It's a funny thing, because the inverse is falsifiable (testable) whereas the positive version is not. The inverse proof (I would say a hypothesis) is simply application of the scientific method.

There is a way to disprove that the statement: "There is no god" by simply showing a counterfactual god.

There is however no way to disprove the statement: "There is a god."

Likewise, there is a way to disprove the statement: "LLMs cannot be successfully used for X application." By showing that LLMs have been used in X application.

Again, there is no way to disprove the statement: "LLMs can (eventually) be used in X application."

The meat of my question was meant to demonstrate a failure to apply the scientific method.