We derive the first closed-form condition under which artificial intelligence (AI) capital profits could sustainably finance a universal basic income (UBI) without additional taxes or new job creation. In a Solow-Zeira economy characterized by a continuum of automatable tasks, a constant net saving rate $s$, and task-elasticity $σ< 1$, we analyze how the AI capability threshold--defined as the productivity level of AI relative to pre-AI automation--varies under different economic scenarios. At present economic parameters, we find that AI systems must achieve only approximately 5-6 times existing automation productivity to finance an 11\%-of-GDP UBI, in the worst case situation where \emph{no} new jobs or tasks are created. Our analysis also reveals some specific policy levers: raising public revenue share (e.g. profit taxation) of AI capital from the current 15\% to about 33\% halves the required AI capability threshold to attain UBI to 3 times existing automotion productivity, but gains diminish beyond 50\% public revenue share, especially if regulatory costs increase. Market structure also strongly affects outcomes: monopolistic or concentrated oligopolistic markets reduce the threshold by increasing economic rents, whereas heightened competition significantly raises it. Overall, these results suggest a couple policy recommendations: maximizing public revenue share up to a point so that operating costs are minimized, and strategically managing market competition can ensure AI's growing capabilities translate into meaningful social benefits within realistic technological progress scenarios.