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Hope Springs Eternal is not necessarily an ending trope. It can take place after a Darkest Hour in the middle of the story.
"Ray of Hope" Ending is a definite Downer Ending that has some upbeat element to show all is not darkness and despair — that there may be a future generation that has a better fate, for instance.
They can intersect, but not always. ("Ray of Hope" Ending may have everyone dead but going to heaven, for instance.)
EDIT: Oh, for future reference, the trope talk forum would really be the proper place for this query.
Edited by DracMonster
Firstly, I would like to say that if this isn't the right place to put this, my apologies. Secondly, however, I was curious as to what exactly the difference between these two tropes is. I know that the trope is referenced in ray of hope ending, but I don't think the difference is directly stated.