Hint
Your hint is in yesterday's puzzle.
A pompous personage of reduplicative cognomen is insufficiently tenacious of a mural perch. Notwithstanding the intervention of a royal entourage—equids included!—ooclasm proves irremediable.
This is a highfalutin synopsis of what familiar children's rhyme?
Humpty Dumpty sat on a wall / Humpty Dumpty had a great fall / All the king's horses and all the king's men / Couldn't put Humpty together again.
Fun fact: There's nothing in "Humpty Dumpty" to indicate that Humpty is an egg. It certainly makes sense if you think of the rhyme as a riddle, but the first person to interpret it this way seems to have been John Tenniel, illustrator of Lewis Carroll's Through the Looking-Glass, in which Humpty Dumpty memorably says "When I use a word, it means just what I choose it to mean—neither more nor less."
In the spirit of Lewis Carroll's Humpty Dumpty, I coined the word ooclasm for this very puzzle—from Greek roots oo- ('egg') and -clasm ('breaking'). Maybe that's cheating, but you can't make an omelet without ooclasm!
The cartoon at left isn't mine, but it's been posted so widely on the internet that I don't know whom to credit it to. Enjoy!
Solution
Congratulations to yesterday's solvers Anna J., Jacob C., Leo S., Nico, Maddy, Jason, Kate, Zachary S., Graham, and Dr. Yetman. Thanks to everybody who made a guess!
Ten years later, Dr. Shapiro revived Puzzle of the Day at Proof School, writing each day's puzzle on a name tag. After 600 puzzles or so, he was just starting to feel normal about students reading his chest all the time when campus closed and the puzzle, like the rest of our lives, moved online. New puzzles are posted daily on school days.
Want to catch up on old PotDs? There's an archive currently containing puzzles from March to June 2020.