We study an inverse problem for the viscoacoustic wave equation, an integro-differential model describing wave propagation in viscoacoustic media with memory in the leading order term. The medium is characterized by a spatially varying sound speed and a space-time dependent memory kernel. Assuming that waves are generated by sources supported outside the region of interest, we consider exterior measurements encoded by the source-to-solution map. To study this inverse problem, we construct solutions concentrating near fixed geodesics and establish a corresponding propagation of singularities result for the semiclassical wave front set. These results are valid without any restriction on the underlying sound speed. Then, under certain geometric conditions, we prove that the exterior data uniquely determine not just the sound speed inside the domain but also all time derivatives at zero of the memory kernel. This involves a reduction to the lens rigidity and geodesic ray transform inverse problems. As an application, we establish uniqueness for the recovery of variable parameters in the extended Maxwell model.

Advances in hybrid quantum architectures hinge on topological materials that can be synthesized with precise stoichiometric and structural control at the nanoscale. While $Bi_4Te_3$ is a promising candidate due to its dual topological phases, acting as both a strong topological insulator and a topological crystalline insulator, high-quality growth remains challenging due to a narrow stoichiometric window and high sensitivity to surface kinetics. Here, we establish a reproducible molecular beam epitaxy (MBE) process to produce stoichiometric, twin-free $Bi_4Te_3$ thin films with ultra-smooth surfaces and atomically sharp van der Waals stacks. By employing selective area epitaxy (SAE), we realize laterally confined $Bi_4Te_3$ nanostructures that exhibit a feature-dependent stoichiometric deviation. This phenomenon, which we term the selective stoichiometric shift, arises from the unequal lateral diffusion of Bi and Te adatoms, revealing a direct coupling between adatom kinetics and nanoscale compositional stability. Atomic-resolution imaging further uncovers asymmetric van der Waals gaps within the stacking sequence, identifying an intrinsic structural asymmetry between the quintuple and bilayer units. These findings provide fundamental insights into the crystallization of Bi_4Te_3$ and demonstrate a scalable route for integrating functional topological materials into next-generation superconducting hybrid quantum circuits.

While the mathematical foundations of score-based generative models are increasingly well understood for unconstrained Euclidean spaces, many practical applications involve data restricted to bounded domains. This paper provides a statistical analysis of reflected diffusion models on the hypercube $[0,1]^D$ for target distributions supported on $d$-dimensional linear subspaces. A primary challenge in this setting is the absence of Gaussian transition kernels, which play a central role in standard theory in $\mathbb{R}^D$. By employing an easily implementable infinite series expansion of the transition densities, we develop analytic tools to bound the score function and its approximation by sparse ReLU networks. For target densities with Sobolev smoothness $α$, we establish a convergence rate in the $1$-Wasserstein distance of order $n^{-\frac{α+1-δ}{2α+d}}$ for arbitrarily small $δ> 0$, demonstrating that the generative algorithm fully adapts to the intrinsic dimension $d$. These results confirm that the presence of reflecting boundaries does not degrade the fundamental statistical efficiency of the diffusion paradigm, matching the almost optimal rates known for unconstrained settings.

Atomically-thin moiré superlattices offer an optically accessible platform for interacting bosons, where strong onsite repulsion $U_{xx}$ suppresses double occupancy and supports excitonic Mott states at unit filling. However, moiré confinement also enhances phonon- and disorder-assisted relaxation, challenging the robustness of these correlated states under dissipation. Here we show that strengthening the intersite exciton repulsion $V_{xx}$ between neighboring moiré cells offers a distinct route to stabilizing unit-filling excitonic Mott states. In H-stacked WSe2/WS2, moiré confinement endows interlayer excitons with an out-of-plane dipole and a pronounced in-plane quadrupolar charge distribution. Helicity-resolved transient photoluminescence, supported by first-principles-informed modelling, reveals that this quadrupolar geometry increases $V_{xx}$ at unit filling by at least a factor of two relative to the dipolar R-stacked excitons. Despite a slight reduction in $U_{xx}$, the enhanced $V_{xx}$ yields a long-lived, valley-polarized excitonic Mott state at unit filling that persists for ~12 ns - more than twice as long as in R-stacks - and remains robust up to ~50 K. Beyond unit filling, the same geometry supports valley-polarized doublons with fourfold longer lifetimes than in R-stacks. These results establish moiré-geometric control of intersite interactions as a route to stabilizing excitonic Mott states and doublons against dissipation in solids.

A large number of NP-hard graph problems can be solved in $f(w)n^{O(1)}$ time and space when the input graph is provided together with a tree decomposition of width $w$, in many cases with a modest exponential dependence $f(w)$ on $w$. Moreover, assuming the Strong Exponential-Time Hypothesis (SETH) we have essentially matching lower bounds for many such problems. They main drawback of these results is that the corresponding dynamic programming algorithms use exponential space, which makes them infeasible for larger $w$, and there is some evidence that this cannot be avoided. This motivates using somewhat more restrictive structure/decompositions of the graph to also get good (exponential) dependence on the corresponding parameter but use only polynomial space. A number of papers have contributed to this quest by studying problems relative to treedepth, and have obtained fast polynomial space algorithms, often matching the dependence on treewidth in the time bound. E.g., a number of connectivity problems could be solved by adapting the cut-and-count technique of Cygan et al. (FOCS 2011, TALG 2022) to treedepth, but this excluded well-known path and cycle problems such as Hamiltonian Cycle (Hegerfeld and Kratsch, STACS 2020). Recently, Nederlof et al. (SIDMA 2023) showed how to solve Hamiltonian Cycle, and several related problems, in $5^τn^{O(1)}$ randomized time and polynomial space when provided with an elimination forest of depth $τ$. We present a faster (also randomized) algorithm, running in $4^τn^{O(1)}$ time and polynomial space, for the same set of problems. We use ordered pairs of what we call consistent matchings, rather than perfect matchings in an auxiliary graph, to get the improved time bound.

We discuss the extraction of heavy-light pseudo-scalar to light pseudo-scalar decay form factors from finite time correlation functions. We place particular emphasis on the contamination from excited states employing summed ratios and input from chiral perturbation theory. The analysis is performed on four CLS ensembles with $N_f = 2+1$ flavours of $\mbox{O}(a)$-improved Wilson fermions (presently) at the $\mathrm{SU}(3)$-symmetric point with relativistic heavy-quark masses in the charm region and above. The study presented here is part of the analysis aimed at the computation of the $B \to π\ell ν$ and $B_s \to K \ell ν$ semileptonic form factors, combining the continuum-limit relativistic results with static-limit calculations.

We study cyclic adjoint modules arising from the relative locally finite part of the adjoint action of a quantum Levi subalgebra on a quantized enveloping algebra. We analyze the realization of irreducible modules inside the quantized enveloping algebra via cyclic generators and describe embeddings of a fixed type. This leads to a natural map to isomorphism classes, whose fibers reflect the non-uniqueness of such realizations. We further introduce a partial order on cyclic adjoint modules and relate its minimal elements to irreducible submodules. In addition, we show that every cyclic adjoint module is generated by finitely many irreducible submodules.

The Pattern Effect describes the dependence of top-of-atmosphere radiation anomalies on changes in the pattern of sea surface temperatures. The emerging consensus in the field explains the impact of Pacific warm pool temperature on radiation using Convective Quasi-Equilibrium Weak Temperature Gradient (QE-WTG) theory: warm pool warming leads to increase in free-tropospheric temperatures across the tropics, a strengthening of inversion, increased cloud cover in the East Pacific low cloud decks, and negative radiative anomalies. Here we call on overlooked past results and new simulations from the Energy Exascale Earth System model to show that Rossby waves dominate the low-cloud response over the subtropical East Pacific low cloud decks, leading to decrease cloud cover in the low cloud decks. While the global radiative response is negative and consistent with QE-WTG, it is dominated by the response of the deep tropics, rather than the subtropical low cloud decks.

We introduce the order-separated Grassmann higher-order tensor renormalization group (OS-GHOTRG) method for QCD with staggered quarks in the strong-coupling expansion. The method allows us to determine the expansion coefficients of the partition function, from which we can obtain the strong-coupling expansions of thermodynamical observables. We use the method in two dimensions to compute the free energy, the particle-number density, and the chiral condensate as a function of the chemical potential up to third order in the inverse coupling $β$. Although near the phase transition the expansion is only a good approximation to the full theory at small $β$, we show that the range of applicability can be greatly extended by fits to judiciously chosen transition functions.

We present a comprehensive first-principles investigation of defects in 4$H_b$-TaS$_2$. In this layered transition metal dichalcogenide, charge transfer between alternating Mott-insulating 1T and metallic 1H layers gives rise to exotic quantum phases such as the Kondo effect and topological superconductivity. Motivated by recent defect manipulation in 4$H_b$-TaS$_2$ via STM, we address their microscopic nature and impact on interlayer charge transfer. To this end, we systematically analyze over 90 defects using large-scale density functional theory (DFT) calculations. Our extensive dataset, compiled from STM simulations, defect formation energies, work functions, and charge transfer, establishes a foundational resource for future theoretical and experimental studies on defect engineering in 4$H_b$-TaS$_2$.

The Photometric Observations of Exoplanet Transits (POET) is a proposed micro-satellite mission dedicated to the characterization and discovery of transiting exoplanets. POET has been identified as a top priority small-sat space mission in the Canadian Astronomy Long Range Plan 2020-2030. POET is being proposed as Canada's next astronomy space mission, with launch possible in late 2029. POET is an iteration on the designs of the Canadian MOST and NEOSSat space missions, which had 15 cm-sized telescopes and observed only in the visible band pass. POET will have a larger 20 cm telescope aperture and three band passes: near-ultraviolet (nUV; 300-400 nm), visible near-infrared (VNIR; 400-900 nm), and short-wavelength infrared (SWIR; 900-1700 nm). All mission components either already have significant space heritage or are seeing rapid adoption in commercial space missions. POET's simultaneous tri-band 300-1700 nm photometric monitoring will allow it to separate the impact of star spots on the transmission spectrum of extended atmospheres on super-Earth or larger exoplanets. POET's SWIR band is optimally sensitive to the emission peak of ultracool dwarf stars and would enable a systematic search for Earth-sized planets around them. POET aims to discover some of the nearest potentially habitable Earth-sized exoplanets that could be scrutinized for biosignatures with JWST or future telescopes. Herein we present the assembly of the POET Input Catalog of Ultracool Dwarfs and simulations of the expected yield of rocky planets with POET.

We study a variational problem modeling equilibrium configurations of charged liquid droplets resting on a surface under a convexity constraint. In the two-dimensional case with Coulomb interactions, we establish the validity of Young's law for the contact angle for small enough charges.

Determining transition states (TSs) of surface reactions is central to understanding and designing heterogeneous catalysts but remains computationally prohibitive with density functional theory (DFT). While machine learning potentials (MLPs) offer significant speedups, task-specific models have limited transferability across catalytic systems, and universal MLPs (uMLPs) lack the accuracy needed for reactive configurations. Here, we present a workflow based on active learning to iteratively fine-tune uMLPs for DFT-quality TS search. Using 250 TSs from the CO2 hydrogenation reaction network on metal and single-atom alloy surfaces, we first benchmark TS search algorithms, identifying the Sella algorithm as most robust, and propose a modification (Bond-Aware Sella) that substantially improves its success rate. We then explore sequential and batch active-learning strategies for fine-tuning and show that DFT-quality TS structures can be found using only 8 DFT single-point calculations on average per structure. This demonstrates the viability of fine-tuned uMLPs for high-throughput catalyst screening.

We prove a Liouville Theorem for ancient solutions of the parabolic Monge-Ampère equation with smooth periodic data, generalizing Caffarelli-Li's result \cite{cl04} in 2004 to the parabolic background. To achieve this, we obtain a necessary and sufficient condition for the existence of the smooth periodic solution of the equation $\left(1-u_t\right)\det \left(D_x^2u+I\right)=f$ in $\mathbb{R}^{n+1}$, where $f$ is smooth and periodic in both spatial and temporal variables. This parabolic existence theorem parallels the elliptic counterpart established by Li \cite{l90} in 1990.

Clinical imaging is routinely used for cochlear implant surgical planning yet lacks the resolution and contrast necessary to visualize the fine intracochlear structures critical for individualized intervention. To address this limitation, an ensemble deep learning model was developed to automatically segment cochlear micro-anatomy from standard clinical scans. The model was trained and validated using an independent internal dataset comprised of paired synchrotron and clinical scans of the same cochlea across various acquisition protocols. Performance was evaluated quantitatively on an unseen internal test dataset and a multi-institutional external test dataset. The deep learning model achieved accurate segmentation of intracochlear anatomy across all tested modalities, outperformed all previously published models, and demonstrated strong viability on the multi-institutional external dataset. Furthermore, anatomical measurements on the automatic segmentations closely matched those obtained from high-resolution ground truth segmentations, confirming reliable estimation of clinically relevant metrics. By bridging the gap between high-resolution imaging and routine clinical imaging, this work provides a practical solution for patient-specific cochlear implant surgical planning and postoperative assessment, advancing the goals of atraumatic insertions and more effective hearing restoration.

Forecasting infectious disease outbreaks is hard. Forecasting emerging infectious diseases with limited historical data is even harder. In this paper, we investigate ways to improve emerging infectious disease forecasting under operational constraints. Specifically, we explore two options likely to be available near the start of an emerging disease outbreak: synthetic data and genetic information. For this investigation, we conducted an experiment where we trained deep learning models on different combinations of real and synthetic data, both with and without genetic information, to explore how these models compare when forecasting COVID-19 cases for US states. All models are developed with an eye towards forecasting the next pandemic. We find that models trained with synthetic data have better forecast accuracy than models trained on real data alone, and models that use genetic variants have better forecast accuracy compared to those that do not. All models outperformed a baseline persistence model (a feat only accomplished by 7 out of 22 real-time COVID-19 cases forecasting models as reported in [38]) and multiple models outperformed the COVIDHub-4_week_ensemble. This paper demonstrates the value of these underutilized sources of information and provides a blueprint for forecasting future pandemics.

The conformal dimension of a metric space $(X, d)$ is equal to the infimum of the Hausdorff dimensions among all metric spaces quasisymmetric to $(X, d)$. It is an important quasisymmetric invariant which lies non-strictly between the topological and Hausdorff dimensions of $(X, d)$. We consider the conformal dimension of the Brownian sphere (a.k.a. the Brownian map), whose law can be thought of as the uniform measure on metric measure spaces homeomorphic to the standard sphere $\mathbf S^2$ with unit area. Since the Hausdorff dimension of the Brownian sphere is $4$, its conformal dimension lies in $[2, 4]$. Our main result is that its conformal dimension is equal to $2$, its topological dimension.

We develop a theory of polarized photoluminescence of interface excitons localized at lateral heterojunctions between transition metal dichalcogenide monolayers. We show that the circular selection rules governing interband optical transitions exactly at the band extrema are modified at finite wave vectors. The corresponding wave-vector-dependent corrections to the optical matrix elements result in a net linear polarization of excitonic photoluminescence. We identify two microscopic mechanisms responsible for linear polarization$-$trigonal warping of the electron and hole dispersions and the energy dependence of the effective masses. Their interplay controls both the magnitude and the angle of the emitted light polarization, with distinct dependences on the crystallographic orientation of the interface. Using a microscopic variational approach, we demonstrate that the degree of linear polarization can reach values exceeding 10% in realistic heterostructures. Furthermore, due to the large built-in dipole moment of interface excitons, their optical response can be tuned by an external in-plane electric field, enabling control over the strength and direction of the polarization.

We describe the design, laboratory manufacture, and on-sky testing of the grating vector apodizing phase plate (gvAPP) coronagraph for the Enhanced Resolution Imager and Spectrograph (ERIS) on the Very Large Telescope. We used both laboratory measurements and on-sky observations to characterise the gvAPP in several different filters, from the K to the L band. In testing, the gvAPP reaches its design specification in the transmission of the optic with 90% in the K bands and 60% in the L band. While the gvAPP reaches its designed raw contrast performance of $1 \times 10^{-5}$, it does not reach the post-processed contrast of $5 \times 10^{-5}$ in on-sky observations. Electronic detector noise, due to the Airy core of the coronagraphic point spread function inducing cross-talk between the readout amplifiers, produces a repeated pattern within the coronagraphic regions of the gvAPP. Despite these limitations, we recommend the gvAPP as a tool for characterising substellar companions with known separations and position angles, which allow them to be placed in the coronagraphic dark holes for observations. The ERIS gvAPP's leakage term can also be used as a photometric reference for time series observations; however, we caution that the contrast performance may limit such studies to only the brightest targets. ERIS gvAPP data quality may be improved further with better modelling of detector electronic noise. This work is a pathfinder for Extremely Large Telescope instruments including METIS, which will include gvAPP coronagraphs with improved designs based on these results.

Data words with binders formalize concurrently allocated memory. Most name-binding mechanisms in formal languages, such as the $λ$-calculus, adhere to properly nested scoping. In contrast, stateful programming languages with explicit memory allocation and deallocation, such as C, commonly interleave the scopes of allocated memory regions. This phenomenon is captured in dedicated formalisms such as dynamic sequences and bracket algebra, which similarly feature explicit allocation and deallocation of letters. One of the classical formalisms for data languages are register automata, which have been shown to be equivalent to automata models over nominal sets. In the present work, we introduce a nominal automaton model for languages of data words with explicit allocation and deallocation that strongly resemble dynamic sequences, extending existing nominal automata models by adding deallocating transitions. Using a finite NFA-type representation of the model, we establish a Kleene theorem that shows equivalence with a natural expression language. Moreover, we show that our non-deterministic model allows for determinization, a quite unusual phenomenon in the realm of nominal and register automata.

This work presents the computation of nuclear magnetic shielding and magnetizability tensors for paramagnetic molecules, using a magnetically induced current density framework to account for orbital and spin contributions. We demonstrate that the methodology proposed by Soncini[1] is physically equivalent to the formalisms of Pennanen and Vaara[2] and Franzke et al.[3], provided that scalar and spin-orbit relativistic effects are included within the ground-state spin density. In our model, these corrections are implemented through a Zeroth-Order Regular Approximation (ZORA) formulation of the current density. The resulting magnetizability tensor is fully consistent with the general Van Vleck formulation, recovering the temperature-dependent Curie contribution through the explicit integration of the magnetically induced spin current density. This methodology offers a straightforward computational route that bypasses the complex evaluation of g-tensors and Zero-Field Splitting (ZFS) Hamiltonians, requiring only a ground-state spin density incorporating relativistic effects. Notably, scalar relativistic effects are shown to be essential for capturing the Heavy-Atom Light-Atom (HALA) effect in 1H and 13C shieldings. To maintain efficiency, relativistic effects on the orbital contribution are neglected as they are negligible for light atoms. This approach represents an optimal compromise for paramagnetic complexes involving transition metals up to the second row, where the HALA effect is primarily driven by scalar relativistic corrections within the ground-state spin density. Neglecting spin-orbit terms in the orbital contribution significantly streamlines the calculation without loss of accuracy, providing the pNMR community with a robust tool for characterizing open-shell systems.

Optical turbulence, driven by fluctuations of the atmospheric refractive index, poses a significant challenge to ground-based optical systems, as it distorts the propagation of light. This degradation affects both astronomical observations and free-space optical communications. While adaptive optics systems correct turbulence effects in real-time, their reactive nature limits their effectiveness under rapidly changing conditions, underscoring the need for predictive solutions. In this study, we address the problem of short-term turbulence forecasting by leveraging machine learning models to predict the atmospheric seeing parameter up to two hours in advance. We compare statistical and deep learning approaches, with a particular focus on probabilistic models that not only produce accurate forecasts but also quantify predictive uncertainty, crucial for robust decision-making in dynamic environments. Our evaluation includes Gaussian processes (GP) for statistical modeling, recurrent neural networks (RNNs) and long short-term memory networks (LSTMs) as deterministic baselines, and our novel implementation of a normalizing flow for time series (FloTS) as a flexible probabilistic deep learning method. All models are trained exclusively on historical seeing data, allowing for a fair performance comparison. We show that FloTS achieves the best overall balance between predictive accuracy and well-calibrated uncertainty.

We prove that the intersection cohomology of the Baily-Borel compactification of a complex Shimura variety is identified with the top weight quotient of the mixed Hodge structure on the reductive Borel-Serre compactification. This yields canonical cup products and functorial pullbacks on the intersection cohomology. As an application, we introduce canonical cycle classes associated to special cycles, relating analytic geometric volumes of non-compact Shimura varieties to topological terms.

We study Schrödinger operators on the real line whose potentials are generated by the Fibonacci substitution sequence and a rule that replaces symbols by compactly supported potential pieces. We consider the case in which one of those pieces is identically zero, and study the dimension of the spectrum in the large-coupling regime. Our results include a generalization of theorems regarding explicit examples that were studied previously and a counterexample that shows that the naïve generalization of previously established statements is false. In particular, in the aperiodic case, the local Hausdorff dimension of the spectrum does not necessarily converge to zero uniformly on compact subsets as the coupling constant is sent to infinity.

Sterile Insect Technique (SIT) is widely regarded as a promising, environmentally friendly and chemical-free strategy for the prevention and control of dengue and other vector-borne diseases. In this paper, we develop and analyze a spatio-temporal reaction-diffusion model describing the dynamics of three mosquito subpopulations involved in SIT-based biological control of Aedes aegypti mosquitoes. Our sex-structured model explicitly incorporates fertile females together with fertile and sterile males that compete for mating. Its key features include spatial mosquito dispersal and the incorporation of spatially heterogeneous external releases of sterile individuals. We establish the existence and uniqueness of global, non negative, and bounded solutions, guaranteeing the mathematical well-posedness and biological consistency of the system. A fully discrete numerical scheme based on the finite element method and an implicit-explicit time-stepping scheme is proposed and analyzed. Numerical simulations confirm the presence of a critical release-size threshold governing eradication versus persistence at a stable equilibrium with reduced total population size, in agreement with the underlying ODE dynamics. Moreover, the spatial structure of the model allows us to analyze the impact of spatial distributions, heterogeneous releases, and periodic impulsive control strategies, providing insight into the optimal spatial and temporal deployment of SIT-based interventions.

The Abelian sandpile model serves as a canonical example of self-organized criticality. This critical behavior manifests itself through large cascading events triggered by small perturbations. Such large-scale events, known as avalanches, are often regarded as stylized representations of catastrophic phenomena, such as earthquakes or forest fires. Motivated by this perspective, we study strategies to reduce avalanche sizes. We provide a first rigorous analysis of the impact of interventions in the Abelian sandpile model, considering a setting in which an external controller can perturb a configuration by removing sand grains at selected locations. We first develop and formalize an extended method to compute the expected size of an avalanche originating from a connected component of critical vertices, i.e., vertices at maximum height. Using this method, we characterize the structure of avalanches starting from square components and explicitly analyze the effect of interventions in such components. Our results show that the optimal intervention locations strike an interesting balance between reduction of largest avalanche sizes and increasing the number of mitigated avalanches.

In this paper, we obtain optimal asymptotic behavior of parabolically convex $C^{2,1}$ solution to the parabolic Monge-Ampère equation $-u_t\det D_x^2u=f$, where $f$ converges to $1$ at infinity with a slow rate. This result extends the elliptic estimate in \cite{lb5} to the parabolic setting.

Irreversible transport is generally attributed to vorticity, nonlinear forcing, or explicit symmetry breaking. We show that it can arise even in strictly time-periodic and locally irrotational flows through a purely geometric mechanism. By reconstructing the velocity gradient through causal self-transport over a finite memory time, deformation acquires the structure of a geometric connection whose holonomy generates a finite Lagrangian drift over one forcing cycle. The resulting contribution admits a closed-form, parameter-free expression. A quantitative consistency analysis using independently published experimental measurements shows that the predicted scaling and magnitude agree with observations without fitting or normalization. These results identify geometric memory as a minimal and generic source of irreversible transport.

A mixed accuracy framework for Runge--Kutta methods presented in Grant [JSC 2022] and applied to diagonally implicit Runge--Kutta (DIRK) methods can significantly speed up the computation by replacing the implicit solver by less expensive low accuracy approaches such as lower precision computation of the implicit solve, under-resolved iterative solvers, or simpler, less accurate models for the implicit stages. Understanding the effect of the perturbation errors introduced by the low accuracy computations enables the design of stable and accurate mixed accuracy DIRK methods where the errors from the low-accuracy computation are damped out by multiplication by \dt at multiple points in the simulation, resulting in a more accurate simulation than if low-accuracy was used for all computation. To improve upon this, explicit corrections were previously proposed and analyzed for accuracy, and their performance was tested in related work. Explicit corrections work well when the time-step is sufficiently small, but may introduce instabilities when the time-step is larger. In this work, the stability of the mixed accuracy approach is carefully studied, and used to design novel stabilized correction approaches.

The NUCLEUS experiment aims to measure coherent elastic neutrino-nucleus scattering (CE$ν$NS) at unprecedentedly low nuclear recoil energies using gram-scale cryogenic calorimeters operated at the Chooz nuclear power plant in France. Access to recoil energies at the $\mathcal{O}(10~\mathrm{eV})$ scale enables CE$ν$NS studies at extremely low momentum transfer and provides enhanced sensitivity to new physics. In this work, we present sensitivity projections for the upcoming NUCLEUS technical and physics runs, incorporating a data-driven treatment of the low-energy excess (LEE) observed during commissioning. We develop a likelihood framework that exploits reactor-power variation to disentangle signal and background in a low signal-to-background regime and to assess the impact of the dominant systematic uncertainties. For the Technical Run with a 7 g CaWO$_4$ target, we find competitive sensitivity to several scenarios beyond the Standard Model, which do not require a CE$ν$NS observation. For the Physics Run, assuming complete suppression of the LEE, we project a 4.7 $σ$ observation of CE$ν$NS with a statistical precision of about 20 % in 1 year, enabling a determination of the weak mixing angle at the lowest momentum transfer probed to date with CE$ν$NS and leading CE$ν$NS-based constraints on the neutrino charge radius and new mediator models.