fractional Sobolev embedding as well as the algebra property of the spaces, which rely solely on dyadic techniques and in particular bypass the Fourier transform. Specific counterexamples are constructed to verify the failure of the algebra property in low-regularity ranges.

We measure the Eddington ratio distribution of local optical narrow-line active galactic nuclei (AGN) as a function of host galaxy properties, as a potential test of galaxy formation theories of AGN feedback. We extract central emission-line fluxes using data from the Mapping Nearby Galaxies at APO (MaNGA) sample of the Sloan Digital Sky Survey IV Data Release 17. Using the line ratio diagnostic techniques of Ji & Yan (2020), we identify AGN galaxies and determine their H$β$ and [OIII] line luminosities. For all galaxies not identified as AGN, we determine the threshold line luminosity they would have needed to be identified as AGN. These luminosity thresholds allow us to account for selection effects that otherwise would lead to strongly biased results. From the H$β$ luminosities and luminosity detection thresholds, accounting for selection effects, we measure the luminosity and Eddington ratio distributions of Seyferts as a function of specific star formation rate (sSFR) and stellar mass. Defining $F_{\rm AGN}$ as the occurrence rate of AGN above a fixed Eddington ratio of $10^{-3}$, we find that $F_{\rm AGN}$ is constant or increasing with stellar mass for star forming galaxies and declines strongly with stellar mass for quiescent galaxies. At stellar masses $\log_{10} M_\ast > 10.25$, the occurrence rate increases monotonically with sSFR. At low statistical significance, in our lowest mass bins $9.25 < \log_{10} M_\ast < 10.25$, $F_{\rm AGN}$ peaks at intermediate sSFR. These patterns reveal a complicated dependence of AGN activity on galaxy properties for theoretical models to explain.

The fidelity estimation between two quantum states is crucial for quantum computation and information science. However, an efficacious method for this, especially for mixed states and higher-dimensional density matrices, remains elusive. While there are many existing algorithms on computing the fidelity between two pure states, there is not much work on how to obtain the fidelity between two mixed states. Here, an efficient quantum algorithm for the fidelity estimation is proposed, based primarily on the density matrix exponentiation and interferometeric scheme for mixed states, with a time complexity of $O(κ^2N^2/ε^7)$, where $N$ is the system size, $κ$ is the condition number of the density matrices and $ε$ is a precision error. This algorithm may serve as a resource-efficient technique to deduce fidelity of any two (pure or mixed) unknown or known quantum states, when the density matrices of the quantum states commute with each other.

Achieving long-term safety in uncertain/extreme environments while accounting for human preferences remains a fundamental challenge for autonomous systems. Existing methods often trade off long-term guarantees for fast real-time control and cannot adapt to variability in human preferences or risk tolerance. To address these limitations, we propose a language-guided adaptive probabilistic safety certificate (PSC) framework that guarantees long-term safety for stochastic systems under environmental uncertainty while accommodating diverse human preferences. The proposed framework integrates natural-language inputs from users and Bayesian estimators of the environment into adaptive safety certificates that explicitly account for user preferences, system dynamics, and quantified uncertainties. Our key technical innovation leverages probabilistic invariance--a generalization of forward invariance to a probability space--to obtain myopic safety conditions with long-term safety guarantees. We validate the framework through numerical simulations of autonomous lane-keeping with human-in-the-loop guidance under uncertain and extreme road conditions, demonstrating enhanced safety-performance trade-offs, adaptability to changing environments, and personalization to different user preferences. Code is available at https://github.com/hoshino06/adaptive_lane_keeping.

We clarify the complexity of answering unions of conjunctive queries over knowledge bases formulated in the description logic $\mathcal S$, the extension of $\mathcal{ALC}$ with transitive roles. Contrary to what existing partial results suggested, we show that the problem is in fact 2ExpTime-complete; hardness already holds in the presence of two transitive roles and for Boolean conjunctive queries. We complement this result by showing that the problem remains in coNExpTime when the input query is rooted or is restricted to use at most one transitive role (but may use arbitrarily many non-transitive roles).

As generative artificial intelligence (GenAI) automates coding tasks and expands access to technical resources, this paper examines how GenAI-enabled coding automation, colloquially known as "vibecoding," affects digital entrepreneurial entry and venture performance. We exploit ex-ante variation in ventures' exposure to vibecoding based on the product characteristics of their initial launches and estimate difference-in-differences models around the diffusion of GenAI coding tools. Vibecoding increases first-time launches and shortens time to launch, but economically viable entry rises only where vibecoding augments, rather than fully automates, product development. In these partially exposed product segments, viable entry increases by 11%, driven entirely by ventures founded by individuals with STEM education or work experience, especially those whose most recent employment was outside middle management. Among ventures launched before GenAI became widely accessible, performance gains similarly concentrate among partially exposed ventures with engineering-intensive initial teams. Together, these results suggest that GenAI-enabled coding automation does not eliminate the value of technical expertise. Instead, vibecoding creates the greatest value when it complements internal engineering capabilities, allowing ventures to delegate lower-level coding tasks to GenAI while shifting human effort toward higher-level problem solving and dynamic adaptation.

Large language models (LLMs) have shown growing potential in software engineering, yet few benchmarks evaluate their ability to repair software during migration across instruction set architectures (ISAs). Cross-ISA migration, such as between x86_64 and aarch64, requires handling complex dependencies, heterogeneous toolchains, and long build logs while ensuring executable verification. To address this challenge, we present Build-bench, an end-to-end benchmark that systematically evaluates the capability of LLMs to repair build failures in cross-ISA settings. Build-bench collects 268 real-world failed packages and integrates auxiliary tools including Structure Extraction, File Content Extraction, Content Modification, and Build Verification to support autonomous, tool-augmented reasoning. The repair process operates in an iterative loop where, upon failure, the model receives updated build logs and previous repair outcomes to refine subsequent attempts. Through a comparative evaluation across the studied models, Build-bench reveals that current models achieve a maximum build success rate of 63.19% and tool usage patterns differ significantly across models. By coupling real build environments with verifiable outcomes, Build-bench establishes the first architecture-aware benchmark for studying LLM-based software build and repair.

Molecular dynamics simulations are an integral tool for studying the atomistic behavior of materials under diverse conditions. However, they can be computationally demanding in wall-clock time, especially for large systems, which limits the time and length scales accessible. Coarse-grained (CG) models reduce computational expense by grouping atoms into simplified representations commonly called beads, but sacrifice atomic detail and introduce mapping noise, complicating the training of machine-learned surrogates. Moreover, because CG models inherently include entropic contributions, they cannot be fit directly to all-atom energies, leaving instantaneous, noisy forces as the only state-specific quantities available for training. Here, we apply a knowledge distillation framework by first training an initial CG neural network potential (the teacher) solely on AA-to-CG mapped forces to denoise those labels, then distill its force and energy predictions to train refined CG models (the student) in both single- and ensemble-training setups while exploring different force and energy target combinations. We validate this framework on a complex molecular fluid, a deep eutectic solvent, by evaluating two-, three-, and many-body properties and compare the CG and all-atom results. Our findings demonstrate that training a student model on ensemble teacher-predicted forces and per-bead energies improves the quality and stability of CG force fields.

Classical software verification and validation techniques, such as procedural audits, formal methods, or model documentation, are the traditional mechanisms used to achieve the verifiable accountability now required by regulations like the EU AI Act. These methods are either expensive or heavily manual, and ill-suited for the opaque, "black box" nature of most Artificial Intelligence (AI) models. A conflict arises: high auditability and verifiability are required by law, but such transparency conflicts with the need to protect the assets being audited (e.g., confidential data and proprietary models). This paper introduces ZKMLOps, an \ac{MLOps} verification framework that operationalizes Zero-Knowledge Proofs (ZKPs) within Machine-Learning Operations lifecycles; a ZKP allows a prover to convince a verifier that a statement is true without revealing any information about the statement itself. By integrating ZKP with established software engineering patterns, ZKMLOps provides a modular and repeatable process for generating verifiable cryptographic evidence-proofs of well-defined computational statements about the audited model and its inputs-that auditors can use as input to a regulatory compliance determination. We evaluate the framework along two dimensions. First, framework viability: orchestration overhead is bounded and stable across architecturally heterogeneous ZKP backends and models of increasing size. Second, cost-versus-assurance trade-offs: the audit-on-demand setting is the regime in which full zero-knowledge auditing is the appropriate tool, where it provides confidentiality and integrity guarantees that lighter-weight alternatives cannot match.

We study the out-of-equilibrium dynamics of entanglement fluctuations in the $ν=1$ Quantum Symmetric Simple Exclusion Process, a free-fermion chain with hopping amplitudes that are stochastic in time but homogeneous in space. Previous work showed that the average entanglement growth after a quantum quench can be explained in terms of pairs of entangled quasiparticles performing random walks, leading to diffusive entanglement spreading. By incorporating the noise-induced statistical correlations between the quasiparticles, we extend this description to the full-time probability distribution of the entanglement entropy. Our generalized quasiparticle picture allows us to compute the average time evolution of a generic function of the reduced density matrix of a subsystem. We also apply our result to the entanglement asymmetry. This allows us to investigate the restoration of particle-number symmetry in the dynamics from initial states with no well-defined particle number. Regarding the possible existence of the quantum Mpemba effect, our analysis indicates that its occurrence is an extremely fine-tuned phenomenon, requiring very specific conditions and therefore being rather difficult to observe in practice.

In this paper, we construct families of polynomials defined by recurrence relations related to mean-zero random walks. We show these families of polynomials can be used to approximate $z^n$ by a polynomial of degree $\sim \sqrt{n}$ in associated radially convex domains in the complex plane. Moreover, we show that the constructed families of polynomials have a useful rapid growth property and a connection to Faber polynomials. Applications to iterative linear algebra are presented, including the development of arbitrary-order dynamic momentum power iteration methods suitable for classes of non-symmetric matrices.

In this paper, we develop new theory connected with resonant vector bundles that will allow for the use of validated numerics to rigorously determine the stability of pulse solutions in the context of the Swift-Hohenberg equation. For many PDEs, the stability of stationary solutions is determined by the absence of point spectra in the open right half of the complex plane. Recently, theoretical developments have allowed one to use objects called conjugate points to detect such unstable eigenvalues for certain linearized operators. Moreover, in certain cases these conjugate points can themselves be detected using validated numerics. The aim of this work is to extend this framework to contexts where the vector bundles, which control the existence of conjugate points, have certain resonances. Such resonances can prevent the use of standard (though involved) techniques in computer assisted proofs, and in this paper we provide a method to overcome this obstacle. Due to its length, the analysis has been divided into two parts: Part I in the present work, and Part II in [BJPS25].