Papers

As robotic hardware and learning methods advance, humanoids need tools to perform tasks beyond their inherent physical limits. Successful tool use requires selecting a suitable tool and coordinating manipulation and, when needed, locomotion to complete the task. Existing benchmarks do not jointly evaluate these capabilities on a humanoid. We introduce HumanoidToolBench, an 18-task benchmark spanning three scenarios, three execution levels, and two tool-set modes, together with ToolBook, a dataset of 3.1k demonstrations collected in simulation and on a real Unitree G1. Evaluation of seven policies in simulation and three on the real robot reveals substantial gaps between selecting a suitable tool and completing the task. Focused GR00T N1.7 probes show reduced selection accuracy on unseen tools and continued task execution under unrelated instructions. Code and data are available at https://snu-pi.github.io/HumanoidToolBench/.

Current all-sky surveys such as Vera Rubin's Legacy Survey of Space and Time and the Zwicky Transient Facility (ZTF) promise a wealth of scientific gains that are contained in the millions of astrophysical transient candidates produced each night. Kilonovae, one such transient of interest, will be challenging to identify in the alert stream and will require efficient artificial intelligence models to parse the large, real-time influx of data. In order to build these large models, comprehensive multimodal datasets are necessary for training. Due to a lack of numerous kilonova observations, we propose $\texttt{kilonova-multimodal-emulator}$ – a simulation pipeline for realistic, multimodal kilonova observations comprised of photometry, spectra, and images. We demonstrate this pipeline for ZTF-type observations, based on historical cadence and limiting magnitude information from the Bright Transient Survey (BTS) and using the latest radiative transfer kilonova models to produce a comprehensive dataset meant for the training of large artificial intelligence model.

We construct and study the Nash manifold of four-point configurations in the real plane modulo the action of a Nash subgroup of the group of similarity transformations. We do so by using the finer notion of a quadrangle in place of that of a four-point configuration, since this retains limiting line data in degenerations. We prove that the space ${\mathcal Q}$ of quadrangles is an 8-dimensional Nash manifold and that, for every Nash subgroup $G$ of the similarity group, the orbit space ${\mathcal Q}/G$ is a Nash manifold. We define a geometric invariant, called aspect, with values in $[-1,1]$, and prove that, over each of the intervals $(-1,0)$ and $(0,1)$, the corresponding part of ${\mathcal Q}/G$ is Nash diffeomorphic to the product of the interval with a fixed fiber. Thus the nonexceptional part of the moduli problem reduces to the analysis of two model fibers, each having a natural geometric interpretation in terms of the quadrangles themselves.

We undertake a systematic comparison of cluster algebras associated to Legendrian knots of the same smooth knot type. We describe a cluster structure on the m-graded augmentation variety of Legendrian representatives of a family of double twist knots that includes all orientably Lagrangian fillable twist knots. For different Legendrian representatives of the same smooth knot type, we show that these cluster structures are equivalent. Along the way, we characterize the representatives that admit Maslov-m exact Lagrangian fillings, construct a Catalan number of such fillings for each representative, and show that any Maslov-0 fillable representative is Legendrian isotopic to the (-1)-framed closure of a positive braid. Finally, we give two Legendrian representatives of a cabled twist knot that yield equivalent cluster structures of infinite type.

We introduce a Monte Carlo approach for estimating isolated polynomial zeros through their contour integral representations. For a simple zero enclosed by an isolating contour, we show that the zero can be expressed exactly as the expectation of a complex-valued random variable obtained by uniformly sampling the contour. This yields an unbiased estimator without perturbing the polynomial coefficients. We establish its variance and derive finite sample probabilistic error bounds based on concentration inequalities, with the bounds explicitly reflecting the geometry of the contour and its separation from the zeros. The framework also provides a stochastic formulation of contour based root counting and root isolation. Numerical results illustrate the estimator's behavior under varying polynomial degrees, and we further provide comparisons to existing methods.

We study free-boundary problems within a physics-informed framework using Kolmogorov-Arnold network (KAN) approximations. The proposed approach incorporates obstacle constraints, partial differential equation (PDE) inequalities, complementarity conditions, and boundary conditions through residual-based loss functions. We consider a linear elliptic obstacle problem, a nonlinear $p$-Laplacian obstacle problem, and a time-dependent one-phase Stefan problem. The proposed KAN solver is compared with physics-informed neural network (PINN) and residual-network baselines. Numerical experiments show that KANs achieve low relative $L^2$ and $L^\infty$ errors while accurately resolving contact regions and moving interfaces. The results indicate that KAN representations provide an effective alternative for solving free-boundary PDEs.

Monopoles have been a subject of much theoretical and experimental research since they were proposed to symmetrize Maxwell's equations. No experimental signature of their existence has been detected. A model generalizing QED with dark massless photons and dark monopoles was proposed not long ago. In this model, by introducing a mixing interaction between the two sectors, leptons and monopoles are transformed into into dyons. This particular behavior of the electrons determines a new path to study the physics of monopoles. Here we generalize the Electroweak theory in the same way by incorporating a dark sector with massive photons and monopoles, and analyze experimental consequences for electrons in both models. The mass of the dark photon plays an important role regarding observability.

Tensor networks are powerful tools for simulating quantum many-body systems, but the growth of spatial entanglement severely limits the direct evolution of pure states under chaotic unitary circuits. Here we address the task of evaluating specified output probabilities of one-dimensional circuits at finite relative precision. We introduce a sweeping algorithm that contracts the circuit transversely and compresses its two temporal boundaries jointly through reduced transition matrices (RTMs), preserving their overlap rather than either boundary separately. At fixed target precision, the retained bond dimension exhibits a slowdown consistent with subexponential growth over the accessible time window, in a regime where the circuit is already deeply anticoncentrated. A noisy random-operator model illustrates how finite precision can induce a crossover from linear to logarithmic temporal entropy growth. These results open a route to probability queries for benchmarking and learning without representing the full quantum state.

There has been a proliferation of sampling algorithms based on Wasserstein gradient flows (WGF) and forward-only diffusion processes (FODP), often accompanied by theoretical guarantees of exponentially fast convergence to the target distribution. These guarantees are frequently interpreted as evidence that such methods can efficiently sample complex multimodal distributions, often supported by empirical results. In this work, we argue that this interpretation is fundamentally misleading. By invoking the Jordan-Kinderlehrer-Otto (JKO) scheme and Otto calculus, we establish that the canonical WGF sampling dynamics and overdamped forward diffusion share the same density evolution and therefore inherit the same metastability and slow-mixing phenomena long understood in nonequilibrium statistical physics. We analyze this family of samplers using two complementary tools – spectral analysis and mean first-passage time (MFPT) analysis – and show that well-separated multimodality can induce exponentially long mixing times associated with small spectral gaps and rare inter-mode transitions. For the commonly adopted log-linear annealing schedule studied here, we find that introducing intermediate distributions does not remove the exponential scaling of the total transport time. The limitation is structural rather than implementation-specific: purely local, gradient-driven transport mechanisms can require exponentially long times to transport probability mass across well-separated modes. We argue that this represents a fundamental limitation of WGF- and FODP-based sampling in their standard forms, and motivates future development of fundamentally nonlocal mechanisms for efficient multimodal sampling.

Epitaxial quantum dots (QDs) are widely recognised as one of the best quantum light sources, given their good stability, brightness, quantum efficiency and coherence. To reach such properties, QDs are protected from potentially detrimental surface states by a relatively thick capping layer, typically exceeding 50nm. This prevents the implementation of near-field effects, like plasmonic-based ones, that can dramatically increase the light-matter interaction, since the control of the spontaneous emission dynamics and directionality of the emission can only occur if the emitter is in close proximity (typically tens of nanometers or less) from the metallic structures. In this work, we report the growth and optical characterisation of InAs/GaAs QDs with GaAs capping layer thickness of 10nm, 20nm, and a more standard 95nm. Remarkably, we observe emission linewidths up to 5 times smaller than previously reported, with unperturbed excitonic lifetimes. These results demonstrate that the epitaxial QD's high optical quality can be maintained even when the emitters are at reduced distances from the surface, opening the path for the exploration of near-field light-matter interactions with coherent and stable emitters, suitable for quantum technology applications.

We study the transverse field Ising model, defined by the Hamiltonian $H =\frac{1}{2}\sum_{i, j\in [n]} J_{ij} Z_i Z_j +\sum_{i=1}^n h_i^z Z_i + \eta\sum_{i} X_i$ where $J $ is the symmetric interaction matrix, and $\eta$ is the transverse field strength. Let $\Delta(J)=\lambda_{\max}(J)-\lambda_{\min}(J)$ be the spectral width of $J.$ When the inverse temperature $\beta\geq0$ satisfies $\Delta(J)\cdot\frac{\tanh(\beta\eta)}{\eta}\leq1$, we give a randomized classical algorithm that approximates the partition function $Z(\beta)=\operatorname{Tr}(e^{-\beta H})$ to a given relative error $\epsilon\in(0,1)$ in time polynomial in $n$, $\beta$, the model parameters, and $\epsilon^{-1}$. When $ \Delta(J) \cdot \frac{\tanh(\beta \eta)}{\eta} > 1 ,$ we show that approximating $ Z(\beta)$ within an $\exp(o(n))$-multiplicative factor is $\textbf{NP}$-hard, and thus unlikely to admit an efficient classical or quantum algorithms under standard complexity theoretic assumptions. Furthermore, in the regime $\Delta (J)\cdot \frac{\tanh(\beta \eta)}{\eta}\leq 1,$ we provide an efficient randomized classical algorithm that approximates Pauli string observables of the Gibbs state $ \rho_\beta = \frac{e^{-\beta H}}{\operatorname{Tr}(e^{-\beta H})}$ within an arbitrarily small additive error. In the special case when the observable is also diagonal in the $X$-basis, i.e. $P \in \{I, X\}^{\otimes n}$, the algorithm further achieves arbitrarily small relative error.

For integer $b\ge3$, let $\Sig\subset\Ball$ be an embedded free boundary minimal surface (FBMS) of genus zero with $b$ boundary components. Suppose that $\Sig$ is invariant under the prismatic group of order $4b$, the rotation of order $b$ permutes the boundary components cyclically, and the reflection in the equatorial plane preserves each of them. We show that the Morse index of $\Sig$ is $2b$ and its nullity is three; moreover, we determine the negative space of the index form as a representation of the symmetry group. Examples include the $b$-noids of Karpukhin, Kusner, McGrath, and Stern and, for large $b$, the genus-zero surfaces of Folha, Pacard, and Zolotareva. To our knowledge, apart from the equatorial disc and the critical catenoid, these are the first FBMS in the ball whose Morse indices are known precisely. There are also lower bounds for the index of surfaces with rotational or polyhedral symmetry. The main new ingredient is an equivariant comparison, using the representation theory of finite groups, between area and energy, in which the Teichmüller directions are counted by symmetry type using circle domains.

We study quadratically regularized optimal transport with quadratic cost in the regime of small regularization $\varepsilon$, where the support of the optimal coupling is sparse. For smooth marginal densities in $\mathbb{R}^d$, we show that the support contains the graph of the Brenier map. After centering by the Brenier image and rescaling by $\ell=\varepsilon^{1/(d+2)}$, the sections of the support have explicit ellipsoidal limits for $\varepsilon\to0$, except at boundary points, where the limits are half-space profiles. The optimal dual potentials admit expansions at order $\ell^2$, uniformly up to the boundary. We identify the leading coefficients, which consist of a common local profile and opposite global corrections determined by a linear Neumann problem. Finally, we analyze two approximations to the Brenier map, namely the gradient of the dual potential and the conditional mean of the coupling. We obtain the sharp $L^p$ rate $\ell^{1+1/p}$ with exact leading constants and further identify the leading interior and boundary biases. Taken together, our results illustrate that quadratic regularization induces an accurate sparse approximation of classical optimal transport.

Jev-style decision models have gained surging traction by making decision-making dramatically faster and more efficient, returning categorical probability distributions over predefined options without generating free-form text. However, because Jev's architecture and training methodologies remain proprietary, how to replicate these capabilities effectively remains an open question. In this work, we investigate the extent to which general-purpose LLMs already possess this capability out of the box, and when fine-tuning is actually necessary. We present LLM-as-Jev, an architecture-preserving framework offering both a training-free inference recipe and a fine-tuning objective with KL-divergence anchors. Across Qwen3.5-4B and Qwen3-0.6B, we find that modern LLMs are inherently effective decision models: off the shelf, frozen models match or exceed existing training-free readouts on the same 4B backbone, scale to arbitrary option counts, and natively handle multimodal decisions. They also seamlessly inherit the backbone's thinking mode: while Jev was introduced as a System 1 model, "Thinking Jev" leverages budgeted test-time reasoning to boost accuracy on challenging tasks without training. Fine-tuning provides targeted gains for smaller models and complex routing, while our KL anchors prevent behavioral degradation in general text generation.

Fermionic non-Gaussianity is a resource for universal quantum computation that can be generated by interactions in quantum many-body systems. Using the magic Rényi entropy (MRE) as a measure of non-Gaussianity, we derive its universal upper bound and Haar mean, proving that typical Haar-random states attain the maximal MRE density of $\ln(4/3)$ per Majorana in the large-system limit. We also show that the MRE can differ extensively between states with identical covariance matrices. To investigate how non-Gaussianity grows toward this maximal density, we evolve a Gaussian state in imaginary and real time under the Sachdev-Ye-Kitaev Hamiltonian. By varying the imaginary- and real-time durations, we identify a first-order transition marked by a kink in the MRE density and spontaneous breaking of the permutation symmetry among the four copies used to evaluate the MRE. This transition represents a qualitative change in the non-Gaussianity of the state that is not reflected in the thermal free energy.

Privacy-preserving machine learning presents significant deployment challenges on the cloud for intelligent systems with confidential data. Fully Homomorphic Encryption (FHE) offers a compelling solution for secure computation, preserving data confidentiality of cloud computations. However, applying FHE to reinforcement learning (RL) requires replacing non-linear operations with polynomial approximations, which diverge catastrophically due to a unique recursive error phenomenon known as the Bellman drift. This article introduces the Homomorphic Advantage Operator (HAO), a stabilization framework designed to prevent polynomial approximation divergence in FHE-based deep RL. HAO adapts the zero-mean centering projection from advantage-based value estimation directly to temporal-difference (TD) targets. This linear projection annihilates the uniform state-value baseline that drives the Bellman drift, maintaining per-state action rankings while requiring zero additional non-linear multiplicative depth and avoiding expensive ciphertext bootstrapping. The proposed HAO framework was evaluated using a three-tier experimental methodology, including a tabular Markov Decision Process (MDP), an encrypted CartPole environment using real CKKS cryptographic operations, and a 20-node logistics routing benchmark with dense continuous features. The results demonstrate that the proposed HAO strictly bounds network pre-activations within the safe polynomial approximation domain. The proposed HAO RL agents achieved 0% boundary breaches across all random seeds used, whereas regularization alone (L2 weight decay and gradient clipping) breached the bound on 3 of 5 seeds and the unstabilized baseline did so in 83.8% of episodes. Finally, HAO agents improve optimal policy accuracy by 18.0 percentage points in tabular domains and remain stable when DP-SGD-style Gaussian noise is added to the clipped gradients.

We construct the first examples of quasi-contact metric structures that are not contact metric in every odd dimension greater than or equal to $5$. In dimension $5$, we construct a one-parameter family of such examples with a Killing characteristic vector field by deforming Sasaki-Einstein $\mathrm{SU}(2)$-structures. This yields compact, homogeneous examples on $S^5$. In arbitrary dimensions, we investigate quasi-contact metric structures with a Killing characteristic vector field and derive a Ricci curvature criterion for the structure to be K-contact.

The study of ceramic materials constitutes a cornerstone of archaeological research, yet the post-production workflow for pottery documentation remains labor-intensive and creates significant publication bottlenecks. This paper presents PyPottery, an open-source, AI-powered suite designed to semi-automate the complete ceramic documentation pipeline. The suite comprises four integrated modules: PyPotteryScan for automated image extraction and handwriting recognition; PyPotteryInk for automatic inking of pencil drawings; PyPotteryTrace for semantically-aware vectorization; and PyPotteryLayout for automated layout generation. Evaluated on 50 hand-drawn sheets containing 240 pottery drawings from the Terramara di Montale (Italy), the framework achieved substantial time savings confirmed by usability study participants, who reported a median perceived speedup of 40$\times$ over traditional workflows (range: 17.5$\times$–120$\times$). These results highlight the potential of AI-assisted tools in archaeological documentation, while the paper addresses the strategic redistribution of cognitive labor toward augmentation rather than automation.

We establish interior and global Hölder estimates for linearized Monge-Ampère equations in divergence form in all dimensions $n\geq 3$, when the Hessian determinant of the convex Monge-Ampère potential is bounded above and below by positive constants and the vector field on the right-hand side is bounded. The estimates use the $L^p$ norm of the solution for any $p>1$. A key ingredient is an $L^1$ estimate, with appropriate decay rates, for the gradient of the Green's function of the linearized Monge-Ampère operator in small sections, uniform with respect to the pole. As applications, we obtain uniform Hölder estimates for the time derivatives of the primal and dual potentials in the three-dimensional periodic dual semigeostrophic system when the initial density is bounded away from zero and infinity, and establish a linear convergence rate for periodic homogenization of the Monge-Ampère equation.

Memory-augmented large language models must decide which memories to retain, and recent systems do so by estimating each memory's effect on task performance. However, these estimates rely entirely on retrieved memories. When a memory is never retrieved, store-level interventions produce identical outcomes, leaving its utility unidentified. This is a retrieval-level positivity violation, invisible to diagnostics that examine only memory operations. We introduce Causal Memory Policy (CMP), a causal framework that restores identification by intervening on retrieval itself, reserving a fixed number of context slots for memories sampled with known propensities. CMP estimates memory utility by self-normalized inverse propensity weighting under a balanced assignment design. We prove the causal factorization of memory utility through retrieval, the unbiasedness and exact variance of the estimator, and the optimal decision rule under irreversible operations. Empirically, identification fails for 54% of required memories on LongMemEval and 67% on LoCoMo, and the failure persists in a deployed memory system. CMP improves discrimination between required and non-required memories from 0.54 to 0.66 AUC. Finally, we show that identified memory utility alone is insufficient for retention decisions: per-query utility reaches 0.78 AUC on the query for which it is estimated, yet no aggregation available to a retention policy predicts a memory's value on unseen queries. Code is available at: https://anonymous.4open.science/r/cmp-release-D0C3/.

Rare weather regime transitions pose a challenge for data-driven modeling due to class imbalance. In this study, we develop a probabilistic deep learning emulator for a prototypical system with regime transitions, the stochastic Holton–Mass model of stratospheric variability, and analyze the structure of its learned latent space. The Holton–Mass model exhibits two metastable regimes, a strong and a weak polar vortex, maintained by nonlinear wave–mean flow interactions, with weak stochastic forcing intermittently triggering rare transitions between these regimes that qualitatively represent SSW events. We employ a ResNet-inspired Conditional Variational Autoencoder with six-layer encoder and decoder layers and explicit current-state conditioning to model the distribution of the system's state at the next time step (one day). The emulator accurately reproduces short-term dynamics, steady-state probability distributions, regime persistence statistics, rare transition rates, the transition committor function, and the transition expected lead time of the physical model. Beyond emulation fidelity, we interrogate the learned latent representation to understand how the model internalizes the underlying metastable structure of the dynamics. Principal Component Analysis of the 32-dimensional latent space reveals a clear and unsupervised separation into four physically interpretable clusters corresponding to strong versus weak vortex regimes and stable versus transition-prone configurations. Such emergent regime separation in latent space is hard to identify for deep generative models applied to high-dimensional stochastic systems. Our results show that carefully designed probabilistic emulators can uncover physically meaningful manifolds governing extreme-event dynamics, potentially aiding the development of improved operational advanced warning systems.

How complex can the responses of a quantum device become as it runs longer with a fixed internal memory? We quantify this complexity through sequential response capacity: how many adaptive testing stages, each using a fresh run, can continue to separate possible processes by a prescribed gap in response probabilities. For fixed system and memory sizes, we establish a tight law relating this capacity to run length and probability resolution. At fixed resolution, the capacity grows on the order of $K\log K$, where $K$ is the number of time steps in each run. Our construction attains this growth using time-dependent phase rotations on a single visible qubit with no additional internal memory; its tests give response probabilities exactly zero or one. Under the same tests, classical stochastic processes that measure in a fixed basis at every step have only linear capacity at fixed sizes and resolution. For phase sequences selected by a stored classical label, we then quantify how known independent Pauli noise changes this logarithmic enhancement. With ideal controls and weak residual phase noise after correction, we prove matching capacity bounds at a fixed small probability gap. These bounds identify the inverse residual phase-flip probability as the coherence timescale that limits the extra logarithmic growth.

While Low-Rank Adaptation (LoRA) enables efficient task specialization, its learned updates can compromise capabilities beyond the target task. We identify adaptation imbalance: a few singular directions dominate the trained update, leaving its performance sensitive to how gains are allocated. We argue that learning where to adapt does not ensure that adaptation gains are well balanced. This motivates LoRA-Norm, a post-training normalization method that retains learned directions while rebalancing their gains. LoRA-Norm combines spectral rebalancing, a fixed nonlinear transformation of singular values, with nuclear-norm restoration, which preserves the original total spectral mass. It requires no calibration data or additional training and introduces no inference overhead. Across two backbones and three adaptation tasks, LoRA-Norm improves average specialization and capability retention, outperforming the evaluated post-hoc spectral pruning and gradient-guided editing configurations on both measures. Stronger functional equalization brings no consistent additional gains, revealing that balancing adapter gains and equalizing their responses are distinct objectives.

As Large Language Models (LLMs) increasingly serve as foundational reasoning engines, their tendency to hallucinate remains a critical vulnerability. While recent internal state probes offer a promising alternative to slow external retrieval systems, they largely reduce hallucination detection to a token-wise binary classification task, failing to capture the structured, sequential boundaries of semantic drift. Here, we introduce an internal hidden state framework for fine-grained, span-level hallucination detection. By inspecting layer-wise activation patterns, we attempt to detect the exact hallucination onset and continuation tokens in an LLM generation. Our experiments show that this approach successfully isolates hallucination onsets, achieving substantial improvements in Precision-Recall AUC over random baselines despite extreme class imbalance. Ultimately, we propose a novel cross-model detection framework in which one model observes the internal representations elicited by another model's generation. We find that an external observer can match or exceed a generator's self-detection of its own hallucination onsets, including when the observer is the smaller model, suggesting that self-detection is not the ceiling for onset localisation.

The linear arboricity $\la(G)$ of a graph $G$ is the least cardinality of linear forests, that is, forests of maximum degree at most $2$, into which its edge set $E(G)$ can be decomposed. The Linear Arboricity Conjecture asserts that $\la(G)\leq\lceil(\Delta(G)+1)/2\rceil$ for every finite graph $G$, where $\Delta(G)$ denotes the maximum degree of $G$. We extend this conjecture to infinite graphs of finite maximum degree and prove that its finite and infinite versions are equivalent. We introduce topological linear arboricity $\latop (G)$ by requiring the linear forests to contain no topological circle, and show that it differs from linear arboricity by at most one. Finally, we prove that every $2k$-regular graph of girth at least $2k$ has topological linear arboricity at most $k+1$.

The critical Ising model admits several natural scaling limits. For an Ising model on a lattice approximation $D^a$ of a planar domain $D$, the rescaled spin field converges, as $a\to0$, to the critical Ising magnetization field (IMF), a rough random distribution on $D$, while the spin interfaces converge to a nested $CLE_3$ and the FK–Ising representation converges to a colored $CLE_{16/3}$. In this work, we prove that the IMF can be directly reconstructed from the nested $CLE_3$. This complements the previously known construction of the IMF from the colored $CLE_{16/3}$ [CGN15]. Our result also implies that %We further show that if one constructs a colored $CLE_{16/3}$ from the nested $CLE_3$ through CLE percolation [MSW17], the resulting construction recovers the same magnetization field. A main difficulty in this present reconstruction comes from the fact that the renormalization exponent of the IMF, $15/8$, is smaller than the Hausdorff dimension of the $CLE_3$ carpet, $187/96$. Consequently, the field cannot be recovered by a direct Minkowski-content construction from the nested $CLE_3$ as is the case for $CLE_{16/3}$. We conjecture that the converse measurability also holds, namely that the nested $CLE_3$ can itself be reconstructed from the IMF.

An outstanding question in holographic superconductivity is how rotation and non-relativistic scaling symmetries compete to modify the onset of condensation. We address this by comparing the probe superconductor sectors of two exact rotating axionic black holes: a four-dimensional asymptotically anti-de Sitter solution dressed by a non-minimally coupled scalar, and a five-dimensional asymptotically Lifshitz solution with $z \in (2,3)$. Writing both metrics in a common stationary form yields a master scalar equation in which rotation enters through the co-rotating potential $A_t-N^xA_x$. This equation explains why the temporal and spatial components of the probe gauge field cannot be treated independently. The numerical profiles reported in [1,2] indicate that the correlated deformation associated with rotation reduces the condensate amplitude in both backgrounds, whereas increasing the dynamical exponent at fixed $J$ enhances the Lifshitz condensate. Since the parameters $\alpha$ and $J$ also modify the matter fields supporting the corresponding exact geometries, these trends should be interpreted as properties of complete correlated families rather than as universal effects of mechanical rotation alone.

X-ray observatories introduce unwanted instrumental effects into photon count data that must be accurately accounted for. In particular, the spatially variant point spread function (PSF) and shot noise pose a non-trivial inverse problem. We apply Bayesian inference grounded in information field theory (IFT) combined with a fast spatially variant PSF representation to infer the posterior mean and uncertainty of the X-ray flux under a minimal set of physically motivated prior assumptions, thereby removing or reducing these instrumental effects. First, ignoring spatial PSF variability, the IFT-based deconvolution is benchmarked on a synthetic example against three variants of the Richardson-Lucy (RL) algorithm using SSIM, RMSE, and NLL as evaluation metrics. Second, the ability of an interpolated patch-based convolution scheme to accurately represent the complex, spatially variant Chandra PSF is assessed on simulated PSF data. Finally, deblurring and spatially variant PSF representation are combined and applied to real Chandra observations of the supernova remnant Cassiopeia A. The IFT-based deconvolution outperforms all tested RL variants on the synthetic benchmark. The patch-based convolution scheme accurately recovers the complex structure of the Chandra PSF. Applied to Cassiopeia A, the reconstructed image appears visually sharper than the exposure-corrected data, and the data residuals are consistent with pure noise, indicating an accurate model of both the instrument and the sky. A comparison with a highly resolved reference dataset confirms this impression. The presented framework enables the removal of spatially variant PSFs from noisy data and opens avenues for future work on cross-calibration between multiple observations or different X-ray instruments, building on the principled uncertainty quantification inherent to the Bayesian framework.

We build a cohesive topological theory of orientations on moduli stacks for non-compact complex-analytic Calabi-Yau $4$-folds by extending and generalising the work of Joyce and Upmeier, and the work of Bojko. As a result, we get, in particular, canonical family orientations for all moduli spaces of Pandharipande-Thomas pairs. Our proofs use the algebraic topology of mapping spaces representing $KU$- and $KO$- theory, as well as some properties of the Lie groups $E_8$ and, surprisingly, $\operatorname{HSpin}(16)$. This is part of a joint project with Felix Thimm dedicated to the proof of the $GW/PT$-correspondence for Calabi-Yau $4$-folds by implementing Pardon's programme in dimension $4$.

The eight-component relativistic wave equation for spin-$\frac{1}{2}$ particles of Robson and Staudte (FV$\frac{1}{2}$), a Feshbach–Villars linearization of the Feynman–Gell-Mann equation, reproduces the Dirac hydrogenic spectrum with a doubled solution space and an indefinite inner product. Its exact Stark shifts must equal the Dirac ones; whether its indefinite-metric perturbation theory delivers them is not guaranteed. We compute the first- and second-order Stark shifts of the $n=1$–3 levels of hydrogenic ions from H to U$^{91+}$ in the Schrödinger, Dirac and FV$\frac{1}{2}$ theories, evaluating all spectral sums, continuum included, by the Dalgarno–Lewis method. The FV$\frac{1}{2}$ and Dirac shifts coincide level by level, to working precision at first order (closed-form roots) and to 45 significant figures at second order. Both ingredients are needed: without the explicit spin–field coupling the first-order roots become complex, and without the negative-norm states the $(Z\alpha)^2$ coefficient is wrong. Norm sign and parity are locked in the doubled space: a potential lifting the $2s_{1/2}$–$2p_{1/2}$ degeneracy needs its own spin–gradient term, and shifts placed by hand on the $\kappa$-labeled states give a spurious imaginary linear Stark shift. The relativistic linear Stark effect is smaller by finite factors, but a second-moment sum rule shows the non-relativistic linear Stark strength is conserved, the missing part reappearing in singular $(Z\alpha)^{-2}$ second-order terms. For levels with a fine-structure partner, an exact intra-shell/regular split of the second-order shift gives the $(Z\alpha)^0$ coefficients analytically and relates the $\pi^2$ terms of the $(Z\alpha)^2$ coefficients to the singular ones. The Dirac weak/strong-field crossover for $n=2$ of hydrogen lies near 2.8~kV cm$^{-1}$, and the perturbative window set by the fine structure grows as $Z^5$.