We present an independent calibration of the photometric redshift (photo-$z$) distributions for source galaxies in the HSC-Y3 weak lensing survey using small-scale galaxy-galaxy lensing. By measuring the tangential shear around spectroscopic lens galaxies from GAMA, SDSS, and DESI, divided into fifteen narrow redshift bins, we compute shear ratios that are sensitive to the mean redshift of source galaxies. Using a blinded analysis, we derive constraints on the photo-$z$ bias parameters in source bins 2, 3 and 4, achieving signal-to-noise ratios of 59, 75, and 62, respectively. Our constraints for $Δz_2$, $Δz_3$ and $Δz_4$ are consistent with those from HSC-Y3 cosmic shear modeling. We observe a mild shift in the $Δz_3$--$Δz_4$ plane due to the heterogeneous depth of the lens sample, which disappears when using only DESI-DR1 lenses. Combining shear-ratio measurements with cosmic shear data, we obtain joint constraints on cosmological parameters: $Ω_{\rm m} = 0.304_{-0.029}^{+0.03}$ and $S_8 = 0.773_{-0.031}^{+0.031}$, consistent with cosmic shear-only results. This work demonstrates the utility of small-scale lensing as an independent probe for calibrating photometric redshift bias in weak lensing cosmology.

Large Language Models (LLMs) are increasingly being integrated into the scientific peer-review process, raising new questions about their reliability and resilience to manipulation. In this work, we investigate the potential for hidden prompt injection attacks, where authors embed adversarial text within a paper's PDF to influence the LLM-generated review. We begin by formalising three distinct threat models that envision attackers with different motivations -- not all of which implying malicious intent. For each threat model, we design adversarial prompts that remain invisible to human readers yet can steer an LLM's output toward the author's desired outcome. Using a user study with domain scholars, we derive four representative reviewing prompts used to elicit peer reviews from LLMs. We then evaluate the robustness of our adversarial prompts across (i) different reviewing prompts, (ii) different commercial LLM-based systems, and (iii) different peer-reviewed papers. Our results show that adversarial prompts can reliably mislead the LLM, sometimes in ways that adversely affect a "honest-but-lazy" reviewer. Finally, we propose and empirically assess methods to reduce detectability of adversarial prompts under automated content checks.

We study the method to test the unitarity of the lepton mixing matrix by using only the long baseline neutrino oscillation experiments, such as the combination of the T2HK experiment and the one with the $ν_e$ beam from a future neutrino factory at J-PARC. Without a specific parametrization, one can directly extract the elements of the lepton mixing matrix by observing the energy dependence of the oscillation probabilities. A non-trivial test of the unitarity under the three-generation assumption can thus be made possible by examining the orthogonality in a similar manner to the unitarity triangle in the quark sector. As the first trial, we perform the analysis based on the simplified situation where the matter effects in the neutrino oscillation can be neglected. Under this simplified analysis, we demonstrate the observation of the unitarity violation in the $3\times3$ part of the lepton mixing matrix for a parameter set in the four-generation model. The statistically most significant measurement can be provided by the energy dependences of the combination of the CP conjugate modes, $ν_μ\to ν_e$ and $\bar ν_μ\to \bar ν_e$, at T2HK and, independently, by the T conjugate modes, $ν_μ\to ν_e$ and $ν_e \to ν_μ$, with the latter measured at the neutrino factory experiments.

We introduce and analyze the ordered Zeckendorf game, a novel combinatorial two-player game inspired by Zeckendorf's Theorem, which guarantees a unique decomposition of every positive integer as a sum of non-consecutive Fibonacci numbers. Building on the original Zeckendorf game\ -- previously studied in the context of unordered multisets\ -- we impose a new constraint: all moves must respect the order of summands. The result is a richer and more nuanced strategic landscape that significantly alters game dynamics. Unlike the classical version, where Player 2 has a dominant strategy for all $n > 2$, our ordered variant reveals a more balanced and unpredictable structure. In particular, we find that Player 1 wins for nearly all values $n \leq 25$, with a single exception at $n = 18$. This shift in strategic outcomes is driven by our game's key features: adjacency constraints that limit allowable merges and splits to neighboring terms, and the introduction of a switching move that reorders pairs. We prove that the game always terminates in the Zeckendorf decomposition\ -- now in ascending order\ -- by constructing a strictly decreasing monovariant. We further establish bounds on game complexity: the shortest possible game has length exactly $n - Z(n)$, where $Z(n)$ is the number of summands in the Zeckendorf decomposition of $n$, while the longest game exhibits quadratic growth, with $M(n) \sim \frac{n^2}{2}$ as $n \to \infty$. Empirical simulations suggest that random game trajectories exhibit log-normal convergence in their move distributions. Overall, the ordered Zeckendorf game enriches the landscape of number-theoretic games, posing new algorithmic challenges and offering fertile ground for future exploration into strategic complexity, probabilistic behavior, and generalizations to other recurrence relations.

We review all the calculations necessary for the construction of a Lyapunov like functional for nonlinear stability analysis of steady states in thermodynamically isolated/open systems composed of compressible heat conducting fluids.

We investigate the nucleon and pion gravitational $D$-form factors, by fitting a $σ/f_0(500)$-meson pole, together with a background term, to lattice data at $m_π\approx 170\text{MeV}$. We find that the fitted residues are compatible with predictions from dilaton effective theory. In this framework, the $σ$-meson takes on the role of the dilaton, the Goldstone boson of spontaneously broken scale symmetry. These results support the idea that QCD may be governed by an infrared fixed point and offer a physical interpretation of the $D$-form factor (or $D$-term) in the soft limit.

Van der Waals (vdW) materials provide a platform to study and control the physical properties of low-dimensional materials. While strategies developed for two-dimensional (2D) crystals are not directly transferable to one-dimensional (1D) systems, we can benefit from them by creating layers formed by interconnected chains. Here, we develop a molecular strategy to illustrate this concept consisting of assembling 1D materials in 2D metal-organic frameworks (MOFs). Crystals of [FeX(pzX)(bpy)] (X = Cl, F; pz = pyrazole; bpy = bipyridine) consist of iron chains along the b-axis, crosslinked via bpy ligands along the a-axis to form 2D layers, stacked along the c-axis via vdW forces. This structural anisotropy manifests itself in highly-anisotropic optical properties, as demonstrated by optical measurements in the visible and terahertz ranges, results which are supported by DFT calculations. Chemical substitution enables the tuning of the optical properties, as exemplified by the photoluminescence of the Cl-derivative, which is quenched for the F-derivative. Thin-layers are obtained by mechanical exfoliation, and their optical properties are further tuned through the fabrication of orthogonally-twisted vdW heterostructures, enabling to effectively switch-off the optical anisotropy. Our work highlights the chemical flexibility of vdW layered MOFs as a platform for designing and manipulating 1D architectures.

We derive a quantum extension of the thermodynamic uncertainty relation where dynamical fluctuations are quantified by the Terletsky-Margenau-Hill quasiprobability, a quantum generalization of the classical joint probability. The obtained inequality plays a complementary role to existing quantum thermodynamic uncertainty relations, focusing on observables' change rather than exchange of charges through jumps and respecting initial coherence. Quasiprobabilities show anomalous behaviors that are forbidden in classical systems, such as negativity; we reveal that negativity or a non-classically enhanced escape rate is necessary to increase an output-to-dissipation ratio beyond classical limitations and show that the requirements are basis-independent and stronger than quantum coherence. To illustrate these statements, we employ a model that can exhibit a dissipationless heat current, which would be prohibited in classical systems; we construct a state that has much coherence but does not lead to a dissipationless current due to the absence of anomalous behaviors in quasiprobabilities.

We tackle the challenge of integrating large language models (LLMs) with external recommender systems to enhance domain expertise in conversational recommendation (CRS). Current LLM-based CRS approaches primarily rely on zero/few-shot methods for generating item recommendations based on user queries, but this method faces two significant challenges: (1) without domain-specific adaptation, LLMs frequently recommend items not in the target item space, resulting in low recommendation accuracy; and (2) LLMs largely rely on dialogue context for content-based recommendations, neglecting the collaborative relationships among item sequences. To address these limitations, we introduce the CARE (Contextual Adaptation of Recommenders) framework. CARE (a) integrates external recommender systems as domain experts, producing candidate items through entity-level insights, and (b) customizes LLMs as rerankers to enhance the accuracy by leveraging contextual information. Our results demonstrate that incorporating CARE framework significantly enhances recommendation accuracy of LLMs by an average of 54% and 25% for ReDial and INSPIRED datasets. The most effective CARE strategy involves LLMs selecting and reranking candidate items that external recommenders provide based on contextual insights.

We give a condition on a $p$-adic representation of the fundamental group of a curve over $\overline{\mathbb{Q}}_p$ which ensures that under the $p$-adic Simpson correspondence the Higgs field vanishes.

We figure out the explicit expression for the trace of the field equations associated to generic higher derivative theories of gravity endowed with Lagrangians depending upon the metric and its Riemann tensor, together with arbitrary order covariant derivatives of the Riemann tensor. Then an equality linking the Lagrangian density with the covariant divergence of a vector field is put forward in terms of the trace of the field equations. As a significant application, we particularly concentrate on a broad range of higher derivative theories of gravity with the Lagrangian density constructed from the contraction of the product for metric tensors with the product of the Riemann tensors and the arbitrary order covariant derivatives of the Riemann tensor. By utilizing the trace for the equations of motion, such a type of Lagrangian density is expressed as the covariant divergence of a vector field.

In this work, within the framework of path integral Monte Carlo, we construct a pseudo-fermion propagator by replacing the original fermionic determinant with its absolute value. This modified propagator defines an auxiliary system free from the fermion sign problem, enabling efficient simulations of fermionic systems. We found that by shifting the pseudo-fermion energy based on the energy of a non-interacting fermion system, we can efficiently and reliably infer the energy of fermionic systems in various situations, from strong quantum degeneracy to weak quantum degeneracy. We have performed first-principles simulations of quantum dots confined in a two-dimensional harmonic potential and found excellent agreement with benchmark results provided by other established methods. We believe that this pseudo-fermion propagator framework opens up new possibilities for first-principles simulations of fermionic systems.

Primarily motivated by the Hubble tension, we analyze the varying electron mass model and axionlike early dark energy model (EDE) using baryon acoustic oscillation data from DESI DR2 data and including the recent results from ACT DR6. Our analysis indicates that $m_{e} / m_{e0} = 1.0081 \pm 0.0046 $ and the energy fraction of EDE is constrained as $f_\mathrm{EDE} < 0.016$. Since those cosmological models fit with different spectral index $n_s$, we show the posterior of those models on the ($n_s-r$) plane and point out that, for example, Starobinsky inflation works for varying electron mass model while the standard supersymmetric hybrid inflation is preferred in the EDE model.

Observations indicate that high-redshift galaxies undergo episodic star formation bursts, driving strong outflows that expel gas and suppress accretion. We investigate the consequences for metal and dust content of galaxies at $z \geq 5$ using our semi-analytical model, ASHVINI. We track gas-phase and stellar metallicities ($Z_{\rm g}, Z_\ast$) and dust mass (M$_{\rm d}$) in dark matter haloes spanning $M_{\rm halo} = 10^6$-$10^{11}$ $M_\odot$, comparing continuous and bursty star formation scenarios - which reflect underlying assumptions of instantaneous and delayed feedback - and we allow for metallicity-dependent feedback efficiency. Delayed feedback induces oscillations in $Z_{\rm g}$ and $Z_\ast$, with $Z_{\rm g}$ declining sharply at low stellar and halo masses; the mass scale of this decline increases toward lower redshift. Reionization introduces significant scatter in $Z_{\rm g}$, producing an upturn followed by rapid decline. Metallicity-dependent feedback moderates this decline at $ z=7 - 10$, flattening the $Z_{\rm g}$-mass relation to $\simeq 0.03$-$0.04\, Z_\odot$. Dust production tracks $Z_{\rm g}$ but is sensitive to burst history, causing delayed enrichment. Our results show that burst-driven feedback decouples $Z_{\rm g}$ and $Z_\ast$, imprints intrinsic scatter in mass-metallicity relations, and delays dust growth. These effects are strongest in low-mass halos ($M_{\rm halo} \sim 10^7 M_\odot$), where shallow potentials amplify the impact of feedback. Our results are consistent with recent hydrodynamical and semi-analytical simulations and provide context for interpreting JWST (James Webb Space Telescope) metallicity and dust measurements, highlighting the importance of episodic star formation in early galaxy chemical evolution.

In this paper we investigate the spatio-temporal dynamics of obesity rates across Italian regions from 2010 to 2022, aiming to identify spatial and temporal trends and assess potential heterogeneities. We implement a Bayesian hierarchical Beta regression model to analyze regional obesity rates, integrating spatial and temporal random effects, alongside gender and various exogenous predictors. The model leverages the Stochastic Search Variable Selection technique to identify significant predictors supported by the data. The analysis reveals both regional heterogeneity and dependence in obesity rates over the study period, emphasizing the importance of considering gender and spatial correlation in explaining its dynamics over time. In fact, the inclusion of structured spatial and temporal random effects captures the complexities of regional variations over time. These random effects, along with gender, emerge as the primary determinants of obesity prevalence across Italian regions, while the role of exogenous covariates is found to be minimal at the regional level. While socioeconomic and lifestyle factors remain fundamental at a micro-level, the findings demonstrate that the integration of spatial and temporal structures is critical for capturing macro-level obesity variations.

Understanding and minimizing the sources of frequency noise in nanomechanical resonators is crucial for many sensing applications. In this work, we report an ultracoherent perimeter-mode nanomechanical resonator co-integrated with an on-chip optical cavity. This device combines low thermomechanical force noise and low detector noise, allowing us to study its intrinsic frequency fluctuations in detail. We find that the fluctuations of two mechanical modes are strongly correlated. Moreover, we demonstrate the generation of a signal at the frequency difference between the two modes directly on chip via nonlinear optomechanical transduction. This `difference signal' has vastly reduced intrinsic frequency fluctuations and can be used for frequency tracking with high precision, as we establish in a proof-of-principle experiment.

Python third-party libraries (TPLs) are essential in modern software development, but upgrades often cause compatibility issues, leading to system failures. These issues fall into two categories: version compatibility issues (VCIs) and code compatibility issues (CCIs). Existing tools mainly detect dependency conflicts but overlook code-level incompatibilities, with no solution fully automating the inference of compatible versions for both VCIs and CCIs. To fill this gap, we propose PCREQ, the first approach to automatically infer compatible requirements by combining version and code compatibility analysis. PCREQ integrates six modules: knowledge acquisition, version compatibility assessment, invoked APIs and modules extraction, code compatibility assessment, version change, and missing TPL completion. PCREQ collects candidate versions, checks for conflicts, identifies API usage, evaluates code compatibility, and iteratively adjusts versions to generate a compatible requirements.txt with a detailed repair report. To evaluate PCREQ, we construct REQBench, a real-world benchmark with 2,095 upgrade scenarios derived from 34 real-world scientific/ML Python projects. Results show PCREQ achieves a 94.03% inference success rate, outperforming PyEGo (37.02%), ReadPyE (37.16%), and LLM-based approaches (GPT-4o, DeepSeek V3/R1) by 18--22%. PCREQ processes each scenario from REQBench in 60.79 s on average, demonstrating practical efficiency. PCREQ reduces manual effort in troubleshooting upgrades, advancing Python dependency maintenance automation.

The capacity of bandlimited direct-detection channels is challenging to compute or approach due to the receiver non-linearity. A generalized vector approximate message passing (GVAMP) detector is designed to achieve high rates at a reasonable level of complexity. The rates increase by using multi-level coding and successive interference cancellation. The methods are applied to fiber-optic channels with intersymbol interference caused by spectrally efficient pulse shapes, chromatic dispersion, and receiver sampling at twice the baud rate. Bipolar modulation operates within 0.26 bits per channel use (bpcu) of the real-alphabet coherent capacity for optically amplified links, reducing the best-known theoretical gap of 1 bpcu. Remarkably, bipolar modulation achieves 6 dB and 3 dB of power gain over unipolar modulation with and without optical amplification, respectively. Simulations with polar-coded modulation confirm the gains. The GVAMP complexity, measured in multiplications per information bit (mpib), is proportional to the number of iterations and to the logarithm of the block length, and is substantially lower than that of other equalizers. For example, a system with 64-ary bipolar modulation and a root-raised cosine pulse with a 1% roll-off factor was simulated over 4 km of optically amplified standard single-mode fiber in the C-band. The GVAMP receiver requires 93 mpib to achieve 5 bpcu at 300 gigabaud.

Topological order (TO) provides a natural platform for storing and manipulating quantum information. However, its stability to noise has only been systematically understood for Abelian TOs. In this work, we exploit the non-deterministic fusion of non-Abelian anyons to inform active error correction and design decoders where the fusion products, instead of flag qubits, herald the noise. This intrinsic heralding enhances thresholds over those of Abelian counterparts when noise is dominated by a single non-Abelian anyon type. Furthermore, we use Bayesian inference to obtain a statistical mechanics model for fixed-point non-Abelian TOs with perfect measurements under any noise model, which yields the optimal threshold conditioned on measuring anyon syndromes. We numerically illustrate these results for $D_4 \cong \mathbb Z_4 \rtimes \mathbb Z_2$ TO. In particular, for non-Abelian charge noise and perfect syndrome measurement, we find a conditioned optimal threshold $p_c=0.218(1)$, whereas an intrinsically heralded minimal-weight perfect-matching (MWPM) decoder already gives $p_c=0.20842(2)$, outperforming standard MWPM with $p_c = 0.15860(1)$. Our work highlights how non-Abelian properties can enhance stability, rather than reduce it, and discusses potential generalizations for achieving fault tolerance.

In a dynamical system $(X,f)$, with $X$ a compact metric space, the chain components, the fundamental building blocks in the Conley decomposition of dynamics, have a natural partial order induced by the chain relation between points. Although chain components are crucial for understanding the long-term behavior of topological systems, they have not been widely studied from the point of view of poset theory. In this work, we pursue this line of research, considering both the case in which $f$ is a continuous map and the general case in which no regularity assumption is made. Our main results are that, if $f$ is continuous: - the chain components poset cannot be linearly and densely ordered; - every countable well-order with a maximum is the order type of the chain components poset of an interval map. If no regularity assumption is made: - there is a dynamical system on the interval whose chain components poset is countable and densely ordered; - the chain components poset has at least one minimal element. These results, bridging dynamical systems and order theory, highlight both the structural constraints and the possibilities for the chain components posets.

We discuss the issue of defining and measuring conical deficits (conicity) in spacetimes with the torsion singularity such as the Misner string in Taub--NUT spacetime. We propose a geometric definition that generalizes the standard notion of conicity to stationary axially symmetric spacetimes with torsion singularity, where the conical deficit becomes observer-dependent -- it depends on the choice of a timelike Killing vector. This implies the existence of observers who perceive no conical singularity along the symmetry axis. As a result, in any spacetime with a non-vanishing NUT parameter, there are observers for whom the conicity has the same value on both semi-axes. This challenges the usual interpretation of conicity differences as indicators of string/rod-induced acceleration along the axis. We illustrate our definition across the full Plebański--Demiański class, including the recently identified accelerated Taub--NUT solution. Our attempts in determining a canonical observer lead to even less desirable definitions of conicity.

Linear observed systems on groups encode the geometry of a variety of practical state estimation problems. In this paper, we propose an observer framework for a class of linear observed systems by restricting a bi-invariant system on a Lie group to its normal subgroup. This structural property enables a system embedding of the original system into a linear time-varying system. An observer is constructed by first designing a Kalman-like observer for the embedded system and then reconstructing the group-valued state via optimization. Under an extrinsic observability rank condition, global exponential stability (GES) is achieved provided that one global optimum of the reconstruction optimization is found, reflecting the topological difficulties inherent to the non-Euclidean state space. Semi-global stability is guaranteed when input biases are jointly estimated. The theory is applied to the GES observer design for two-frame systems, capable of modeling a family of navigation problems. Simulations are provided to illustrate the implementation details.

This paper investigates the failure of certain metric measure spaces to be infinitesimally Hilbertian or quasi-Riemannian manifolds, by constructing examples arising from a manifold $M$ endowed with a Riemannian metric $g$ that is possibly discontinuous, with $g, g^{-1} \in L^\infty_{\mathrm{loc}} $ and $ g \in W^{1,p}_{\mathrm{loc}}$ for $ p \leq \mathrm{dim} M - 1 $.

This paper studies the oscillatory behavior of solutions to linear nonautonomous impulsive differential equations with piecewise constant arguments, including both advanced and delayed cases \[ x'(t) = a(t)x(t) + b(t)x([t-k]), \quad k \in \mathbb{Z}. \] By exploiting the hybrid structure of these systems, we reduce the problem to an associated difference equation whose coefficients explicitly incorporate both the continuous dynamics and the impulsive effects. Classical oscillation criteria for difference equations do not account for impulsive phenomena. Through the proposed reduction, we extend these criteria to a class of impulsive and non-impulsive equations (IDEPCA and DEPCA), obtaining explicit sufficient conditions for oscillation in terms of the original system data. An example is provided to illustrate the applicability of the results.

We construct and analyze the "syntomic Steenrod algebra", which acts on the mod $p$ syntomic cohomology (also known as etale-motivic cohomology) of algebraic varieties in characteristic $p$. We then apply the resulting theory to resolve the last open cases of a 1966 Conjecture of Tate, concerning the existence of a symplectic form on the Brauer groups of smooth proper surfaces over finite fields. More generally, we exhibit symplectic structure on the higher Brauer groups of even dimensional varieties over finite fields. Although the applications are classical, our methods rely on recent advances in perfectoid geometry and prismatic cohomology, which we employ to define a theory of "spectral syntomic cohomology" with coefficients in motivic spectra. We then organize the resulting cohomology theories into a category of "spectral prismatic $F$-gauges", generalizing the prismatic $F$-gauges of Drinfeld and Bhatt--Lurie, for which we establish a ``spectral Serre duality'' extending classical coherent duality. These abstract constructions are leveraged to explicitly compute the syntomic Steenrod operations.

We prove that, modulo rigid motions, the Hopf map is the unique minimizer of the Faddeev--Skyrme energy in its homotopy class, provided that the radius of the target 2-sphere is not smaller than the radius of the domain 3-sphere.

This paper considers the problem of estimating a matrix that encodes pairwise distances in a finite metric space (or, more generally, the edge weight matrix of a network) under the barycentric coding model (BCM) with respect to the Gromov-Wasserstein (GW) distance function. We frame this task as estimating the unknown barycentric coordinates with respect to the GW distance, assuming that the target matrix (or kernel) belongs to the set of GW barycenters of a finite collection of known templates. In the language of harmonic analysis, if computing GW barycenters can be viewed as a synthesis problem, this paper aims to solve the corresponding analysis problem. We propose two methods: one utilizing fixed-point iteration for computing GW barycenters, and another employing a differentiation-based approach to the GW structure using a blow-up technique. Finally, we demonstrate the application of the proposed GW analysis approach in a series of numerical experiments and applications to machine learning.

We report that moderately eccentric flows around supermassive black holes (SMBHs), formed via either circumnuclear gas accretion or tidal disruption events, generate eccentricity cascades (from >0.8 to 0.2 outward), explaining multiwavelength emission and variability in active galactic nuclei (AGNs). The flows' non-axisymmetric temperature structure explains non-axisymmetric dust sublimation fronts, distinct broad emission-line components, and their radial motions. The innermost broad-line region (BLR) links to the SMBH vicinity through highly eccentric streams that produce soft X-rays at periapsis. General relativistic precession further compresses these flows, generating a hard X-ray continuum near the SMBH. Precession of the eccentric flow drives optical/X-ray variability, reproducing the observed X-ray power spectral density and occasional X- ray quasi-periodic eruptions. We thus propose eccentric accretion disks as a physical AGN model that unifies the elusive BLRs and X-ray corona. This model will enable detailed anatomy of AGNs and maximize their potential as cosmological standard candles.

The confinement of molecules within the van der Waals (vdW) gap between a two-dimensional 2D material and a catalytic substrate offers a promising route toward the development of molecule-selective catalysts with increased reaction rates. However, identifying the kinetic limitations of such confined reactions remains challenging. Here, we employ an inverted wedding-cake configuration of multilayer graphene on platinum to study the dynamics of graphene etching in the vdW gap by various molecules (O2, H2, and CO), using in situ scanning electron microscopy. Under the experimental conditions explored (up to p = 1.4x10-3 Pa and T = 1000 °C), the etching reaction rates are limited by mass transport within the confined space. This limitation persists even for CO, despite its anomalously enhanced transport resulting from a significant lifting of the vdW gap. Reactive molecular dynamics simulations further reveal multiple etching pathways for CO, enabled by confinement within the vdW space. Once mass-transport limitations are overcome, the vdW gap acts as an effective nanoreactor, facilitating reaction pathways that would be otherwise inaccessible on a pristine surface without spatial confinement.

The interplay of spin and orbital degrees of freedom offers a versatile playground for the realization of a variety of correlated phases of matter. However, the types of spin-orbital interactions are often limited and challenging to tune. Here, we propose and analyze a new platform for spin-orbital interactions based upon a lattice of Wigner molecules in moiré transition metal dichalcogenides (TMDs). Leveraging the spin-orbital degeneracy of the low-energy Hilbert space of each Wigner molecule, we demonstrate that TMD materials can host a general spin-orbital Hamiltonian that is tunable via the moiré superlattice spacing and dielectric environments. We study the phase diagram for this model, revealing a rich landscape of phases driven by spin-orbital interactions, ranging from ferri-electric valence bond solids to a helical spin liquid. Our work establishes moiré Wigner molecules in TMD materials as a prominent platform for correlated spin-orbital phenomena.