Vision Language Models have demonstrated remarkable proficiency in interpreting static visual artifacts, but modern data analysis is inherently dynamic, requiring the active interrogation of interactive environments. Existing benchmarks are predominantly constrained to static imagery and one-shot question answering and fail to capture the epistemic demands of this domain, where evidence is frequently occluded, distributed across linked views, or conditionally revealed through user agency. In this paper, we introduce InSight, a benchmark for agentic claim verification over interactive visualizations. The dataset consists of 21,349 claims derived from human-authored analytical narratives and grounded in fully interactive web-based environments. Agents must navigate these environments to determine whether a natural language claim is supported, refuted or not verifiable given the available evidence. Unlike traditional evaluations, InSight treats interaction traces as intrinsic proxies for reasoning, enabling a rigorous audit of how models seek and synthesize visual evidence. We evaluate state-of-the-art models, revealing that interactive verification remains a non-trivial challenge. We release InSight at https://github.com/maevehutch/insight.

Controlling the output of Large Language Models (LLMs) through context-sensitive constraints has emerged as a promising approach to overcome the limitations of Context-Free Grammars (CFGs) in guaranteeing generation validity. However, such constraints typically require manual specification—a significant barrier demanding specialized expertise. We introduce a framework that automatically learns context-sensitive constraints from LLM interactions through a two-phase process: syntactic exploration to gather diverse outputs for constraint learning, followed by constraint exploitation to enforce these learned rules during generation. Experiments demonstrate that our method enables even small LLMs (1B parameters) to learn and generate with perfect constraint adherence, outperforming larger counterparts and state-of-the-art reasoning models. This work represents the first integration of context-sensitive grammar learning with LLM generation, eliminating manual specification while maintaining generation validity.