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The exponential proliferation of Artificial Intelligence (AI) workloads, particularly Large Language Models (LLMs) and generative AI systems, has precipitated a critical operational trilemma in modern cloud computing: the simultaneous minimization of financial expenditure and carbon emissions while maintaining strict Service Level Objectives (SLOs). Traditional Financial Operations (FinOps) frameworks, which prioritize unit economic efficiency, often inadvertently exacerbate carbon footprints by incentivizing the usage of low-cost but carbon-intensive cloud regions. Conversely, emerging Green Operations (GreenOps) initiatives can incur prohibitive costs or introduce unacceptable latency penalties, creating friction with business objectives. This paper introduces Predictive Green FinOps, a comprehensive, formal methodology for the joint optimization of cost, carbon, and reliability in AI-intensive cloud environments. By integrating advanced predictive analytics for spot instance interruption risks and grid-level carbon intensity forecasting, the framework enables dynamic, policy-driven workload placement and temporal shifting. A rigorous simulation using 2024 data across ten major cloud regions demonstrates that this approach can achieve a 28% reduction in carbon emissions ($tCO_2eq$) and a 19% reduction in training costs compared to baseline schedulers, while maintaining reliability scores above 99.9%. Furthermore, the research addresses the "hidden costs" of reliability failures, providing a mathematical basis for evaluating the trade-off between checkpointing overhead and spot market volatility. These findings provide a robust foundation for enterprise-grade, sustainable AI infrastructure strategies in the 2025–2035 era, ensuring compliance with emerging regulations such as the EU Corporate Sustainability Reporting Directive (CSRD) and SEC climate disclosure rules.