Most arguments for "hard takeoff" involve some form of inductive proof: if we have "intelligence level" n, and we know that it can be improved n + 1, then there

Most arguments for "hard takeoff" involve some form of inductive proof: if we have "intelligence level" n, and we know that it can be improved n + 1, then there is a guarantee of eventual performance n + k where k is arbitrarily large. Aside from not actually having any reliable way to quantify n or the concept of "intelligence" in the first place, this proof falls apart if improvement is n + ε, where ε changes at each step, and there is no reason to assume ε will escape the pattern of every other process in the known universe of trending to zero in the face of natural limiting factors.