Papers

We prove that the maximal cp-rank $p_n$ satisfies $p_n\ge n(n-3)/2$ for odd $n\ge5$ and $p_n\ge n(n-3)/2-1$ for even $n\ge6$. Together with the known upper bound $p_n\le\binom{n+1}{2}-4$, this gives $p_n=n^2/2+O(n)$.

Deploying Vision Transformers (ViTs) on low-power edge devices is challenging due to high computational demands. Conventional pruning frameworks require a separate compiled binary for each sparsity level, increasing storage overhead and limiting runtime adaptability. This paper presents an end-to-end deployment pipeline that transforms pretrained ViTs into a single runtime-configurable binary, enabling dynamic compute-budget switching on embedded CPUs. This is achieved by restructuring generated C kernels with modified loop bounds and binary-mask control logic, allowing execution to switch across discrete sparsity levels via compact external configuration files. Compared to multi-binary deployment, the proposed runtime-adaptive approach reduces on-device storage by up to 4.86x, requiring only 163 MB for ViT-Base instead of nearly 800 MB. To maximize pruning efficiency, we introduce a hardware-aligned block pruning strategy for Multi-Layer Perceptron (MLP) layers. In addition, a custom ISA extension is proposed to exploit input-reuse patterns in linear projection kernels. On a Synopsys TRV32P3FX RISC-V processor, the full system achieves up to 2.8x speedup at 65% MLP and 50% attention-head pruning for ViT-Base. The ISA extension alone provides a 1.56x speedup and 33% lower inference energy, with a 24.7% area overhead in a TSMC 28 nm implementation.

Local fermionic mode (LFM) based quantum computation has pure state spaces that are isomorphic, via the Jordan-Wigner representation, to qubit state spaces, but its compositional structure is subtly different. Indeed, the algebraic formalism specifying how to embed fermionic systems underlies a notion of parallel composition, for which, in general, the usual interchange law of monoidal categories fails. We give a categorical account of this phenomenon. Starting from the CAR-algebraic semantics of fermionic gate application, we show that LFM processes form a symmetric premonoidal category whose centre is precisely the even, parity-preserving subcategory. The same processes can alternatively be organized with the ordinary tensor product and a fermionic presymmetry that is natural exactly on even maps. To reason diagrammatically in this latter presentation, we introduce pronaps, a relaxation of the notion of prop, in which permutations are not-necessarily-natural, and use them to organize a hierarchy of fragments of the fermionic ZW calculus relevant to the study of fermionic circuits on physically motivated gate-sets. We then extend scalable diagrammatic notation to pronaps. In this setting, syntactic sugars inspired by SZX and GSA are defined, with the specificity that matrix arrows encode minors and determinants, while graph triangles encode pfaffians of principal submatrices. These constructions yield elegant normal forms and completeness proofs for our presentations of the EoW, FoW, EW, and FW fragments. In particular, the FW presentation gives a new normal form for the matchgate fragment, distinct from the rWGS-X normal form of the planar-W (pW) presentation of the same category. The resulting framework connects fermionic circuit semantics, diagrammatic rewriting with scalable notations, and the algebra of determinants and pfaffians.

Much of quantitative science involves difficult mathematical computations: a Feynman integral, evidence for an evolutionary tree, or a regulatory statistic. These are often estimated by Monte Carlo or heuristic search, or computed in floating point without tracking rounding error. Exact methods often exist, but in a field far from the problem. The reduction methods of collider physics also work on a Bayesian evidence, and a regulatory threshold can be settled with interval arithmetic from numerical analysis, but few people know both fields. A large language model knows both, and every other field, and with working programs in hand it can carry a method from one to another. BootLoops is a pool of such programs: integral reduction, differential equations and high-precision evaluation from particle physics, integer-relation fitting from experimental mathematics, exact enumeration, and ball arithmetic, in one toolkit that any agentic large language model can operate and extend. These tools can compute multi-loop Feynman integrals by bootstrap methods rather than direct integration, across the function classes that arise, including polylogarithms, elliptic functions, and periods of K3 surfaces and Calabi-Yau manifolds. They can also evaluate Bayesian evidence integrals as exact rational numbers, enumerate finite configuration spaces with proof of completeness, show when a sought closed form cannot exist, and redo floating-point calculations with guaranteed digits. In practice this brings the tools of mathematics, computer science and physics to problems in genomics, statistics, ecology and other fields. The toolkit and its documentation are at https://bootloops.ai.

We prove the Koumandos–Ruscheweyh conjecture for every $0<\rho\leq1$. If $\nu(\rho)$ is the unique root in $(0,1]$ of $\int_0^{(1+\rho)\pi}t^{\mu-1}\sin(t-\rho\pi)\,\mathrm{d}t=0$, then $(1-z)^\rho s_n^\mu(z)\prec((1+z)/(1-z))^\rho$ for every $n\geq0$, $0<\mu\leq\nu(\rho)$ and $z\in\mathbb{D}$, where $s_n^\mu(z)=\sum_{k=0}^n(\mu)_kz^k/k!$. The parameter $\nu(\rho)$ is optimal. The proof combines an all-parameter gamma-coefficient comparison with an exact beta integral and a binomial variance estimate, reducing the infinitely many degrees to finitely many continuous interval inequalities. The remaining inequalities are certified by 256-bit ball arithmetic; rational covers, source code and alternative-formula verifiers are provided. The weak positive-real-part conjecture follows as a sharp corollary. We also derive sharp consequences for starlike functions and Gegenbauer polynomial sections: a full-parameter convolution subordination, the optimal starlike order for uniform partial-sum sectors, and the optimal Gegenbauer parameter and sector angle. The necessary bounds and angular sharpness are obtained from explicit kernels and interior scaling limits.

Temporal context may aid automated bat-species classification, but the contribution of specific features remains unclear. We investigated whether variation in the interpulse interval (IPI)-the time between consecutive call onsets-provides species-discriminative information and whether transformer-based models are more sensitive to this information than convolutional neural networks. We created two matched datasets from European bat recordings: a natural-IPI condition retaining the original call timing and a normalized-IPI condition in which call onsets were spaced at 50-ms intervals. EfficientNet-B0 and PaSST were fine-tuned and evaluated within each condition. In an additional experiment, each architecture was trained separately on natural-IPI and normalized-IPI recordings, and evaluated on the same natural-IPI test set. Finally, the pretrained classifiers BatDetect2 and BAT were evaluated on both conditions. Within-condition IPI normalization had model-dependent effects. PaSST accuracy differed little between the natural-IPI ($71 \pm 2.3\%$) and normalized-IPI ($70 \pm 6.3\%$) conditions, whereas EfficientNet accuracy increased from $47 \pm 4.7\%$ to $57 \pm 3.9\%$. PaSST exceeded EfficientNet under both conditions. In the cross-condition evaluation, models trained on natural-IPI recordings outperformed those trained on normalized-IPI recordings on the natural-IPI test set: accuracy decreased from 54% to 50% for EfficientNet and from 65% to 57% for PaSST. BatDetect2 and BAT differed little between IPI conditions. Overall, we found limited support for the hypotheses that natural IPI variation contributes substantially to bat-species classification and that it is used more effectively by transformer-based than CNN-based models. Nevertheless, the cross-condition performance decrease shows that results obtained under normalized conditions may not transfer fully to natural recordings.

Low-rank adaptation (LoRA) provides a parameter-efficient approach to continual learning, but its nominal rank can conceal a loss of effective adaptation capacity. We identify spectral plasticity collapse: during sequential adaptation, update energy becomes concentrated in a small subset of singular modes, leaving much of the available low-rank space underutilized. This exposes a limitation of interference avoidance alone: protecting historical representations does not ensure that the remaining adaptation capacity is responsive to new tasks or effectively utilized. To address this problem, we propose MuLoRA, which jointly controls capacity allocation and utilization. First, historical whitening identifies input directions with strong current-task response relative to accumulated historical response, yielding a task-adaptive basis that remains fixed during training. Second, approximate polar orthogonalization of momentum updates reduces spectral concentration within theselected space. An orthonormal basis connects these mechanisms by transferring the factor-update spectrum exactly tothe induced weight update. We establish a max–min characterization of exact subspace selection and derive cumulative spectral bounds under controlled cross-step anisotropy. Across five class-incremental benchmarks and eight incremental settings, MuLoRA achieves the highest mean accuracy in 15 of 16 reported metrics.

Let $P$ be a Markov operator on a general state space $(S,\Sigma)$ with invariant probability $m$, assumed ergodic. When $P$ is normal in $L_2(m)$, we prove the necessity of uniform ergodicity, under some spectral conditions (satisfied when $P$ is symmetric), for the annealed CLT to hold for every centered $ f \in L_2(m)$. We also obtain results without assuming normality under some Ritt-type conditions.

Societal resilience research relies on access to useful and actionable data, which motivates our main research question: Can annual reports, processed at scale with LLMs, provide a useful signal about how companies disclose their response to AI? We test this by applying a reproducible two-stage classification pipeline to 9,821 annual reports from 1,362 UK listed companies (2020-2025, with partial 2026 data). We first validate the method against 474 human-annotated passages, finding high recall and moderate label-level agreement. We then report three empirical patterns: (i) between 2020 and 2025, the share of reports mentioning AI risk rose from 2.8% to 41.2%, while AI adoption disclosure also rose, from 13.8% to 45.2%, and named vendor mentions cluster around a small set of major providers led by Microsoft; (ii) disclosure varies substantially by Critical National Infrastructure sector and market segment: AIM reports disclose AI risk at far lower rates than Main Market reports, and sectors such as Energy and Data Infrastructure lag behind the rest in AI risk disclosure; and (iii) harm disclosures are near-absent (seven reports across the entire corpus). We develop a substantiveness classification to assess the quality of the disclosure and find that most AI risk disclosure is not substantive: in 2025, 41.2% of all reports mention AI as a risk, but only 4.3% contain AI risk disclosure we classify as substantive.

Electromagnetic properties of materials are commonly inferred from how they scatter light. Here we show that this diagnostic is fundamentally incomplete. We predict the existence of an intrinsic electromagnetic property that remains invisible in any scattering experiment independently of geometry, polarization, far-field or near-field excitation. Still, this property affects the radiation emitted by the sources embedded inside the material. We reveal this phenomenon for a specially designed nonreciprocal magnetoelectric material combining axion and dual axion responses. Our results establish intrinsic electromagnetic properties as a distinct class of material responses and show that scattering experiments alone are insufficient to characterize matter.

For edge-disjoint graphs meeting along a finite boundary, we construct relative shellings of NBC faces with prescribed boundary connectivity. The construction adds paths in spanning forests to the restriction faces of auxiliary rooted-forest complexes. When every edge of the first graph precedes every edge of the second, these relative shellings combine into a shelling with consecutive connectivity blocks. The resulting nonnegative decomposition of the $h$-polynomial realizes the chromatic specialization of the Negami splitting identity. We also give path formulas for gluing graphic Orlik–Terao Gröbner bases across boundaries of size two or three, and a virtual-clique elimination formula for arbitrary boundaries. An exponential output lower bound at treewidth two distinguishes explicit basis enumeration from coefficient computation at fixed treewidth. Finally, state polytopes and their Cayley lifts describe an obstruction to linear block orders in the original coordinates and a lifted realization of these orders.

Cognitive diagnosis (CD) is a fundamental task in intelligent education that profiles learner proficiency over knowledge concepts. In real-world learning platforms, newly added items continually introduce previously unseen concepts, necessitating dynamic expansion of the underlying concept space. Yet existing incremental CD models assume a fixed concept space, allowing gradients from new items to overwrite historical pathways and induce catastrophic forgetting. More critically, these methods rely solely on soft constraints to preserve historical diagnoses. Such constraints may fail to satisfy the requirement of diagnostic invariance after incremental updates, a requirement known as psychometric consistency in cognitive diagnosis. Therefore, we propose CLEAN (Continual Learning with Expandable and Architecturally Isolated Networks), a novel incremental CD framework supporting concept-space expansion while providing structural guarantees for pointwise invariance of historical diagnoses. Specifically, CLEAN first introduces a strict topological bipartition protocol, freezes historical diagnostic functions and applies deterministic orthogonal column masking to sever gradient interference. Second, to accommodate concept expansion, expandable full-rank branches with micro-variance initialization are deployed to learn novel concepts. Finally, to verify that this architectural design achieves invariance by construction, we formalize Representation Drift (RD) to quantify the perturbation of historical traits. Extensive experiments on three large-scale educational datasets demonstrate that CLEAN achieves zero RD, preserving old-item metrics identically to static anchors through architectural isolation while remaining competitive with or superior to strong continual-learning baselines on new items.

We study deformations of foliated nonsingular varieties defined by locally free subsheaves of tangent sheaves and locally free subshseaves of cotangent sheaves in the language of functors of Artin rings. We identify first-order deformations and obstructions.

Choose a uniformly random ordering of the vertices of a finite tree and run the usual greedy maximal-independent-set algorithm. We compare the resulting law on maximal independent sets with the uniform law. We prove that exact uniformity occurs only for the one-vertex tree and the single edge. The proof is structural: a diameter endpoint exposes a pendant star, and the remaining one-pendant-leaf case is resolved by a strict injection between exact permutation fibres obtained by swapping the pendant leaf with its support vertex. Exact uniformity is therefore rigid, but it can be approached closely. For an explicit mixed-spider family $T_{k,l}$ we count the maximal independent sets and compute the exact probability of every output. With $l=2^k-k$ the total-variation bias is positive and satisfies [ b(T_{k,2^k-k})=O!\left(\frac{\sqrt{k}}{4^k}\right) =O!\left(\frac{\sqrt{\log n_k}}{n_k^2}\right), \qquad n_k=2^k+k+1. ] The theorem package has also been formalised in Lean and registered with Palomar. These records document machine-checked formal verification and the checked axiom boundary; they are not peer review or a certificate of novelty.

An edge open packing is a set of edges whose endpoints induce a disjoint union of stars. We study the problem of finding a maximum edge open packing in AT-free, chordal, and convex bipartite graphs. For AT-free graphs, we use the oriented star-conflict graph $A_G$, introduced by Das and Santra, for which $\rho_e^o(G)=\alpha(A_G)$, and prove that $A_G$ is AT-free whenever $G$ is AT-free. Consequently, a known maximum independent set algorithm for AT-free graphs yields an $O(n^2+m^4)$-time algorithm for Maximum Edge Open Packing, where $n$ and $m$ denote the numbers of vertices and edges of $G$, respectively. For chordal graphs, we develop an $O(n^4)$-time dynamic programming algorithm over a nice tree decomposition derived from a clique tree, exploiting the structural fact that the endpoint set of an edge open packing intersects every clique in at most two vertices. Finally, for convex bipartite graphs, we obtain an $O(n^4)$-time dynamic programming algorithm based on two boundary indices that separate consecutive star components. This result extends the previously known algorithm for biconvex bipartite graphs and, in particular, provides an improved explicit running-time bound for that class.

Generating useful robot-training data requires more than visually plausiblescenes: objects must support interaction, placements must remain physicallyvalid, and tasks must admit repeatable execution. We presentAwomo-SimDataEngine, an agentic system that connects asset and scenegeneration to robot demonstration synthesis. Shared asset services providerigid and articulated objects, including structure-grounded part and jointgeneration with ISArt. Scene generation supports two complementary routes:Unravel reconstructs editable scenes from images, while SimForge buildssingle-room and multi-room environments from text. A graph-native harnesscoordinates construction, validation, andbounded repair, routing failures to the responsible module while retainingunaffected scene state. PolicyForge binds validated worlds to tasks and robotembodiments to produce replayable demonstrations. Evaluations cover assetgeometry, scene quality, and downstream policy learning. On MuJoCo-basedLIBERO-Plus, co-training with Isaac Sim demonstrations improves the overallsuccess rate of a World-Action Model (WAM) from $77.17\%$ to $89.43\%$. Goal and spatialsuccess improve by $31.66$ and $6.25$ percentage points, respectively.These results support the utility of the generated data for cross-simulatorpolicy training, with more limited gains on long-horizon tasks.

I argue against the proposal of arXiv:2609.35787 to divide arXiv astrophysics submissions into two streams: peer-reviewed papers and pre-review discussion drafts. I contend that such a division would threaten open scientific discussion and could lead to increasingly restrictive categories based on journal impact factor or mainstream versus non-mainstream papers. Instead, readers who want only peer-reviewed work can use AI or other tools to filter papers, while others can continue monitoring the broader literature. I emphasize three benefits of posting papers on arXiv before journal acceptance. First, public posting lets researchers beyond the referee provide feedback, including identifying possible errors, missing references, and overlooked prior work, improving manuscripts beyond what a single referee may catch. Second, arXiv can help address weaknesses in the traditional anonymous peer-review system, such as lengthy delays, inconsistent reports, or unjustified rejection. Public posting establishes a transparent record of priority and allows the community to access results immediately. Finally, I propose that astrophysics journals require authors to post their papers on arXiv for at least a week before acceptance, making arXiv the central component of the astrophysical publication process.

Let $G$ be an $n$-vertex connected nonregular graph with maximum degree $\Delta$ that attains the maximum spectral radius. For $3\le\Delta\le n-2$, Liu and Li (2008) conjectured that $G$ has the degree sequence $(\Delta,\ldots,\Delta,\delta)$ with $\delta<\Delta$ as large as possible. In this paper, we study the dense case where $\Delta=n-k$ with $k\ge4$ fixed and $n$ sufficiently large. We prove that \[ d(G)= \begin{cases} (\Delta,\ldots,\Delta, 1), & \text{if $n$ is even and $k$ is odd};\\ (\Delta,\ldots,\Delta,\Delta-1), & \text{if $n$ is odd and $k$ is even};\\ (\Delta,\ldots,\Delta,2), & \text{if $n$ and $k$ have the same parity}. \end{cases} \]

A prime labeling of a graph labels its vertices bijectively with the integers from one to the number of vertices, so that adjacent vertices receive coprime labels. I prove the Entringer-Tout conjecture that every tree admits a prime labeling, extending the result of Haxell, Pikhurko, and Taraz from sufficiently large trees to all orders. My proof combines structural decompositions of trees, arithmetic organization of the labels, and matching arguments. I reduce the labeling problem to a structured bipartite matching problem by separating a small part of the tree, distributing labels according to their divisibility properties, and applying Hall's theorem to complete the labeling. I handle the finite and intermediate ranges using exact computer-assisted certificates whose validity is justified mathematically, while the remaining range is covered by analytic estimates. Together, these ingredients yield a proof for trees of every order.

Medical vision-language model (VLM) evaluation is sensitive to workflow design, prompting strategy, and benchmark construction, yet most studies treat these factors in isolation. We introduce a reliability stress test for chest X-ray interpretation built on two balanced datasets (a private report-backed set and a curated MIMIC subset). Three medical VLMs (CheXagent, MedGemma-4B, and MedGemma-27B) are evaluated across three prompt styles and two workflows (single-VLM and multi-agent), producing 36 configurations. We show that exact-match accuracy alone can overstate the effectiveness of conservative models that default to "Normal" predictions. Diagnostic reliability also depends heavily on model family and scale: multi-agent reasoning helps some configurations but hurts others. Building on these observations, we propose a decision-time routing framework that selectively escalates to multi-agent inference only when beneficial, improving the cost-quality trade-off over fixed workflows. Our results highlight the need for evaluation protocols that jointly consider prompt sensitivity, failure-mode diversity, and workflow choice before clinical deployment.

Emergency chest X-ray (CXR) triage has a structural modality gap: reports arrive after triage decisions, yet multimodal foundation models require image-text inputs. We present Variational Risk Minimization (VRM), a distillation framework that treats LVLM-generated report variants as Monte Carlo samples of latent clinical interpretations. Rather than distilling from a single teacher target, VRM learns from a variationally marginalized teacher distribution, enabling uncertainty-aware supervision under missing-modality constraints. Under matched encoder families, VRM outperforms direct fine-tuning baselines and improves calibration with strong recovery from hallucinated supervision. Marginalized supervision reduces report-selection instability. In our compact edge-student instantiation, a confidence-gated cascade reaches AUC 0.941 at 103ms average latency with 20.3% cloud escalation, yielding an explicit reliability-latency operating point for cloud-edge clinical workflows.

Machine unlearning is needed when data must be removed because of deletion requests, outdated records, or data-quality concerns, while retraining from scratch can be costly. Certified machine unlearning methods provide mathematical guarantees, while deployed systems release concrete finite-precision artifacts produced by software. To bridge the gap between mathematical guarantees and practical deployment, we introduce Executable Release Certification (ExecCert), a release-time layer that certifies the candidate artifact considered for release. ExecCert either closes a method's native certificate for the executed candidate or applies Retraining-Reference Release Verification (RRV) to certify fidelity to current retain-set retraining. Sequential deletion makes the latter nontrivial because the exact retain-set reference and the stored numerical state evolve separately. For frozen representations with a mutable ridge head, we develop an incremental realization of RRV that maintains certified evidence across deletion requests rather than reconstructing it at each release. On four published unlearning implementations, ExecCert preserves valid certificates, changes release decisions, tightens conservative bounds, and identifies the retraining-reference fidelity supported by concrete outputs. In sequential-service experiments, RRV eliminates false releases caused by stored-equation verification while closely tracking realized error, and incremental certification remains cheaper than both fresh and maintained verified-factor alternatives once release checks become sufficiently frequent.

Agent harnesses make many small, typed decisions per task: which model to call, which tool to use, whether retrieved text is relevant, whether an input carries an injection. System-1 decision models answer such questions in a single forward pass with class probabilities, promising large cost and latency savings over LLM calls. We present a paired evaluation of an open-weight (Laya) and a hosted (Jev) System-1 model on 11 agent decision points built from 18 public sources: 7,283 base cases plus 6,640 robustness variants, with byte-identical inputs, paired tests, and cross-hardware and cross-day reproducibility checks. Jev is significantly more accurate on 9 of 11 decision points (+10.8 to +46.0 pp). Neither model beats chance on zero-shot model routing, and they tie on RAG relevance gating. Laya changes 30% of its answers when the option order is reversed and degrades sharply with many or similar candidates (31% at 50 nearest-neighbour tools, vs. 98% for Jev on items with a unique correct tool). We also audit our own pipeline. Three analysis errors and one design confound distorted headline deployment claims: an omitted pre-screen cost (reported 23.9% saving, actual 4.3%), gate accuracy reported as end-to-end quality (58% vs. 98%), in-sample thresholds (5% target, up to 17% held-out misses), and a "channel effect" on injection false positives that vanishes with channel-native content. Two other suspected confounds did not change the conclusions. All cases, raw outputs and analysis code are available at https://github.com/David-DL-Space/sys1-eval.

We study the spherical geometry from the Simpson-Visser prescription $r\to\sqrt{r^2+a^2}$ to the quantum Oppenheimer-Snyder-Datt (qOSD) black hole of loop quantum cosmology, whose lapse function $A=1-2M/R+\alpha M^2/R^4$ carries the correction $\alpha$. This family interpolates between the qOSD black hole ($a\to0$), the Simpson-Visser black bounce ($\alpha\to0$), and Schwarzschild. The geometry is regular for every $a>0$, the Ricci scalar at the throat is blind to $\alpha$ while the Kretschmann scalar is not, and the null energy condition is violated at the throat by $-1/(4\pi a^2)$. The $(\alpha,a)$ plane splits into a two-horizon regular black hole, a single-horizon black hole with a hidden bounce, no Cauchy horizon, and a traversable wormhole; the middle region removes the mass-inflation channel of regular black holes without fine tuning. Because circular orbits are governed by the areal radius, photon sphere, shadow, ISCO and azimuthal epicyclic frequency depend on $\alpha$ only, whereas the Lyapunov exponent and radial epicyclic frequency carry the factor $\sqrt{1-a^2/R^2}$: shadow constrains only $\alpha$, while ringdown damping and the twin-peak quasiperiodic-oscillation ratio are sensitive to $a$. We derive horizon thermodynamics with a first-law entropy that departs from the area law, show that the bounce changes the thermodynamic topological charge from $W=-1$ to $W=0$ at every $\alpha$, and compute scalar, electromagnetic and Dirac quasinormal spectra using third-order WKB, Chebyshev spectral collocation and time-domain integration, which agree with one another and with eikonal scaling. Greybody bounds, Hawking emission, cascade sparsity and ringdown quality factor complete the analysis, and in the wormhole region the double-peaked potential produces echoes.

Neural video codecs derive motion and temporal contexts mainly from RGB frames, leaving room for cross-modal guidance from complementary temporal observations. Event streams can provide such observations by recording brightness changes between frames. In this work, we propose an Event-guided Neural Video Codec (ENVC) that uses events shared by the encoder and decoder to improve RGB compression efficiency. For motion coding, ENVC forms an event-guided motion prior and codes the remaining motion residual. For frame coding, an event-conditioned predictor supplies multi-scale features for gated temporal context refinement. To support training and evaluation on standard video datasets, we synthesize paired RGB-event data and assess its predictive utility through comparisons with real events. Across six benchmarks, ENVC achieves average BD-rate savings of 39.13% using PSNR-RGB and 67.63% using LPIPS relative to DCMVC. Further analyses show that our gains persist on large-motion sequences and that ENVC effectively learns to integrate event information. These results demonstrate the potential of events as a complementary modality for reducing the RGB coding rate. Our model and code are available at https://github.com/kjungwoo03/ENVC.

For a positive integer $n$, let $S(n)=\{\sigma(d): d\mid n\}$ be ordered by divisibility, and let $a(n)$ be its width. We study how much of the divisor lattice of $n$ survives under the map $d\mapsto \sigma(d)$. For every fixed prime-exponent pattern, the largest possible value of $a(n)$ is the width of a corresponding product of divisibility posets, and this maximum is attained infinitely often. In the squarefree case with $\omega(n)=r$, every width from $1$ to $\binom{r}{\lfloor r/2\rfloor}$ occurs, again infinitely often. For repeated exponent patterns, Engel's limit theory reduces the asymptotic width problem to optimal representation variance. We show that for the divisibility poset $D_m=(\{1,\ldots,m\},\mid)$, the prime directions $2$ and $3$ already determine the exact variance. The structural reason is the critical dilation $4$, lying strictly between $3$ and the next prime $5$. This yields an exact finite formula, the limit $V(D_m)\to 1.0367934201598\ldots$, and asymptotics for repeated exponent patterns. The complementary minimum-width problem remains largely open.

Patient-level lung ultrasound (LUS) diagnosis requires integrating findings acquired across multiple anatomical zones, yet clinical examinations frequently involve variable and incomplete scanning protocols with missing zones. Existing diagnostic AI methods primarily analyze individual frames or video loops, relying on heuristic aggregation strategies such as max or mean pooling that ignore inter-zone relationships for patient-level inference. This paper presents ZAGNet, a Zone- Aware Graph Neural Network that represents temporally tracked pathology findings as graph nodes connected by anatomical zone adjacency. A graph transformer network propagates contextual information across neighboring lung regions, while a virtual global node aggregates graph-level features to predict patientlevel consolidation and pleural effusion using only patient-level supervision. ZAGNet accommodates missing zones by computing on a graph structure without fixed input format or size. We evaluate ZAGNet on a multicenter dataset of 714 subjects (20,256 LUS video loops) with exams varying from 4 to 16 zones across anterior, posterior, and lateral thoracic regions. For consolidation diagnosis, ZAGNet achieved an AUC of 0.803 compared to 0.677 (max pooling) and 0.674 (mean pooling). For pleural effusion, AUC increased to 0.893 from 0.804 (max pooling) and 0.815 (mean pooling). These represent improvements of up to 19% and 11% for consolidation and pleural effusion respectively. The results demonstrate that graph-based inter-zone reasoning provides an effective and clinically consistent framework for automated patient-level LUS assessment.

A binary word is a shuffle square if its positions can be partitioned into two identical subsequences. A finite word over $\{1,2\}$ is smooth if repeated finite run-length differentiation remains binary until the empty word is reached. This paper proves that every smooth binary word in which both letters have even multiplicity is a shuffle square once its length is at least $18$. There are exactly $34$ exceptions, all of length at most $16$. The infinite part of the proof is a parity-return atom induction: a smooth even-Parikh word factors uniquely into smooth parity atoms, and every such atom has length at most $8$. A finite, independently checkable certificate handles lengths $18$ through $40$. As a consequence, a nonempty prefix of the classical Kolakoski sequence is a shuffle square if and only if both symbols occur an even number of times and its length is not $4$ or $8$. There are infinitely many such prefixes. In fact, the entire infinite Kolakoski sequence can be partitioned into two identical infinite subsequences with monochromatic blocks of uniformly bounded length.

We give explicit positive factorizations for a genus-19 Lefschetz fibration over the torus and a family of genus-4 Lefschetz fibrations over a genus-2 surface. All these examples have only three nodes, and their total spaces are symplectic 4-manifolds of general type on the Bogomolov-Miyaoka-Yau line. In the course of deriving the monodromy relation in genus 19, we obtain several interesting identities, including a factorization of a Dehn twist into a product of three order-3 mapping classes. We show that the total space of this fibration is topologically s-cobordant to the Cartwright-Steger surface, and we determine its real quotient up to homeomorphism. The genus-4 family consists of variations of the Bauer-Catanese fibrations and contains 65 isomorphism classes of Lefschetz fibrations on 45 diffeomorphism types of compact complex ball quotients. We describe their topology and that of the real quotients of the three original surfaces. The handle monodromies in our main factorizations are pseudo-Anosov, and we show more generally that for holomorphic genus-4 Lefschetz fibrations on the BMY line, this holds for every choice of handle generators.

I don't think I have ever done anything as peculiar in my life. Among other things, it shows a young man looking with interest at a print on the wall of an exhibition that features himself. How can this be? Perhaps I am not far removed from Einstein's curved universe.” So wrote M.C. Escher about his 1956 lithograph Print Gallery. Nearly half a century later, a mathematical analysis related its geometry to an untwisted source image through a conformal power map $z \mapsto z^\alpha$, $\alpha \in \mathbb{C}$. Building on this construction, we use a frozen text-to-image diffusion model to generate new self-referential scenes. Prompting alone does not enforce the recursion, while a post-hoc transformation can leave structures poorly connected. Applying the transformation during sampling is also insufficient: the denoiser may "repair" the intended distortion or drift out of the prescribed geometry. We construct a generalized inverse $T^\dagger$ of the non-invertible image transformation $T$, adapted to its recursive constraint. In the idealized formulation, the Penrose identity $TT^\dagger T = T$ makes $TT^\dagger$ an idempotent projection onto geometrically admissible images. Yet denoising only the transformed image remains an out-of-distribution task, even with projection. We therefore braid denoising steps with $T$ and $T^\dagger$: source-space steps develop the untwisted scene, while transformed-space steps refine its appearance and connections in the final geometry. We generate Print Gallery-like compositions and explore further transformations. Rather than distorting a finished image, we let the scene and its distortion develop together.