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We determine sharp class-size estimates for multiset codes over $q$ symbols that correct up to $t$ deletions. For fixed $q,t\ge2$, we prove that the syndrome classes defined by Nathanson's geometric-sum labels modulo $M=1+t+\cdots+t^{q-1}$ differ from their average size by $O(n^{q/p-1})$, where $n$ is the blocklength and $p$ is the least prime divisor of $q$. This exponent is sharp, and every prescribed class has size asymptotic to the average. For prime $q$, all classes have equal size when $n\equiv-1,\ldots,-(q-1)\pmod M$. We also present our radix construction, whose labels are zero and the first $q-1$ powers of $t+1$, starting at $1$, modulo $t(t+1)^{q-2}+1$. We determine its sharp class-size deviation exponent. For even $q\ge4$, its syndrome classes have a smaller deviation exponent, although their average size is smaller. Over four symbols and every $t\ge2$, we show that the radix labels still correct up to $t$ deletions modulo a reduced modulus, which gives bounded deviations and eventually larger classes than either unmodified construction. For a fixed number $k\ge2$ of surviving symbols, we prove that some maximum code contains all constant words, reducing the remaining optimization to multisets with multiplicities below $k$. Finally, for one deletion at length four over $q\ge4$ symbols, we show that the best bound from fractional weights on deletion outputs, rounded down, equals the maximum code size exactly when $q\equiv2,4\pmod6$.

Unsupervised video object-centric learning aims to decompose dynamic scenes into persistent, object-level representations without supervision. However, existing slot-based methods struggle to maintain stable object identity in challenging settings such as rapid motion and partial occlusion. First, they typically encode both the per-frame appearance of an object and its identity across frames in a single slot vector, creating an objective conflict that leads to slot swapping: reconstruction requires sensitivity to transient visual changes, whereas temporal consistency requires invariance to them. Second, the token renormalization used in Slot Attention can amplify weakly attending slots, allowing them to absorb tokens from other objects and destabilize slot-to-object correspondence. We propose Dual-State Slot Attention (DSSA), a fully self-supervised framework that addresses these limitations by separating appearance from identity and by reducing spurious updates from weakly matching slots. DSSA decomposes each slot into a local state for per-frame appearance and an identity state for temporally stable object information, thereby aligning reconstruction and temporal consistency with separate representations. The identity state is updated through a learned recurrent transition that acts as a temporal filter on the local state, while competition-modulated aggregation (CMA) down-weights updates from weakly matching slots and prevents them from absorbing tokens from other objects. Experiments on MOVi-C, MOVi-D, and YouTube-VIS demonstrate that DSSA consistently improves segmentation quality and temporal consistency over prior methods, while also yielding stronger downstream object recognition and video dynamics prediction. Code is available at https://uark-aicv.github.io/dssa/.

This paper determines the accepting-state spectrum of the right-quotient operation for permutation automata, solving an open problem posed by Rauch and Holzer. They showed that in the unary case in which the languages whose right quotient we consider are languages over an alphabet with but one symbol, the accepting state complexities which could ensue for automata recognizing their quotient is exactly all integers between $1$ and $mn$ inclusive when $m,n \ge 1$; we show that for arbitrary alphabets the accepting-state spectrum is $g^{\operatorname{asc}}_{-1,\mathrm{PFA}}(m,n)=\{0\}$ if $m=0$ or $n=0$, and $g^{\operatorname{asc}}_{-1,\mathrm{PFA}}(m,n)=\mathbb{N}_{>0}$ if $m,n\ge 1$. Our result shows that every positive accepting-state complexity ensues for right quotient as long as both of the languages of which we consider the right quotient are nonempty. $0$ becomes the only unavoidable magic value. Moreover, we show that an alphabet with just two letters can witness any arbitrary $α\ge 1$ when $m,n \ge 1$, in that languages can be defined over a binary alphabet with accepting-state complexity $m$ and $n$ respectively for which their right quotient has accepting-state complexity $α$. What ultimately happens in our proof is that we construct $(A^{\mathrm q}_{m,α},B^{\mathrm q}_{n,α})$ such that the words accepted by $B^{\mathrm q}_{n,α}$ induce a point stabilizer in a symmetric group, and the standard quotient construction giving an automaton recognizing the quotient language then saturates the original final set of $A^{\mathrm q}_{m,α}$ to a full orbit; this gives a minimal quotient automaton with exactly $α$ final states. The complete spectrum which is our result follows from this, solving the open question of what the right quotient's accepting-state complexity for permutation automata is.

Concept vectors aim to enhance model interpretability by linking learned internal representations with human-understandable semantics, but their practical utility is often limited by noisy and inconsistent activations. In this work, we discover the SuperActivator Mechanism: a transformer dynamic that amplifies concept activation gaps, concentrating the most reliable concept evidence into a small set of highly activated tokens, which we call SuperActivators. We theoretically characterize this mechanism, proving that concept-aligned attention produces two competing effects: a tail amplification effect, where gaps among already-high activations grow disproportionately fast, and an equalization effect that eventually dominates as attention concentrates on extreme tokens. We validate the predicted tail amplification empirically, finding that in-concept activation distributions develop well-separated upper tails consistently across models, image and text modalities, and concept types. Moreover, we demonstrate the practical implications of SuperActivators for concept detection and localization. First, using only high-activation tokens as concept indicators improves detection by up to 14 F1 points over competing baselines. Second, restricting detection to the extreme activation tail is not merely sufficient but essential: including activations beyond the top 5-10% of the in-concept distribution introduces noise that degrades performance. Finally, using SuperActivators as instance-specific attribution targets yields more accurate and faithful concept localization heatmaps than using global concept vectors.

We study the query complexity of envy-free division for cake, chores, and resources with both positive and negative values. For arbitrary nonnegative nonatomic chore costs, we give a deterministic exact protocol using $2^{n\log_2 n+O(n)}$ queries and cuts. For both cake and chores, a Las Vegas protocol uses $O(n^{12}\log(2n))$ expected queries and $O(n^{11}\log(2n))$ expected cuts. Conversely, every exact protocol with fixed positive success probability requires $Ω(n^2)$ expected queries in general, even with arbitrarily many output cuts at no query cost. For mixed cake and chores, no uniform finite expected-query bound exists even when every Cut returns the leftmost root or every Cut returns the rightmost root. This holds for any fixed positive multiplicative approximation factor and success probability, already when all net totals are one. An additive impossibility holds on the same normalized domain.

Politically biased AI can shift users' opinions, yet it remains unclear whether model disclaimers protect against such persuasion. We report a preregistered experiment (N=1,586) examining how disclaimer design and user characteristics shape susceptibility to politically biased AI. We varied whether disclaimers identified political bias versus potential inaccuracies and whether they included an explanation. Participants completed two decision-making tasks with an LLM biased against their political views, with outcomes assessed immediately and one week later. Disclaimers identifying political bias and including an explanation reduced persuasion, but effects varied across contexts and did not persist. Individual differences were more consistent predictors: Republicans were more persuaded and reported higher trust than Democrats across tasks. Similarly, participants who used AI more frequently were also more persuaded, whereas those with greater self-reported AI knowledge were less persuaded. These findings suggest that generic disclaimers are not enough: effective safeguards must account for user heterogeneity and context.

Many open mixture-of-experts (MoE) language models train with Muon instead of AdamW. Most balance the load with a bias on each expert's routing score, adjusted after every step. Their recipes disagree on whether Muon should also train the router. In models with 55M and 108M active parameters, we measure how the optimizer changes the span of these biases, the largest minus the smallest in a layer. The span grows when all tokens favor the same experts, so it shows how hard a layer is to balance. At learning rates tuned on shorter runs, Muon raised the span of the first MoE layer at every seed, roughly doubling it on average, while the other layers changed much less. The rise came from Muon on the rest of the network, whichever optimizer trained the router. In those runs, the inputs to the first router shared a direction, and soon after the warmup its busiest expert averaged four to seven times its fair share. Over the first 3,000 steps, subtracting the batch mean from these inputs removed this overload while barely changing the loss. On the smaller model, halving Muon's learning rate removed most of the rise and slightly improved the loss. Moving only the router to AdamW made the loads shift more during the warmup and slightly raised the loss, so at this scale the results favor keeping the router on Muon.

Geometry estimation from perspective images has greatly advanced, maturing to the point where off-the-shelf foundation models are able to reconstruct 3D scene structure not only from multi-view imagery, but even from a single view. A natural extension is 3D reconstruction from panoramas, with the exciting prospect of recovering a full 360-degree scene from a single panoramic image. In this work, we introduce PaGeR (Panoramic Geometry Reconstruction), a framework to lift powerful 3D foundation models designed for perspective imagery to the panorama domain. Our strategy is to start from a pre-trained transformer for 3D reconstruction and turn it into a unified high-performance model that predicts scale-invariant depth, metric depth, surface normals, and sky masks from both perspective and omnidirectional images, in a single forward pass. By keeping architectural changes to a minimum and mixing perspective and panoramic images during training, PaGeR retains the rich 3D prior of the underlying foundation model while learning to also estimate geometrically consistent 360-degree scenes from single panoramas. We extensively test our method in both indoor and outdoor environments and find that it delivers state-of-the-art performance and excellent zero-shot performance across a wide range of scenes. Code, data and models are available $\href{https://github.com/prs-eth/PaGeR}{\text{here}}$.

We describe SCARAB--Swarm-Capable Autonomous Robotic Aquatic Bridging. Using distributed swarm control and multi-model sensing of agents and docking targets, agents can localize themselves, join into formations and proceed to desired target locations. Our technologies would eventually allow the Army to perform unpredictable, dispersed river crossings, enhance crew survivability, and minimize the logistics footprint compared to the current Improved Ribbon Bridge. Our methods operate without GPS or RF communications, though these can be used in non-contested environments (e.g., civilian disaster relief for flooding, etc.). We demonstrate our system via physics-engine-based simulation of several agents using unmanned surface vehicles (USVs) in the presence of currents and wind.

Long-horizon embodied tasks require agents to learn, reason, and generalize robustly across complex real-world environments. However, existing approaches often struggle with partial observability, fragmented spatial reasoning, and inefficient integration of heterogeneous memories, limiting their capacity for long-horizon adaptation. To address this, we introduce RoboMemory, a brain-inspired framework that unifies Temporal, Spatial, Semantic, and Episodic memory for efficient long-horizon planning and interactive learning. Its core innovations are a comprehensive memory system designed for embodied tasks, and a dynamic spatial knowledge graph for scalable, consistent memory updates. Extensive experiments on EmbodiedBench show that RoboMemory improves the average success rate of Qwen2.5-VL-72B-Ins by 26.5 percentage points and achieves performance comparable to strong closed-source VLM agents. Real-world trials further suggest RoboMemory's potential for interactive environmental learning by reusing accumulated experience across repeated task attempts. These results demonstrate RoboMemory as a scalable memory-augmented framework for embodied agents, improving long-horizon planning and cumulative learning in complex interactive environments.

We study integer programs in standard form $\min\{\mathbf{c}^{\mathsf T}\mathbf{x}:A\mathbf{x}=\mathbf{b},\mathbf{l}\leq\mathbf{x}\leq\mathbf{u},\ \mathbf{x}\in\mathbb{Z}^n\}$, where $A\in\mathbb{Z}^{d\times n}$, all entries of $\mathbf{b},\mathbf{c},\mathbf{l},\mathbf{u}$ are integers, and every variable bound is finite. Let $N$ be an upper bound on the absolute values of the entries of $A$, and $c_{\max}$ one on the absolute values of the costs. For every fixed $d$ and every fixed $\varepsilon>0$ we give a deterministic algorithm running in time \[ T_{\textrm LP}+O_d(n)+O_{d,\varepsilon}\bigl(N^{2d+\varepsilon}\log^2(c_{\max}+2)\bigr), \] where $T_{\textrm LP}$ is the time to solve one linear programming relaxation exactly. Prior to our work, the best known running time for bounded variables was $N^{O(d^2)}$ [Eisenbrand \& Weismantel'20], and it has been a fundamental open problem whether the exponent $O(d^2)$ can be improved to one linear in $d$. We answer this question affirmatively, with the exponent $2d+\varepsilon$.

Reinforcement learning from human feedback (RLHF) is widely used to align large language models with human preferences, but it remains vulnerable to \emph{reward hacking}, whereby a policy exploits flaws in the reward function at the expense of the intended behavior. Reward shaping can stabilize training and partially mitigate reward hacking, yet its design principles remain underexplored. Our systematic study of common shaping methods suggests that shaped rewards should be bounded and should rise rapidly before gradually saturating. We introduce \emph{Preference as Reward} (PAR), which applies a sigmoid to centered rewards to produce a signal that reflects the reward model's latent preferences. We show that PAR has a variance-reduction property that stabilizes training and substantially widens the window for effective early stopping. Using Proximal Policy Optimization (PPO) and Gemma2-2B, PAR consistently outperforms the other shaping methods evaluated and demonstrates high data efficiency and robustness. Comparisons with unshaped rewards across four base models and four reinforcement learning algorithms show that PAR is effective across models and particularly beneficial for actor--critic methods when value estimates become unstable. The code is available at https://github.com/PorUna-byte/PAR.

Choosing a file format for tabular data is a routine engineering decision, yet published comparisons often time reads served from the operating-system page cache, use one session, or rely on a single synthetic dataset. This article benchmarks fifteen format and access-path configurations (CSV, JSON Lines, Pickle, Feather, Parquet, ORC, DuckDB, and SQLite, with several access paths) on two real public datasets, New York City taxi trips and United States airline on-time records, at 50,000 to 3 million rows. Every measurement uses an explicitly cold cache, randomized execution order, and three independent sessions (with restarts on the laptop), and is analyzed with bootstrap intervals and Holm-adjusted permutation tests. Parquet answered a filtered aggregation 2.6 to 11 times faster than CSV, while JSON Lines was 10 to 27 times slower. A secondary index made SQLite queries 2.6 to 4.8 times slower on randomly ordered data and helped only on data sorted by the filtered column, at low selectivity. A row filter inside the Parquet reader was significantly faster than a projected scan on flight data from one million rows and on both datasets at three million, and row-group skipping needs several row groups, which the default writer produced only at 3 million rows. A 4 GB Linux host ran out of memory on configurations that a 24 GB laptop completed and reversed some orderings, such as Pickle versus Parquet. Code, derived data, and raw timings are released.

Reproducing software vulnerabilities is fundamental to security researchers, open-source maintainers, and educators. Yet, vulnerabilities are often hard to reproduce, and even when they can be reproduced once, recreating a software environment where the vulnerability can be exploited becomes harder and harder over time. We present NICE, the NIx CvE reproduction framework, which uses declarative recipes to build and automatically validate vulnerable environments. In NICE, a reproduced CVE comprises one or more NixOS virtual machine configurations, a scripted exploitation scenario, and machine-checkable assertions that provide factual evidence of exploitation. This design facilitates sharing, validation, review, and long-term reproducibility. Building on NICE, we construct a public archive of 56 reproduced vulnerabilities spanning multiple CWE categories, attack vectors, and target types (user-space, system software, kernel, and graphical applications), combining manually authored recipes with recipes produced by an LLM-based automated pipeline and merged after human review. We evaluate the feasibility of the automated pipeline on 110 real-world CVEs, achieving fully automated reproduction in 54.5% of the cases, at an average cost of $0.73 per CVE. The evaluation demonstrates that NICE provides a practical target for LLM-assisted recipe generation, while also identifying where human refinement or additional automation remains necessary.

Research on bias in large language models (LLMs) has predominantly focused on third-person audits, which study how models represent or evaluate demographic groups as external subjects. However, this paradigm overlooks a structural blind spot because the user is absent from the audit. In practice, LLMs are used in open-ended, personal interactions, during which the model implicitly represents the user and adjusts its responses accordingly. When identical requests yield different responses depending on who is asking, bias manifests not in how the model describes others but in how it treats its interlocutor. Empirical first-person bias studies are multiplying, but they use different cues, outcomes, and validation choices, which makes their results hard to compare. We propose Situated Interaction Auditing (SIA), a shared diagnostic framework for this literature. It introduces a typology of user profile signals---names and surface markers, writing style and dialect, and stated identity---together with design principles and outcome measures that let studies be compared and accumulated. We instantiate the framework in a case study of Chilean first and last names across nine task domains and three models, and we close with guidance for practitioners and open questions for researchers.

Predictive Coding (PC) is an influential account of cortical learning. Much of recent work has focused on comparing PC to Backpropagation (BP) to find whether PC offers any advantages. Small scale experiments show that PC enables learning that is more sample efficient and effective in many contexts, though a thorough theoretical understanding of the phenomena remains elusive. To address this, we quantify the efficiency of learning in BP and PC through a metric called ``target alignment'', which measures how closely the change in the output of the network is aligned to the output prediction error. We then derive and empirically validate analytical expressions for target alignment in Deep Linear Networks. We show that learning in PC is more efficient than BP, which is especially pronounced in deep, narrow and pre-trained networks. We also derive exact conditions for guaranteed optimal target alignment in PC and validate our findings through experiments. We study full training trajectories of linear and non-linear models, and find the predicted benefits of PC persist in practice even when some assumptions are violated. Overall, this work provides a mechanistic understanding of the higher learning efficiency observed for PC over BP in previous works, and can guide how PC should be parametrised to learn most effectively.

An attributed hypergraph comprises nodes with attributes and hyperedges that connect varying numbers of nodes. Attributed hypergraph node and hyperedge embedding (AHNEE) maps nodes and hyperedges to compact vectors for use in important tasks such as node classification, hyperedge link prediction, and hyperedge classification. Generating high-quality embeddings is challenging due to the complexity of attributed hypergraphs and the need to embed both nodes and hyperedges, especially in large-scale data. Existing solutions often fall short by focusing only on nodes or lacking native support for attributed hypergraphs, leading to inferior quality, and struggle with scalability on large attributed hypergraphs. We propose SAHE, an efficient and effective approach that unifies node and hyperedge embeddings for AHNEE computation, advancing the state of the art via comprehensive embedding formulations and algorithmic designs. First, we introduce two higher-order similarity measures, HMS-N and HMS-E, to capture similarities between node pairs and hyperedge pairs, respectively. These measures consider multi-hop connections and global topology within an extended hypergraph that incorporates attribute-based hyperedges. SAHE formulates the AHNEE objective to jointly preserve all-pair HMS-N and HMS-N similarities. Direct optimization is computationally expensive, so we analyze and unify core approximations of all-pair HMS-N and HMS-N to solve them simultaneously. To enhance efficiency, we design several non-trivial optimizations that avoid iteratively materializing large dense matrices while maintaining high-quality results. Extensive experiments on diverse attributed hypergraphs and 3 downstream tasks, compared against 11 baselines, show that SAHE consistently outperforms existing methods in embedding quality and is up to orders of magnitude faster.

UAV-ground visual tracking (UGVT) aims to simultaneously track the same target from both UAV and ground views. However, existing two-stream methods suffer from isolated feature extraction and rely heavily on implicit appearance matching, which often fails to establish reliable cross-view correspondence under drastic viewpoint discrepancies, thereby leading to unstable tracking. To address these issues, we propose VL-UniTrack, a unified framework enhanced by visual-language prompts. By encoding features from both views with a single shared encoder, our method reduces feature isolation between the two views and enables sufficient cross-view interaction. To alleviate the ambiguity caused by appearance-only matching, we further design a visual-language viewpoint prompting module that fuses language descriptions with visual features to generate learnable prompts. These prompts are then injected into a prompt-guided cross-view adapter to facilitate cross-view interaction and guide the learning of view-specific feature representations. In addition, we introduce a confidence-modulated mutual distillation loss to regularize training by suppressing noise propagation. Extensive experiments demonstrate that VL-UniTrack achieves state-of-the-art performance on the latest benchmark. The code is available at https://github.com/xuboyue1999/VL-UniTrack.git.

Implicit neural representations (INRs) have emerged as powerful tools for encoding signals, yet dominant MLP-based designs often suffer from slow convergence, and how their accuracy depends on their size is known only empirically. We introduce FUTON (Fourier Tensor Network), which models signals as generalized Fourier series whose coefficients are parameterized by a low-rank tensor decomposition, read out by a linear or MLP head. FUTON combines complementary inductive biases: Fourier bases capture smoothness and periodicity, while the low-rank parameterization enforces low-dimensional spectral structure. Beyond universal approximation, we show that with a linear head, the best FUTON's error splits exactly into the truncation of the signal's series and the best low-rank error of its coefficients, which we bracket from the samples before training, with a FUTON that attains the upper bound. On image fitting, occupancy volumes, and novel view synthesis, against twelve INRs of matched size, FUTON achieves the best average quality, reaches the final quality of each task's strongest baseline 4-34x sooner than that baseline, infers fastest, and trains in the least memory.

Parametric retrieval enables LLMs to retrieve tools implicitly by assigning each API a unique virtual token and training the model to generate it via constrained beam search. Toolsense shows that this regime has two critical drawbacks: it destroys parametric tool knowledge during training, and its beam-search decoding is too slow for real-time deployment. We introduce TRACE (Tool Retrieval via Augmented Chain-of-thought and Enterprise rules), a two-stage curriculum that resolves this dissociation. Stage 1 reuses the multi-format memorization SFT from ToolSense to seed tool knowledge with LoRA. Stage 2 is our core contribution: the model is trained to emit a thinking trace before producing a JSON list of tool tokens, using two data sources -- RRB pairs from ToolSense and queries synthesized to target business rules curated by domain experts -- both augmented with reasoning traces. This training objective preserves Stage 1 MCQ and QA probing accuracy while enabling single-beam greedy decoding at production latency. Evaluated on a combined enterprise catalog of 8,300+ tools across two enterprise product lines, TRACE training for Stage 2 not only preserves but improves tool understanding: MCQ accuracy gains +3.2 pp and QA probing gains +9 pp over Stage 1. On retrieval, TRACE achieves ~86% recall on Domain A and ~60% on Domain B -- compared to embedding baseline performance of ~27% & ~52% -- both with single-beam greedy decoding, making it directly deployable at production latency.

The paper presents a statistical geometric duality between the spherical squared-Hellinger distance and a hyperbolic isometric invariant of the Poincare disc under the action of the general Mobius group. Motivated by the geometric connection, we propose the usage of the L2-embedded hyperbolic isometric invariant as an alternative way to quantify statistical divergence between Gaussian measures as a contribution to information theory and machine learning.

Autonomous SaR requires robots to explore unknown or partially known environments, avoid hazards, coordinate team activities, and guide discovered victims to safety. Improving individual capabilities of robots does not necessarily improve mission outcomes: safer motion can restrict the exploration behavior, faster discovery can increase the rescue demand, and effective guidance depends on victim responses. We introduce PA-SaR, a mathematical and architectural framework for coordinating heterogeneous local control, learning-based exploration, supervisory task allocation, and adaptive approach-lead victim guidance around mission-level objectives. PA- SaR distinguishes environmental ground-truth, robot-specific perceptions, and simulated victim behavior. This provides a common basis for integrating complementary mechanisms and analyzing their interactions across mission stages. We demonstrate the framework through single- and multi-robot simulation studies incorporating model-predictive and learning-based exploration, perception-based safety shielding, event-driven task allocation, and adaptive victim guidance. The results show trade-offs between collision avoidance and search completion, indicate that supervisory allocation improves search completion beyond information sharing alone, and demonstrate that increased exploration does not necessarily yield more rescues. Adaptive guidance increases rescues relative to continuous leading, without a detected improvement in mean or final victim trust or a reduction in mortality. These findings support evaluating local control, team coordination, and evacuation as related but distinct aspects of autonomous SaR within the considered simulation environments.

A strong conflict-free vertex connection coloring of a graph is a graph coloring for which, for every two distinct vertices of the graph, there exists a shortest path between them in which some color in said coloring appears only once. The strong conflict-free vertex connection number, denoted by $\operatorname{svcfc}(G)$, is the number of colors in the strong conflict-free vertex connection coloring with the smallest amount of colors out of all such colorings of $G$. We study the problem of deciding whether this number is at most some threshold together with the parameter twin cover. Our first result is that when we have given $(G,k)$ in which $k$ is some target number of colors and given some twin cover $X$ of $G$ which is of size $t$, there is a polynomial time reduction to an equivalent instance which has at most $\max\{2,t+k2^t\}$ vertices. This result allows us to give a kernel for Strong CFVC Number which is parameterized by $\operatorname{tc}(G)+k$, which implies that Strong CFVC Number is FPT w.r.t. the twin-cover number of $G$ so long as $k$ is fixed. Our second result is that if $G$ is a connected graph for which there exists a twin cover of size $t$, then $χ(G)\le \operatorname{svcfc}(G)\le χ(G)+t$. Our last result is an an exact expression for the chromatic number of graphs whose twin cover number is bounded. Namely, we show that for every proper coloring $\varphi$ of $G[X]$, the minimum number of colors needed to extend $\varphi$ to all of $G$ is $K_\varphi=\max_{S\subseteq X}(|\varphi(S)|+m(S))$, and hence $χ(G)=\min_{\varphi\text{ proper on }G[X]} K_\varphi$. Our results indicate that twin cover is a meaningful parameter in the context of strong conflict-free vertex-connection, and show that once we have a fixed twin cover, the only difficulty left is in some additive threshold above the chromatic number.

Modern LLM deployments use a number of implementation choices and inference optimizations (e.g., batching, custom kernels, and quantization) on top of fixed weights, so two engines serving "the same model" can produce meaningfully different distributions. We study the problem of estimating the total variation (TV) distance between two length-$n$ autoregressive distributions to additive error $\varepsilon$, under three access models. (1) Under sample access, we use $\widetilde{O}(n^2 K/\varepsilon^2)$ queries, where $K$ is the maximum support of the next-token distribution. This improves upon the $\widetilde{O}(n^3 m/\varepsilon^5)$-query estimator of Meel et al. (2025), where $m \geq K$ is the total size of the token alphabet. (2) Under logit access, we use $O(n/\varepsilon^2)$ queries, and this is tight. (3) Under noisy logit access, we smoothly interpolate between the above two guarantees: if probability values are given to relative error $σ$, we use $\widetilde{O}((n+n^2σ^2)/\varepsilon^2)$ queries. We complement our theoretical results with an empirical evaluation of our algorithms, for example measuring the distance between SGLang and vLLM serving identical weights. Our experiments highlight the robustness and practicality of estimating the total variation distance, which remains estimable where the KL divergence is infinite. Our code is available at https://github.com/XunZhiyang/llm-tv-estimation.