I would presume that entirely different methods would be needed to prove that the constant can't be positive. There are many types analytical proofs in math tha

I would presume that entirely different methods would be needed to prove that the constant can't be positive. There are many types analytical proofs in math that break down to "negative", "zero" and "positive" cases and often a couple of those are relatively easy whereas others are extremely difficult. The Calabi conjecture is an example. So really it's hard to say. You might even say this proof made things harder. It guarantees that you have to prove the constant is 0 if you want to prove the conjecture. Before you might have hoped prove negativity.