I think it's meant to represent one of the forms of closure at the end of the story.
To explain in full... I found that the "Coda" was the final time he played, but the edit didn't hang on that closing scene and left me feeling unsatisfied like it would have if it were the "Closure" for his character, instead it was like an afterthought or "Coda" to a composition, that makes you want to go back and listen (Watch) the main story again...
All the substance seems to be in the first 2 acts and the conversations Patrick and Katie have seem to foreshadow the movies story...
One of these conversations is how a happy memory can wipe away many sad or painful ones... So when he remembered her, went to the hotel for her pianist friend and saw the boulder he remembered her pulling up his sock and that helped give him the real closure to the movie... Which then led to the "Coda" or epilogue of the movie... his final performance.
Apologies for a big answer. The movie wasn't perfect... But 10 years of Avengers and only one great film per year, the subtle nuances of this are the best thing I've seen this year.
