Mother believes my birthday party happened out on the deck and she wasn’t invited. There was no party. How do I respond?

Tell her the party was not for her. That a neighbor borrowed your deck to have a party. Sorry, Neither she or you were invited.

Other than telling her over and over again that it never happened, why don't you throw yourself a birthday party out on the deck. and make sure she attends. Even if your birthday was a while back, that might just satisfy her. And who doesn't like a good party any time of the year?

Or, with a " therapeutic fib".

This is a delusion. If this is a change in mental status, please talk to her doctor about it. Could be a UTI.

With the truth. And gently. And likely over and over and over again. I am so sorry. No answer to a broken mind.