Static Application Security Testing (SAST) tools are integral to modern software development, yet their adoption is undermined by excessive false positives that weaken developer trust and demand costly manual triage. We present ZeroFalse, a framework that integrates static analysis with large language models (LLMs) to reduce false positives while preserving coverage. ZeroFalse treats static analyzer outputs as structured contracts, enriching them with flow-sensitive traces, contextual evidence, and CWE-specific knowledge before adjudication by an LLM. This design preserves the systematic reach of static analysis while leveraging the reasoning capabilities of LLMs. We evaluate ZeroFalse across both benchmarks and real-world projects using ten state-of-the-art LLMs. Our best-performing models achieve F1-scores of 0.912 on the OWASP Java Benchmark and 0.955 on the OpenVuln dataset, maintaining recall and precision above 90%. Results further show that CWE-specialized prompting consistently outperforms generic prompts, and reasoning-oriented LLMs provide the most reliable precision-recall balance. These findings position ZeroFalse as a practical and scalable approach for enhancing the reliability of SAST and supporting its integration into real-world CI/CD pipelines.