Papers

We extend the spherical-integral approach to $\mathcal{R}$-transforms of non-Hermitian random matrices to random matrices with a fixed $K \times K$ block structure. We introduce a rank-$K$ spherical integral defining a scalar $H$-transform on $2K \times 2K$ overlap matrices and an associated matrix-valued $\mathcal{R}$-transform encoding the joint block structure. Using the replica method, we derive a conjectural subordination relation for sums of independent block matrices. This framework provides a method for determining the spectral boundaries of large non-Hermitian random matrices with a fixed block structure.

Let $(\mathcal{X},\mathcal{Y})$ be an admissible balanced pair in a locally presentable abelian category $\mathcal{A}$, and let $\mathcal{E}$ be its induced exact structure. We study the class $G(\mathcal{X})$ of cycles of two-sided complexes with terms in $\mathcal{X}$ that are both right and left $\mathcal{X}$-acyclic. If $\mathcal{X}$ is closed under direct summands and finite direct sums and is $\kappa$-Kaplansky for every sufficiently large regular cardinal $\kappa$, we prove that $(G(\mathcal{X}),G(\mathcal{X})^{\perp *})$ is a hereditary cotorsion pair relative to $\mathcal E$. For relative purity determined by finitely presented modules, we give two criteria for functorial completeness and show that relative periodic modules generate a functorially complete hereditary cotorsion pair. We also compute the relative Gorenstein objects, their right orthogonal, and explicit approximations for a family of exact structures over Dedekind domains.

Weak, non-covalent interactions such as electrostatics, induction, and dispersion are commonly treated as distinct physical mechanisms, often using separate models and approximations. Here, we present a unified quantum-electrodynamical framework in which all weak interactions between neutral or charged subsystems emerge from a single interaction Hamiltonian and are expressed entirely in terms of electromagnetic Green functions. Starting from macroscopic QED in arbitrary linear environments, we derive compact master formulas that generate permanent electrostatic, mixed induction, and fully induced (dispersion) interactions on equal footing. Monopoles, dipoles, and quadrupoles are treated consistently within the same multipole expansion, leading to a common structural form in which all interaction channels are obtained from contractions of multipole moments and response functions with Green tensors. Environmental effects enter universally through dressed Green functions. The standard non-retarded free-space limits, including Keesom, Debye, and London/van der Waals interactions, are recovered as a consistency check of the formalism. Beyond this validation, the formulation shows that once the general expressions are established, different physical scenarios, such as interfaces, confinement, or polarisable continua, can be treated by direct substitution of the corresponding Green function, without reformulating the interaction for each case. This elevates Green functions to a universal organising principle for weak interactions in complex environments.

Oshiwambo – a cluster of mutually intelligible Bantu languages spoken by over a million people across northern Namibia and southern Angola, and the home language of roughly half of Namibian households – has, to our knowledge, no published machine-translation evaluation benchmark. Major commercial services (Google Translate, DeepL, Microsoft Translator), open multilingual MT models (NLLB-200, MADLAD-400), and the open Masakhane checkpoint collection all lack coverage of either standardised dialect, Oshindonga or Oshikwanyama. We announce Ongiini-Eval-OW, a planned 600-item English-Oshindonga and English-Oshikwanyama benchmark with native-speaker references from two independent translators, a 30-item inter-translator agreement set, an 11-tag phenomenon-tagged stratification (at least 30 items per tag), a deterministic 30% blind split, and a reproducible scoring protocol over chrF++, BLEU, and COMET-22, supplemented by a 50-item human-evaluation round. We document the empirical coverage gap, the dataset composition, the launch-leaderboard model matrix across American, European, and Chinese frontier and open-weight systems, and the contribution pipeline. The dataset is targeted for first public release in Q4 2026; this v1.0 concept paper announces the design and the call for participation. Data and code will be released under CC-BY-4.0 and MIT respectively.

Collaborative medical AI platforms allow researchers to train models on sensitive imaging data while restricting data export. However, trained models can serve as covert carriers of patient information: medical images may be encoded within model parameters and reconstructed outside the secure environment. Existing defenses rely on lightweight sanitization (e.g., fine-tuning, pruning, quantization) and limited statistical auditing, creating a realistic insider exfiltration risk. We introduce a high-capacity neural steganography attack that encodes medical images as continuous latent representations embedded into model initialization. A StyleGAN2-based adversarial autoencoder learns compact latent codes regularized to match standard weight initialization statistics, keeping embedded parameters statistically consistent with clean models. Noise injection during training improves robustness to export-time mitigation. The carrier model remains functional on its intended task and hidden images can be reconstructed directly from its weights after export. This continuous encoding enables robust and scalable exfiltration, allowing up to 99 brain MRI volumes to be embedded within a 30MB model, and remains recoverable under mitigations that disrupt prior bit-level schemes. While reconstructions are approximate rather than pixel-exact, embedded content remains anatomically recognizable and recoverable at scale, exposing a privacy risk distinct from prior bit-level approaches. Experiments on MIMIC-CXR, BraTS, and LiTS demonstrate effectiveness across modalities, tasks, and architectures, highlighting the need for structural defenses beyond parameter-level sanitization. Code is available at https://github.com/ElieThellier/high-capacity-robust-medical-image-exfiltration.

Video Object Segmentation (VOS) aims to accurately segment target objects from consecutive video frames and track the changes of the objects in each frame of the video. Conventional VOS methods typically demand substantial quantities of pixel-level labeled video sequences for fully supervised learning, which limits the performance of the model in sparse video scenes, while existing VOS methods have limited adaptability to global changes in objects. Based on this observation, in this paper, we propose self-supervised VOS with Sliding window, Wavelet transform and optimal Transport (SWT), a self-supervised VOS framework entirely trained on static dataset using contrastive learning. Firstly, a rolling sample buffer reuses overlapping groups of independently sampled images across successive updates. Secondly, to address the long-distance modeling difficulty caused by simple convolutional structures, we introduce wavelet transform to expand the receptive field of convolutional kernels, thus improving the model's representational capability. Finally, we incorporate optimal transport to assist the model in finding the globally optimal match between the target across two frames, improving the model's ability to handle nonrigid deformations of objects. SWT only requires training on the COCO dataset once and achieves excellent results on five VOS datasets as well as an additional body part propagation dataset. The code will be released soon at [https://github.com/machine928/SWT.git](https://github.com/machine928/SWT.git).

This paper studies weak mixing for two classes of dynamical systems with the \(\mathscr{M}_{\alpha}\)-shadowing property using the generic factor characterization of weak scattering. First, for every \(\alpha\in[0,1)\), every \(M\)-system with the \(\mathscr{M}_{\alpha}\)-shadowing property is weakly mixing, thereby answering [P. Oprocha et. al., On partial shadowing of complete pseudo-orbits, J. Math. Anal. Appl., Question~1] negatively. Second, for every \(\alpha\in(0,1)\), the \(\mathscr{M}_{\alpha}\)-shadowing property implies weak mixing for \(E\)-systems and, in particular, for dynamical systems whose measure center is the whole space.Finally, an \(E\)-system with the \(\mathscr{M}_{0}\)-shadowing property which is not weakly mixing is constructed, showing that the parameter range \(\alpha\in(0,1)\) in the \(E\)-system result is optimal.

With the rapid progress of AI, the number of AI-generated images has increased significantly in recent years. However, the increasing variety of image generation models makes detection more difficult. In this work, we use Fast Fourier Transform (FFT) representation with EfficientNet-B0 for AI-generated image detection. EfficientNet-B0 provides a lightweight architecture that can be useful for resource-limited applications. Most frequency-domain detectors use a standard encoder to extract features from the FFT spectrum and directly pass them to a classifier. We explored a different approach by investigating ECA, U-Net, and Attention U-Net as alternatives to this direct encoder-to-classifier approach. ECA applies channel attention, while U-Net and Attention U-Net use decoder-based architectures to recover and refine spatial information in the extracted frequency features. We used a balanced subset of the MS COCOAI dataset that includes AI-generated images from five different models. Three runs were carried out for each experiment, and the average values were recorded. Experimental results indicate that EfficientNet-B0 obtained an accuracy of 84.64%, which is 4.50 percentage points higher than the ResNet-50 baseline reported in the dataset paper. EfficientNet-B0 with U-Net provided a small improvement, achieving an accuracy of 84.85%, while ECA did not increase the overall performance. EfficientNet-B0 with Attention U-Net achieved the best overall performance, with an accuracy of 85.51% and an ROC-AUC of 92.99%. This represents an improvement of 0.87 percentage points in accuracy compared to the EfficientNet-B0 baseline and 5.37 percentage points over the ResNet-50 baseline reported in the dataset paper.

Fixed image distortions do not cover an attacker that learns from paired clean and watermarked images. We study this paired-training threat with single-image inference: deployment uses neither the clean reference nor the watermark key, payload, or decoder. An encoder maps each image to a structural latent $g$ and an auxiliary residual latent $u$. Push supervision reconstructs the watermarked image from $D(g_w,u_w)$. Pull supervision trains the zero-auxiliary output $D(A_g(g_w;k),0)$ toward the paired clean image. At $k=1.10,u=0$, the four-method sweep gives an average BER of $0.3958$, PSNR of $31.07$ dB, and SSIM of $0.9554$. Restoring $u$ from $0$ to $0.15$ moves average BER from $0.3893$ to $0.3357$, while PSNR falls from $30.99$ to $28.23$ dB. The intervention supports decoder dependence on the auxiliary input in the evaluated setting. The accompanying theory is a conditional, post-hoc account of this behavior rather than an experimentally verified information-relocation result.

A ball spinning about a vertical axis is pushed sideways by the Magnus force, so its ground track curves. We show that when the lift and drag coefficients are constant, the horizontal part of this problem is exactly solvable in elementary functions. The step that makes this work is to write the horizontal velocity as a complex number and to use path length rather than time as the independent variable. Both aerodynamic forces are quadratic in speed, so the speed cancels and what remains is a linear equation with constant coefficients in which gravity does not appear. Three exact results follow from it. The heading turns at a fixed rate per unit path length, so a half turn always costs the same distance whatever the launch conditions and whatever the drag. The horizontal speed decays exponentially in the turn angle at a rate fixed by the drag-to-lift ratio, which makes the horizontal hodograph a logarithmic spiral. The radius of curvature of the ground track equals the lift length times the cosine of the flight-path angle. The whole flight problem also collapses to one scalar differential equation controlled by two dimensionless numbers. We use this to settle a concrete question. Two identical balls thrown in opposite horizontal directions with opposite spin can meet head-on when drag is absent, on a one-parameter family we give in closed form, and they fail to meet once drag acts. Every analytic claim is checked against machine-precision numerics, and the code and data are openly available.

We study rainbow spanning configurations in bijumbled graphs whose edges are coloured independently and uniformly from a prescribed palette. For $n$-vertex $(p,\beta)$-bijumbled graphs with minimum degree at least a fixed positive multiple of $pn$, we obtain rainbow perfect matchings and Hamilton cycles with a sufficient palette surplus of order $(\log n)/p$, assuming $pn=\omega(\log n)$ and $\beta\le cpn$ for a sufficiently small constant $c$. For each prescribed spanning tree of fixed maximum degree $\Delta\ge2$, a surplus of order $L_{n,\Delta}(\log n)/p$ suffices under $\beta\le cpn/L_{n,\Delta}$, where $L_{n,\Delta}=\Delta^{5\sqrt{\log n}}$. The palette surplus is sublinear under these hypotheses. These results use a McDiarmid-type coupling and retain the discrepancy scales of the relevant deterministic embedding theorems. We also prove exact-palette results, using precisely as many colours as the number of edges of the target configuration. After independent edge percolation at rate $\rho$, it is shown that a rainbow perfect matching or Hamilton cycle exists asymptotically almost surely when $\rho pn \ge C(\log n)^2$ and $\beta\le\gamma pn$ for appropriate constants $C$ and $\gamma$. We obtain corresponding results for each prescribed bounded-degree spanning tree and for clique factors under appropriate stronger hypotheses. The exact-palette proofs construct spread measures from uncoloured containment estimates and apply the rainbow threshold theorem of Han and Yuan.

In the literature, several graph matrices based on vertex degrees have been introduced, and the study of their spectral properties has attracted considerable attention in recent years. Motivated by these developments, in this paper, we investigate the spectral radius of the weighted adjacency matrix $A_f(G)$, where the function $f$ satisfies property $P^{*}$. More precisely, we characterize the graphs that attain the maximum spectral radius of $A_f(G)$ among graphs with a vertex cut set of prescribed order, graphs with exactly $s$ cut edges, and graphs with given vertex connectivity and independence number. Our results further develop the unified framework for studying extremal spectral properties of degree-based weighted adjacency matrices and extend this framework to several classes of graphs with prescribed structural parameters.

We analyze a simple stochastic inertial Krasnosel'skii–Mann (iKM) method for finding a fixed point of a nonexpansive operator in a real Hilbert space. Our method is obtained simply by adding two inertial extrapolations to stochastic KM [Bravo and Cominetti, 2024], and it retains one call to a possibly biased stochastic oracle per update and achieves sharp rates in both the stochastic and deterministic regimes. Specifically, with our proposed parameter schedule, we prove the following last-iterate fixed-point residual bound: \[ {O}\!\left(\frac{1}{K} +\frac{\sigma\log K}{\sqrt K} +\frac{B_K\log K}{K}\right), \] where $K$ is the horizon, $\sigma$ is the noise level and $B_K$ is the accumulated root-mean-square bias. When $B_K=O(\sqrt K)$, this yields $\widetilde O(\epsilon^{-2})$ sample complexity that matches, up to a logarithmic factor, the stochastic-oracle lower bound given under the unbiased subclass of our model [Foster et al., 2019, Theorem 2]. It also improves the best-known $O(\epsilon^{-4})$ random-iterate guarantee for stochastic KM [Bravo and Cominetti, 2024, Corollary 5.4]. To our knowledge, this is the first single-loop method for general nonexpansive fixed-point problems to attain this near-optimal sample complexity without variance reduction or batching. When the oracle is exact, the same method attains the worst-case-optimal $O(K^{-1})$ last-iterate residual rate [Park and Ryu, 2022, Theorem 4.6], improving the $O(K^{-1/2})$ rate of classical KM [Cominetti et al., 2014; Bravo and Cominetti, 2018].

We present the canonical quantization of Weyl-invariant gravity in a specific conformal gauge. Fixing the scale factor as $a(\eta) = 1/(H_0\eta)$ yields an effective geometric cosmological term $\mathcal{M}(\eta) = \alpha/\eta^2$ on the left-hand side of the Einstein equations, guaranteeing an equation of state $w = -1$ by construction. The Bianchi identity requires a compensator scalar field; we show that this compensator is the conformal mode of the metric, a gauge artifact that does not propagate ghost instabilities in the full quantum theory. We develop the complete ADM Hamiltonian formulation and prove that the constraint algebra closes in standard first-class form. Canonical quantization yields a well-defined Wheeler-DeWitt equation where the $\mathcal{M}(\eta)$ term acts as a potential barrier dynamically suppressing the quantum creation of small-scale universes. The physical reduced Hamiltonian is demonstrated to be bounded from below, time evolution is strictly unitary in the physical Hilbert space, and microcausality is preserved. In the pure vacuum regime, the model admits an exact analytical solution with $\Omega_{\text{DE}} = 1$, resolving the cosmic coincidence problem without fine-tuned parameters. In the matter-dominated era, the comoving horizon scales as $R_H \propto \eta$, so our result $\mathcal{M} \propto \eta^{-2} \propto R_H^{-2}$ reproduces the central Compton Mass Dark Energy (CMaDE) hypothesis from first principles and establishes $n = -2$ as the theoretical benchmark for the Conformal Holographic Dark Energy (CHDE) parametrization $\mathcal{M} \propto \eta^n$. From a pure conformal symmetry principle, the model delivers an effective geometric cosmological term that drives the accelerated expansion with $w = -1$, providing the missing action principle for CMaDE.

Phase unwrapping estimates the missing multiples of $2\pi$ in measured phase images. For large images, tiling limits the size of local reconstruction problems and enables parallel processing. Adaptive tiling could further reduce the number of local problems and boundaries by retaining large tiles where little refinement is needed. We investigate whether this reduction makes reconstruction faster. We compare complete reconstruction time and accuracy for a regular grid, quadtree, and kd-tree partitions. We also evaluate nine criteria for deciding where quadtree tiles should be subdivided, including residue count, fringe density, and measures of phase variation, at different tile sizes and budgets. In single-threaded experiments on a heterogeneous image dataset, optimized adaptive partitions use fewer tiles but remain slower than the optimized grid, and some reconstructions lose substantial accuracy. Stage measurements explain why: constructing the partition and solving larger retained tiles outweigh the savings at tile boundaries. The criterion comparison also shows that more refinement does not consistently improve accuracy. These results motivate evaluating adaptive partitions by the complete time needed to reach a chosen reconstruction accuracy, including whether limited refinement can provide a faster approximate result.

We show, analytically, that a static sine-Gordon soliton cannot exist in 1 + 1 non-dynamical de Sitter spacetime if $\alpha := (m/H)^2 < 2$, where $m$ is the mass parameter of the sine-Gordon theory and $H$ is the Hubble constant. Conversely, we also show that static sine-Gordon solitons exist in 1 + 1 non-dynamical de Sitter spacetime if $\alpha > 2$. The above threshold is explained—qualitatively and to within an O(1) factor—using a heuristic argument involving the interplay of tensile force in the Lorentzian sine-Gordon soliton and the tidal force in de Sitter spacetime. A similar heuristic argument, which remains to be confirmed analytically, also suggests the existence of a threshold, $ (m_V/H)^2 \sim O(1)$, below which the tidal forces are too strong to permit the existence of a static `t Hooft–Polyakov monopole in non-dynamical 3 + 1 de Sitter spacetime; $m_V$ is the mass of the vector boson. Linde has suggested that new inflation could have triggered secondary inflation at the core of a GUT (Grand Unified Theory) monopole even if the Hubble constant at or after the GUT phase transition was significantly smaller than the mass of the $X$ boson. We present a heuristic argument, which suggests that the $SO(3)$ `t Hooft–Polyakov monopole does not allow secondary inflation at its core when the inflationary background is weak. Based on the above, as yet analytically unconfirmed, heuristic argument for the $SO(3)$ `t Hooft–Polyakov monopole, we conjecture that secondary inflation at the core of a GUT monopole is infeasible.

Gaussian head avatars typically model intrinsic facial appearance as temporally static, omitting subtle cardiac-induced skin-color variation. We propose Heartian, a physiology-aware modulation framework that learns cardiac-cycle-dependent per-frame albedo modulation of facial skin-region Gaussians within a relightable head avatar to encode remote photoplethysmography (rPPG) signals. Using synchronized contact PPG supervision, Heartian models the prescribed cardiac waveform as the sum of two Gaussian functions and learns per-frame spatial residuals via a lightweight MLP. Across 152 stationary recordings from UBFC-rPPG, PURE, and MMPD, attribute-space recovery of the supplied signal achieves a pooled recording-level heart-rate MAE of 0.29 bpm and MAPE of 0.38%. The signals remain detectable after rendering by benchmark rPPG methods, with the best tested configuration - a motion-augmented TS-CAN decoder pretrained on UBFC-rPPG - recovering heart rate from the rendered MMPD avatars at 0.97 bpm MAE and 1.21% MAPE. Meanwhile, Heartian maintains reconstruction quality comparable to the baseline, with negligible average PSNR degradation of 0.005 dB. Overall, our work embeds recoverable rPPG signals as controllable material attributes to subject-specific Gaussian head avatars while retaining the reconstruction quality.

Microwave cavity haloscopes search for dark-matter axions through the weak electromagnetic field generated by axion-photon conversion in a static magnetic field. Realistic synthetic signals are essential for end-to-end calibration of the receiver, validation of the analysis pipeline, measurement of signal recovery efficiency, and blind-injection studies. Here, we present a general method for producing stochastic synthetic axion waveforms by shaping Gaussian white noise in the frequency domain. The target axion power spectral density is imposed directly on the Fourier coefficients, and arbitrarily long continuous time streams are generated efficiently using an overlap-save implementation. The method reproduces both the ensemble-averaged spectral line shape and the time-domain fluctuations expected for a classical axion field, while providing direct control of the total injected power. Because the spectral model enters only through a replaceable transfer function, the routine can accommodate the Standard Halo Model as well as nonstandard velocity distributions and narrow substructure.

Over the past thirty years, the online advertising industry built a large-scale data collection ecosystem, with the goal of tracking a user's online activity to infer their demographics and interests. Traditionally, the ecosystem relied upon the collation and analysis of highly-structured text data like user IP addresses, GPS coordinates, e-commerce purchase histories, and visited URLs. However, recent ML models can parse not only structured text, but also multimedia files and unstructured text inputs—meaning a user's photos, videos, inboxes, and calendars are now ripe for automated analysis. The privacy risks are particularly acute in the context of smartphone apps. A user's phone already acts as a natural collation point for sensitive user information, but users may not understand that permitting an app to, for example, access a user's photo does not just give the app access to the bytes in the photo: the app also receives access to inferences about the user that are enabled by the photo. To explore these privacy risks, we built Priva-See, an LLM-based inference system for app-collected user data; Priva-See reflects our best understanding of how real-life adtech companies would leverage machine learning to build user profiles. Through an IRB-approved user study, 465 participants deployed Priva-See on their phones; Priva-See made privacy-invasive inferences despite having access to only a subset of a user's data. We see the experience significantly impacted participant willingness to share permissions data moving forward. Based on the observed privacy violations, we suggest changes to how smartphone OSes should gather user consent for data access, to better inform users about downstream data usage capability.

For type $A$ braid varieties, including Richardson varieties and positroid varieties, we prove that the standard torus has a dense orbit exactly when the associated link is smoothly slice. More generally, we relate the smooth slice genus to the codimension of a generic standard-torus orbit. For knots, this codimension equals twice the slice genus. We also characterize algebraic tori among open affine Richardson varieties and positroid patches by 2-crown avoidance in their Bruhat intervals. Finally, positroid links have equal Seifert and slice genera, so a positroid link is smoothly slice exactly when it is an unlink.

Distributed systems are hard to implement correctly, and subtle bugs can go undetected using traditional testing. Formal verification offers an alternative for proving the correctness of complex distributed systems. Despite previous efforts to automate and facilitate formal verification, it is still hard to integrate formal verification in software development. Programmers need to query the theorem prover repeatedly to find the correct proof of their system. This cycle of going back and forth with the theorem prover involves a lot of human intervention and is a barrier to adopting formal verification in software development. In this paper, we address the challenges of scaling formal verification in practice by reducing the human intervention required to find the correctness proof of a distributed system. We propose ProofSaX, an automated tool that shards large verification tasks and explores the proof search space for each shard independently to synthesize correctness proofs. We use controlled exploration when adding possible proof annotations to handle the search-space explosion problem. We evaluate ProofSaX on a variety of safety proofs for distributed systems. We show that ProofSaX can find the last proof annotation in 86 out of 103 proof-completion tasks, with runtimes ranging from one minute to two hours.

A weak form of the Hellinger conjecture of Anantharam, Bogdanov, Chakrabarti, Jayram, and Nair for the binary symmetric channel is proved: dictator functions maximize Hellinger $\Phi$-entropy among all Boolean functions of the input and all one-bit statistics of the output of a noisy channel. The technical heart of the matter is an explicit inequality in three real parameters, which is proved using explicit polynomial approximations and computer-assisted positivity checks. The results are also formally verified in Lean 4.

This article provides an overview of potential alternatives for providing wireless backhaul in regions that suffer from the lack of fiber optic-connectivity to the core network. These regions can be rural and remote locations, low-income neighborhoods in urban and suburban regions, and post-disaster locations suffering from the destruction of cellular infrastructure. For these scenarios, extending fiber optic cables to such locations might be extremely expensive, impractical, or simply not feasible. Hence, in order to enhance the backhaul connectivity in these scenarios, we study the potential and applicability of the integrated access and backhaul (IAB) technique, and a hybrid combination of IAB and non-terrestrial networks (NTN) that includes high/low altitude platforms (HAPs/LAPs) and low earth orbit (LEO) satellites. We conclude this article by discussing the design considerations and potential research problems that would enable efficient deployment of such solutions.

Sample return from South Pole-Aitken Basin (SPA) on the Moon has been recognized as the highest priority lunar science mission for well over two decades, appearing as a recommended New Frontiers class mission in the two previous Planetary Science Decadal Survey documents (2003-2012 New Frontiers, 2013-2022 Visions and Voyages), and as a prioritized mission for the Lunar Discovery and Exploration Program (LDEP) in the 2023-2032 Origins, Worlds, and Life Decadal Survey. The South Pole-Aitken basin presents the ideal target destination to test nearly 60 years of lunar science hypotheses owing to a unique combination of its size, antiquity, and location on the lunar farside. Convened by NASA in response to OWL Decadal Survey recommendations, the SPARX report provides analysis on prioritized science objectives, measurement requirements, implementation architectures, and a design reference mission for a South Pole-Aitken basin sample return mission. The mission profiles presented in this report leverage key advancements in remote surface exploration technology, the U.S. commercial space industry, and the programmatic focus on sustained human presence and exploration at the lunar south polar region. The science objectives and measurements presented in this report outline a mission that would transform lunar science, would address multiple high-level Decadal science questions at the only place in the Solar System where they can be credibly addressed, would pioneer a new paradigm for planetary surface exploration in the 21st century, would leverage the burgeoning space economy to pursue ground-breaking science on a more cost-efficient basis, and would unite robotic exploration with human exploration in a model for a future sustained lunar enterprise.

For an infinite connected locally finite undirected graph with positive symmetric edge weights and normalized Laplacian $\Delta$, we prove weak type $(1,1)$ estimates for two Riesz transforms. The edge transform has universal constant $2$, with no volume assumption. The vertex carré-du-champ transform has constant $H_1+1$ whenever $H_1=\sup_x m(B(x,1))/m(x)<\infty$. These endpoint constants apply to real-valued inputs and yield strong $\ell^p$ bounds for $1<p\le2$. The proof uses a discrete obstacle decomposition obtained by a contraction mapping, followed by elementary support estimates for the two gradients.

A general-purpose vision-language model can understand a task goal without knowing how a particular robot's motion and functional parts produce the intended effect. We introduce KnowBody, a harness that makes these action-relevant body relations explicit, queryable, and revisable while keeping the model weights frozen. Initialized from one off-task trajectory, a partial body model guides action selection and the interpretation of past interactions. New evidence refines the model, and knowledge dependent on revised body estimates is rechecked before reuse. Across 32 fixed-budget trials on four real-robot tasks, initialized KnowBody achieves 75% completion versus 25% for the native harness and requires fewer planner rounds on successful trials in tasks completed by both. With persistent updates enabled, planner rounds decrease by 29-53% from the first to the fifth recorded success.

We establish a structural rigidity principle for the nontrivial zeros of the Riemann zeta function by analyzing the asymptotic interaction between discrete Dirichlet oscillations and their continuous analytic envelope. The starting point is a renormalized discrepancy operator, the C-transformation, which isolates the finite structural difference between the Dirichlet series and its integral approximation. Applied to the kernel x^(-s), this operator yields a decomposition of the zeta function into three parts: an oscillatory Dirichlet component X_N(s), an explicit analytic growth envelope Y_N(s), and a polynomially decaying remainder R_N(s). We show that the vanishing condition zeta(s)=0 forces an asymptotic compatibility between the two dominant components. This relation is formalized in a Bridge Lemma, which proves that zeros require asymptotic equality of the squared magnitudes |X_N(s)|^2 and |Y_N(s)|^2. A detailed asymptotic analysis then reveals that the ratio of these magnitudes exhibits three distinct regimes depending on the real part sigma=Re(s): divergence for sigma>1/2, collapse for sigma<1/2, and exact balance for sigma=1/2. This incompatibility of growth regimes produces a rigidity mechanism: the asymptotic balance required by the zero condition can occur only on the critical line. Consequently, every nontrivial zero must satisfy Re(s)=1/2. The result provides a deterministic explanation for the location of the zeros, showing that the critical line emerges from an intrinsic structural constraint of the Dirichlet representation of the zeta function.

The goal of this note is to give a detailed proof, to the best of our understanding, of the recent presentation by Harrison and Leeman (arXiv:2609.17650v01 and arXiv:2609.17650v02) of the proof by Astra on the lower bound for differentially private continual counting. We believe a more natural and easy proof is possible and hope that this note will help in that effort. Prior to the initial preprint by Harrison and Leeman (arXiv:2609.17650v01), Bairaktari and Larsen (arXiv:2607.00876) gave an elegant proof to show a lower bound of $\Omega(\log^{3/2}(n))$ for both pure and approximate-DP continual counting, and in personal communication had informed us that they have a proof of optimal $\Omega(\log^{2}(n))$ for pure-differential private continual counting as well. They have subsequently published their $\Omega(\log^{2}(n))$ bound, which is now a joint work of Bairaktari, Dahl, and Larsen (arXiv:2607.00876v3). Their new result is an elegant extension of their technique for approximate-differential privacy. Although the two proofs are technically different, the Astra argument uses related tree geometry introduced in Bairaktari and Larsen.

Kahn proved that, among bipartite $d$-regular graphs on $n$ vertices, the number of independent sets is maximized by a disjoint union of copies of $K_{d,d}$. Zhao later extended this result to all $d$-regular graphs. We prove a robust version of this theorem in which independent sets are replaced by sets spanning few internal edges. If $G$ is $d$-regular on $n$ vertices, then the number of subsets spanning at most $\gamma dn$ edges is at most $$ 2^{n/2}\exp\left\{ O\bigl(\gamma\log(e/\gamma)n\bigr) + O\bigl(n/d\bigr) \right\}. $$ Both correction terms are sharp up to absolute constants: the $\gamma\log(1/\gamma)n$ term is necessary when $d$ is sufficiently large in terms of $\gamma$, while the $n/d$ term is already necessary for independent sets. Our result answers a question of Seth.

We quantify the entropy-decrement proof of the logarithmically averaged two-point Chowla theorem, uniformly in the affine forms and the averaging window. For L_i(n)=a_i n+b_i with positive integral slopes and nonzero determinant Delta=a_1 b_2-a_2 b_1, put h=max(a_1,a_2,|b_1|,|b_2|,|Delta|,2). For either the Liouville or Mobius function in each position and 0<epsilon<=1/10, the correlation sum over x/omega<n<=x, weighted by 1/n, has absolute value at most epsilon log omega whenever x>=omega>=exp_4(C h epsilon^(-2)), where exp_4 denotes four iterated exponentials. A translation lemma separates the common translation from the tower: for lambda(n+b)lambda(n+b+k), with k nonzero, cancellation is uniform when |k|=o(log_4 omega) and |b|=omega^(o(1)), where log_4 is the fourth iterated logarithm. The same thresholds give logarithmic equidistribution of the four Liouville sign pairs. The constant C is effective but depends on uncomputed constants in two analytic inputs. Separately, writing ell=log log x, we prove fully explicit bounds ell/3-(4/3)log ell-7 <= N_lambda(x) <= ell-log ell+2.53/ell for ell>=64, where N_lambda is the non-pretentiousness level in Tao's hypothesis. Under GRH the lower bound improves to ell-log ell-log 2-2/ell; hence N_lambda(x)=ell-log ell+O(1). The arguments use an explicit zero-free region, Harnack's inequality, a finite resonator, and a comparison of smoothed Euler products. These results concern logarithmic averages, not natural-density two-point Chowla correlations.