AI Turing Machines and Catastrophe Theory - Application to the Extinction Problem: " Will AI Extinguish Humanity ? "

This work introduces the AI Turing Machine (AITM) framework and its probabilistic extension, formalizing adaptive computation, solvability hierarchies, and catastrophe theory mappings. We apply the model to the existential question “Will AI extinguish humanity?” and prove that while deterministic prediction of extinction is unsolvable, probabilistic risk estimation is tractable. Using René Thom’s cusp catastrophe, we show how solvability bifurcates into probabilistic modeling and collapses at singularities, offering a rigorous mathematical lens for existential risk analysis.