I suspect that Wolfram just wanted to reveal the relevant proof himself, first, in his book NKS (A New Kind of Science), and that Matthew Cook probably was contractually obligated to allow Wolfram to do that.
Given that the two parties settled, and that Cook published his paper about his proof in Wolfram’s own journal (Complex Systems), two years after NKS was published, seems to mostly confirm my suspicions.
I now think it is plausible that Wolfram sued “over literary conventions”:
Wolfram Research v Cook—LessWrong
I suspect that Wolfram just wanted to reveal the relevant proof himself, first, in his book NKS (A New Kind of Science), and that Matthew Cook probably was contractually obligated to allow Wolfram to do that.
Given that the two parties settled, and that Cook published his paper about his proof in Wolfram’s own journal (Complex Systems), two years after NKS was published, seems to mostly confirm my suspicions.