The write up serves as documentation to start your termination process and that you were not discriminated or fired without cause, even though your employment is at will.
The comment section for you in the write up is a scheme by HR and lawyers as proof for them that you got the write up, and to make it easier for you to sign it.
Whenever you are putting notes in your write up or writing anything "explaining" your side, is acknowledging that you physically got the write up.
If your leader is new, put up a fight... don't make it easy for him. If you complain to HR, he will think twice about writing you up again, especially if he didn't write up every store who fail to meet same metric. They have other things to worry about. On the other hand, if it is so easy to write you up, he will probably do it continuously until you lose your job.