We investigate the quantum dynamics of a levitated nanoparticle in a structured light rotating saddle-like optical potential consisting of a superposition of Gaussian and Laguerre-Gauss modes with detuned frequencies. This rotating saddle trap offers unique opportunities for quantum experiments, such as reduced decoherence due to photon recoil and absorption, the possibility of large delocalization of the particle's center-of-mass motion, particle recovery protocols, the generation of motional entanglement and momentum squeezing. As an application, we show that this saddle-trap architecture enables force detection with sensitivity in the zepto-Newton regime.

We present the discovery of a second coherent radio burst from the thermally emitting neutron star 2XMM J104608.7$-$594306 in our follow-up observations with the Murriyang Ultra-Wideband Low receiver. This burst shows complex morphology with multiple components and wideband emission spanning from 704 to 4032MHz. We measured a steep spectral index of $α=-2.18\pm0.16$. Our polarimetric analysis demonstrates that the burst is highly polarised with a linear and circular polarisation fraction of 54% and 22%, respectively. We identified an orthogonal jump in the polarisation position angles of the burst, resembling those seen in radio pulsars. We compared this burst with the first radio burst detected from the source with MeerKAT. These two bursts detected in a total of 40 hours on source with MeerKAT and Murriyang, combined, show that 2XMM J104608.7$-$594306 can emit sporadic radio emission with luminosity jumps comparable to those seen in the bright bursts from SGR 1935+2154. This suggests that previously thought radio-quiet neutron stars such as X-ray dim isolated neutron stars and central compact objects could exhibit rare radio bursting activity.

We explore the crucial interplay between climate change and power system planning, highlighting the urgent need to systematically integrate climate information into energy system studies. Climate change impacts the energy sector on multiple fronts. Short-term weather variability drives daily and seasonal fluctuations in supply and demand. Long-term trends and increased frequency of extremes pose risks to infrastructure performance, asset lifetimes, and system adequacy. Representing compound events and spatial correlations across borders is a complex challenge, and uncertainties persist due to uncertainties from different models, scenarios, and downscaling methodologies. The Pan-European Climate Database (PECD4.2), developed in partnership between ENTSO-E and C3S, marks a change in how energy system planning is conducted. The PECD4.2 integrates historical reanalysis and six climate models across four SSP's, providing harmonised, openly available datasets tailored for power system studies. The physical conversion models for wind and solar energy better reflect technological progression than machine learning methods trained on historical data, improving robustness under changing future conditions. Despite these advances, challenges remain. Particularly in hydropower modelling and the lack of public harmonised energy datasets that are required to train these models. Complex processing chains from raw climate data to actionable insights and the lack of standardized integration of climate information lengthen lead times for energy-sector adoption. This leads to diverging approaches and variable consideration of climate risks. Closer, more generalised collaboration and communication between climate service providers and energy stakeholders are therefore necessary, as are the development of user-friendly tools for data manipulation and analysis and robust feedback loops.

Coating thermal noise in high-reflectivity test-mass mirrors is a major limitation for future gravitational-wave detectors, especially in the 10--300 Hz band. ET-Pathfinder therefore requires mirror coatings that combine very high reflectance at 1.55 micrometer with low thermal noise under cryogenic conditions. Conventional dielectric Bragg mirrors provide high reflectance but require thick coatings, whereas metasurface mirrors can reduce coating-related noise but are limited by fabrication tolerances and line-edge roughness. We present a hybrid metasurface--Bragg mirror concept tailored to ET-Pathfinder. The design combines a fabrication-tolerant one-layer metasurface, an anti-resonant Fabry--Perot spacer, and a reduced dielectric Bragg stack. Optical performance is evaluated using full-wave electromagnetic simulations, while fabrication robustness is assessed with a truncated-Gaussian Monte Carlo analysis. Line-edge roughness is included as a systematic edge-smoothing effect. The resulting reflectance distributions are used to determine the minimum Bragg-stack support required to meet system-level specifications. The ideal metasurface exceeds 99.999% reflectance. When fabrication uncertainties and line-edge roughness are included, the metasurface reflectance is limited to about 99.9% at the 95% yield level. The remaining transmission can be compensated by a supporting Bragg stack with as few as seven layer pairs. For this configuration, the hybrid mirror achieves a total thermal displacement noise about one order of magnitude below the projected ET-Pathfinder coating-noise budget. These results show that fabrication-limited metasurface reflectance can be compensated within a hybrid architecture, enabling reduced coating thickness and thermal noise for next-generation gravitational-wave detectors.

We provide an explicit description of the smallest filtered sub-isocrystal generated by the Hodge filtered piece of the crystalline cohomology for an abelian scheme over a $p$-adic ring. Our method is based on the theory of arithmetic jet spaces and delta characters associated to the abelian scheme, introduced by Buium and later studied by Borger and Saha using a functor of points approach. In particular, we prove that the delta isocrystal constructed by Borger and Saha is indeed isomorphic to the fundamental smallest sub-isocrystal of the crystalline cohomology in the category of filtered $F$-isocrystals. As an application, we establish a comparison isomorphism between the delta isocrystal and the crystalline cohomology of abelian schemes, which is governed by the group of order $1$ delta characters of the abelian scheme.

The paper discuss the limit point concept of a subset in a group via ideal of the power set ring. This idea along with anti-ideal give the topological structure in a group. Homomorphic images of both ideal and anti-ideal are played the remarkable role to change the topological structure from one system to another system.

We study decentralized optimization over networks where agents cooperatively minimize a smooth (strongly) convex sum of local losses while communicating only with immediate neighbors. Prevailing decentralized methods require either centralized knowledge of global problem and network parameters for stepsize tuning--hence impractical, or costly per-iteration line-searches that demand access to local function values. We propose line-search-free, fully decentralized algorithms in which each agent adapts its stepsize using only past local iterates and gradients--with no extra function evaluations and no global tuning. The key technical ingredient is a new Lyapunov function, from which a natural adaptive stepsize rule emerges: at each iteration, each agent selects the largest stepsize that guarantees descent, based solely on a local curvature estimate built from successive gradients. The proposed algorithms enjoy strong theoretical guarantees: sublinear convergence rates for merely convex objectives and linear rates under strong convexity. Numerical experiments on standard benchmarks show consistent improvements over the state of the art, both adaptive and non-adaptive.

We present a near-linear-time algorithm that, given a bridgeless cubic graph, finds a perfect matching intersecting every 3-edge-cut in exactly one edge. This improves over a cubic algorithm of Boyd et al. for the same problem, and over our previous algorithm, which worked only for 3-edge-connected graphs. The main ingredient is a cactus representation of the 2-edge-cuts, together with an efficient update procedure under 2-cut reductions.

This paper develops bootstrap procedures for inference in linear regression models with two-way clustered data. We characterize the estimator's asymptotic behavior in five mutually exclusive and exhaustive regimes: three Gaussian and two non-Gaussian. We establish four impossibility results: heterogeneous score components preclude uniform consistency; uniform consistency also fails in one non-Gaussian (infeasible) regime; the infeasible regime is not uniformly distinguishable from a feasible one; and uniform validity over all feasible regimes rules out uniform conservativeness over the infeasible regime. To address the feasible regimes, we propose a data-driven regime classifier and a projection-based wild bootstrap procedure. The procedure delivers uniformly valid inference across the four feasible regimes while allowing serial dependence along the second clustering dimension and spatial dependence along the first. This combination of regime adaptivity and flexible dependence is new to the two-way clustering literature. Monte Carlo simulations confirm the accuracy and flexibility of the proposed methods in settings with complex clustering structures.

We develop a toric topological framework for studying the cohomology of Vietoris--Rips complexes $VR(Q_n;r)$ of hypercube graphs. Using total domination invariants and spectral methods, we establish general lower bounds on connectivity, which leads to infinite families of counterexamples to Shukla's conjecture, and derive first global upper bounds on coconnectivity. Our approach interprets Vietoris--Rips complexes via Stanley--Reisner rings, moment-angle complexes, and Tor algebras, allowing global topological information to be extracted from combinatorial data. In a second direction, we construct explicit cohomology classes using the Koszul resolution and show that they decomposable products of $1$-dimensional classes, and that their representatives can be combimbinatorially realised as the boundary of cross polytopes positively answering the question posed by Adams and Virk. We introduce ghost vertices as a new tool for detecting, extending, and proving linear independence of cohomology classes.

TrueEBSD is an open-source MATLAB program for image alignment and spatial distortion correction of images and electron backscatter diffraction (EBSD) maps. We have re-implemented TrueEBSD as an add-on to MTEX, an established toolbox for EBSD data analysis. Spatial alignment enables correlative analysis methods, such as augmenting EBSD orientation maps with data from other imaging modes. The augmented EBSD maps can then be analysed further using MTEX. We demonstrate TrueEBSD on two example case studies: one for measuring Co phase fraction and WC contiguity in a WC-Co composite, and another for determining the relative susceptibility of grain boundaries to void formation in a copper polycrystal. In both examples, the EBSD map was augmented with scanning electron microscopy (SEM) image data. This enabled quantitative crystallographic measurements which would not be possible from analysing the EBSD maps and images separately.

An improvement of a {\em Global (strong) Gagliardo-Nienberg inequality with a BMO term} is established by replacing local derivatives by {\em and fractional Laplacians.} Local versions are also given.

The surface characterization of rocky exoplanets via emission spectroscopy represents a frontier of current (JWST) and future (HWO) observational efforts. Here, we implement new features in the open-source retrieval code $\texttt{POSEIDON (v1.4)}$ to fully account for an emitting and reflecting planetary surface and an overlying absorbing and scattering atmosphere. We show that realistic rocky surfaces (with wavelength-dependent albedos derived from laboratory measurements) affect emission spectra by imparting mid-infrared diagnostic absorption features, imprinting pseudo-features due to atmospheric transparency windows, and flipping absorption features to emission via surface-atmosphere interface pseudo-temperature inversions. We demonstrate that current JWST spectral data can distinguish between tenuous (low surface pressure, $\leq$ 1 bar) and thick (high surface pressures, $\geq$ 0.1 bar) atmospheres by performing atmosphere + surface retrievals on published JWST emission data of the rocky worlds TOI-1685b and 55 Cancri e. We then explore JWST MIRI LRS's capability to constrain surface geology of rocky worlds, finding that with sufficient SNR retrievals can distinguish between granite-like and basaltic surfaces for synthetic datasets. Finally, we provide an open-source database of lab-derived surface albedos (in the form of directional-hemispherical reflectances), organized by geologic classification and include supplemental tables developed to foster future collaboration between geology and exoplanet science. Our atmosphere + surface retrieval technique provides a pathway to probe geologic processes on rocky exoplanets, showing that upcoming JWST data for terrestrial worlds will enable a deeper exploration of rocky surfaces beyond our Solar System.

We prove an analogue of Morley's categoricity theorem where cardinality is replaced by the recursion-theoretic notion of arithmetic degree. We say that a complete arithmetically definable theory $T$ is $D$-categorical if any two arithmetically extendible models of $T$ of arithmetic degree $D$, considered over a common elementary submodel with arithmetical elementary diagram, are isomorphic over that submodel by an isomorphism which preserves the complexity of sets of degree $D$. Here an arithmetically extendible model means an elementary substructure of a model whose elementary diagram is arithmetical. Our main result is: If $T$ is $D_1$-categorical for some nonzero arithmetic degree $D_1$, then $T$ is $D_2$-categorical for every nonzero arithmetic degree $D_2$. We also show that, assuming ZFC, $D$-categoricity for some nonzero arithmetic degree is equivalent to uncountable categoricity.

Let $d(n)$ denote the number of divisors of a positive integer $n$. A classical problem in analytic number theory is given by the asymptotic behavior of the divisor sum $\sum_{n \leq x} \frac{1}{d(n)}$, with Ramanujan having introduced an asymptotic formula for this sum with an explicit evaluation for the constant $A_1$ for the leading term $A_1 \frac{x}{\sqrt{\log x}}$. Gabdullin et al. recently considered a hybrid of this problem and the Titchmarsh divisor problem concerning $\sum_{p\leq x} d(p-1)$, proving that $$\sum_{p\leq x} \frac{1}{d(p-1)} \asymp \frac{x}{(\log x)^{3/2}}.$$ This result, together with Erdős's asymptotic formula $\sum_{n \leq x} d(d(n)) \sim c \, x \log \log x $ for a constant $c \in (0, \infty)$, lead us to consider the hybrid $\sum_{n \leq x} \frac{1}{d(d(n))}$ of the Erdős and Ramanujan divisor sums. The presence of the reciprocal significantly complicates the analysis, as it amplifies the contribution of integers for which $d(d(n))$ is exceptionally small. In this paper, we prove that $$\sum_{n \leq x} \frac{1}{d(d(n))} \asymp \frac{x}{ \log \log x}, $$ through a combined application of Golomb's estimate for powerful numbers and Turán's quantitative form of the Hardy-Ramanujan theorem.

We establish the first general regularity result for constrained optimal control problems arising naturally in mathematical physics and mathematical biology. Namely, we prove that for a large class of problems of the form ``maximise $\int ψ(Θ_m)-c\int m$ where $-ΔΘ_m=mΘ_m+B(x,Θ_m)$, under the constraint $0\leq m\leq 1$ a.e.", the solution $m^*$ is bang-bang, in the sense that $m^*=χ_{E^*}$, and that $\partial E^*$ is smooth up to a $(d-2)$-dimensional subset. Moreover, we prove that the solutions to the volume constrained problem ``maximise $\int ψ(Θ_m)$ where $-ΔΘ_m=mΘ_m+B(x,Θ_m)$, under the constraint $0\leq m\leq 1$ a.e and $\int m=m_0$" are bang-bang in the sense that $m^*=χ_{E^*}$ and that, in the two-dimensional case, $\partial E^*$ is a finite union of smooth curves. This is done via reduction to an unstable free boundary problem, the regularity analysis of which was pioneered by Monneau \& Weiss and Chanillo, Kenig \& To. In our case, the free boundary is not minimising, and the laplacian of the state function is sign-changing, which creates significant difficulties, in particular regarding the non-degeneracy of blow-ups. This requires a new approach blending tools from optimal control theory, free boundary and measure theory to establish the regularity of the free boundary.

The foundation for modeling the coupling of the internal structure of compact objects in binary systems to their dynamics and emitted gravitational waves is a systematic effective field theory (EFT) framework, where each compact object is replaced by a worldline endowed with a set of internal degrees of freedom. These degrees of freedom encode finite-size effects and thereby distinguish between different classes of compact objects. Among finite-size effects, tidal interactions play a central role, as they are associated to various kinds of deformations of a body under the influence of external tidal fields. In this work, we analyze the dynamical tidal response of Kerr black holes to generic-spin perturbations, focusing primarily on the scalar and gravitational cases, and working to linear order in frequency. We establish an EFT description of the perturbed black hole that accounts for the couplings between the spin, gravitoelectric and -magnetic tidal fields. We match this to wave-like solutions to the full black hole perturbation equations in order to determine the tidal response coefficients. In particular, we obtain the dynamical Love number, which appears at linear order in frequency for spinning black holes, and derive an approximate expression for the dynamical tidal response, including both dissipative and conservative pieces. We also examine several technical subtleties that arise in the matching procedure, with special emphasis on the mixing of multipolar modes induced by the spin of the compact object, which proves to be essential for a consistent treatment.

This study investigates the properties and stability of the Multiplicative Kantian Equilibrium (MKE) in symmetric games. We first demonstrate that MKE lacks strategic equivalence: the Kantian best-response function is not invariant under monotonic strategy rescaling. This strategic non-equivalence implies that the choice of measurement scale - a subjective interpretation of the game - materially impacts equilibrium outcomes. Exploiting this non-equivalence, in a game where players may be Kantian or Nasher, we propose an efficient strategy rescaling that allows Kantians to neutralize the free-rider advantage of Nashers, while preserving Pareto-efficient outcomes among themselves. In a dynamic framework, we show that the subgame-perfect Nash equilibrium with endogenous choice of optimization type leads all players to prefer Kantian optimization over Nash optimization. In an evolutionary setup, we show that Kantian optimization is an evolutionarily stable strategy (ESS). Our results suggest that the inherent strategic non-equivalence of Kantian optimization provides a robust pathway to stable cooperation.

Distributed blackbox consensus optimization is a fundamental problem in multi-agent systems, where agents must improve a global objective using only local objective queries and limited neighbor communication. Existing methods largely rely on handcrafted update rules and static cooperation patterns, which often struggle to balance local adaptation, global coordination, and communication efficiency in heterogeneous nonconvex environments. In this paper, we take an initial step toward trajectory-driven self-design for distributed black-box consensus optimization. We first redesign the agent-level swarm dynamics with an adaptive internal mechanism tailored to decentralized consensus settings, improving the balance between exploration, convergence, and local escape. Built on top of this adaptive execution layer, we propose Learning to Act and Cooperate (LACMAS), a trajectorydriven framework in which large language models provide sparse highlevel guidance for shaping both agentinternal action behaviors and agentexternal cooperation patterns from historical optimization trajectories. We further introduce a phased cognitive scheduling strategy to activate different forms of adaptation in a resource-aware manner. Experiments on standard distributed black-box benchmarks and real-world distributed tasks show that LAC-MAS consistently improves solution quality, convergence efficiency, and communication efficiency over strong baselines, suggesting a practical route from handcrafted distributed coordination toward self-designing multi-agent optimization systems.

This paper is concerned with portfolio selection for an investor with exponential, power, and logarithmic utility in multi-asset financial markets allowing jumps. We investigate the classical Merton's portfolio optimization problem in a Volterra stochastic environment described by a multivariate Volterra--Heston model with jumps driven by an independent Poisson random measure. Owing to the non-Markovian and non-semimartingale nature of the model, classical stochastic control techniques are not directly applicable. Instead, the problem is tackled using the martingale optimality principle by constructing a family of supermartingale processes characterized via solutions to an original Riccati backward stochastic differential equation with jumps (Riccati BSDEJ).The resulting optimal strategies for Merton's problems are derived in semi-closed form depending on the solutions to time-dependent multivariate Riccati-Volterra equations, while the optimal value is expressed using the solution to this original Riccati BSDEJ. Numerical experiments on a two-dimensional rough Heston model illustrate the impact of both path roughness and jumps components on the value function and optimal strategies in the Merton problem.

A kinematic analysis of the young stellar association TWHya has been performed. The components of the displacement matrix in the Ogorodnikov-Milne linear model have been estimated both graphically and by solving the basic kinematic equations. The association's volume expansion with a coefficient of $K_{xyz}=103\pm9$ km s$^{-1}$ kpc$^{-1}$ was confirmed, which yields a dynamical age estimate of $t=9.7 \pm0.8$ Myr. Using the graphical method, estimates of the association's proper rigid-body rotation parameters $ω$ around the galactic axes x and y have been obtained for the first time, with velocity values in the range of 50-70 km s$^{-1}$ kpc$^{-1}$ and errors in their determination of 14-19 km s$^{-1}$ kpc$^{-1}$. However, these values are not confirmed by another method. For example, when solving kinematic equations only using proper motions, all three components of rigid body rotation do not differ significantly from zero, $(ω_x,ω_y,ω_z)=(4,7,11)\pm(5,5,5)$ km s$^{-1}$ kpc$^{-1}$.

Recent megakernel designs for Mixture-of-Experts (MoE) inference fuse expert computation with fine-grained, GPU-initiated communication into a single persistent GPU kernel, and outperform collective-based MoE on a single node by overlapping data transfer with compute at tile granularity. This benefit does not carry over cleanly to multi-node inference, where experts span many nodes connected by an RDMA fabric. Communication-bound MoE models regress by up to $10\times$ on 8 nodes, and the regression worsens with node count. We trace this regression to hidden serialization in proxy-based RDMA transports. The ordering requirement between each tile transfer and its completion signal forces a fence that drains the NIC pipeline, and its cost grows with the number of concurrent transfers. As a result, models whose per-expert compute is too small to absorb this inflated network latency expose communication on the critical path. We present \emph{Perseus}, which eliminates this serialization through two techniques. \emph{Decoupled signaling} batches fences at per-destination granularity, reducing fence count by $8\times$. \emph{NIC-side ordering} replaces proxy stalls with hardware fence flags, so the proxy never blocks. On proxy-based transports, Perseus achieves up to 10.3$\times$ end-to-end speedup. Perseus on IBRC matches or exceeds IBGDA GPU-direct by up to 1.2$\times$, which shows that serialization, rather than the choice between proxy-based and GPU-direct transport, is what bounds multi-node megakernel performance.

The color structure of two-loop gluon amplitudes is examined both from a color trace basis expansion and an alternative based upon structure constants. We use use this as a vehicle for systemising relations between the partial amplitudes of the color trace formalism.

We develop a quantitative mathematical theory that offers new perspectives on nonlinear harmonic generation in plasmonic structures arising from symmetry breaking. Focusing on second harmonic generation--the most fundamental process and the most extensively studied owing to its practical significance--we establish a theoretical framework that can be readily extended to higher-order harmonics. We investigate the plasmonic system in the static regime using a columnar nanowire with \(n\)-fold rotational symmetry (\(n \in \mathbb{N}\)) and construct a phenomenological model in which the second harmonic response originates from nonlinear sources confined to a selvedge region near the surface. By introducing a notion of symmetry degree grounded in group theory, we precisely quantify the second harmonic generation in terms of multipolar contributions. Our theory complements existing physical descriptions of this practically important phenomenon and provides a rigorous account of how nonlinear optical efficiency depends on shape, size, symmetry, and defects in plasmonic structures.

The accurate estimation of observables is a crucial task in quantum computing. Recent advances have highlighted the need for (a) specialized protocols for qudit-based devices, that include (b) error-aware strategies. Here, we present AQUIRE, the first protocol that can (a) accurately estimate both the mean and the error of an observable on qudit-based quantum computers. AQUIRE achieves this by constructing a Bayesian model to accommodate generalized Pauli operators. It is designed to continuously monitor the estimated average and the associated error of the observable, adjusting the subsequent measurements in real-time. Additionally, AQUIRE is (b) device- and experiment-specific error-aware, and accounts for hardware imperfections and experimental noise during the estimation process. We demonstrate AQUIRE's advantage via numerical simulations and showcase its ability to quantify the noise affecting the estimation by implementing it on a trapped-ion qudit quantum processor. By exploiting general commutation relations and overlap grouping measurements, our protocol is state-of-the-art when restricted to qubit-based quantum computers and extends this advantage to the qudit case.

Accurately forecasting the bursty and non-stationary power demand of AI data centers has become increasingly important, as abrupt workload-driven variations at the GPU-node level can affect real-time operational efficiency, power management, and grid-data center coordination. However, high-capacity forecasting models are often difficult to deploy at scale because of their memory and latency requirements, while lightweight predictors may fail to capture short-horizon temporal dynamics. To address this accuracy-deployment tradeoff, this paper proposes a deployment-efficient knowledge distillation framework for short-term load forecasting in AI data centers. The proposed framework first trains a high-capacity sequence teacher model for multi-step load trajectory prediction, where residual learning is used to improve robustness under non-stationary operating conditions. A lightweight point-wise student model is then developed for low-latency rolling inference using a compact neural network architecture. To transfer temporal knowledge from the teacher to the student, a sequence-to-point distillation strategy is introduced by aligning near-term predictive behavior and temporally pooled representations. Case studies on the MIT Supercloud dataset demonstrate that the proposed student model improves forecasting accuracy over recent deep learning baselines while reducing the deployment footprint by over 10x in parameter memory and model size.

We prove the Riemannian Penrose inequality in arbitrary dimension for smooth complete asymptotically flat manifolds with nonnegative scalar curvature and compact outer-minimizing minimal boundary, where the boundary is allowed to have a singular set of Hausdorff dimension at most \(n-8\). Moreover, the equality holds exactly when the manifold is isometric to the Riemannian Schwarzschild exteriors. Our proof extends Bray's conformal-flow method to higher dimensions, where the outer-minimizing enclosures along the flow may be singular.

We study the matter and entropy transport between two ultra-cold neutral Fermi-gas reservoirs linked by a quantum point contact under a chemical-potential gradient. We describe the two leads with a BCS mean-field model and derive the current-bias characteristics for both particle and entropy transport. We compute the out of equilibrium steady-state currents by using the Keldysh formalism. In accordance with previous works in the literature, we confirm the well-known behavior for the particle current and extend the computation to the entropy current in the BCS regime. The entropy current shows an oscillatory behavior at low voltage in the ballistic junction limit. We analyze the results for a wide range of values of the junction's transparency. We also compare our findings with experimental results in cold atomic gases in the unitary regime.

Databases, and datasets more generally, evolve continuously through updates, transformations, versioning, schema changes, streaming operations, and other mechanisms. While prior work has noted connections among some of these areas, they have traditionally been studied in isolation, each with its own abstractions, algorithms, and system implementations. In this paper, we argue for unifying these diverse functionalities under a single abstraction and a common set of computational primitives. We present such an abstraction, powerful enough to encompass existing use cases and to support new ones. Going beyond previous approaches, our framework seamlessly integrates provenance tracking for system-visible operations, conditional propagation of updates, and configurable alerts on change events. It also offers a principled treatment of dependent objects such as views and derived artifacts like machine learning models, by providing declarative mechanisms to control their evolution. Finally, we sketch a prototype implementation in a relational-like database system based on an adaptation of the "Prolly Tree", a Merkle tree-inspired data structure with tunable parameters to meet varying performance requirements, and present some initial experimental results.

We revisit Beukers' modular-form proof of the irrationality of $ζ(3)$ from the point of view of the auxiliary weight two modular form. For the Fricke group $Γ_0(6)^\star$, we show that Beukers' choice is not isolated: it belongs to a one-parameter affine family. These approximations have the same exponential decay as the classical Apéry approximations and satisfy the same denominator-growth estimate needed in Beukers' irrationality argument. We then apply the same construction to several other genus-zero Fricke groups.