It’s not, but I can understand your confusion, and I think the two are related. To see the difference, suppose hypothetically that 11% of the first million digits in the decimal expansion of π were 3s. Inductive reasoning would say that we should expect this pattern to continue. The no-coincidence principle, on the other hand, would say that there is a reason (such as a proof or a heuristic argument) for our observation, which may or may not predict that the pattern will continue. But if there were no such reason and yet the pattern continued, then the no-coincidence principle would be false, whereas inductive reasoning would have been successful.
So I think one can view the no-coincidence principle as a way to argue in favor of induction (in the context of formally-defined phenomena): when there is a surprising pattern, the no-coincidence principle says that there is a reason for it, and this reason may predict that the pattern will continue (although we can’t be sure of this until we find the reason). Interestingly, one could also use induction to argue in favor of the no-coincidence principle: we can usually find reasons for apparently outrageous coincidences in mathematics, so perhaps they always exist. But I don’t think they are the same thing.
Isn’t this just the problem of induction in philosophy?
E.g., we have no actual reason to believe that the laws of physics won’t completely change on the 3rd of October 2143, we just assume they won’t.
It’s not, but I can understand your confusion, and I think the two are related. To see the difference, suppose hypothetically that 11% of the first million digits in the decimal expansion of π were 3s. Inductive reasoning would say that we should expect this pattern to continue. The no-coincidence principle, on the other hand, would say that there is a reason (such as a proof or a heuristic argument) for our observation, which may or may not predict that the pattern will continue. But if there were no such reason and yet the pattern continued, then the no-coincidence principle would be false, whereas inductive reasoning would have been successful.
So I think one can view the no-coincidence principle as a way to argue in favor of induction (in the context of formally-defined phenomena): when there is a surprising pattern, the no-coincidence principle says that there is a reason for it, and this reason may predict that the pattern will continue (although we can’t be sure of this until we find the reason). Interestingly, one could also use induction to argue in favor of the no-coincidence principle: we can usually find reasons for apparently outrageous coincidences in mathematics, so perhaps they always exist. But I don’t think they are the same thing.