This can't be a theorem of ID because an intelligent designer could also produce a "structure" with no discernable pattern. (I will be happy to produce one if you like.) If both A and not-A are compatible with your axioms that neither can be entailed by them.
But it doesn't stop there. ID purportedly claims that no naturalistic theory can explain nature fully. This is what you really need evidence for. Based on history, humans are pretty darn good at coming up with naturalistic explanations.
Given the possibilities you are most correct in so saying. But we are not dealing with possibilities. We are dealing with what IS, and what IS is organized matter that behaves according to predictable laws, and as such, intelligent design, while not an infallible explanation, is a good one.