The integration of large language model (LLM) technology into judicial systems is fundamentally transforming legal practice worldwide. However, this global transformation has revealed an urgent paradox requiring immediate attention. This study introduces the concept of ``consistency-acceptability divergence'' for the first time, referring to the gap between technical consistency and social acceptance. While LLMs achieve high consistency at the technical level, this consistency demonstrates both positive and negative effects. Through comprehensive analysis of recent data on LLM judicial applications from 2023--2025, this study finds that addressing this challenge requires understanding both task and stakeholder dimensions. This study proposes the Dual-Track Deliberative Multi-Role LLM Judicial Governance Framework (DTDMR-LJGF), which enables intelligent task classification and meaningful interaction among diverse stakeholders. This framework offers both theoretical insights and practical guidance for building an LLM judicial ecosystem that balances technical efficiency with social legitimacy.