Papers

We present the first $\textit{ab initio}$ many-body application of the Efros method for scattering utilizing the harmonic oscillator expansion of the continuum wave function. We select single-channel $n\alpha$ scattering and, using low-lying eigenstates from no-core shell model (NCSM) calculations, we obtain well-converged scattering results that are in reasonable agreement with experiment. The results demonstrate that the Efros method is computationally practical and promising for further $\textit{ab initio}$ many-body studies of nuclear reactions.

Galaxy clustering encodes information about the host galaxies of binary black hole (BBH) mergers even when individual hosts remain unidentified, because the distribution of neighboring galaxies depends statistically on the host-galaxy population. We construct event-level host-type likelihoods from host-conditioned galaxy-count distributions within gravitational-wave localization regions and combine them to infer the fractions of BBH mergers hosted by different galaxy populations. The framework applies to common and rare galaxy populations without prescribing formation efficiencies or merger delay-time distributions. Using measured star-forming and quenched galaxy clustering, idealized one-year forecasts assuming all mergers occur in quenched galaxies yield median one-sided 90% lower credible limits of 0.19, 0.48, and 0.92 on the quenched-host fraction for O5, post-O5, and third-generation networks. These forecasts illustrate the potential for host-galaxy population inference before the third-generation detector era.

Micrometre spatial resolution is achieved for X-rays with the MÖNCH detector, a charge-integrating hybrid pixel detector with a 25 {\mu}m pixel pitch, by exploiting charge sharing between neighbouring pixels. The conventional analytical interpolation method, however, relies on a global mapping derived from the charge-sharing statistics and is applicable only to isolated single-photon clusters, which limits both the achievable resolution and the usable photon flux. To overcome these limitations, we trained deep learning models to localize single-photon and pile-up events based on high-fidelity simulation data. For knife-edge measurements at 12 keV and a low occupancy of 0.74% (photons per pixel and frame), where single-photon events dominate, the proposed method reaches a of the edge spread function of 1.36 {\mu}m, compared with 1.54 {\mu}m for the conventional eta interpolation. We demonstrate experimentally that resolving up to three photons per cluster raises the reconstruction efficiency from 59% to 93% at a 2.4{\times} higher occupancy of 1.81%, while preserving a of 1.39 {\mu}m.

Given a matrix $A\in M_{k\times N}(\Bbb{F}),$ we examine finite frames $\{f_j\}_{j=1}^N$ with vanishing $A$-moments and investigate their connections with the frame dilation property. We establish several equivalent characterizations for the existence of vanishing $A-$moment dilations, including conditions involving dual frames, operator ranges, and linear independence properties associated with the matrix $A$. In particular, we show that every frame with redundancy $k$ admits a dual frame with vanishing $A$-moments for some totally nonsingular matrix $A$. A key takeaway of this is that by invoking such a dual frame as the encoding frame, any frame with redundancy $k$ (erasure robustness is not needed) allows perfect signal reconstructions for any number of $\ell$-erasures ($\ell \leq k$) which only involves inverting a matrix of small size ($\ell$) instead of inverting the subframe operator which is a matrix of large size ($N-\ell$). A special case of this type of frames are $k$-th vanishing moment frames which have natural connections with graph frames. We characterize the existence of dual frames with $k-th$ vanishing moments through polynomial cancellation identities and operator theoretical conditions. We further discuss some applications to problems in erasure recovery and graph frames. These results provide a unified framework linking vanishing moment conditions, dual frame theory, dilation theory, and graph theoretic constructions of frames.

We study general maximal hard-core models on periodic lattice-type graphs. While for standard models, phase coexistence may arise for high activity, for maximal models this behavior may occur also for sufficiently low activity values. Relying on the concept of volume allocation, we develop a unified set of assumptions that imply the Peierls condition, a convergent cluster expansion for the partition function, and hence the conclusions of Pirogov-Sinai theory for this class of models. We further check these assumptions for a number of examples including standard lattice-type periodic graphs, proving crystallization for both high and low activity. We also derive estimates for the values of activity that bound the phase coexistence regions in the phase diagram, rewriting for this purpose the proof of a technical result in the derivation of Pirogov-Sinai theory.

Developers frequently insert temporary print or log statements, known as **ad-hoc logs**, to better understand program behavior at runtime, particularly when facing unexpected issues or complex control flows. Despite being a nearly universal practice, systematic study has been limited because these logs are ephemeral: they usually remain only in local environments and are removed before code is committed, making them difficult to capture. In this work, we addressed this challenge by mining accidental commits where developers unintentionally left ad-hoc logs and later deleted them, and by analyzing live-streamed programming sessions to observe their use in practice. Using these methods, we constructed a large dataset across three major programming languages (Java, JavaScript, and Python), enabling the large-scale investigation of logging practices. Our analysis reveals both common and language-specific patterns in where and how developers rely on ad-hoc logs. Across languages, ad-hoc logs tend to appear in program regions that are harder to reason about at runtime. We also identify distinctive language-level patterns, such as frequent use in asynchronous and callback functions in JavaScript and in thread-related classes in Java. In addition, functions containing ad-hoc logs generally have higher cyclomatic complexity than the overall function population. Production logs show a similar association, consistent with logging serving as a means of observing runtime behavior in structurally complex functions. Together, these findings provide empirical insight into developers' runtime comprehension practices and offer a valuable dataset for researchers and tool builders seeking to better support debugging and logging.

In this paper, we first classify complete four-dimensional shrinkers under a pinching condition on the $2$nd-Ricci curvature. We then classify closed four-dimensional shrinkers with (weighted) integral parameter-pinching of the half Weyl curvature under several pinching assumptions on the biorthogonal curvature. Finally, we establish a Hitchin-Thorpe type inequality for closed non-trivial four-dimensional shrinkers under some assumptions on the half Weyl curvature and the biorthogonal curvature.

Active robot vision requires controlling the camera to reveal task-relevant information that is hidden from the current viewpoint. For example, determining what is inside a box may require raising the camera and looking down into it. For viewpoint-dependent question answering, the challenge is to select camera motions that expose the visual evidence needed to answer the question. Although vision-language models (VLMs) can interpret observed images, selecting such motions requires anticipating the usefulness of unseen views. We quantify this usefulness as answerability, a VLM's estimate that a view suffices to answer the question, and present Rendering-Free Lookahead (RFL), a viewpoint-selection policy that ranks candidate camera motions by predicted future answerability. RFL transfers visual lookahead from deployment to offline training. At training, a privileged teacher renders candidate future views in 3D Gaussian Splatting (3DGS) scenes and uses a frozen VLM to compute one- and two-step answerability targets. Through two-stage distillation, a student learns to predict these action values from the question, recent visual observations, and a candidate camera motion. At deployment, RFL uses these predicted values to select camera motions without rendering future views. On 377 E3VS-Bench test episodes in unseen environments, RFL improves the mean judge score by 43% over a direct-action baseline using the same VLM. These results support learning camera-control policies from privileged visual lookahead for viewpoint-dependent question answering.

We investigate the consequences of requiring a posterior approximation to preserve a specified downstream decision problem. A target posterior $P$ and loss determine a regret geometry on actions, a baseline approximation $Q_0$ determines the forward-Kullback-Leibler information required to induce action changes, and a restricted approximation family $\mathcal{Q}$ determines which such changes are available. Contracting KL divergence over Bayes-action fibers gives exact distances to decision adequacy and decision failure together with the least-informative posterior deformations that reach either side of the decision boundary. In regular finite-dimensional problems, the target and baseline constructions have quadratic local limits: a target regret Hessian $G$ and a baseline information metric $J_I$ . Their generalized eigenproblem $Gv = \gamma J_Iv$ orders local decision directions by regret consequence per unit information cost and induces a tolerance-dependent effective dimension. For restricted approximation families, the tangent image separates decision coverage from information efficiency: a family may miss consequential decision directions, or it may realize reachable directions only at excess Fisher cost. The resulting framework provides decision-relative criteria for comparing and designing posterior approximation families.

Recent image editing systems have achieved impressive semantic understanding, visual fidelity, and instruction-following ability, while video editing remains substantially more difficult and costly. In this paper, we present a simple alternative to end-to-end video editing: instead of training a monolithic video editor, we transform a strong image editor into a video editor through anchor-based generation. Our key insight is that video editing can be decomposed into two subproblems: editing a sparse set of keyframes and propagating those edits across time. Based on this observation, we propose Anchor-based Video Editing (AVE), a two-stage framework in which a powerful image editor first performs composed editing on selected keyframes, and a motion-guided image-to-video diffusion model then generates the final video by treating the edited keyframes as fixed anchors. This design directly inherits the strengths of modern image editors while avoiding expensive end-to-end video editing training. Experiments on IVEBench and VIE-Bench show that AVE achieves strong performance in instruction following, temporal consistency, and content fidelity. Further ablations reveal that final video editing quality is strongly correlated with the quality of the image editor, suggesting that future progress in video editing may come from stronger image editing foundations and lightweight transfer to video. Code is available at https://github.com/wangf3014/AVE.

Appearance-based gaze estimation offers a low-cost alternative to infrared eye tracking for screen-based behavioral and clinical applications, however existing models are typically developed for wide ranges of gaze angle and head pose. Prolonged screen viewing presents a distinct regime in which gaze remains near the screen center, head motion is limited, and calibration drift accumulates over time. In this work, we benchmark six published estimators along with a proposed differential-gaze model on 140 long-duration facial video recordings with synchronized eye tracking under subject-disjoint evaluation and a common budget of calibration frames. Differential estimation was the only static method that significantly improved upon a baseline predictor of each recording's mean gaze, and was further boosted by addition of temporal context. It also yielded the strongest agreement with reference fixations and saccades, demonstrating that low angular error alone could not validate eye movement reconstruction. These findings establish differential estimation as a promising foundation for reliable gaze tracking in long-duration, narrow-range settings.

I propose a Hamiltonian framework for chiral gauge theories (CGT) based on an existing Euclidean formulation, that uses 2n dimensional chiral fermions on the boundary of a 2n+1 dimensional disk. In the original Euclidean formulation, boundary gauge fields were extended into the bulk using $2n + 1$ dimensional gauge field equations of motion (EOM). This creates a bottleneck for constructing a Hamiltonian. I present an alternate proposal for extending the gauge fields into the bulk that is compatible with both a Hamiltonian framework and a Euclidean path integral. Applying this to Abelian gauge fields produces exact expressions of interior fields as a functional of the boundary fields, which can be directly used in a Hamiltonian formulation. Euclidean analysis of the new gauge field extension shows that it can preserve the non-perturbative content of the original construction, including the behavior of topological charge, associated chiral fermion zero modes and an absence of the strong CP problem when applied to the Standard Model. This construction opens up a route to a Hamiltonian treatment and future quantum simulation of CGTs on a disk boundary.

We study the convergence of minimising movement approximations to weak solutions of a fourth-order viscoelastic phase separation system with degenerate concave mobility. The underlying semi-implicit variational scheme combines a dynamic transport distance for the phase-field variable with a weighted $L^2$-distance for a transformed stress variable. We establish subsequential convergence of the scheme and show that the limiting solutions satisfy the energy-dissipation inequality. To this end, we derive a coercivity estimate for the metric slope, which yields the required space-time regularity for the De Giorgi interpolant, and construct constitutive fluxes whose dissipation is controlled by the squared metric slope. These estimates allow us to pass to the limit in the discrete energy-dissipation inequality. The weak formulation of the evolution equation for the phase-field variable is derived using the flow interchange technique. Under additional assumptions on the data, we further show that the limiting solutions satisfy an entropy-dissipation inequality

Multiple institutions may each hold their own private knowledge edits and wish to integrate them into a single large language model without sharing raw edit requests. Null-space-constrained editing methods such as AlphaEdit mathematically guarantee that each update leaves unrelated knowledge intact, while collaborative frameworks such as CollabEdit aggregate edits from multiple clients without data sharing. Combining the two appears trivial. However, we show that this naive combination fails structurally, and we identify its cause. Guided by this analysis, we propose FedAlphaEdit. To our knowledge, this is the first collaborative knowledge editing framework that aligns both local editing and the server-side merging rule under a single null-space principle for preserving existing knowledge. FedAlphaEdit builds on null-space-aligned merging, in which clients share projected statistics and the server provably recovers the result of editing everything in one place under a one-shot idealization. Empirically, the proposed method repairs the collapse and brings edit success and preservation simultaneously close to the level of centralized editing across two architecture families. FedAlphaEdit thus lets institutions that cannot share raw edit data, such as hospitals and financial firms, jointly maintain a shared model that closely approximates editing all facts in one place.

Let $K$ be a function field over an $F$-finite field $k$ of characteristic $p > 0$ and let $\nu$ be a valuation of $K/k$. We show $\nu$ is an Abhyankar valuation of $K/k$ if and only if the corresponding valuation ring is Frobenius split. The proof proceeds by analyzing when ideals of a valuation ring are uniformly $F$-compatible and establishing a general ramification-theoretic characterization of Frobenius split valuation rings of $F$-finite fields. In addition, when the ground field $k$ is not $F$-finite, we show that the equivalence between Frobenius splitting and the Abhyankar property can fail by constructing an example of an excellent Frobenius split DVR of a function field whose corresponding valuation is not divisorial. Our methods do not rely on local uniformization results.

A long-standing hypothesis in artificial intelligence and neuroscience posits that intelligence is closely related to compression: the ability to compress information efficiently intuitively reflects capacities associated with intelligence and learning. Indeed, recent experimental works verify this intuition by showing connections between the capabilities of large language models (LLMs) and their ability as compressors: for instance, Deletang et al. (ICLR'24) demonstrate that LLMs can be used as powerful compressors, and Huang et al. (COLM'24) show that the compression ability of LLMs is highly correlated with their performance on benchmarks for knowledge and reasoning. In this work, we initiate a theoretical study of this connection. Our main result is that LLMs (formally modeled as next-token predictors) are equivalent to monotone (a.k.a. order-preserving) compression algorithms—namely, compression algorithms where the encoding process preserves the ordering of the inputs—in the sense that the one can be constructed from the other while preserving the same error up to an additive gap of 2. We next show that the monotonicity is required for this equivalence to hold if and only if cryptographic (infinitely-often) one-way functions exist. As a direct corollary, we get a cryptographic result of independent interest: the notion of next-bit pseudoentropy (a computational analogue of entropy) of a distribution is equivalent to monotone incompressibility of the distribution. (Previously, it was only known (Haitner et al., ITCS'23) that incompressibility implies next-bit pseudoentropy.)

Tool-using LLM agents violate the policies they are deployed to enforce, often silently. Prior defenses hand-write rules, query an LLM verifier per action, or compile policies through heavyweight formal machinery. Naive compilation fails: extracted rules block the tool satisfying their own precondition, or read arguments their tool lacks. NOMOS, a four-pass compiler, turns a natural-language policy into a deterministic tool-call gate; static verification with tool-schema-level checks alone (no prover, solver, or LLM) repairs or rejects 37% (airline) and 13% (retail) of candidates, without which most shipped rules are inoperable. Replaying compiled rules over undefended transcripts flags bindings that refuse legitimate work (a development binding refused 95.9% of task-passing calls); no evaluation binding is flagged. On $\tau^2$-bench the gate cuts violations of reference-encoded clauses among state-changing calls from 66.3% to 2.6% (airline) and 30.8% to 6.9% (retail), raising airline task success significantly for $2 \le k \le 4$; a 26B on-premise compilation is not significantly worse than hand-written or frontier-compiled rules. Unlike AgentDojo's shipped defenses, it reaches a zero attack success rate (ASR) on banking, where nine attack families collapse onto three structural rules. On the other three suites its ASR is at most 3.6%, from goals with no tool call to govern and one write admitted by a binding weaker than its clause; a second agent model, Llama-3.3-70B, reproduces the effect on both benchmarks. Decisions take microseconds without an LLM call, at a domain-dependent benign-utility cost; compilation runs on-premise on open-weight gemma-4-26B.

How the local energy transfer across scales in turbulence is encoded in the instantaneous deformation of fluid elements remains a fundamental question. We exploit the spherical geometry of the real-space energy cascade at length scale $\ell$. An exact derivation shows that it can be decomposed locally by strain self-amplification and vortex stretching, with a coefficient equal to the specific moment of inertia $2/5(\ell/2)^2$ of a spherical fluid neighborhood. Direct numerical simulations show that this contribution captures the dominant local transfer and defines a statistical boundary separating forward- and inverse-cascade events. A further longitudinal and transverse decomposition reveals that the longitudinal transfer underlying Kolmogorov's $4/5$ law is governed solely by strain self-amplification, whereas vortex stretching enters through the transverse component. These results establish a purely mechanical connection between turbulent energy cascade and the local rotational and deformational dynamics of fluid elements.

Synchronization among coupled oscillators – from neurons to power grids – is a universal signature of collective behavior, classically measured by the Kuramoto order parameter. Applied identically across five asset classes and 61 financial instruments, a Kuramoto-style cross-timeframe coherence measure detects structure that a calibrated, heteroskedasticity-robust variance-ratio test of the random-walk hypothesis does not: the two diagnostics are statistically independent even on the same price windows. Here we explain why, and extract the physics behind it. The four-state regime alphabet underlying the measure is a Z4 clock model whose exact factorization into two Ising sectors is precisely the order parameter's decomposition into a direction channel and a volatility channel, and the sign-flip symmetry of returns acts on the regime circle as a reflection, yielding a parity selection rule: the direction channel is provably independent of the variance ratio and of every statistic built from squared returns – confirmed by controlled simulation on processes with and without genuine persistence. Boltzmann inversion of the regime statistics then yields effective couplings between adjacent timescales, benchmarked against matched random-walk, GARCH, and fractional-Gaussian-noise nulls: the direction sector's coupling never exceeds its random-walk null in a single instrument, and its deficit independently re-measures the panel's Hurst exponent, while the volatility sector's coupling exceeds every null in every instrument and survives across scale separations where GARCH cannot – a volatility cascade. A closed-form mean-field analysis places markets deep below the synchronization transition, with all genuine cross-scale structure confined to the even, volatility sector.

This paper is concerned with the long-time dynamics of a coupled nonlinear suspension bridge system with past history, in which the deck and the main cable interact through the nonlinear restoring force \(k[u-v]^{+}\). Building on the well-posedness and the solution semigroup established in \cite{Mukiawa2025}, we first show the exponential stability of the associated linear semigroup by means of the Gearhart–Prüss theorem together with a detailed resolvent estimate. We then combine this exponential stability with the local Lipschitz property of the nonlinearity, the fact that the nonlinearity acts only on the elastic component, the smoothing of the elastic component, and the exponential decay of the memory kernels to conclude that the solution semigroup is quasi-stable in the sense of Chueshov–Lasiecka, and therefore asymptotically smooth. Together with the dissipativity recalled from \cite{Mukiawa2025}, this leads to a compact global attractor in the natural energy space. In particular, our approach may be viewed as closing a gap in the proof of asymptotic smoothness in \cite{Mukiawa2025}, where a non-decaying initial-difference term was absorbed into an arbitrarily small constant without justification.

Accurate excited state energies are essential for interpreting fluorescence and photochemical processes, yet variational quantum eigensolvers (VQEs) on noisy intermediate-scale quantum (NISQ) hardware have largely been confined to ground states. Subspace-search VQE (SSVQE) offers a NISQ-friendly route to excited states without ancilla-based overlap measurements, however it has lacked an experimental demonstration since it requires the coherent preparation and manipulation of multiple orthogonal basis states under a shared variational unitary. Here we realize SSVQE on a photonic platform by encoding orthogonal qudit states in the orbital angular momentum of single photons and optimizing a shared variational circuit. We estimate excited state energies for a four-dimensional $\rm{HeH^+}$ molecule and an eight-dimensional Heisenberg chain, thereby demonstrating scalable excited state eigensolving in a high-dimensional Hilbert space. Our results establish photonic qudits as a resource-efficient pathway to excited state quantum simulation.

Exploratory data analysis (EDA) is rarely open-ended in practice: analysts work from high-level domain questions toward the concrete analyses that can answer them, prioritizing directions with domain knowledge and prior hypotheses. Large language models (LLMs) can supply such knowledge, but their responses are unstructured, leaving analysts no way to see what has been explored, what is missing, or why one direction was chosen over another. We present DAG-EDA, a system that lets analysts and an LLM co-navigate the space of possible analyses through two linked structures. An intent graph, governed by a grammar of analytical intent, decomposes an ambiguous natural-language question into progressively concrete analysis tasks, keeping alternative framings open and letting analysts branch, backtrack, and compare paths. A multi-layered knowledge graph externalizes the LLM's domain knowledge, linking domain concepts to the dataset variables that can measure them, so analysts can inspect and contest how their question is grounded in the data. Both graphs are constructed from only the dataset and the analyst's question, and the analyses the analyst reaches are rendered as interactive dashboards. We illustrate the system through a usage scenario and describe a user study design for examining whether the system scaffold analysts' reasoning and navigation.

The real-axis jump of the prime-zeta function gives an integral form of Gram's series for Riemann's $R$-function. Applying the same operation to the classical almost-prime generating functions defines a smooth family $R_k$, with $R_1=R$. We give the construction and a convergence argument, and explain why its expansion at every fixed logarithmic order is the Selberg–Delange expansion. The distinction lies in the smaller real-axis contributions retained by the definition, rather than in new asymptotic coefficients. A finite Perron decomposition also identifies the additional contour terms required for exact recovery of the count.

Identity-preserving video generation aims to maintain a subject's identity while synthesizing realistic videos. Yet a single reference portrait captures the subject's appearance under only one facial configuration. As expressions change, facial appearance can vary in highly identity-specific ways, leaving the subject's appearance under unseen expressions underdetermined by the reference alone. This expression-dependent variation also complicates evaluation: similarity to a neutral reference may decrease under strong expressions even for real images of the same person. We investigate this limitation from both generation and evaluation perspectives. First, we quantify how face-recognition similarity varies with expression intensity using controlled photographs and MEAD videos. We then construct a compact yet expressive reference gallery that captures diverse expression-dependent facial configurations. Matching against this gallery provides a more robust measure of identity similarity under expressive motion. To further expose performance degradation with expression intensity, we report identity similarity separately for mild, intense, and extreme expressions. For generation, we extend Stand-In to condition on our expression-diverse reference sets and develop a data-curation pipeline that extracts consistent yet diverse face crops from training videos. In practical settings where only a single portrait is available, we construct the reference set by synthesizing additional expressions with a pretrained facial reenactment model. On our controlled benchmark, both real and synthesized reference sets outperform the evaluated baselines in identity similarity across all three expression-intensity regimes, with the largest improvements for extreme expressions.

Skillful prediction of biomass-burning activity several days in advance is important for air-quality forecasting and aerosol prediction. Two operational constraints motivate this work. First, the GBBEPx satellite fire radiative power (FRP) product used to initialize NOAA's GEFS-Aerosols is available with about a 1.5-day latency, so each forecast cycle relies on the most recently available, but already outdated, fire observations. Second, these fire inputs are then held fixed throughout the subsequent 5-day operational forecast, or 7 days in the GSL experimental system, effectively assuming no evolution in fire activity. We develop a data-driven model that predicts global FRP one to seven days ahead from the most recent available observations. The model adapts a spatiotemporal graph neural network using reanalysis meteorology, land-cover and vegetation information, recent fire history, and GBBEPx FRP as the training target. It is trained on 2020-2022 data and evaluated for 2023-2024. The model reproduces the global seasonal cycle and substantially outperforms persistence. At 0.1$^\circ$ resolution, mean squared error is reduced by 32% at one-day lead and 43% at seven days in 2023, and by 24% and 40% in 2024. At 1$^\circ$ resolution, the critical success index ranges from 0.32 to 0.60. Detection skill declines only modestly with lead time, whereas intensity skill degrades more rapidly. Large fires are detected reliably, but their radiative power is systematically underestimated. These results demonstrate useful predictability of fire activity several days ahead and identify intensity calibration and small-fire placement as the main remaining challenges before predicted FRP can support operational aerosol forecasts.

Let $(\mathcal N,\nu)$ be a finite tracial von Neumann algebra and set $\mathcal{K}_r(P,Q)=\nu((P^{1/2}QP^{1/2})^r)$. We prove that $$ [\mathcal{K}_r(P_i,P_j)]_{i,j=1}^3\ge0, \qquad 1/2\le r\le1. $$ The range $1/2<r<1$ is new even for matrices, and order three is sharp for $1/2\le r<1$. This single mechanism resolves two problems. First, if $(\mathcal R,\omega)$ is finite tracial and $H=[H_{ij}]_{i,j=1}^3\in M_3(\mathcal R)_+$, then $$ [\omega(|H_{ij}|^p)]_{i,j=1}^3\ge0, \qquad 1\le p\le2, $$ settling the Lin–van den Driessche conjecture on Schatten norm compression. Second, for a semifinite tracial von Neumann algebra $(\mathcal M,\tau)$, define $$ d_t(A,B)^2=\frac{\tau(A)+\tau(B)}2- \|B^{t/4}A^{t/4}\|_{2/t}^{2/t}. $$ We prove that $d_t$ is a complete metric on $L^1(\mathcal M,\tau)_+$ for $1\le t\le2$, including the previously open range $1<t<2$ between the Hellinger and Bures endpoints, and that $$ d_t\text{ is a metric on }L^1(\mathcal M,\tau)_+ \quad\Longleftrightarrow\quad t\in[1,2]\ \text{or}\ \mathcal M\text{ is abelian}. $$ Its topology and Cauchy sequences agree with those of $L^1$. The same classification holds for unital tracial $C^*$-algebras and, with the known $t=0$ obstruction, resolves Problem~2 of Komálovics and Molnár.

We identify all radial interaction potentials that generalize Newton's exterior shell theorem, for which a uniformly distributed spherical shell is indistinguishable from a point-source up to strength renormalization and radial rescaling. We call these shape-preserving shell interactions, extending Gurzadyan's cosmological theorem which considers strength renormalization alone. We further expand the theory from monopolar to multipolar sources and establish the corresponding classification in arbitrary spatial dimensions. We find potential families beyond the spherical property subclass and show that the classifications for multipolar shape-preserving interactions form a nested hierarchy, with each lower-rank class being a subset of the next higher-rank class.

Multimodal large language models learn mostly from paired image-text data or annotated video, and raw web video is rarely used to further train an existing language model. We study whether raw video, with no captions and no text loss, can serve as mid-training data for a pretrained language model. Frames are encoded into continuous visual tokens, and the language model learns to predict the next visual token. We mid-train Qwen3-1.7B on raw clips from YT-Temporal-1B and then apply the same image-text instruction tuning to it and to the model without mid-training, so that the two differ only in mid-training. The mid-trained model scores 2.9 points higher on average across four video benchmarks and 5.1 points higher across ten image benchmarks, spanning perception, document, and chart tasks. Text performance is preserved even though mid-training includes no text, with an average of 48.9 across 14 text benchmarks compared with 48.0 for the model without mid-training. Analyses across training show that the image and video gains emerge within 30% of training and plateau thereafter, varying by less than 0.5 points. Predicting captions fails to outperform next-visual-token prediction, demonstrating that video mid-training can remain purely self-supervised without the computational overhead or labeling noise of automated captioning.

We give a necessary and sufficient homological criterion for smoothing projective Calabi-Yau threefolds with nodes and anticanonical cones over smooth del Pezzo surfaces of degrees five, six and seven, or over a smooth quadric surface. For some choice of local smoothing components, the criterion requires a relation on a crepant resolution among link classes whose associated local parameters avoid every discriminant. The complexified relation space describes first-order deformations modulo locally trivial ones. Necessity follows from local period and vanishing-cycle arguments; sufficiency applies Friedman's unobstructedness theorem to a crepant partial resolution. Whenever the criterion holds, the deformation space on the chosen components is smooth. A nonsmoothable example, whose deformation space is reduced with four smooth components, shows why nonvanishing at every singular point is insufficient. We also construct a smoothable Batyrev hypersurface admitting no smoothing induced by proper ambient families obtained from Minkowski decompositions at a single lattice degree.

Let $\mathcal{M}$ be a positroid. We prove that the positroid variety $\Pi_{\mathcal{M}}$ is toric with respect to some algebraic torus if and only if $\mathcal{M}$ is regular. In that case, $\Pi_{\mathcal{M}}$ is toric with respect to the standard diagonal torus. Thus, there are no unexpected toric positroid varieties.