There are a number of techniques you could try to improve your results.
But I get the distinct feeling that you are not interested in improving your results as you have a motivation to "prove" something about LLMs.
This becomes a problem when people start to treat LLMs as authoritative on facts.
So, it's sort of like driving a car based not on a complete map of the area, but a partial map where the missing pieces are filled in from statistical averages of things that appear generally in the world-- but not necessarily in that location. We all may be impressed at how close the predictions come to reality, but the only truly reliable thing is that those predictions, measured with precision, will be wrong. And if you rely on them without verifying them yourself, you might have a serious problem.
The “true” way to answer a proposition is not even remotely as simple as you’ve described.
The problem is that an LLM can only be as reliable as its training material.
For example I wrote a blog post on the 2 generals problem. In the comments there is more text from inexperienced people asserting that a solution exists than there is text from my original article.
An LLM trained on that article and comments will always be wrong.
LLMs can be trained on smaller vetted training sets sure, but they also currently require massive amounts of data and there’s no guarantee that just waiting a few months or years will improve reliability enough to fix these issues.