Generative Large Language Models (LLMs) are increasingly used in non-generative software maintenance tasks, such as fault localization (FL). Success in FL depends on a models ability to reason about program semantics beyond surface-level syntactic and lexical features. However, widely used LLM benchmarks primarily evaluate code generation, which differs fundamentally from semantic program reasoning. Meanwhile, traditional FL benchmarks such as Defect4J and BugsInPy are either not scalable or obsolete, as their datasets have become part of LLM training data, leading to biased results. This paper presents the first large-scale empirical investigation into the robustness of LLMs fault localizability. Inspired by mutation testing, we develop an end-to-end evaluation framework that addresses key limitations in existing LLM evaluation, including data contamination, scalability, automation, and extensibility. Using real-world programs with specifications, we inject unseen faults and ask LLMs to localize them, filtering out underspecified programs where localization is ambiguous. For each successfully localized program, we apply semantic-preserving mutations (SPMs) and rerun localization to assess robustness and determine whether LLM reasoning relies on syntactic cues rather than semantics. We evaluate 10 state-of-the-art LLMs on 750,013 fault localization tasks from over 1,300 Java and Python programs. We find that SPMs cause LLMs to fail on previously localized faults in 78% of cases, and that reasoning is stronger when relevant code appears earlier in context. These results indicate that LLM code reasoning is often tied to features irrelevant to semantics. We also identify code patterns that are challenging for LLMs to reason about. Overall, our findings motivate fundamental advances in how LLMs represent, interpret, and prioritize code semantics to reason more deeply about program logic

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