Generative Artificial Intelligence (GenAI) represents a rapidly expanding digital infrastructure whose energy demand and associated CO2 emissions are emerging as a new category of climate risk. This study introduces G-TRACE (GenAI Transformative Carbon Estimator), a cross-modal, region-aware framework that quantifies training- and inference-related emissions across modalities and deployment geographies. Using real-world analytics and microscopic simulation, G-TRACE measures energy use and carbon intensity per output type (text, image, video) and reveals how decentralized inference amplifies small per-query energy costs into system-level impacts. Through the Ghibli-style image generation trend (2024-2025), we estimate 4,309 MWh of energy consumption and 2,068 tCO2 emissions, illustrating how viral participation inflates individual digital actions into tonne-scale consequences. Building on these findings, we propose the AI Sustainability Pyramid, a seven-level governance model linking carbon accounting metrics (L1-L7) with operational readiness, optimization, and stewardship. This framework translates quantitative emission metrics into actionable policy guidance for sustainable AI deployment. The study contributes to the quantitative assessment of emerging digital infrastructures as a novel category of climate risk, supporting adaptive governance for sustainable technology deployment. By situating GenAI within climate-risk frameworks, the work advances data-driven methods for aligning technological innovation with global decarbonization and resilience objectives.