We derive the first closed-form condition under which artificial intelligence (AI) capital profits could sustainably finance a universal basic income (UBI) without relying on new taxation or the creation of new jobs. In a Solow-Zeira task-automation economy with a CES aggregator $σ< 1$, we introduce an AI capability parameter that scales the productivity of automatable tasks and obtain a tractable expression for the AI capability threshold -- the minimum productivity of AI relative to pre-AI automation required for a balanced transfer. Using current U.S. economic parameters, we find that even in the conservative scenario where no new tasks or jobs emerge, AI systems would only need to reach only 5-7 times today's automation productivity to fund an 11%-of-GDP UBI. 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 automation 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. These results therefore offer a rigorous benchmark for assessing when advancing AI capabilities might sustainably finance social transfers in an increasingly automated economy.