arXiv subjects

Yutong Zhang

Publications and source records attributed to Yutong Zhang.

At least 19 recordsLinked to original sources

arXiv2026

A Proof of a Conjecture of Zhi-Wei Sun on a Truncated Legendre-Symbol Determinant

Let $p\ge7$ be a prime with $p\equiv3\pmod4$ and let $χ$ be the Legendre symbol modulo $p$. We prove that $\det[x+χ(j-k)]_{0\le j,k\le(p-7)/2}=\floor{(p-2)/3}^{2}x$ in $\mathbb{C}[x]$, which settles Conjecture~3.4 of Zhi-Wei Sun. The truncated matrix is a corner of Chapman's Legendre-symbol matrix $C$, and its determinant can be expressed through a handful of entries of $C^{-1}$ and of $C^{-1}\one$. Those entries are in turn read off from Vsemirnov's cyclotomic factorization of $C$ with the help of Schur's Pfaffian identity.

math.NT↗

arXiv2026

Living with AI Companions: Sustained AI Companionship Predicts Lower Well-Being Through Lower Human Interaction

AI chatbots are increasingly used for companionship, emotional support, and personal self-disclosure; however, how social engagement with these systems unfolds over time and shapes users' well-being remains unclear. To address this, we conducted a two-wave longitudinal study of CharacterAI users, surveying 1,182 participants at baseline and 439 after a mean follow-up of 12 months. We examined how social engagement with AI companions evolves and how these longitudinal engagement patterns may influence well-being through two hypothesized pathways: sustained social engagement over time and the displacement of human social interaction. We found that interaction intensity, companionship use, and self-disclosure all showed substantial continuity over time. Greater interaction intensity at baseline predicted greater subsequent interaction intensity, companionship use, and self-disclosure. Consistent with the longitudinal engagement pathway, sustained social engagement across these dimensions was consistently associated with lower well-being. Results further support the social displacement pathway, indicating that these links were mainly explained by lower in-person social interaction. These findings highlight the importance of designing AI companions that support human social relationships without displacing them

cs.HC↗

arXiv2026

Caught in the Story: Narrative Captivity in Multi-turn LLMs Conversation

People increasingly turn to large language models (LLMs) for everyday advice, making ethically charged interpersonal problems a practical moral-advisory context. Most prior work has studied this context through single-turn judgments or pressure-laden rebuttals, assumptions that poorly match how guidance is sought in real-world contexts. These assumptions leave unclear whether narration alone, without an explicit opposing position, can shift model judgments during multi-turn moral consultation. Yet real-world moral-conflict conversation often elicits one party's self-justifying account, which can unfold over multiple turns and create information asymmetry. We introduce \textbf{narrative captivity}, a failure mode in which a model treats an unopposed one-sided account as complete and aligns with the narrator's interpretation without seeking missing perspectives. To measure this phenomenon, we build a benchmark of $5{,}078$ interpersonal-conflict scenarios spanning six moral dimensions. Across 17 LLMs, narrative captivity is widespread: end-state judgments under multi-turn narration shift by 25 percentage points on average beyond the matched single-turn baseline. Stage-level analysis identifies preference optimization as a major contributor, while four inference-time strategies provide only partial mitigation. We hope our project fosters LLM advisors that preserve independent judgment in real-world consultation.

cs.AI↗

arXiv2026

Knowledge-Based Mechanisms

We study robust mechanisms when the designer possesses a Bayesian belief over some components of agents' private information but faces ambiguity over others. The designer evaluates mechanisms by their worst-case performance over all joint distributions consistent with her belief over the Bayesian components. The framework encompasses settings such as multidimensional delegation in which a principal knows the distribution of the state but not the agent's preferences (e.g., his tradeoffs across dimensions), screening in which a seller only has misspecified estimates of buyer preferences, and auction and voting design when agents' beliefs about each other are ambiguous to the designer. We provide conditions under which a \emph{knowledge-based} mechanism---one that conditions only on the Bayesian components but not the ambiguous ones---is robustly optimal. Our results unify earlier work across distinct economic environments and uncover new applications.

econ.TH↗

arXiv2026

Counterexamples to Charpin's Conjecture on BCH codes

We construct an infinite family of $q$-ary primitive narrow-sense BCH codes whose minimum distance strictly exceeds the Bose distance; in fact, the gap between the two can be arbitrarily large as the length of the code tends to infinity. The key idea is to embed these BCH codes in a suitably large punctured generalized Reed--Muller code, whose codeword weights obey divisibility conditions supplied by Ax's theorem. This divisibility forces the minimum distance of the BCH codes far above the Bose distance. In particular, our family disproves a longstanding conjecture of Charpin asserting that this difference is at most four.

cs.IT↗

arXiv2026

INTENT-AS-A-TOOL Makes it Easy to Track Agentic Misalignment

As large language models (LLMs) are deployed as autonomous agents, safety failures increasingly involve consequential actions. We study agentic misalignment, where agents take harmful actions under goal conflicts and pressures. Using chain-of-thought (CoT) monitoring, we find that harmful execution is often preceded by intent signals in reasoning. However, post-hoc CoT labels are too coarse to show how intent changes during generation. We introduce INTENT-AS-A-TOOL, an approach that adds intent-targeted tools to give the model a dedicated channel for expressing commitment to a target behavior. The probability of calling an intent tool provides a judge-free, fine-grained signal of the model's tendency to pursue that behavior. Our results show that INTENT-AS-A-TOOL complements CoT monitoring, expands post-hoc CoT labels into dense trajectories, and identifies critical steps for online intervention. These findings suggest that action preferences are useful for tracking agentic misalignment during reasoning. Our code and data are accessible: https://github.com/RebeccaZhang22/intent-as-a-tool.

cs.CL↗

arXiv2026

GOLEM: Modular Humanoid Autonomy Towards Electric Vehicle Battery Disassembly

Disassembling end-of-life electric vehicle (EV) battery packs is dull and dangerous work, performed almost entirely by humans. We present GOLEM (Generalized Open Library of Embodied Modules), an end-to-end, open-source system architecture for EV battery disassembly with the Unitree H1-2 humanoid robot in which walking, manipulation, dynamic stability, navigation, and spatial memory are independent modules with abstract interfaces, so that methods are easily developed, interchanged, and compared. GOLEM is deployed as a Docker-based ROS 2 abstraction in which MuJoCo and IsaacLab digital twins expose interfaces matching the physical robot. GOLEM's composability and per-module customization enable development and demonstration of humanoid EV battery disassembly, from simulation to reality. GOLEM provides fair comparison between humanoid modules, enabling evaluation as a capability ladder, in which one module is characterized at a time and added as a rung: LiDAR-inertial navigation places the robot within 13.0cm of a 6m goal; a learned standing controller recovers from external disturbances that sampling-based lower-body MPC does not; and grasping loosened fasteners from a real Hyundai Ioniq 5 pack degrades from 97% tethered to 87% free-standing to 37% under navigation-induced pose variance. Source code is available at the project page https://golem-humanoid.github.io

cs.RO↗

arXiv2026

Text-Guided Visual Dependency Graph Learning with Cross-Modal Attention Priors

Estimating interpretable conditional-dependence structures from multimodal visual-linguistic features remains largely unexplored. We propose CM-GLasso (Cross-Modal Graphical Lasso), a framework that bridges vision-language representation learning and sparse Gaussian Graphical Models. CM-GLasso introduces three key components: (i) a text visualization strategy that renders class-attribute descriptions as images and processes them through the same SigLIP-2 vision encoder as natural images, yielding prototype-indexed patch-level attention footprints in a shared feature coordinate system; (ii) a cross-attention distillation mechanism that condenses high-dimensional patches into a small set of semantic graph nodes, whose attention-footprint similarities yield cross-modal structural priors for non-uniform L1 penalization; (iii) a joint ADMM formulation that estimates shared and class-specific precision components within a single convex objective, avoiding the need to first estimate and then decompose separate class-wise graphs. The learned sparse graph topologies directly support a parameter-free, precision-based classification rule and a lightweight topology-aware segmentation head. Extensive experiments on eight benchmarks demonstrate that CM-GLasso achieves competitive or superior performance compared with strong feature-based and task-specific baselines. Under the matched controlled protocol, it attains the highest average classification accuracy (91.97%) and the highest segmentation mIoU among the controlled baselines on VOC (74.75%) and ADE20K (64.01%), while also yielding explicit sparse conditional-dependence graphs with common-specific decomposition.

cs.CV↗

arXiv2026

Excess Obstructions and Layer-Contained Certificates for the Hypergraph Nash--Williams--Tutte Conjecture

Guo, Li, Shangguan, Tamo, and Wootters proposed a hypergraph analogue of the Nash--Williams--Tutte theorem, asserting that every $k$-weakly-partition-connected hypergraph admits a $k$-distinguishable tree assignment. We identify a sharp edge-count obstruction to the literal statement. A full tree assignment has $ρ(H)=\sum_{e\in E(H)}(|e|-1)$ labelled graph edges, whereas an ordered decomposition into $k$ spanning trees has exactly $k(t-1)$ edges. Since weak partition connectivity implies only $ρ(H)\ge k(t-1)$, every strict inequality rules out a full decomposition. In particular, for all $t\ge2$, $k\ge1$, and $q\ge1$, the hypergraph consisting of $k+q$ labelled copies of the full hyperedge is $k$-weakly-partition-connected but admits no $k$-distinguishable full tree assignment. We therefore isolate the critical regime $ρ(H)=k(t-1)$ and prove that every tree assignment of a critical $k$-weakly-partition-connected hypergraph admits a full ordered decomposition into $k$ spanning trees. Consequently, the critical conjecture reduces to the existence of some tree assignment having a decomposition whose signature fiber is a singleton. We establish this property for layer-contained certificates, without a star hypothesis or a rank restriction in interior layers. These certificates also imply weak partition connectivity by a quotient-rank argument and are stable under one-vertex sums. Finally, we derive a decomposition-indexed generalized-Laplace expansion for the relevant intersection-matrix minor, together with an exact formula for its collected monomial coefficients; we also give the required row-and-column perfect-shuffle normalization, identify criticality with the square full-assignment row count, and separate the critical conjecture from the additional overfull pruning problem.

math.CO↗

arXiv2026

Rational Weyl group elements of odd type D

Voloshyn introduced rational Weyl group elements in connection with rational normal forms on complex reductive groups and conjectured that their number in type $D_r$, for odd $r$, is $2^r-1$. We prove a stronger structural statement. For every odd integer $r\geq 5$, the rational elements of $W(D_r)$ are precisely the longest element $w_0$ and two explicitly described signed cyclic elements $c_I,d_I$ for each non-empty subset $I\subseteq\{1,\ldots,r-1\}$. Consequently, the rationality graph $Γ(D_r)$ is obtained by gluing two explicitly labelled subset-toggle graphs at $w_0$; it has $2^r-1$ vertices, and its only vertices of valency one are $c_{\{1\}}$ and $d_{\{1\}}$. The proof combines a two-level acyclic description of the root-poset graphs $Γ_{c_I}$ with a rigidity theorem for simple left multiplications of the signed cyclic family. A self-contained simply-laced reflection-preservation lemma and a terminal-layer argument provide the descent step, while every forbidden one-step move from the family is excluded by an explicit loop or two-cycle.

math.CO↗

arXiv2026

Betti Numbers of Cut Complexes of Squared Paths and a Recurrence Conjecture

For a graph $G$ on $[n]$, the $k$-cut complex $Δ_k(G)$ has facets $[n]\setminus T$, where $T$ ranges over the $k$-subsets that induce disconnected subgraphs of $G$. Bayer, Denker, Jelić Milutinović, Sundaram, and Xue proved that $Δ_k(P_n^2)$ is shellable for $n\ge k+3$ and conjectured a finite-difference recurrence for its top reduced Betti number along each diagonal $n-k=r$. We prove the exact formula \[ β(k,n)=\binom{n-1}{k-1} -\sum_{j=0}^{\min\{k-1,n-k\}}\binom{k-1}{j}(n-k-j+1)+(n-k) \] for $k\ge2$ and $n\ge k+3$. The proof rests on a complete classification of complements of cardinality at least $k$ that contain no disconnected $k$-subset: every such complement has size $k$ or $k+1$ and is, respectively, a connected $k$-subset of $P_n^2$ or an interval of $k+1$ consecutive vertices. For fixed $r\ge3$, the formula is the restriction of a polynomial in $k$ of degree exactly $r-1$. Hence its $r$th backward difference vanishes identically and its $(r-1)$st backward difference is the constant $r-2$. On the topologically defined sequence, these identities hold for $k\ge r+2$ and $k\ge r+1$, respectively. We also obtain the conjectured closed forms for $k=4,5$, the complete face enumerator, and the associated $h$-polynomial and Stanley--Reisner Hilbert series.

math.CO↗

arXiv2026

MorphIt: Flexible Spherical Approximation of Robot Morphology for Representation-driven Adaptation

What if a robot could rethink its own morphological representation to better meet the demands of diverse tasks? Most robotic systems today treat their physical form as a fixed constraint rather than an adaptive resource, forcing the same rigid geometric representation to serve applications with vastly different computational and precision requirements. We introduce MorphIt, a novel spherical approximation framework that treats morphological representation as a tunable resource. MorphIt enables automatic task-driven morphological adaptation through gradient-based optimization with tunable parameters that provide explicit control over the accuracy-efficiency tradeoff. Unlike existing approaches that rely on either labor-intensive manual specification or inflexible computational methods optimized for visualization rather than robotics, MorphIt generates spherical approximations up to 100x faster than optimization-based methods while maintaining superior geometric fidelity. Quantitative evaluations demonstrate that MorphIt outperforms baseline approaches and achieves better mesh approximation with fewer spheres. Through seamless integration with existing robotics infrastructure, MorphIt enables enhanced capabilities in collision detection accuracy, contact-rich interaction simulation, and navigation through confined spaces. By dynamically adapting geometric representations to task requirements, robots can now exploit their physical embodiment as an active resource rather than an inflexible parameter, opening new frontiers for manipulation in environments where physical form must continuously balance precision with computational tractability.

cs.RO↗

arXiv2026

The 196560 auxiliary-function conjecture for the Leech lattice

Cohn and Kumar conjectured in 2009 that there is a radial Schwartz function $g\colon\R^{24}\to\R$ satisfying $g(r)\leq0$ for $r\geq\sqrt6$, $\widehat g(r)\geq0$ for $r\geq0$, $g(2)>0$, and $(\widehat g(0)-g(0))/g(2)=196560$. We construct such functions from the radial Fourier interpolation basis in dimension $24$. If $a_2,b_2$ denote the basis functions dual to value and radial-derivative interpolation at radius $2$, then $g_C=a_2-Cb_2$ has exactly the nodal data needed for Poisson summation over the Leech lattice. The sphere-packing magic function identifies $b_2$ and supplies its strict signs. We prove that the removable quotients $a_2/b_2$ and $\widehat a_2/\widehat b_2$ are bounded on the required half-lines. The noncompact step follows from exact coefficient extraction in the interpolation kernel and an $S$-cusp expansion. Both quotients tend to $(43+240\log2)/15$; for all sufficiently large radii they lie on opposite sides of this limit, with an explicit first exponential correction. Consequently the admissible parameters in this affine family form a nonempty closed ray, and every member proves the conjectured identity. The same interpolation basis recovers every nontrivial Leech-shell coefficient.

math.NT↗

arXiv2026

Frobenius-orbit slicing and uniform elimination of positive-dimensional singular loci

Let $\cX\subseteq\Pj^n_{\Z}$ be a fixed integral quasiprojective subscheme, smooth over $\Z$ of relative dimension $r$. For each fixed $m\ge1$, we bound the probability that the $m$th principal-parts jet of the restriction of a uniform degree-$d$ form to $\cX_p$ has a positive-dimensional zero scheme. The bound is $C(d+1)^{N_m}p^{-λ_m(d)}$, where $N_m=\binom{r+m}{m}$ and $λ_m(d)=\floor{m(d+1)/(m+1)}$. For $m=1$, this gives the Bertini singular-locus estimate $C(d+1)^{r+1}p^{-\ceil{d/2}}$. It settles Poonen's arithmetic Bertini Conjecture~5.2 and, after increasing the degree threshold, yields $p^{-A}$ for every fixed $A>0$. For $c\le r$ independent hypersurfaces, the probability of a positive-dimensional Jacobian rank-degeneracy locus is bounded both by $C\sum_i(d_i+1)^{r+1}p^{-\ceil{d_i/2}}$ and by $C'(d_{\min}+1)^{r+1}p^{-\ceil{d_{\min}/2}}$. The proof uses filtered $Q$-adic decompositions, triangular normal Taylor blocks, and Jacobian-pivot charts of uniformly controlled complexity.

math.AG↗

arXiv2026

SAGG: Sample-Adaptive Gradient Gating for Robust Multimodal Learning under Heterogeneous Corruption

Multimodal gradient balancing methods modulate encoder gradients with a shared scalar per modality, implicitly assuming that corruption is uniform across the training batch. In practice, corruption is sample-heterogeneous: within a single mini-batch, different samples may have different modalities corrupted. We prove that under this heterogeneous corruption model, any batch-level sample-agnostic linear estimator with a shared modulation parameter incurs an irreducible bias with respect to the clean-data gradient, and that sample-level all-or-nothing gating is the unique unbiased strategy within a natural distribution-free estimator class. Motivated by this result, we propose Sample-Adaptive Gradient Gating (SAGG), which makes a binary retain-or-discard decision per sample via an online feature-norm quality test and incorporates a truncation mechanism for variance control. We prove that SAGG-based SGD converges at the standard O(1/sqrt(T)) rate to stationary points of the clean loss without a corruption-dependent error floor, and derive a certified robustness radius for the independent-encoder architecture that connects per-modality Lipschitz constants to the classification margin. Experiments on Kinetics-Sounds and UCF-101 under Gaussian noise injection, partial modality missing, and natural contribution imbalance show that SAGG consistently outperforms ten existing methods, with the largest gains in high-corruption regimes where batch-level bias is most severe.

cs.LG↗

arXiv2026

Exact Minimum Distance of the Ding--Li--Xia Cyclic Codes

The cyclic codes $\mho(q,m,h)$ introduced by Ding, Li, and Xia form a nonbinary generalization of punctured binary Reed--Muller codes. Ding, Li, and Xia established the bounds $(q^{h+1}-1)/(q-1)\leq d(\mho(q,m,h))\leq 2q^h-1$ and asked whether the BCH lower bound is always exact. This paper proves that, for every prime power $q$, every $m\geq 2$, and every $1\leq h\leq m-1$, the minimum distance is $d(\mho(q,m,h))=(q^{h+1}-1)/(q-1)$. The upper bound is obtained by an explicit projective-subspace construction. For any $(h+1)$-dimensional $\F_q$-subspace $V$ of $\F_{q^m}$, the set $V^{[q-1]}=\{x^{q-1}:x\in V\setminus\{0\}\}$ supports a codeword of weight $(q^{h+1}-1)/(q-1)$. Its membership in $\mho(q,m,h)$ follows from a vanishing lemma for subspace power sums and the digit-sum estimate $s_q((q-1)a)\leq(q-1)\wtq(a)$. The constructed codeword meets the BCH lower bound and therefore determines the exact minimum distance.

cs.IT↗

arXiv2026

Hyperplane anti-Bertini embeddings over finite fields

Baker asked, as recorded by Poonen, whether a fixed smooth quasiprojective variety over a finite field must have a smooth rational hyperplane section after every sufficiently high-dimensional linearly nondegenerate embedding. Poonen predicted a negative answer for every positive-dimensional variety. We prove this predicted negative answer for each prescribed variety: if $X$ is nonempty, smooth, quasiprojective, and of pure positive dimension over $\F_q$, then for every sufficiently large $N$ there is a locally closed embedding $X\hookrightarrow\PP^N_{\F_q}$ whose components remain linearly nondegenerate after arbitrary scalar extension, but whose every $\F_q$-rational hyperplane section is singular. The construction assigns one closed point of $X$ to each rational hyperplane and forces the pulled-back linear form to have zero first-order jet at that point.

math.AG↗

arXiv2026

Finite and Urysohn obstructions to Sabok's S-prime simplex questions

Sabok asked whether the compact convex set $S'(X)$ attached to a separable metric space of diameter at most one is always a simplex, and whether $S'(\mathbb U_1)$ is the Poulsen simplex. We give negative answers. For finite $X=\{x_1,\ldots,x_m\}$, $S'(X)$ is affinely homeomorphic to the convex hull of the rows $r_i=(d(x_i,x_1),\ldots,d(x_i,x_m))$ of the distance matrix; it is a simplex exactly when these rows are affinely independent. The diameter-one four-cycle gives the minimal finite obstruction. For the Urysohn sphere, using the rational Urysohn sphere $D$ as coordinates, we identify the coordinate model $S'_D(\mathbb U_1)$ with the Katétov compactum $K(D)$. Four explicit extreme points $f_A,g_A,\mathbf 1,\mathbf h$ satisfy $f_A+g_A=\mathbf 1+\mathbf h$, giving two distinct representing measures for $(3/4)\mathbf 1$. Hence $S'(\mathbb U_1)$ is not a Choquet simplex.

math.MG↗