From Uncertainty to Lucidity: Awareness Neutrosophic Kähler-Einstein Innovative Evaluator Methodological in Era of Green Artificial Intelligence

The rapid development of generative artificial intelligence (Gen AI) is a double-edged sword. On the positive side, Large Language Models (LLMs) of Gen AI as chatbot considered intelligent friend. Due to its potential to stimulate the maturation of ideas and cultivate fundamental general abilities like problem-solving and critical thinking. The advancement of Gen AI continued after that, moving from “chatbots” to “AI agents” that carry out multi-step activities in addition to responding to queries. Regarding the downside, the terminology of “Red AI” era brought about by generative AI is marked by a performance at any expense that puts pressure on the world’s water and energy resources. This downside is seen as an incentive for the shift to a green AI (Gr AI) paradigm, which is essential to creating an ecosystem that is sustainable. Due to Gr AI deploys synonyms of green in AI and green by AI. That is why this paper clarified the role of Gr AI to achieve sustainability by deploying it in various domains. Thereby, we analyze and gauge the greenness of healthcare institutions that leveraging Gr AI in its operations cycle. We propose a mathematically rigorous and computable model for evaluating the greenness of healthcare empowered by Gr AI. The framework blends single-valued neutrosophic evidence-truth/support T, indeterminacy I, falsity/opposition F-with Kähler geometry to form a Neutrosophic Kähler-Einstein (NKE) model. A policy vector α=(αT,αI,αF) turns the triplet of (1,1)-forms ωN=(ωT,ωI,ωF) into an aggregated Kähler form ωα=αTωT+αIωI+αFωF. We then link indicators to neutrosophic evidence measures (μT,μI,μF) and calibrate a neutrosophic Monge-Ampère equation that controls global consistency. Ultimately, the constructed model applied in empirical case study.