How to navigate convincing professor to submit a letter of recommendation without falsehood My professor has submitted a letter of recommendation for me for graduate program admissions, and has listed that I did X and Y (specific activities) under him, when I was only involved in doing X. He shared the letter with me after he submitted it, that's how I know.

I emailed him saying that I only did X and gave him info on how to submit a new letter, and his response was that if the admission committee ever calls him to check, he will clarify. 

I really worry about this because 1) committees seldom check, 2) it's a falsehood that I did Y under him, 3) I think he is forgetful, and may just not remember to correct himself if they ever check, and 4) if I don't convince him to change this it would be like I am complicit in the falsehood.

He's a chill, old dude, my professor, and I don't want to rub him the wrong way by saying "We would be lying, please submit a new letter". 

What's a gentle, tactful way to convince him to submit a new letter of recommendation with the correct information? Or is it okay if he clarifies if they ever call him to check?
I would say that this would be a different situation if the recommendation was for a potential employer, rather than for graduate school admissions. If this were for a potential employer and you might be expected to do Y, this would be a bigger issue, in my opinion. 

Although the letter does not represent your abilities 100%, calling it a lie suggests that your professor knowingly stated falsehoods about your abilities on the letter, rather than it be an oversight as you describe. Sometimes these things happen. His writing a recommendation for you is outside of his other academic duties, so you are asking him to volunteer more work on top of what he has already volunteered to do. 

In my many years on graduate school admission committees, we have only contacted references of candidates in cases where there is a questionable part of the application that suggests that the candidate is not prepared for the program or would be a potential problem. If everything else in your application looks fine, this may go unnoticed. Plus, the graduate admissions may find it strange that your reference submitted a revised letter. In my opinion, I would just wait to see if this comes up. Then, you can simply state, "Yes, my professor said in my letter that I did Y under his supervision, but I really was more involved with X and can only speak about X." This is very different than if YOU stated in your application that you could do Y. Your reference making a mistake would be more forgivable than if you knowingly made the same statement (if all else is fine with your application). If your professor didn't give you a copy of the letter, you would have never known that this is an issue.