> I read through it and it says part of the document has been admitted to be true.
I don't understand: You asked whether Twitter had any basis for marking it false. I said the basis was (at least in part) a major fact-checking organization. I'm not sure how the above is a response to that. Also, can you provide some basis to your claim? Every lie is partly true; someone who says things that are 'partly true' is a liar.
>> Should government be the arbiter of truth? About it's own political candidates?
> Elected governments shouldn't be, but isn't this why other arms of the government exist, like judiciary.
I strongly disagree: In democracies the judiciary is not at all there to decide truth. It's there to apply the law, which at times involves finding facts of specific situations before it, and then only to a degree sufficient to apply the law. For example, the judiciary is not there to decide the facts of climate change. It is there to apply emissions laws, and to find the facts about what someone may be emitting to the degree needed to apply the law.
In a democracy, it is up to citizens - you and I included - to decide what we believe. There is no authority or mechanism that will save you from that challenging duty. Democracy is a solution, in a sense, to the problem of there being no source of truth. It is governemnt of the people, by the people, for the people.