Chiral-induced spin selectivity (CISS) has been shown to enhance magnetic sensitivity in radical pair mechanism (RPM) models under specific Hamiltonian conditions, yet whether these enhancements persist across a broader parameter space remains untested. We incorporate the CISS effect as a spin-dependent initial state and recombination operator and systematically evaluate the spin dynamics of a model radical pair across a comprehensive parameter sweep of the RPM Hamiltonian. We characterise the orientational response through symmetric and antisymmetric decomposition of the yield distribution under field reversal, providing a direct quantitative signature of CISS-induced symmetry breaking. Our analysis demonstrates that CISS does not function as a generic amplifier of magnetic sensitivity. Claimed enhancements are conditional on the relative alignment of the internal hyperfine and dipolar interaction axes, arising specifically under conditions of non-collinear internal interactions. Extension to a two-nucleus model confirms that these enhancements are sensitive to nuclear spin. CISS-induced effects observed in the single-nucleus model are substantially suppressed when a second collinear nucleus is introduced, with the exception of the hyperfine axis rotation sweep where non-collinear tensor misalignment drives a robust antisymmetric response. These findings indicate that the conditions for CISS-enhanced magnetoreception are more stringent than previously demonstrated, requiring highly ordered and rigid molecular geometries to sustain the effect.

A weave is a type of textile that consists of vertical and horizontal threads, and typically it has a periodic structure. In this paper, we regard a weave as a link in the thickened torus with a diagram consisting of closed geodesics. As main results, we characterize isotopies and hyperbolicity of weaves to determine them from diagrams. Moreover, we show that there does not exist an essential Conway sphere for a weave. We use normal positions of essential surfaces of weave complements to describe them.

Binding energy, which quantifies pair formation, is a key factor in the emergence of superconductivity. Here, we show that even when multiple spins are complexly coupled, hole-doped systems, which can be mapped onto the universal hardcore boson model in the strong-binding-energy limit, exhibit promising signatures of superconductivity. We analytically and numerically demonstrate this concept in the double Kondo lattice model. Using the density-matrix renormalization group method, we show that a pairing state is maintained via a crossover even for parameters away from the strong-coupling regime. Additionally, we find that once binding energies are sufficiently generated, pair correlations develop similarly regardless of the details of local spin correlations. Our findings suggest useful guidelines for research on superconductivity.

Let $G$ be a finite group and $T(G)$ be the sum of the degrees of its irreducible complex representations. We investigate the relationship between $T(G)$ and the number of twisted involutions $m_σ= |\{g \in G \mid σ(g) = g^{-1}\}|$ for an automorphism $σ$. While it is known that $T(G) = m_e$ for the identity automorphism $e$ in certain cases (e.g., real characters), we analyze this relation for non-identity automorphisms of groups of order $p, 2p, p^2$. We prove that for the family of Dihedral groups $D_n$, the inequality $T(D_n) \geq m_σ$ holds for all $σ\in \mathrm{Aut}(D_n)$. We provide a complete classification of $m_σ$ using number-theoretic properties of the automorphism parameters.

The identification and classification of different magnetic states are essential for understanding the complex behavior of magnetic systems. Traditional approaches that rely on handcrafted features or manual inspection often fall short, particularly when dealing with subtle or topologically complex spin textures. In this study, we present an automated deep learning model that employs an EfficientNetV1B0 Convolutional Neural Network to classify nine distinct magnetic states, including both ferromagnetic (FM) and antiferromagnetic (AFM) spin textures such as AFM skyrmions and AFM stripe domains. The spin configurations are generated through atomistic spin dynamics simulations using the Spirit code, then visualized with VFRendering to produce RGB images, which serve as inputs to the classification model.

In this paper, we study tensor spaces beyond the boundary format and analyze whether the general critical space coincides with the general span of singular vector tuples. For all tensor spaces exceeding the boundary format by one in an arbitrary number of factors, we relate the codimension of this span within the critical space to the dimension of the kernel of a map in cohomology. Furthermore, we exhibit an infinite family of order-three tensors with a defective behavior: the general span of singular vector tuples achieves the maximum possible codimension rather than the expected minimum. Finally, we conjecture a classification of the behavior of critical spaces in this regime and draw a connection to Koszul cohomology.

Extremely large aperture arrays (ELAAs) and millimeter-wave (mmWave) technologies are essential for achieving high data rates in future wireless communication systems. To perform precise beamforming, these systems require accurate channel estimation, in which the near-field wavefront curvature effect must be taken into account. Existing channel estimation methods rely on the spherical wavefront channel (SWC) model, which is suitable for near-field propagation with point sources, scatterers, and reflection planes. However, when a near-field curved reflecting surface exists, the wavefront of the reflected wave becomes anisotropic rather than spherical, causing the SWC model to become inaccurate. To address this problem, in this paper, we formulate a parameterized model for the anisotropic wavefront channel (AWC). Using this model, we propose a channel estimation algorithm based on physical parameter recovery for the AWC. Simulation results reveal that the AWC no longer retains sparsity in the angle-distance domain. Furthermore, the results demonstrate how different physical characteristics of the propagation scenario affect the degree of wavefront anisotropy, and confirm the effectiveness of our proposed algorithm in AWC scenarios.

The Ramsey number $\mathrm{R}(G_1,G_2)$ is the smallest integer $N$ such that any red-blue coloring of the edges of the complete graph $K_N$ contains either a red copy of $G_1$ or a blue copy of $G_2$. In 2022, the third author and others gave lower and upper bounds of the Ramsey number $\mathrm{R}(W_n,W_n)$, where $W_n$ is the wheel graph with $n$ vertices. In this paper, we improve their bounds by showing that $3n-2\leq \mathrm{R}(W_n,W_n)\leq 6n-6$ for even $n\geq 8$ and $2n\leq \mathrm{R}(W_n,W_n)\leq \frac{9n-7}{2}$ for odd $n\geq 7$. Furthermore, we give recursive bounds for the $k$-colored Ramsey number for $W_n$.

The Monge-Ampère equation is a fundamental fully nonlinear elliptic partial differential equation that finds extensive applications across multiple disciplines. This study proposes a novel physics-informed neural network integrated with attention feature expansion (PINN-AFE) for its numerical solution. A multi-head attention enhanced feature pool is constructed to enable adaptive nonlinear feature representation, and input convex neural networks are adopted to impose strict convexity of solutions with rigorous theoretical guarantees. Meanwhile, a dynamically weighted loss function combined with hybrid optimization is formulated to accelerate training convergence. Comprehensive numerical experiments validate the accuracy and computational efficiency of the developed framework. The PINN-AFE paradigm is further extended to image processing tasks, delivering high-quality and physically consistent results in both image enhancement and medical image registration scenarios.

The Internet of Medical Things (IoMT) marks a shift toward decentralized healthcare, enabling continuous monitoring and personalized care through connected wearable and implantable devices. However, ensuring the trust and integrity of these devices themselves remains a major challenge, as physical compromise or counterfeiting can directly endanger patient safety, privacy, and data integrity. This work presents QT-PUF, a gate-tunneling-leakage-based physical unclonable function (PUF) that leverages quantum-mechanical gate leakage resulting from process-induced variations in standard CMOS devices. A differential readout circuit with a pseudo-resistor I-to-V frontend is proposed to convert the picoampere-level leakage variations into digital responses. Unlike existing PUFs such as those based on memory, ring oscillators, or arbiters, which are less suitable for ultralow-power IoMT devices (due to additional circuitry, power overhead, or poor stability), QT-PUF requires no external excitation or stabilization and operates under static bias. Simulation-based measurements for a $\mathbf{65}$~nm CMOS process demonstrate an entropy of $\mathbf{0.9999998}$, an FHD of $\mathbf{0.5001}$, and an average power (energy) consumption of $\mathbf{96.04}$~nW/bit ($\mathbf{19.21}$~fJ/bit, respectively) at $\mathbf{1.2\,V}$ and $\mathbf{35\,^{\circ}C}$ for the proposed PUF. It operates reliably across $\mathbf{0.9}\text{--}\mathbf{1.3}$~V and $\mathbf{0}\text{--}\mathbf{100\,^{\circ}C}$ with an average BER below $\mathbf{0.000163}$ across $\mathbf{1.0}\text{--}\mathbf{1.3}$~V and $\mathbf{10}\text{--}\mathbf{70\,^{\circ}C}$ within the operating conditions of typical implantable devices.

We propose a computationally simple framework for clustering functional data based on Gaussian-process-generated random projections. In this approach, each curve is first projected onto a large collection of independent Gaussian process realizations. The resulting high-dimensional representations are clustered using the Mean Absolute Difference of Distances (MADD), a dissimilarity measure well suited for high-dimensional settings. A population-level analysis of this dissimilarity provides insight into how random projections help capture distributional differences between functional populations. We introduce a second stage of clustering to additionally leverage on data-driven projection directions. Thus, in Stage I, an initial clustering is obtained using a set of prespecified projection families. In Stage II, this partition is refined by constructing Gaussian random projections based on an estimated covariance operator that uses the first stage of cluster labels. Finally, a normalized cost function is used to select the optimal clustering among candidate solutions. The proposed clustering algorithm is broadly applicable to diverse functional data regimes including irregular and partially observed data. Through extensive simulations and real-data applications, we show that the proposed method achieves a high degree of accuracy and outperforms many of the state-of-the-art methods across a wide range of functional data settings.

We present an implementation of the MAcNLOPS matching prescription for $pp \to ZZ$ production in a MadGraph5_aMC@NLO + Pythia8 setup. Starting from a standard MC@NLO event sample, negative H events are removed and compensated by a veto applied to the first shower emission of the S events. The implementation is validated against MC@NLO for radiation-sensitive and inclusive diboson observables. Agreement is found up to a rather small power-suppressed contribution affecting the very low-pT region. The method removes all negative H weights with negligible additional computational cost, while negative S weights are left unchanged, showing that MAcNLOPS is a promising alternative to MC@NLO with a reduced fraction of negative weights.

A Full Turbulence Simulation (FTS) of turbulent channel flow at friction Reynolds number (Re_tau) approx 1000 was performed by resolving the Kolmogorov wavenumber in all spatial directions. At this Reynolds number, the intermediate layer attains a physically meaningful width and is fully resolved in the present computation, providing the reference dataset that captures its turbulence and dissipation characteristics with high fidelity. The wall-normal grid spacing of the FTS also confirms that, when the Kolmogorov length scale is sufficiently resolved, the second-order central-difference scheme introduces no adverse numerical effects in the wall-normal direction. In the wall-parallel directions, two resolution criteria were identified based on the present FTS: a first-approximation DNS resolution that resolves more than 99 percent of the turbulent kinetic energy and dissipation rate (Delta x+ approx 19, Delta y+ approx 8, where Delta x+ and Delta y+ denote the streamwise and spanwise spatial resolutions in wall units) and a full dissipation-resolution criterion (Delta x+ approx 7.5, Delta y+ approx 5.0). The first-approximation resolution by means of a spectral method reproduces the essential turbulence statistics within 1 percent accuracy while requiring only one-eighth of the grid points used in the FTS, demonstrating its practical efficiency. In contrast, even the highest-resolution second-order central-difference case (Delta x+ approx 5.0, Delta y+ approx 4.5) fails to match the accuracy of the first-approximation spectral resolution. These findings provide important resolution guidelines for high-Reynolds-number DNS, particularly for simulations at Re_tau = O(10^4).

As first shown by Shafranov, toroidal plasmas in magnetohydrodynamic equilibrium tend to expand in major radius when the pressure is increased. Here, an average measure of the resulting Shafranov shift is introduced, and its properties are discussed for various classes of optimised stellarator configurations. It is shown to be particularly small in quasi-helical and quasi-isodynamic stellarators with a large number of field periods, which are thus particularly robust to variations in the plasma pressure.

We study site percolation on the diamond hierarchical lattice, a finite-dimensional fractal network, using an exact generating-function analysis. In contrast to bond percolation, site percolation on this lattice does not undergo a transition from a nonpercolating phase to a percolating phase. Instead, the system exhibits a nonpercolating phase for $p<p_{\rm c}$ and a critical phase for $p>p_{\rm c}$. In the critical phase, the size of the largest cluster remains subextensive, scaling as $N^{ψ(p)}$, where the fractal exponent $ψ(p)$ varies continuously with $p$. By analyzing the renormalization-group recursion relation in the vicinity of $p_{\rm c}$, we show that the correlation length exhibits a Berezinskii-Kosterlitz-Thouless-type essential singularity, $ξ(p)\sim \exp \left({\rm const}/\sqrt{p_{\rm c}-p}\right)$ for $p \to p_{\rm c}^-$, which is further confirmed by finite-size scaling analyses showing excellent data collapse. These results demonstrate that critical phases in percolation can emerge even on finite-dimensional networks and that exponential volume growth is not necessary for such phases to appear. We argue that the critical phase on the diamond hierarchical lattice stems from site dilution remaining relevant under renormalization.

In 2020, Peter Larsen reported flaws in the methods for centrosymmetry parameter computation in the existing molecular dynamics and analysis packages. He proposed an intuitive an mathematically rigorous formulation for centrosymmetry parameter in terms of minimal-weight matching (MWM) on a fully-connected graph of atomic neighbors. He proposed using Edmonds' blossom algorithm for computing such a matching. In this paper, we investigate an alternative algorithm for MWM CSP computation using path finding approach and A* algorithm.

Mainstream film is one of the richest sources of cultural content that AI systems learn from. Yet we have few tools for measuring the gender values it encodes. We present a proof-of-concept framework that turns fictional film characters into surveyable LLM agents. Using 160 U.S. films (1990--2019), we build 734 character agents from script dialogue and scene descriptions, condense their personas via expert-style reflections, and simulate World Values Survey gender-attitude responses. Agents reproduce systematic gender differences without explicit demographic prompting, suggesting attitudes emerge from behavior rather than identity labels. Benchmarked against historical survey data, agents exaggerate gender gaps and show greater decade-to-decade volatility than real populations. Narrative sharpens rather than homogenizes gender contrasts, complicating the consistent-input assumption underlying cultivation theory's mainstreaming mechanism. AI systems trained on such corpora may inherit this stylization before any model-level amplification occurs.

We present a "black box" proof of mean-field near-critical behaviour for a family of functions on $\mathbb Z^d$ (${d>2}$) satisfying a short list of assumptions. The functions represent two-point functions of a lattice statistical mechanical model in the subcritical or critical regimes, and are proved to have decay of the form $|x|^{-d+2+\varepsilon}\exp[-c|x|/ξ]$, for any $\varepsilon>0$. The black box applies to several models for which commonplace methods can be used to verify the assumptions. Applications include models of self-avoiding walk, percolation, spins (Ising, XY, $|φ|^4$), and lattice trees, all above their upper critical dimensions. The proof is based on random walk techniques, and provides a new, unified, probabilistic, and relatively simple proof of mean-field near-critical behaviour.

Navigation mesh (Navmesh) inconsistencies affect the player experience by directly impacting the navigation systems used by non-playable characters (NPCs) in game environments. While navmeshes are generated from world geometry using well-established algorithms, environments change throughout development as terrain is adjusted and assets are moved or replaced, resulting in mismatches between the navmesh and the actual environment. Existing automated approaches attempt to detect navigation issues using exploration agents and reinforcement learning techniques. However, since these methods rely on the navigation data itself or evaluate navigation behavior indirectly, they do not explicitly verify whether the navigation representation reflects the walkable space defined by underlying geometry. This paper presents a framework for validating navigation meshes through an independent, geometry-driven analysis of navmesh correctness. The approach reconstructs walkable space directly from environment geometry using a voxel-based representation, followed by constraint-aware traversal and connectivity evaluation. Validation is formulated as a prioritized search problem over the voxel space, where reinforcement learning guides sampling toward regions more likely to exhibit inconsistencies. At each sampled location, reachability derived from the voxel representation is compared against reachability obtained from the navmesh via engine-level queries. Experiments across multiple large-scale open-world game environments show that the approach consistently lowers exploration effort while maintaining similar defect detection coverage. The framework runs offline within the game engine and can be integrated into automated quality assurance pipelines. Since the method relies on geometry, it can be adapted across game engines with minimal changes, making it suitable for production deployment.

Since 2016, Apple has claimed that device analytics collected to improve user experience are protected by differential privacy (DP). Apple's DifferentialPrivacy framework is deployed across its operating systems and handles sensitive signals such as Safari domains, keyboard events, photo attributes, and health-related reports. Because Apple has not open-sourced its privatization algorithms, these privacy claims have been difficult to verify independently. We present a client-side audit of Apple's DP framework on macOS Sonoma 14.2 and Sequoia 15.6. We reverse engineer the shipped binaries, recover Objective-C interfaces, build runtime harnesses that execute Apple's deployed mechanisms, and test whether their outputs match the advertised privacy guarantees. Our audit covers nearly all active deployed mechanisms, including Count Median Sketch, Hadamard-CMS, randomized-response mechanisms, and Prio-style secure aggregation. We find multiple implementation bugs and misconfigurations. Every audited mechanism that relies on floating-point noise fails to meet its advertised DP or zero-knowledge proof guarantee, due to insecure samplers with known floating-point vulnerabilities. We also find secure-aggregation configurations with local DP disabled, exposing pre-aggregation records to any party with access to those logs. Overall, we find DP violations in 5 of 9 audited mechanisms, affecting 87% of data collection in macOS Sonoma and 68% in Sequoia. We also identify public leaked iPhone logs that can be decoded to recover private information, including Safari domains and keyboard emoji signals.

Over the last decades, several independent studies have shown the need for large convective boundary mixing (CBM) and convective core sizes in massive stars to reproduce a variety of their observed properties. Yet, stars more massive than 20Msun lack a quantitative prescription for CBM as well as an unequivocal constraint on the internal mixing mechanisms acting in them. We use the apsidal motion observed in the twin binary HD152248 - linked to the internal stellar structure constants k2 of the stars - to constrain massive stars' internal density stratification and CBM. We build GENEC stellar models assuming two different angular momentum transports: purely hydrodynamic (hydro) and magneto-diffusive (magnetic). We confront single- and binary-star models to assess the impact of tidal locking on the star's evolution. We investigate the impact of CBM (overshooting), metallicity, initial helium abundance and mass, mass-loss rate, and mixing length parameter on the evolution of stellar parameters. We highlight that k2 from the models are systematically larger than observed ones, the so-called k2-discrepancy. Models predict stars with too low a density contrast between their core and external layers. Both hydro and magnetic models require large step-overshoot of 1.2 to reproduce stellar parameters, including k2. Other parameters have almost no impact. Given the efficiency of tides to synchronise systems, the assumption of pseudo-synchronisation is sound for this system. It sets an upper limit on the misalignment angle of stellar rotation axes of ~50°. Even with such unexpected large angles, the k2-discrepancy is not solved. Even if the mass-loss rate was underestimated by a factor two, it would have no impact on stellar parameters evolution, including k2. It demonstrates that the apsidal motion is a powerful, robust means to probe stellar interiors.

Distributed optimization is widely used in large-scale and privacy-preserving machine learning, where each agent stores a local objective and communicates only with its neighbors in a connected network. We study decentralized second-order optimization and focus on consensus procedures that approximately average local iterates, gradients, and Hessians through neighbor-to-neighbor communications. We propose a general Decentralized Cubic Newton method for convex optimization under $L_1$-smoothness of gradients and $L_2$-Lipschitz continuity of Hessians, and develop a theory that accurately tracks the inaccuracies caused by consensus and by disagreement between local iterates. Under these assumptions, the method matches the iteration complexity of the exact Cubic Newton method and requires only additional polylogarithmic communication-round overhead to reach the necessary consensus accuracy. We further propose an Accelerated Decentralized Cubic Newton method for strongly convex objectives and show that it matches the iteration complexity of the exact Accelerated Cubic Newton method, again with only additional polylogarithmic communication-round overhead. Finally, although the general method requires exchanging full $d \times d$ Hessian matrices, we show how it can be implemented for generalized linear models by transmitting only vectors, making the approach substantially more practical in high dimensions.

Time synchronization is a crucial requirement in quantum key distribution (QKD)8 protocols, ensuring accurate key generation via the correct assignment of bits of raw key and9 enabling eavesdropping detection via the precise recording of photon statistics. State-of-the-art10 experiments typically use an extra channel to synchronize the clocks of the transmitter and receiver11 via classical signals. In this work, we study the possibility of performing clock synchronization12 via the signals used for the key generation, which are already present in decoy-state-based BB8413 protocols.14 Without altering the protocol in any way, we use the different mean photon numbers of the15 signal and decoy states for time synchronization without a dedicated physical channel capable of16 clock synchronization. The proposed method relies only on the photons sent and received for17 key generation and does not require any change to the QKD protocol. The only change in the18 experiment is on the software level, thus making it very simple to implement.19 We demonstrate clock synchronization method in a simulation of a specific fiber-based QKD20 experiment. Like other decoy-state-based BB84 protocols, it is based on weak coherent pulses.21 In this simulation, we investigate the parameter space to find limits and optimal choices of our22 proposed method.23 In addition to the non-protocol-altering clock synchronization method, we also discuss an24 approach that significantly improves performance in lossy channels by introducing an additional25 decoy state with a very high mean photon number.26 By eliminating the need for an extra channel capable of clock synchronization, both methods27 proposed potentially reduce the complexity and cost of QKD systems and improve their agility

We show that for every foliation $\mathcal{F}$ of rank $r$ on a normal projective variety, the optimal constant in the bend-and-break inequality for tangent rational curves is $r+1$. The proof combines the method of Bogomolov--McQuillan and the bend-and-shatter method developed by Jovinelly--Lehmann--Riedl. The proof of the main result of this paper substantially uses generative AI, particularly the Rethlas system.

Boltzmann transport theory, the standard framework for predicting thermal conductivity, assumes that every vibrational mode eventually scatters, acquiring a finite lifetime that yields a convergent, length-independent thermal conductivity: Fourier's law. Here we show that this assumption fails in a real molecular system. Through atomistic simulations of Au-alkane-Au single-molecule junctions spanning five orders of magnitude in length (0.5 nm to 4 $μ$m), we find that thermal conductivity never converges. Transport is ballistic for up to one hundred nanometers at room temperature, extending nearly two orders of magnitude beyond existing single-molecule measurements. Past this window, conductivity diverges as $L^{1/3}$, the scaling predicted by the Kardar-Parisi-Zhang universality class for momentum-conserving systems. Frequency-resolved decomposition of the heat current reveals the mechanism behind the divergence. Low-frequency acoustic modes never thermalize: protected by momentum conservation, they remain ballistic at every chain length, still carrying 50% of the total heat current at $L = 2 μ$m. All other modes thermalize collectively as discrete vibrational states merge into scattering-active phonon bands with increasing length. Hence, the diverging conductivity emerges from the boundary between these coexisting transport regimes: as $L$ grows, the onset of scattering shifts progressively toward lower frequencies, suppressing the ballistic channel at a rate that sustains the $L^{1/3}$ divergence, leaving a finite contribution at every length. This coexistence of permanent ballistic and well-behaved diffusive transport, anticipated in abstract one-dimensional lattice models, survives the structural and chemical complexity of real micrometer-sized junctions.

The advent of Extremely Large Telescopes ELTs, ground-based optical or infrared observatories with primary mirrors exceeding 20 m heralds a transformative epoch in observational astronomy. This article examines the dawn of this new era and the three upcoming facilities in the optical infrared band the Giant Magellan Telescope GMT, the Thirty Meter Telescope TMT, and the European Extremely Large Telescope ELT. This article will focus on the ELT, while a sequel will cover GMT and TMT. We describe the key technological breakthroughs enabling its construction, most notably the segmented mirror design, advanced adaptive optics AO, and laser guide star systems. These innovations will deliver more than an order of magnitude leap in light-gathering area and spatial resolution, providing image sharpness exceeding that of spacebased telescopes for widefield observations. The scientific impact of the ELT is profound and multifaceted. We discuss its inception and construction milestones and explore its potential to directly image and characterize the atmospheres of Earth like exoplanets, searching for biosignatures, and trace the formation of the first stars, galaxies, and supermassive black holes. This paper concludes that ELTs are not mere incremental improvements but foundational instruments that will redefine the frontiers of astrophysics, address some of science's most enduring questions, and inevitably lead to discoveries beyond current prediction.

Contributions from connected quintuple excitations in coupled cluster theory can reach the 0.5 kcal/mol range, important enough to matter in accurate computational thermochemistry, yet the very steep $\propto N^{12}$ CPU time scaling impedes routine evaluation. We show that for the differential contribution of quintuples, convergence of a frozen natural orbital (FNO) expansion with respect to the NO cutoff is rapid enough to make FNO-CCSDTQ(5)$_Λ$ with cutoffs of 0.0025 or 0.001 viable alternatives. A naive extrapolation to zero cutoff from \{0.005,0.0025\} works surprisingly well as a low-cost option. Interestingly, FNO convergence is definitely slower for second-row than for first-row compounds.

We show that $E$-theory for locally compact Hausdorff spaces constitutes a six-functor formalism which is equivalent to the six-functor formalism of $\mathrm{E}$-valued sheaves. We furthermore show that the $E$-theory category for locales that can be written as unions of finite open sublocales is equivalent to the category of $\mathrm{E}$-valued cosheaves.

Exploiting newly available data on Gopakumar-Vafa invariants at high genus for one-parameter hypergeometric Calabi-Yau threefolds, we study numerically the growth of the 5D indices, stable pair (PT) invariants and rank one Donaldson-Thomas (DT) invariants at large charges. For the 5D index $Ω_{5D}(d,m)$, below a critical value of the angular momentum $m$, we find perfect agreement with the Bekenstein-Hawking-Wald entropy of rotating 5D BMPV black holes, including the subleading correction from 4-derivative interactions. When $m$ exceeds the critical value, the 5D index is instead dominated by black rings with the smallest possible dipole charge. The stable pair invariant $PT(d,m)$, which is determined by 5D indices, has a similar black ring/hole transition at negative $m$ (now interpreted as the D0-brane charge) but surprisingly exhibits two other phase transitions at positive $m$: first, to a plateau and then to a polynomial growth $\sim m^{2d-1}$. In each phase, we derive an approximate expression for the invariant. Finally, the rank one DT invariant $DT(d,m)$ is similar to $PT(d,m)$ at negative $m$, and then transitions to a phase dominated by D0-branes, with entropy of order $m^{2/3}$. Along the way, we determine the fixed genus, large degree behavior of GV invariants (including the overall $g$-dependent constant), extend it to an approximate formula valid also for large $g$, point out the unreasonable effectiveness of a simple PT/MSW relation, and study the growth of topological free energies at fixed degree, confirming a conjecture of Mariño.

The number of finite additive 2-bases is known to grow exponentially. While this fact has been established by Marzuola and Miller (2010) using complex analytic techniques embedded in the study of numerical sets, we provide a direct, short proof using elementary probabilistic arguments.