Papers

We study the complexity of the Code Equivalence problem and show that it is polynomially equivalent to several computational automorphism problems for codes. These problems ask for the cardinality (ACOUNT), an orbit partition (APART), and a generating set (AGEN) for the permutation automorphism group of a code. We present deterministic, polynomial-time reductions between Permutation Code Equivalence (PCE) and each of these problems, including a one-shot reduction from search-PCE to AGEN that makes a single oracle call. We present similar reductions between Linear Code Equivalence (LCE) and analogous problems for the monomial automorphism group of a code. All of our reductions work for any linear codes.

Large language model (LLM) pretraining conventionally returns the raw final iterate. This couples two design choices: the learning-rate schedule that generates the parameter trajectory and the estimator that constructs the deployed model (e.g. the raw final iterate or a checkpoint average). A schedule that promotes optimization progress may differ from one that minimizes variation in the raw final iterate. Separating these choices creates an opportunity to maintain progress late in training while reducing variation in the returned model. To this end, we propose Terminal Shrinkage Averaging (TSA), which interpolates between the raw final iterate and the average of recent checkpoints to balance recent progress against terminal variation. We analyze how TSA changes the preferred terminal learning-rate schedule under a local quadratic approximation and test this interaction through a sequence of controlled NanoChat experiments. Finally, we demonstrate that the resulting gains transfer to depth-22 NanoChat, where the combined schedule and estimator improve validation quality. A qualifying time-to-GPT-2 run also finishes faster than the public baseline used in our experiments, providing preliminary evidence of benchmark acceleration.

Institutions often face constraints richer than a single capacity. Protecting the priority of a student who cannot be accommodated may then block a lower-priority student who fits. We introduce COP-and-Choose, a cumulative offer process followed by a choice step, and show that its greedy version, which skips such blockers, leaves no student worse off than any fair matching or Knapsack Deferred Acceptance (KDA). Greedy is the unique consistent rule preserving this guarantee. With substitutability and size monotonicity at every school, greedy is strategy-proof and Pareto-undominated among strategy-proof mechanisms. If either fails at a school, one additional capacity-one school yields a market where, under every proposal order, greedy is manipulable and no strategy-proof mechanism weakly Pareto dominates it. In a three-resource refugee-resettlement calibration with simulated preferences, 19–33 percent of families gain relative to KDA and none lose, while 21–83 percent strongly envy another family.

Magnetic reconnection between emerging magnetic flux and the ambient coronal field fundamentally drives solar jets. We use three-dimensional resistive-MHD simulations of eruptive flux emergence to investigate how the large-scale current sheet formed at the interface fragments during recurrent jet activity. The simulations show that jet formation and current-sheet disruption are coupled elements of a single multiscale process. The current structures developing along open field lines and within the jet are relatively weak, exhibiting a predominantly filamentary morphology. In contrast, the larger structures linked to the closed-field topology are noticeably stronger, and the eruptive development of the emerging flux further increases both their number and strength. The parallel-current distributions within the closed magnetic field lines region exhibit Gaussian central cores with pronounced non-Gaussian tails, implying a small subset of strong, intermittent structures. Magnetic spectra follow a power law with a Kolmogorov-like slope in the large-scale range, becoming much steeper in the small-scale range—consistent with reconnection-mediated magnetic structuring. Box-counting analysis shows that the strongest current structures have a fractal dimension $D_{\mathrm{f}} \simeq 2$, indicating a fragmented population of sheet-like coherent structures. Cluster analysis further reveals broad, power-law distributions of cluster volumes and their Ohmic dissipation energies, with individual events reaching $\sim 10^{20}$–$10^{25}$ erg over their estimated lifetimes. These results demonstrate that eruptive flux emergence can self-consistently transform a global reconnecting current sheet into a localized, intermittent, sheet-dominated dissipative network, providing a natural environment for impulsive coronal heating and motivating future particle-acceleration studies.

Unmanned Aerial Vehicles (UAVs) performing deadline-bound analytics over large rural areas cannot reliably offload workloads to sparse cellular base stations. We propose an approach that uses scheduled public buses as mobile fogs: a UAV hands off data to a bus during a halt, and the bus carries it until its route enters cellular coverage. Because a public bus follows a fixed route and timetable, a handover depends on where and when the bus next reaches a cellular zone, rather than how near the stop is. We formulate a Mission Scheduling Problem over this model, co-scheduling UAV routes with the placement of each analytics task on the UAV edge, a stationary fog, or a bus, under deadline, energy, and cost constraints. Our Divide and Assign (DA) heuristic selects the cheapest halt that still meets a task's deadline. Across 36 workload configurations in a rural region, derived from real cellular and transit data, DA achieves up to 20% higher utility than the strongest heuristic and up to 41% higher utility than the strongest adapted-prior scheduler, while incurring the lowest aggregate cost. The transit tier handles up to 71% of drop-offs, raises task completion to 100%, and reduces recharging cycles by up to 31%. Finally, in the presence of traffic variability, the adaptive variant recovers 93% of the utility the delays cost and maintains completion rates above 97%.

We show how genuine three-body phase information can enter the invariant mass of a relativistic three-particle Dirac system in $(1+1)$ dimensions. As a solvable reference system, we consider the Sakamoto–Munakata–Ino model with pairwise contact interactions $g_{ij}(1-\alpha_i\alpha_j)\delta(x_i-x_j)$. These singular interactions can be transferred into sector-dependent phases and matching conditions by a discontinuous unitary transformation. Although the explicit contact terms are thereby removed, the nonzero constituent-mass operator is rotated and retains nontrivial spectral information. We introduce a genuine three-body holonomy generated by $Q_3=\alpha_1\alpha_2\alpha_3$. The kinetic and pair-interaction parts commute with $Q_3$, while the constituent-mass operator anticommutes with it. Consequently, the massless system separates into the $Q_3=\pm1$ sectors, which acquire opposite holonomy phases $e^{\pm i\theta_3}$, whereas nonzero constituent masses mix the two sectors. This phase-sector-mixing mechanism makes the relative three-body phase dynamically accessible to the bound-state spectrum and establishes an operator-level mechanism through which the three-body holonomy generates a $\theta_3$ dependence of the physical three-body invariant mass. We further emphasize that the topological three-body holonomy is not automatically equivalent to a bare triple-contact potential; such an equivalence requires a regulated self-adjoint realization and a compatible interaction-dependent boost satisfying the Poincaré algebra. The resulting framework therefore connects genuine three-body phase information to the mass spectrum of a relativistic composite system while clearly separating the controlled holonomy construction from the unresolved short-distance triple-contact realization.

Inspired by black hole thermodynamics and other results suggesting that general relativity has some intrinsic connection with thermodynamics, this work aims to establish a quasi-local thermodynamic formalism in general spacetimes. With this purpose, we generalize the Bondi-Sachs formalism to past light cones associated with an arbitrary worldline. From this generalization, we demonstrate the emission of Hawking radiation in a quasi-local context through a few examples, suggesting that it may occur under mild conditions. In addition, we argue that an "irreversible inequality" is satisfied in typical gravitational systems, indicating that an analogue of the second law may be established. Together with a definition of internal energy derived from the Landau-Lifshitz pseudotensor, these results are used to construct a postulational framework that generalizes classical thermodynamics in order to include the dynamics of spacetime. As a consequence, it is possible to define the rest-mass density from thermodynamic variables, decoupling it from the dynamics of constituent numbers.

A recent article explores some respective contributions of the initial velocity and effective mass of a striking implement to the ultimately induced, post-collision velocity of a stationary strike target, in a combat sports context. Therein, it was found that multiplicative increases in the effective mass associated with a punch to the human body can have a greater quantitative effect on target velocity than that of its initial, pre-collision speed. Given that fold-increases in effective mass may be limited to low-integer multiples of a modest, literature-supported value that spans martial arts styles – and the fact that increases in the strike velocity are likely bounded by a factor of two, even for trained fighters – translating the aforementioned findings into incremental changes in either variable may be of more benefit to coaches who intend to apply the theoretical physics of fighting to develop practical training programs. Here, we build on previous work, studying the effects of marginal increases in effective mass on target velocity, and relating them to equivalent, single-unit changes in strike speed.

Quantum tunneling of particles across a black hole (BH) horizon provides an intuitive interpretation of Hawking radiation (HR), yielding a conceptually simple derivation of the thermal spectrum within a semiclassical framework. However, for BHs obeying the no-hair theorem, usual implementations of the complex-path (Hamilton-Jacobi) method typically rely on approximating the metric near the BH horizon. In this work, we demonstrate that the Hawking spectrum can be derived without such approximations by exploiting gauge invariance in quantum field theory and the separability of the Hamilton-Jacobi (HJ) equation arising from hidden symmetries of these spacetimes. This perspective shows explicitly why the BH behaves as an effective $(1+1)$-dimensional blackbody at the event horizon, provides the conditions needed to extend the HR to higher-spin particles, and offers a framework that can naturally incorporate back-reaction and quantum-gravity effects.

Know Your Client (KYC) onboarding in capital markets requires analysts to manually classify documents, extract structured data from heterogeneous sources, and validate compliance against complex regulatory policies. This process requires significant analyst time per client, with end-to-end onboarding often stretching to multiple weeks due to sequential handoffs. In this work, we analyze an on-boarding process and find that it comprises repeatable components well-suited to AI automation. We therefore propose a restructured workflow to be amenable to automation: we consolidate the traditional four-party process into two parties that share most of the work and can be automated together, eliminating intermediate handoffs that compound delays. To automate the remaining steps, we introduce Spectra, an AI-assisted document processing platform that combines a structured rules engine with LLM-based classification, extraction, and validation agents. The rules engine encodes compliance policy as a queryable database, enabling focused context injection that reduces token usage while improving extraction precision. Rather than a single monolithic prompt, the system decomposes document processing into isolated, auditable stages, each optimized independently and traceable to specific policy clauses. In evaluation on real KYC documents, Spectra achieves 100% classification accuracy and 89.4% extraction accuracy. Human review burden dropped by 96%.

We present a simple analysis of the energy references and of the magnetic-field-dependent and system-dependent features of the Landau energy levels. The energy reference used in describing Landau quantization is usually taken as given, with the Fermi levels of the electron system and the coupled reservoir aligned in thermal equilibrium. Nevertheless, the dependence of the Landau-level spectrum on magnetic field can make the relation between these energy references less transparent. In this work, we show that when a magnetic field is applied and the confined Landau levels $E_n$ are established, both the Fermi level $E_F$ and the bottom energy $E_0$ remain fixed when the magnetic field varies. We show that the bottom energy $E_0$ is independent of $B$, and determined by a system-dependent field $B_1$, defined by the condition that the first Landau level equals the Fermi energy. In contrast, the minimum of the parabolic well depends explicitly on the magnetic field. The resulting picture provides a simple way of displaying the relative motion of the Landau levels and the fixed Fermi level without introducing a magnetic-field-dependent energy zero.

We study dynamical sampling for graph signals using the spectral theory of the normalized graph Laplacian. Graph Paley-Wiener spaces GPW_w are defined as spectral subspaces associated with a bandwidth parameter w, and reconstruction is studied from measurements generated by iterates of bounded operators leaving these spaces invariant. For the lattice graph Z, we give an explicit spectral decomposition of the normalized Laplacian, identify the corresponding Graph Paley–Wiener spaces with classical bandlimited spaces on the torus, and recover the classical sampling theorem in this setting. Using frames of operator orbits, we construct dynamical frames for GPW_w(Z) and derive reconstruction formulas. We also discuss the extension to Z^n, where the bandwidth region is described by a sublevel set of the Laplacian symbol. Finally, for finite graphs, we formulate dynamical recovery in GPW_w(G)\) in terms of finite frames generated by iterates of the graph operator. Using the path graph P_N as a numerical model, we study how the number of iterates and the spatial distribution of the sampled vertices affect the optimal frame bounds and the conditioning of the resulting dynamical frame.

Semi-supervised federated learning (SSFL) trains models on clients' unlabeled data using a teacher to generate pseudo-labels, with a small labeled seed dataset on the server. Automatic Speech Recognition (ASR) is particularly fragile here: pseudo-label errors compound across the output sequence and across training rounds into divergence, leaving a large gap to fully-supervised FL. We show that closing this gap turns on two coupled design axes – the teacher (which model generates the pseudo-labels) and the anchor (the server-side updates on labeled data that stabilize training). On the teacher axis, a per-client online teacher (each client's own evolving model) diverges on its own, but once stabilized it matches or beats the broadcast global teacher (one server model, fixed within a round) – decisively in-domain and competitively under domain shift. As the seed grows stronger and the online teacher's advantage narrows, a transitioning teacher (global $\rightarrow$ online at round $r$) matches or beats both. On the anchor axis, the server must keep training on labeled data between rounds – otherwise the online teacher drifts – and this interleaving, more than the seed model, governs convergence. The two axes are inseparable: aggressive teacher choices pay off only once the anchor stabilizes training, which is highly sensitive to data augmentation and batch size – the settings that govern how much input and gradient noise the server injects. How much stabilization is needed is domain-dependent, governed by the dispersion of the seed data and its overlap with client data. These findings yield guidelines for SSFL in ASR training, improving over the strongest prior method on 9 of 11 pairs, by $20.8\%$ on average in-domain and $10.0\%$ cross-domain, narrowing the gap to fully-supervised FL.

Bioinspired feedback controls for pursuit, tracking, and collective motion are often expressed in terms of the relative configuration between interacting agents. In practice, however, onboard sensors may not directly provide all quantities required for feedback control, necessitating estimation of unobserved quantities. This paper develops a bioinspired internal model-based estimator for reconstructing those quantities from partial sensory observations and known self-motion. State reconstruction is posed as an optimization problem that treats the relative kinematics as constraints and minimizes the disagreement between the internal model outputs and measurements from onboard sensors. Pontryagin's Maximum Principle is used to derive the necessary optimality conditions. A forward-backward algorithm is used to provide a numerical solution and a moving horizon formulation is employed for online implementation. The estimator is evaluated numerically against classical state estimators. Real-time implementation of the proposed framework on robotic hardware is demonstrated through two pursuit strategies.

Retrieval augmented generation (RAG) systems have emerged as the dominant architecture for grounding large language model (LLM) outputs in verifiable external knowledge, yet their structural reliance on a dynamic retrieval pipeline introduces a largely unexplored class of adversarial vulnerability. Existing knowledge-base poisoning attacks are fundamentally static. Adversarial documents are pre-computed and injected without any awareness of what the victim system will actually retrieve for a given query, leaving the attack blind to the competitive documentary landscape that surrounds its payload in the generator's context window. Unlike traditional static poisoning attacks that are blind to the retrieved context, we introduce RAG-NAROK (Retrieval-Anchored Generation Negation And Response Quality Collapse), a RAG attack framework that adapts to the query text. RAG-NAROK exploits the transparency inherent in RAG pipeline to first extract the legitimate source identities, then generate Anchor-Specific Refutation documents that explicitly name and devalue retrieved sources while leveraging recency and authority biases to steer the text generation toward a target answer. Our results demonstrate that RAG-NAROK significantly outperforms static baselines across diverse domains, revealing a fundamental tension between RAG transparency and AI security.

We study a logarithmic re-gauging of the visual boundary, arising from the logarithmic metric on the sublinear Morse boundary introduced by Garg-Jana-Qing. For every proper geodesic hyperbolic space $X$, we prove that $d_{\mathrm{log}}(\xi,\eta)\asymp (1+(\xi\mid\eta)_o)^{-\theta}$. As a consequence, under the natural identification of boundaries, $d_{\mathrm{log}}$ is bi-Lipschitz equivalent to the polynomial Floyd boundary metric associated with the radial density $g_\theta(t)=(1+t)^{-1-\theta}$. Every quasi-isometry between proper geodesic hyperbolic spaces induces a bi-Lipschitz map between their logarithmic boundaries with a fixed admissible exponent. For every non-elementary hyperbolic group, the logarithmic boundary is nondoubling and has infinite Hausdorff and Assouad dimensions. Nevertheless, the logarithmic re-gauging preserves Nagata (capacity) dimension and continues to recover the asymptotic dimension of the group. It also detects one-endedness through linear connectedness (bounded turning) and preserves uniform perfectness. Thus the logarithmic and visual metrics are not quasisymmetrically equivalent, although several coarse-geometric features remain visible in the logarithmic boundary.

We often teach a colleague by showing the work and explaining the decisions as we go. How can we check what an agent understood from the same lesson? We introduce ShowTellArena, a benchmark protocol and public dataset for comprehension after narrated business demonstrations. The v1.0 release contains 50 business workflow tasks, with recordings, screenshots, narration, fixture seeds, and 502 questions. Tasks span finance, hiring, procurement, customer decisions, inventory, and logistics. The protocol holds the business scenario and quiz fixed while allowing each product to capture the lesson through its own teaching interface. Questions test operational rules, boundaries, exceptions, and errors in proposed automations. We analyze 218 selected pilot attempts across 39 workflow cases, including 28 cases attempted by all three evaluated systems. These exploratory results expose both answer errors and failures to complete the teaching experience. We describe the release's verification gaps and the pilot's uneven coverage, exclusions, and grading provenance. The contribution is an inspectable dataset and assessment workflow that others can extend; the selected pilot is not a controlled product ranking.

Hand gesture recognition is a key component in human-computer interaction (HCI), enabling intuitive interfaces for applications in gaming, virtual reality (VR), robotics, and more. This study integrates transformer-based machine-learning models for real-time hand gesture recognition, using hand-tracking data captured through the OpenXR standard in Unity. We leverage positional data of hand joints and wrist rotation angles to train a custom gesture recognition system. By utilizing the sequential modeling capabilities of transformers, the system captures temporal dependencies within short gesture windows and classifies gestures robustly across hand orientations and sizes. The results show a significant improvement in gesture classification accuracy. Building on this, we outline how the approach can be extended toward detecting the flow of movement, i.e., the transitions between gestures, as future work.

We show that any stable hypersurface of a symplectic $4$-manifold, on which the cohomology class of the symplectic form restricts to a multiple of a rational class, can be $C^\infty$-approximated by (possibly unstable) hypersurfaces whose closed characteristics equidistribute. The cohomological condition is necessary due to a famous example of Herman. The proof combines KAM theory with recent quantitative closing lemmas for Reeb flows and area-preserving maps. As a further application, we prove that every geodesible volume-preserving vector field on a closed three-manifold can be $C^\infty$-approximated by volume-preserving vector fields with equidistributed periodic orbits.

{O2 stars define the hottest end of the normal O-star sequence, but their identification requires blue-optical nitrogen diagnostics absent from many surveys.} {We reassess three luminous Large Magellanic Cloud (LMC) sources with multi-epoch Sloan Digital Sky Survey V (SDSS-V) spectroscopy obtained with the Baryon Oscillation Spectroscopic Survey (BOSS) spectrographs: \wsi, \lh, and \sk.} {We measure visit-resolved radial velocities and H, He, and N equivalent widths (EWs), compare robust combinations with LMC O2 templates at a common effective resolution, and use OSTAR2002 spectral energy distribution (SED) fits and a Gaia colour–magnitude diagram (CMD) as broad-band checks.} {The weighted $\log|{\rm EW}(\niv~\lambda4058)/{\rm EW}(\niii~\lambda4640)|$ values are $0.64\pm0.29$, $0.48\pm0.23$, and $0.34\pm0.44$ for \wsi, \lh, and \sk. The detection of \nv~$\lambda\lambda4604,4620$ supports the O2~If$^\ast$ and O2~V–III classifications adopted for \wsi\ and \lh, respectively. For \sk, the uncertain \niv/\niii\ ratio and non-detection of \nv\ motivate O2–3~V–III. Weak \hei~$\lambda4471$ may indicate unresolved later-type contamination in \wsi\ and \sk. The radial velocities of \sk\ show suggestive but non-significant variability. OSTAR2002 fits over 45–55~kK and the Gaia CMD place all three sources in the luminous very-early-O regime; \wsi\ shows a strong mid-infrared (MIR) excess, \lh\ is nearly photospheric, and \sk\ shows only a modest $3$–$8\,\mu$m excess.} {The BOSS spectra add \wsi\ and \lh\ to the LMC very-early-O population and identify \sk\ as an additional O2–3 source. These cases show how incomplete blue coverage and ambiguous catalogue associations can conceal the hottest massive stars.}

Reasoning-oriented reinforcement learning enables large language models to solve mathematical, coding, and other multi-step tasks, but shifts a substantial portion of the training cost to rollout, where trajectories are generated for policy updates. Efficient rollout mechanisms are therefore essential to reduce this cost while maintaining the freshness, consistency, and statistical validity of training data. This survey provides a systematic taxonomy of recent research on rollout efficiency for reasoning-oriented reinforcement learning, classifying existing approaches from both mechanism and bottleneck perspectives. Based on this taxonomy, we analyze how different technique families address distinct sources of rollout inefficiency, examine opportunities and potential conflicts for combining them, identify gaps in the evaluation and reporting of efficiency gains, and discuss open challenges and future research directions.

We show that $1.4<K_G^{\mathbb{C}}<1.404898554746$, where $K_G^{\mathbb{C}}$ is the complex Grothendieck constant. The upper bound improves on Haagerup's bound of $1.40490913\ldots$ from 1987, and the lower bound improves on Davie's bound of $1.33807$ from 1984. The upper bound combines ideas from the real and noncommutative complex cases of Grothendieck's inequality.

The evolution of wireless networks toward sixth-generation (6G) is driving demand for intelligent, flexible, and energy-efficient propagation control. Reconfigurable holographic surfaces (RHSs) enable fine-grained wavefront manipulation through densely packed programmable elements. This paper introduces shape-adaptive RHSs, which dynamically change surface geometry to provide extra spatial degrees of freedom. The adaptive design responds to environmental dynamics, mitigates blockages, and improves performance across urban communications, aerial coverage, and joint sensing and communication scenarios. We propose a network architecture that jointly designs the surface shape at the RHS and optimizes transmission. Simulation results show that the proposed shape-adaptive RHSs significantly increase data rates. These findings establish shape-adaptive RHS as a foundational technology for intelligent, context-aware wireless environments in 6G and beyond wireless networks.

Autonomous spacecraft guidance and control requires a fast solution to non-convex trajectory optimization, which can be accelerated by providing a near-optimal initial guess to an optimization protocol, i.e., warm-starting. A robust warm starting method is especially useful for rendezvous, proximity operations, and docking (RPOD) in cislunar space, where the underlying dynamics become severely nonlinear and chaotic compared to those in Earth orbit, especially at perilune. This paper extends the Autonomous Rendezvous Transformer (ART), a transformer-based warm-start trajectory generation method, to cislunar RPOD scenarios for the first time. To accurately and reliably solve the nonconvex optimal control problems (OCPs) posed by these scenarios, a new and enhanced version of ART, ART-TWIN (Two-Way INference), is introduced. Inspired by forward-backward shooting methods used in other trajectory design applications, ART-TWIN autoregressively generates two arcs, one from the initial state and one from the desired terminal state, that are patched together at the midpoint of the timeseries. When evaluated on a set of simulated rendezvous scenarios that are initialized at perilune, ART-TWIN is demonstrated to substantially accelerate convergence and increase feasibility guarantees when used as a warm-start to sequential convex programming (SCP), compared to convex relaxations and the original ART. These results illustrate the necessity of ART-TWIN's dual-arc generation to enable the viability of and gain benefits from using transformer-based warm-start methods in the most challenging areas of the cislunar dynamical regime.

We investigate analytically and numerically the floral deformation of particle rings induced by higher-order polynomial modes of pp-waves. The transverse tidal eigendirections determine $m$ interlaced sectors of radial stretching and compression, while a weak-pulse solution provides the complete first-order response of the ring. Direct geodesic integration for $m=2,3,4$ confirms the predicted orientation and discrete symmetry. A Fourier analysis shows that the response remains overwhelmingly dominated by the angular mode $n=m$; in particular, the loops found for $m=4$ arise from geometrical folding of the fundamental mode rather than strong harmonic mixing. The temporal moments of the pulse further distinguish displacement from velocity memory without changing this angular signature.

Strong entanglement limits the reach of tensor-network descriptions of quantum many-body systems. We study Gaussian-augmented matrix product states (GAMPS), formed by applying a free-fermionic (Gaussian) unitary to a matrix product state. To explore the capabilities of this family, we consider random GAMPS}, a minimally structured ensemble obtained by independently sampling random matrix product states and Gaussian unitaries. Exploiting the algebraic structure of Gaussian unitaries, we develop a replica tensor-network method that computes averages over the random GAMPS ensemble for hundreds of qubits. Random GAMPS exhibit nearly maximal volume-law entanglement already at modest bond dimension, while other quantum resource quantifiers rapidly approach Haar-random state values as the bond dimension increases. The entanglement scaling suggests an exponential reduction in the bond dimension needed to approach the volume-law entropy compared with unaugmented matrix product states. We find a state two-design behavior of random GAMPS at fixed accuracy with a bond dimension independent of system size. Our results reveal how Gaussian augmentation expands the expressive power of tensor networks.

For fixed integers $d,r\ge1$ with $dr\ge2$, the $r$-free points of $\mathbb{Z}^d$ are those whose coordinate gcd is not divisible by the $r$th power of any prime. Fix $1\le q\le \infty$. A lattice point is $R$-deep if every lattice point in the closed $\ell_q$-ball of radius $R$ centred there is non-$r$-free. We mark each finite nearest-neighbour component of non-$r$-free points containing an $R$-deep point by its lexicographically least such point. Uniformly for $1\le q\le\infty$, the densities of $R$-deep points and of these representatives both have the asymptotic form $\exp\left\{-\frac{v_{d,q}R^d}{\zeta(dr)}\left[d(dr-1)\log R+dr\log\log R-\Upsilon_{d,r,q}+r\frac{\log\log R}{\log R}-\frac{\Lambda_{d,r,q}}{\log R}+O_{d,r}\left(\frac{(\log\log R)^2}{(\log R)^2}\right)\right]\right\}$ as $R\to\infty$, where $v_{d,q}$ is the volume of the unit $\ell_q$-ball and $\Upsilon_{d,r,q},\Lambda_{d,r,q}$ are explicit constants. For deep points, this sharpens the positive-density hole constructions of Baake, Moody, and Pleasants and of Pleasants and Huck, and the latter authors' upper bounds for sparse-pattern frequencies. For $d=1$, consider the densities of $r$-free integers followed by at least $g-1$ consecutive non-$r$-free integers, or by exactly $g-1$ such integers and then another $r$-free integer. Both have the asymptotics $\exp\left\{-\frac{g}{\zeta(r)}\left[(r-1)\log g+r\log\log g-\widehat{\Upsilon}_r+r\frac{\log\log g}{\log g}-\frac{\widehat{\Lambda}_r}{\log g}+O_r\left(\frac{(\log\log g)^2}{(\log g)^2}\right)\right]\right\}$ as $g\to\infty$, where $\widehat{\Upsilon}_r=\Upsilon_{1,r,1}+(r-1)\log2$ and $\widehat{\Lambda}_r=\Lambda_{1,r,1}+r\log2$. The exact-gap expansion improves Grimmett's leading asymptotic and refines the formula in Jiang's recent preprint. For fixed $r$, we also obtain estimates uniform in growing dimensions $d=O_r((\log R)^r)$.

We present the Quantum Atom Control Kit (QuACK), a low-cost FPGA-based control system that provides memory-efficient precise signal timing and low-delay measurement-based decision making for operating neutral atom optical tweezer array experiments. The system consists of an FPGA module connected to a host computer via USB 3.0 and multiple cable-connected modular daughterboards enabling a myriad of configurable I/O formats. In this work, digital input and output as well as analog outputs are demonstrated. Multiple experimental sequence segments can be encoded with 5 ns timing resolution. On-chip condition kernels enable fast branching between segments based on digital inputs, eliminating host intervention for time-critical decisions, while high-speed USB communication allows rapid sequence updates and host-level orchestration. We demonstrate initial system loading and rearranging neutral ytterbium-171 atoms in optical tweezers. The combination of high-resolution timing, on-chip logic, and distributed orchestration architecture makes this system well suited for atomic, molecular, and optical (AMO) experiments pursuing applications in quantum information science.

To comprehensively understand the environmental effects on galaxy evolution, it is crucial to precisely map the distribution and thoroughly examine the properties of galaxies, including low-mass building blocks. Deep narrow-band imaging observations conducted with Hyper Suprime-Cam (HSC) on the Subaru Telescope systematically probed the distribution of low-mass star-forming galaxies as H$\alpha$, [O {\footnotesize II}], and [O {\footnotesize III}] emission-line galaxies at $0.4\leq z\leq1.6$. We perform a comprehensive spectroscopic follow-up observation of these emission-line galaxies with narrow-band-based emission-line fluxes $\geq 2\times10^{-17}$ erg s$^{-1}$ cm$^{-2}$ within $\approx1.8$ deg$^2$ area centered on the COSMOS field using `Ōnohi`ula - Prime Focus Spectrograph (PFS) on the Subaru Telescope. In this paper, we outline the data product and present the large-scale structures at $z>1$. 9079 out of 9917 of the targets are observed, and the redshifts of 92%~of the observed targets are successfully confirmed. We survey groups, filamentary large-scale structures, and voids at $z=1.19$, $1.47$, and $1.60$ based on the three-dimensional distribution of $520-920$ [O {\footnotesize II}] emission-line galaxies with stellar masses $M_\star\geq10^{9.5}~M_{\odot}$ at each redshift. Groups with halo masses $1\times10^{12}-6\times10^{13}~M_{\odot}$ are identified along the filaments, whereas the most massive groups are located at the densest intersections of filaments as predicted by cosmological simulations. Voids hidden on the surface density maps are robustly extracted on the volume density map. Our spectroscopic sample provides a powerful basis for quantifying the evolution of galaxies along the diverse environments back in time toward cosmic noon epoch in forthcoming papers.

Geographic measurements are often sparse, leaving large areas without labels for the quantities we want to map. Geographic implicit neural representations (INRs) provide coordinate-based embeddings that can be combined with sparse labels to predict at unsampled locations without satellite imagery at inference. Yet existing INRs are largely evaluated with random holdouts, leaving their ability to generalize across larger geographic gaps unclear. We introduce Matryoshka Implicit Neural Distillation (MIND), a geographic INR whose spatial granularity can be adjusted after training. MIND distills several pretrained geospatial models using nested supervision at increasing embedding dimensions, dividing the representation into contiguous chunks. Early chunks capture broad spatial patterns, while later chunks add increasingly local variation. Downstream models can retain only the leading chunks or use our Chunked Penalty to reduce reliance on later chunks without retraining the INR. We also introduce CoordBench, comprising $52$ datasets and $78$ targets with both random and regional holdouts at multiple spatial scales. Across CoordBench, fine-scale features help most when labels are nearby, while smoother representations generalize better across larger geographic gaps. MIND with the Chunked Penalty achieves the highest aggregate regression and classification performance among tested INRs and the highest overall performance under regional holdout. These results show that geographic representations should be evaluated and adapted according to the spatial separation between labeled and prediction locations.