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> Oh yeah? What's the word for accusing someone for murdering without an iota of proof?

A murder accusation. What's your point?

This is a distortion of the statement that was actually made / what is actually claimed.

The prime minister put his credibility that of the government on the line. They obviously have evidence and are convinced of this being factual as the price to pay for being wrong will be very steep.

By the way, the word you are looking for would be the same for someone accusing of lying or deception without an iota of proof.

Slander, defamation?