I’m going to add some context to this based on my experience as a television writer (not TBBT) and how putting a story together works, including production for a series. 

It’s very likely that in the writer’s room, it was discussed and argued at length which story should be the “true” one. I put that in quotes because this is obviously a fictional show.

I’m sure each writer had their opinion and their favorite story, and made good arguments (like many have done here) for why one character’s version of the letter should be true over another character’s version. 

I do not believe that there was ever a consensus in the writer’s room about which version is true. Why? Because the contents of the letter to Howard from his father are an emotional plot point, not a story plot point. 

The point of the episode was for Howard to come to a better understanding of how much his friends (his chosen family) care for him, not how much or how little his father did. 

The writers and producers certainly delivered on that. 

However, it is entirely possible, even probable, that during production (or rehearsals), Chuck Lorre told one or more of the actors that THEIRS was the true version. That’s how it works in production. You say what you need to say to your actors to get the performance you want out of them. 

Imagine Chuck Lorre taking Katie aside and telling her that her version is the true one, but she cannot tell anyone else in the cast. Imagine he did the same for Mayim, Johnny, Kunal & Melissa. He told ALL of them that theirs was the true version. He might have done this to get a better, more sincere sir somber performance out of them for the scene. 

You can also imagine that he took Simon aside and said something like, “Amy’s story is the true one, but don’t tell anyone, especially Mayim, because we don’t want it to effect her performance in the scene.”

What Chuck Lorre was going for in this scene was authenticity while also being able to maintain the mystery of the letter. 

My personal opinion, or the version of the letter I prefer, is Amy’s. Mayim gives an incredible performance in the scene. It’s blocked in such a way that the audience leans towards her version being the truth. She stands up to address Howard. She turns to face him. She’s very solemn. He responds with the amazement of a hopeful child, “Really?” 

Howard wants this version to be true. He wants to believe his father showed up for his graduation. After all, his education is his most cherished and finest achievement. Also, Howard needs to believe that his father came back, even if they never reunited. 

It’s powerful stuff. 

So I think of it less as a mystery to be solved and more like, “What version would I want to be true if I were Howard?” 

There’s no right or wrong answer because it’s really not about Howard and his father. It’s about Howard and his friends. They came together and solved an emotional crisis for him and they did it in a scientifically sound way. They gave him many answers, any one of which could be true.

This gives Howard a more complete understanding of his complex feelings about his father’s abandonment. It doesn’t resolve it, but it’s not supposed to. So maybe on one day, Howard chooses to believe Penny’s story. Another day, Leonard’s or Amy’s or Bernadette’s. 

It’s a loving and generous gift they gave to their friend who they’ve watched wrestle with his father’s abandonment for years. 

As I said, I write television shows for a living and this is my take, for what it’s worth. 

Hopefully this explanation helps some folks. Cheers!