arXiv preprints from January 1, 2026 through September 23, 2026 — 19:19:34 EST
This document reports over 15 counterexamples found using GPT Pro over the course of 12 months, for problems in combinatorics, number theory, convexity, analysis, and other areas. The counterexamples concern both (reasonably) well-known problems as well as lesser known instances. The counterexamples are accompanied by the repo...
Let $p$ be an odd prime and let $K/\mathbb{Q}_p$ be a finite unramified extension. From a finite presentation by roots of unity of order prime to $p$, we construct a localized Iwasawa pseudomeasure on $\mathbb{Z}_p^\times$. Although the pseudomeasure depends on the chosen presentation, its image modulo bounded measures depends only on...
Let $J_r^{SU(n)}(K;q)$ denote the reduced $SU(n)$ quantum invariant of a zero-framed knot $K$, colored by the $r$th symmetric power of the defining representation and normalized to be $1$ for the unknot. For every fixed $n\ge2$ we prove the Chen--Liu--Zhu cyclotomic expansion conjecture: there are unique coefficients...
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a right Auslander-Reiten-Serre (ARS for short) duality $(τ, η)$ in the sense of Iyama, Nakaoka and Palu. We show $(τ, η)$ induces right ARS dualities on the relative theories of $(\mathcal{C},\mathbb{E},\mathfrak{s})$. Under relative structures, we show...
Let $A, T \in M_n(\mathbb{C})$ and let $Δ_A(T) = AT - TA$ denote the inner derivation induced by $A$. We determine when $T$ and $Δ_A(T)$ are nilpotent under the second-order relation $$ Δ_A^2(T) + α\, Δ_A(T) + β\, T = 0, \qquad α, β\in \mathbb{R}, $$ according to the location of the roots of $z^2 + αz + β$. If the roots have...
Fix an integer $p\geq 2$ and $0<s<1$. We construct an ergodic $\times p$-invariant measure $μ$ on $\mathbb T^2$ of dimension $s$, such that every line projection preserves dimension, including when the corresponding projected IFS has exact overlaps. In fact, our measure assigns mass $O(w^s)$ to every planar tube of width $w$. The...
The main focuses of the article are limit theorems of Wasserstein projection in the convex order which are useful for inference tasks. The main results rely on the establishment of dual attainment, stability and several useful observations. The first is a clean criterion to invoke the Wasserstein projection dualities on the classical...
We prove a purely classical, coding-theoretic characterization of Hausdorff and packing dimensions in \(\mathbb R^n\). A \emph{ball code} assigns names to closed balls of \(\mathbb R^n\): it is a partial map from a prefix-free set of finite binary strings to balls. For every nonempty \(E\subseteq\mathbb R^n\), the Hausdorff dimension...
A company with a fixed artificial intelligence (AI) budget must decide which large language model (LLM) handles each recurring workload. What it lacks is the quality table, how well each model performs on each workload. Given that table, the decision is a multiple-choice knapsack problem and is routine to solve, so estimating it is...
Let $u_j $ traverse an orthonormal basis of Hecke--Maass forms for $\mathrm{SL}_2 (\mathbb {Z}) $ with Hecke eigenvalues $λ_j (n)$ and Laplace eigenvalue $1/4+t_j^2$. In this paper, we consider the short-interval variant of the twisted spectral large sieve inequality of Luo for $ λ_j (n) n^{it_j} $ on the range $t_j \leqslant T$ and...
Let $K$ be a field containing $\mathbb{Q}$, let $Γ$ be the ring of all functions from the positive integers to $K$ under Dirichlet convolution, and let $Γ_n$ be its truncation to functions supported on $[1,n]$. In [Snellman, Homology Homotopy Appl. 2 (2000), 17-27; arXiv:math/9904143] it was shown that $Γ_n$ is a polynomial ring...
Salient object detection in optical remote sensing images (ORSI-SOD) requires dense predictions that preserve object completeness and structural continuity under complex backgrounds, scale variation, and irregular object shapes. Existing methods often localize salient regions, but their predictions may still suffer from structural...
Natural Language Processing methods have enabled novel solutions and advances in the field of privacy, particularly in the sub-domain of text-to-text privatization, where the goal is to transform a sensitive input text into a privatized output by ideally masking (in)directly identifiable or otherwise private information. The...
Recent advances in large reasoning models (LRMs) have shown strong performance on complex problems through long chain-of-thought (Long CoT) reasoning. However, distilling such trajectories into smaller student models remains challenging: direct Long CoT supervision often provides limited gains and can be less effective than concise...
Retrieval-Augmented Generation (RAG) improves the factuality of large language models (LLMs), yet existing RAG systems often struggle with complex, multi-step reasoning that requires adaptive retrieval and continuous revision of intermediate contexts. Recent reinforcement learning (RL)-based agentic RAG methods partially alleviate...
Long-horizon story-driven video generation requires a production agent to coordinate narrative decomposition, state tracking, shot design, prompt construction, rendering, and revision across interdependent scenes. Existing adaptive video systems primarily refine requests or reusable skills, leaving recurring production failures...
Multi-access speculative inference (Multi-SPIN) extends SPIN to multi-device edge networks to accelerate cooperative token generation. It allows on-device small language models (SLMs) to autoregressively draft multiple tokens for individual generation tasks, while an edge-server large language model (LLM) verifies them in parallel....
This paper introduces a hybrid fact-checking framework that integrates Knowledge Graph-based semantic memory with adversarial multi-agent reasoning for explainable misinformation detection. The proposed system follows a memory-first, web-fallback architecture, in which input claims are initially evaluated against a dual-index...
Criminal judgment prediction requires models to infer statutory articles, charges, and sentencing outcomes from case facts. Unlike standard classification tasks, it involves a structured reasoning process in which statutes should be matched with facts, charges should be justified by statutes, and sentencing outcomes should remain...
Across industries, machine-learning systems support applications ranging from prediction and anomaly detection to forecasting, optimization, and scheduling, yet operationalizing these systems requires coordinating application development, model pipelines, cloud infrastructure, security, deployment, monitoring, retraining, recovery,...
Function vectors (FVs) have recently emerged as a promising mechanism for steering the behavior of large language models (LLMs) by injecting task-specific latent direction representations derived from in-context demonstrations. While prior studies have shown that FVs can recover task behavior in structured in-context learning...
Sustained scientific work requires a knowledge substrate that carries interpretation across tasks and preserves paths to source evidence. We call this process \emph{scientific knowledge compilation} and implement it in ASKS, the \emph{Agent-Driven Scientific Knowledge System}. For each source, an LLM produces a readable Wiki view and...
Unsupervised action segmentation aims to discover latent action categories and their temporal organization without action annotations. Optimal transport-based methods provide structured frame-to-action assignments, however, their pseudo-label quality is fundamentally conditioned on the representation space used to construct the...
The current alignment tuning paradigm for Large Language Models (LLMs) prioritizes surface-level behaviors -- fluency, safety, and tonal consistency. While effective for casual chat, this thesis argues that such surface alignment masks a lack of grounding, creating models that are stylistically confident but situationally brittle. We...
Semantic segmentation is a crucial task for understanding Mars, the most Earth-like planet in our solar system. However, it is challenging because the Martian surface is highly unstructured and complex, making accurate pixel-level prediction and fine-grained annotation difficult. Recent advancements in deep learning have introduced...