The logistic two-gene negative-feedback oscillator is locally asymptotically stable for all biological parameter values, since the trace of the Jacobian is uniformly negative. Real biological oscillators (circadian rhythms, the segmentation clock, Hes1, p53) nevertheless rely on delays. We extend the logistic two-gene model to a delay-differential system with transcriptional delays $τ_1$ and $τ_2$, and prove that the equilibrium loses stability through a Hopf bifurcation as the total delay $τ=τ_1+τ_2$ crosses an explicit critical value $τ_c$. The Hopf frequency $ω_c$ and $τ_c$ are computed in closed form from the logistic derivatives; the loop-gain condition $AB>γ_1γ_2$ is necessary and sufficient; the transversality $\mathrm{Re}(dμ/dτ)|_{τ_c}>0$ admits a parameter-uniform positive lower bound; and the bifurcation persists globally. A sum-of-delays symmetry reduces the analysis to the scalar parameter $τ$. Numerical simulations confirm three regimes (damped, small limit cycle, relaxation), the supercritical amplitude scaling $A\sim c\sqrt{τ-τ_c}$, and the deep-relaxation period asymptote $T\sim 2τ+C_\infty$ with closed-form offset $C_\infty$. For the symmetric-threshold loop, supercriticality is proved by a Lindstedt--Poincaré reduction yielding closed-form amplitude and frequency laws; for the general asymmetric loop it delivers a closed-form first Lyapunov coefficient and an explicit criticality criterion. Calibrated to p53--Mdm2 data, the closed-form Hopf period matches the observed oscillation within $3\%$, and the standard Hill-function model within a few percent. The analysis extends to cyclic $N$-gene loops, with a closed-form transversality rate valid for every $N$ and -- in the symmetric case -- an explicit delay-induced-Hopf window $γ^N<Λ<γ^N\sec^N(π/N)$.

In this work, we develop a constructive method for deriving four structure relations and a fourth-order linear differential equation satisfied by Laguerre-Hahn orthogonal polynomial sequences. The method relies on a combination of structure relations, their successive derivatives, and algebraic elimination techniques. Particular attention is given to semiclassical and classical families, which are recovered as special cases within this general framework. The approach is systematized in the form of an algorithm. Using symbolic computations, we obtain explicit new results for Laguerre-Hahn polynomials of class zero, analogous to the Hermite case. In addition, we present results for a semiclassical example of class 1.

We study geodesics in the Brochette first-passage percolation model, where edges on the same axis-parallel line share a common random passage time, inducing long-range dependence. We focus on the maximal transversal deviation H n of geodesics from the origin to ne 1 . We prove existence of geodesics under mild assumptions and establish the order of magnitude of H n depending on the behavior of the passage-time distribution near its infimum. These results yield explicit wandering exponents in this dependent setting.

In this work, we calculate the transition form factors for the weak decays $Λ_b^0 \to Λ_c(2860)^+$ and $Λ_b^0 \to Λ_c(2625)^+$ using QCD light-cone sum rules, and compute the branching fractions of the corresponding semileptonic decays $Λ_b^0 \to Λ_c(2860)^+ \ell^- \barν_\ell$ and $Λ_b^0 \to Λ_c(2625)^+ \ell^- \barν_\ell$. Our predicted branching fraction for $Λ_b^0 \to Λ_c(2625)^+ \ell^- \barν_\ell$ is consistent with experimental data and other theoretical predictions, validating the reliability of our method. On this basis, we also present the branching fraction of $Λ_b^0 \to Λ_c(2860)^+ \ell^- \barν_\ell$. These results may serve as a theoretical reference for future experimental measurements of this decay channel.

The CH4rLiE (CH4 Livestock Emission) project investigates the technical feasibility of adapting gas recovery systems from high-energy physics to mitigate methane (CH4) emissions in livestock housing. This work presents a proof-of-principle based on the adaptation of CERN's gas recuperation systems for the capture of CH4 at low concentrations. A laboratory-scale prototype was developed to evaluate the performance of various adsorbent materials under realistic conditions, including multi-stage humidity removal and pressurized gas flows. Experimental results obtained with the prototype led to the selection of commercial Z5 zeolite as the primary adsorbent due to its high adsorption capacity and stable regeneration performance through Vacuum Swing Adsorption cycles. The study demonstrates the feasibility of CH4 capture at concentrations down to 0.1%. Furthermore, it was observed that increasing the CH4 partial pressure enhances the adsorption capacity, with tests conducted up to approximately 5 bar. To bridge the gap between laboratory conditions and the representative 10-100 ppm levels found in dairy barn environments, a negative exponential extrapolation was applied to the experimental data. This allowed for the modeling of the adsorption behavior in the ultra-low concentration regime. These results validate the operational principle and provide the necessary parameters for the design of a full-scale system for field installation.

We present a physically consistent multiport framework for stacked intelligent metasurfaces (SIMs) with linear and explicit nonlinear terminations. The model provides closed-form input--output relations in the linear case and fixed-point forward evaluation in the nonlinear case, with adjoint-based gradients for optimization in both settings. Under stage-isolated SIM structure, complexity remains $\mathcal{O}(QK^3)$. In a 28 GHz near-field localization case study, nonlinear terminations improve transfer-function matching and reduce mean localization error, close to the ideal benchmark.

This Letter deals with the gravitational angular momentum carried by axial (odd-parity) perturbations of the Bardeen regular black hole within the teleparallel equivalent of general relativity (TEGR). Using the Hamiltonian definition of conserved quantities in TEGR, we derive a closed expression for the perturbative angular momentum $δJ$ in terms of the axial perturbation function $h_0(r,t)$. The result exhibits a sharp multipolar selection rule: $δJ$ vanishes for odd values of the multipole index $\ell$, while even-$\ell$ modes yield a nonzero contribution. The radial and temporal behavior of $δJ$ is illustrated using the known axial quasinormal modes of the Bardeen spacetime.

We present in this work a very short proof for the existence, uniqueness and smoothness in dimensions $d\leq 3$ of the system of reaction diffusion $ \partial\_t a\_i - d\_i Δa\_i = (-1)^i (a\_1 a\_3 - a\_2 a\_4)$, where $a\_i \geq 0$ model the concentrations of chemical species undergoing a chemical reaction and diffusing (each with its diffusion rate $d\_i > 0$) in a bounded container.

Online services strive to maintain application responsiveness even when the traffic is unpredictable and fluctuating. Today's online services are commonly deployed as chains of microservices, each microservice packaged as one or more containers inside virtual machines (VMs). While performant and affordable when the load is steady, VM-based deployments are known to be slow to scale when the load spikes, resulting in degraded performance for end-users of the service. To avoid such performance degradations, service providers can over-provision their deployments; however, such a strategy is costly and inefficient, leaving resources under-utilized for extended periods. To address the challenge of unpredictable load spikes, we propose Flare, a hybrid microservice architecture that combines VMs with serverless computing. Flare utilizes VMs to cost-effectively handle steady workloads and leverages serverless elasticity to absorb traffic spikes. When a spike occurs, Flare detects which specific service(s) are overloaded and shifts the excess load of only those services to serverless, thus minimizing the cost overhead. Flare seamlessly integrates into existing auto-scaling and serverless infrastructure, requiring minimal changes to the control plane and no modifications to the application.

Online advertising platforms host hundreds of thousands of A/B tests, but the platform's delivery algorithm routes each creative to the audience it predicts will engage. Every two-arm test therefore conflates the creative's effect with the algorithm's targeting response, and adjusting for the realized audience is biased because audience is a post-treatment mediator. We propose a three-arm design that adds an arm exposing the algorithm to the treatment metadata while holding the user-facing creative identical to control, point-identifying the natural indirect (algorithmic) and direct (creative) effects without sequential ignorability. In a live Meta campaign with a women-targeted text fragment, the algorithmic channel raises female impression share by +2.07 ppt while the creative channel moves it by -0.68 ppt; roughly three-quarters of the absolute reallocation is algorithmic, and a conventional two-arm test understates the algorithmic channel by a factor of two. The design isolates the contribution of platform's algorithm to the outcome which is separable from creative content.

The Game Description Language (GDL) is a widely used formalism for specifying general games. Due to their similar syntax and semantics, Answer Set Programming (ASP) and its extensions have been applied to single- and two-player deterministic turn-taking GDL games. This paper presents the first ASP-based approach for solving two-player turn-taking GDL games with uncertainty. We introduce Stochastic An- swer Set Programming (SQASP) to encode the maximally achievable winning probability for a given player in stochastic GDL games, and develop a translation-based solver that evaluates SQASP programs by converting them to Extended Stochastic Satisfiability. Our empirical results show that the proposed approach is competitive with forward search on small stochastic games and can potentially support general game players in endgame evaluation.

We classify connected finite acyclic graded quivers $Q$ for which the graded path algebra $kQ$, regarded as a formal dg algebra, is silting-discrete. We prove that $kQ$ is silting-discrete if and only if it is derived-discrete, and that both conditions are equivalent to the underlying graph of $Q$ being of type ADE, or of type $\widetilde{A}$ with unequal clockwise and counter-clockwise total degrees. The key ingredient is an explicit construction of an infinite pre-simple-minded collection in $\\text{pvd } kQ$ in the non-discrete case.

We study consumer demand in large-scale retail settings with many products, multiple categories and repeated purchase behavior. While inertia and brand loyalty are well documented, existing discrete choice models typically focus on single categories or become computationally infeasible in high-dimensional environments. We propose a dynamic product-level factor model that captures heterogeneity in baseline preferences, price sensitivity and inertia through a shared latent factor structure. By factorizing individual-product coefficients, the model pools information across individuals and categories and allows for correlated heterogeneity. We estimate the model using Bayesian variational inference, enabling scalable estimation with tens of thousands of parameters. In a simulation study calibrated to realistic retail data, we show that the dynamic factor model substantially improves predictive performance relative to static factor models and mixed logit benchmarks, particularly when individual purchase histories are sparse. Accounting for inertia also leads to more elastic demand estimates, underscoring the importance of dynamics for measuring consumer responsiveness. Our results highlight dynamic factor models as a scalable and flexible approach for demand estimation in modern, high-dimensional retail markets.

Personalized discovery systems often train separate models for item ranking, carousel ranking, and search, even though these tasks expose complementary signals from the same viewer journey: watches shape carousel and item ranking, search queries reveal intent even when they do not lead to a catalog match, and watch history helps interpret search as rewatching, continuation, or new discovery. We introduce the user story, a serialized representation that turns a user's cross-surface history - attributes, sessions, watch events with surface and carousel context, and search events - into a single token sequence. By interleaving pretrained language tokens with domain-specific event tokens, user stories let heterogeneous recommendation and search tasks be expressed as prompted next-token prediction over a shared grammar. TubiFM is one instantiation of this approach: a Llama 3.2 1B-based model trained on user stories and prompted to rank items, carousels, or search results without task-specific architectures. In offline evaluation, this single model outperforms specialist baselines across item, carousel, and search ranking. In online A/B tests, TubiFM significantly improves search total viewing time (TVT) by $+3.9\%$ and carousel TVT by $+0.30\%$. Item ranking is statistically neutral on TVT ($+0.14\%$), but matches a mature production stack; across all three tasks, TubiFM serves on L40S GPUs and reduces p99 ranking latency from 500ms to 200ms. These results show that shared user stories can improve discovery while simplifying ranking systems.

Context. The SPHERE High-contrast Imaging survey for Exoplanets (SHINE) represents one of the largest direct imaging campaigns, targeting over 400 young, nearby stars with the goal of detecting and characterizing giant exoplanets and brown dwarfs. This dataset offers a unique opportunity to revisit observations using modern, data-driven approaches, potentially uncovering new substellar candidates that may have been overlooked by classical analysis techniques. Aims. Our study focuses on reprocessing and reanalyzing the so-called F150 sample, a well-defined subset of 150 main-sequence stars within 100 pc observed in the H-band with VLT/SPHERE as part of the SHINE survey. Methods. We apply NA-SODINN, a supervised deep learning model specifically tailored for detecting faint planetary signals in angular differential imaging (ADI) sequences. Designed to model local noise properties and capture spatial context, NA-SODINN is effective at distinguishing real companions from residual speckle noise. To translate the model's pixel-wise confidence maps into actionable detections, we introduce a novel F1-score-based thresholding strategy. This principled approach balances sensitivity and specificity, addressing a key limitation in current deep learning-based methods. Results. NA-SODINN recovers all known companions and some of the debris disks in the F150 sample, and identifies 13 new substellar candidates not reported in previous studies: ten detected in both the H2 and H3 bands, and three in only one band. For the ten sources detected in both bands, we use the H2-H3 color-magnitude diagram to perform a first assessment of their nature. Based on this analysis, we identify two ambiguous cases and three photometrically promising candidates. However, in light of the currently available multi-epoch SPHERE data, only the candidate around Smethells 20 remains a strong target for follow-up.

We prove norm inflation phenomena for KdV and KP equations in negative order Sobolev spaces, in the periodic case, as well as on the whole space, on an arbitrarily large scale of negative order Sobolev spaces as target spaces. The proof relies on WKB analysis for a semiclassical version of the equation, in a weakly nonlinear r{é}gime, and the creation of the zero Fourier mode by resonant interaction. Unlike in previous similar results, this average mode has a smaller order of magnitude than the initial data, which requires a more detailed WKB analysis.

Quantum-selected configuration interaction (QSCI) is a promising hybrid quantum-classical approach in which a quantum device generates configurations for subsequent classical diagonalization. Here, we analyze the performance of QSCI combined with the local unitary cluster Jastrow (LUCJ) ansatz, focusing on the interplay between ansatz expressivity, sampling, noise, and configuration recovery. Using the dissociation of N2 in a large active space as a benchmark, we show that noiseless LUCJ sampling produces compact and biased configurational spaces, limiting the accuracy of the resulting CI energies, particularly in strongly correlated regimes. By introducing a simple noise model, we demonstrate that sampling noise can enhance Hilbert-space exploration by generating additional configurations beyond those supported by the ideal ansatz. When combined with configuration recovery, this leads to systematically improved energies. Moreover, recovery alone (starting from randomly generated configurations) can efficiently construct accurate CI spaces, highlighting its central role in QSCI.

Validating threat modeling results remains difficult because completeness is hard to judge without an external oracle. Existing studies often rely on expert-produced reference models and other human baselines, but these can contain omissions or disagreements. This paper evaluates a complementary, vulnerability-grounded validation approach. We apply threat modeling to intentionally vulnerable applications with a known vulnerability set to measure the number of related vulnerabilities that can be discovered. We compare ThreMoLIA, an LLM-assisted threat modeling solution developed by our team, with the Microsoft Threat Modeling Tool (MTMT) across two vulnerable applications: AzureGoat and the Vulnerable Bank Application (VulnBank). The inputs to both tools are limited to architecture, data flow diagrams, and their descriptions. The results show that ThreMoLIA achieved higher vulnerability coverage on both systems. We show that vulnerable test applications provide a practical benchmark for assessing threat coverage and complement expert-based validation.

The dynamics of two-phase flows out of mechanical and thermal equilibrium are described by a partially dissipative first-order quasilinear system with stiff interaction terms associated with fast relaxation scales. In this paper, we analyze from a mathematical point of view the resulting pressure and temperature relaxation singular limit problem for a one velocity Baer-Nunziato model. This leads to a singular limit problem involving two small parameters. We propose a uniform symmetrization of this system which allows us to justify the strong relaxation limit and to establish a convergence rate for classical solutions.

Stochastic simulators are increasingly used to expand the frontier of scientific knowledge and inform decision-making across real-world contexts. Simulator calibration, a process by which internal model inputs are tuned to match some external criteria, usually in the form of observed data, is a key step in model design and validation. Epidemiological simulators present an especially compelling use case, as evidenced by the recent COVID-19 pandemic. Among several calibration paradigms, trajectory-oriented optimization is an emerging approach that does not require assumptions on the stochastic behavior of the simulator replicates and is particularly effective at identifying trajectories through the lens of errors between the simulator and observed data, especially when combined with Bayesian optimization. We present a tutorial on trajectory-oriented optimization with \texttt{adaptive\_ts}, an open-source Python package. We also provide a series of worked examples on an accompanying webpage.

Randomized clinical trials typically aim to estimate a marginal treatment effect. While covariate adjustment can improve precision, it may change the estimand in nonlinear models due to noncollapsibility, leading to conditional rather than marginal treatment effects. At the same time, identifying prognostic and predictive covariates is important for understanding treatment effect heterogeneity and informing clinical decision-making. Keeping marginal interpretability while allowing efficiency gains and assessment of heterogeneity remains a methodological challenge. In this work, we extend nonparanormal adjusted marginal inference to allow for heterogeneous treatment effects. The proposed framework embeds the marginal treatment effect directly in a joint model for the outcome and baseline covariates. This construction preserves marginal interpretability while adjusting for potentially prognostic and/or predictive covariates. The method applies to continuous, binary, ordinal, and time-to-event outcomes and allows explicit estimation and ranking of prognostic and predictive covariates on a common scale. For continuous outcomes, we show that the asymptotic variance of the marginal treatment effect measured as Cohen's $d$ is never worse and often better under covariate adjustment than without adjustment. Efficiency gains are primarily driven by prognostic effects, with realistic predictive effects contributing little additional improvement. Simulation studies confirm these findings across outcome types and demonstrate unbiased and more efficient estimation of marginal effects for Cohen's d, log-odds ratios, and log-hazard ratios. Application to an acupuncture trial demonstrates that the method reproduces the original trial findings while improving efficiency and allowing ranking of prognostic and predictive covariates.

We present a brief overview of the various methods proposed to derive the s- and r-components of the proto-solar chemical composition and we discuss some recent developments in the field, including the impact of rotating massive stars, nuclear measurements, physics of low mass asymptotic giant branch stars, isotopic composition of presolar SiC grains and new derivations.

Microresonator frequency combs are key components for integrating optical devices into photonic circuits. They provide high stability, coherence, and low noise, even without external stabilization. Yet microcomb design remains largely heuristic: waveguide and resonator parameters are typically swept manually or semi-empirically, and the resulting spectra are evaluated only afterwards. This forward-design workflow is computationally costly, relies heavily on designer intuition, and does not generally identify optimal solutions. Here, we present an adjoint-based inverse-design framework for microresonator frequency combs that directly optimizes the comb spectrum with respect to pre-defined objectives. We demonstrate the power and flexibility of this approach by addressing three challenging problems: designing spectrally flat combs, synthesizing arbitrarily shaped comb spectra, and enforcing several performance metrics simultaneously through multi-objective optimization. Our results show that inverse design offers a systematic and efficient route to compact on-chip light sources with properties tailored to diverse applications.

We prove a tropical analogue of Cartan's second main theorem for holomorphic curves intersecting hyperplanes in general position--a setting that was not fully resolved by previous tropical Nevanlinna theory. Two versions are obtained. The first (Theorem 1.7) requires subnormal growth and involves the tropical Casorati determinant. The second and main version (Theorem 1.9) is completely free of growth conditions and exceptional sets; it replaces the Casorati term by the sum of the counting functions of the curve's components, yielding an inequality valid for every r. The proof uses a tropical Cramer theorem, bypassing the logarithmic derivative lemma. This improves upon previous results by Korhonen-Tohge and Cao-Zheng, where the coefficient could be suboptimal even under the general position hypothesis. We also clarify the relation between different notions of linear independence, and present the first counterexample to the truncated second main theorem in the tropical setting (Example 5.4).

The Hundred-deg$^2$ HI Deep (HD$^2$) survey carried out with the Five-hundred-meter Aperture Spherical Telescope (FAST) is planned to map a contiguous region within the DESI DR1 footprint, achieving an effective integration time of 20 minutes for each pointing and a uniform detection sensitivity of 0.28 mJy beam$^{-1}$ at 4.8 km s$^{-1}$ resolution. We present early results from the pilot HD$^2$ survey: a 10 deg$^2$ field overlapping with HSC-SSP and the DESI EDR SV3, observed with an integration time of 7.3 minutes per beam and the rms of 0.45 mJy beam$^{-1}$ at 4.8 km s$^{-1}$ resolution. We identify 339 HI sources at $z<0.09$, corresponding to $\sim$34 detections per deg$^2$, nearly six times higher than the detection rate of the wide-field surveys. Optical counterparts are primarily identified using DESI redshifts, yielding a matching rate and correctness exceeding 90% for galaxies with $r<19.5$ mag, a substantial improvement over SDSS. Under the constraint of $r < 17.8$ mag and $0.01 < z < 0.05$, nearly 50% of galaxies in the DESI BGS samples have HI detections in this pilot survey. The optical properties of these HI-detected galaxies span nearly the entire parameter range of the DESI sample. The gas fraction scaling relations versus stellar mass, stellar mass surface density, NUV-r, and specific star formation rate are consistent with previous surveys, e.g., ALFALFA, DINGO, and xGASS. These results justify the feasibility of the full HD$^2$ survey, which will build a high-completeness HI census over a contiguous area to probe the cold gas scaling relations of galaxies over different scales.

The growing accessibility of Large Language Models via conversational interfaces capable of responding to users' questions by drawing on, synthesizing, and citing information from the web (i.e., Generative Search Engines) has simplified the information-seeking process for users. However, with the proliferation of AI-generated content on the web, it is unclear whether these engines can reliably omit citing synthetic sources (i.e., AI-generated sources). Should these engines be unable to do so, this puts users at risk of harm by treating information from AI-generated sources synthesized in responses of generative search engines as equivalent to information from authoritative or official sources. In a step towards identifying whether AI-generated sources are being cited by these engines, this work presents an audit of four generative search engines (ChatGPT, Copilot, Gemini, Perplexity) using a total of 712 real-world human-generated queries spanning domains of public importance: politics, health, and the environment. Our findings show evidence of AI-generated sources being cited across all four generative search engines (~16% of cited sources) and identifies key source web domains these sources belong to that are frequently cited across these engines and topics. In addition, we observed that generative search engines include a somewhat narrow set of repeatedly cited domains while predominantly surfacing a large number of minimally cited domains in responses to users' queries. These findings contribute to the growing body of work on assessing the risks of generative search engines with the objective of increasing public awareness of their limitations and encouraging appropriate measures to improve information quality and governance of these systems.

In this paper, we investigate an intelligent reflecting surface (IRS)-assisted multi-user system, where the base station (BS) employs rotatable antennas (RAs) and the IRS can adjust the panel orientation.To alleviate the severe multiplicative path loss of the cascaded channel, the IRS is deployed near the BS, while the user-BS and user-IRS links remain in the far field. We formulate a sum-rate maximization problem by jointly optimizing the receive beamforming, IRS phase shifts, BS antenna boresights, and IRS panel orientation. To tackle the resulting highly coupled and non-convex problem, we first study a single-user case to reveal the structure of the dual-rotation gain, which is shown to be multiplicatively separable in the far field but coupled in the near field. For the general multi-user case, we develop an alternating optimization algorithm, where the receive beamforming is updated in closed form, the IRS phase shifts are optimized by an FP-assisted Riemannian conjugate gradient method, and the BS antenna boresights and IRS panel orientation are updated via projected gradient methods. Simulation results demonstrate the significant sum-rate gains achieved by the proposed coordinated rotation design over fixed-orientation and single-rotation benchmark schemes, and provide useful insights into near-field dual-rotation design.

Optomagnonic modulation techniques are an emerging platform for information transfer from the microwave to the optical domain. However, these techniques focus largely on the spectral domain of the transduced signal. Given the potential of the field to bridge the gap between microwave and optical signals, analyzing and studying the interactions real-time in temporal domain becomes equally essential. In this work, we exploit the optomagnonic modulation in a YIG microsphere to demonstrate and study microwave to optical real-time dynamic transfer on a time scale comparable to the decay of magnons. We inductively excite magnons in the microwave domain and use magnon-based Brillouin light scattering to transduce the signature of excited magnonic waveforms to the optical domain. The square type modulation of the magnons is retrieved in the corresponding optical sidebands. Our work enables real-time measurement of the magnonic dynamics and therefore direct access to lifetime measurements of the magnonic mode. Providing insight into the temporal dynamics of magnons, this work can open up new promising research directions such as in magnon coupled superconducting qubits or magnon-based Brillouin memory.

This work derives exact expressions for the voltage induced into a two conductors dielectrically isolated transmission line by a monochromatic incident plane wave from an arbitrary direction, at a given polarization. The transmission line cross section, consisting of the conductors and the dielectric material, may be of any shape, provided the cross section size is much smaller than the wavelength, so that the waves in radiation mode satisfy the quasi TEM condition. We calculate analytically the voltage along the transmission line for given end loads and compare the results with ANSYS HFSS simulation results. Our calculations are based on the knowledge of the radiation from such a transmission line, derived elsewhere and the radiation-absorption reciprocity.

This paper describes the energy scale procedures and germanium detectors performance in the LEGEND-200 experiment, a critical component for the first unblinding in the search for neutrinoless double beta decay. We detail the digital signal processing pipeline, the methodologies for peak-shape modeling and energy calibration procedures utilizing weekly $^{228}$Th source calibration runs. The optimized energy reconstruction achieves a combined average resolution of $(2.47 \pm 0.08)$~keV at $Q_{ββ} = 2039$~keV. The weekly variation of calibration peak positions are below 0.05~keV for energies up to 2614.5~keV, showing a high stability of the energy scale over time and across detectors. Furthermore, systematic corrections effectively address residual non-linearities and energy bias in the region of interest.