On the Falsification of AI Mathematics and the Meaning of True and False

The deep involvement of artificial intelligence in mathematical discovery is reshaping the ontological boundaries of mathematical knowledge. In October 2026, OpenAI and Anthropic independently claimed to have falsified the Exponential Time Hypothesis (ETH) in subexponential time. However, this is not a simple victory of Popperian binary logic. This paper introduces the original DHT (Defect–Homotopy–Topos) framework and invokes the O1 theoretical reachability diagnostor (see Paper X (in preparation)) to conduct a topological analysis of this event. This paper proposes the Homotopy Probing Theorem (HPT [To Be Constructed]): algorithm search driven by computational power is essentially a boundary probing within a topos, which necessarily induces a geometric morphism, leading to a nontrivial deformation of the original homotopy groups. Under internal topos logic, AI formal verification manifests as a locally self-consistent "open embedding" (), the natural language transcription layer presents as a "thin intersection" (), and the event as a whole is in a state of "transient connection." This paper points out: AI is a local prober within a topos, while human intuition is the constructor of global geometric morphisms.