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This is: Chapter 1: A Day of Very Low Probability, published by Eliezer Yudkowsky on the LessWrong.
Disclaimer: J. K. Rowling owns Harry Potter, and no one owns the methods of rationality.
This fic is widely considered to have really hit its stride starting at around Chapter 5. If you still don't like it after Chapter 10, give up.
This is not a strict single-point-of-departure fic - there exists a primary point of departure, at some point in the past, but also other alterations. The best term I've heard for this fic is "parallel universe".
The text contains many clues: obvious clues, not-so-obvious clues, truly obscure hints which I was shocked to see some readers successfully decode, and massive evidence left out in plain sight. This is a rationalist story; its mysteries are solvable, and meant to be solved.
The pacing of the story is that of serial fiction, i.e., that of a TV show running for a predetermined number of seasons, whose episodes are individually plotted but with an overall arc building to a final conclusion.
All science mentioned is real science. But please keep in mind that, beyond the realm of science, the views of the characters may not be those of the author. Not everything the protagonist does is a lesson in wisdom, and advice offered by darker characters may be untrustworthy or dangerously double-edged.
Beneath the moonlight glints a tiny fragment of silver, a fraction of a line...
(black robes, falling)
...blood spills out in litres, and someone screams a word.
Every inch of wall space is covered by a bookcase. Each bookcase has six shelves, going almost to the ceiling. Some bookshelves are stacked to the brim with hardback books: science, maths, history, and everything else. Other shelves have two layers of paperback science fiction, with the back layer of books propped up on old tissue boxes or lengths of wood, so that you can see the back layer of books above the books in front. And it still isn't enough. Books are overflowing onto the tables and the sofas and making little heaps under the windows.
This is the living-room of the house occupied by the eminent Professor Michael Verres-Evans, and his wife, Mrs. Petunia Evans-Verres, and their adopted son, Harry James Potter-Evans-Verres.
There is a letter lying on the living-room table, and an unstamped envelope of yellowish parchment, addressed to Mr. H. Potter in emerald-green ink.
The Professor and his wife are speaking sharply at each other, but they are not shouting. The Professor considers shouting to be uncivilised.
"You're joking," Michael said to Petunia. His tone indicated that he was very much afraid that she was serious.
"My sister was a witch," Petunia repeated. She looked frightened, but stood her ground. "Her husband was a wizard."
"This is absurd!" Michael said sharply. "They were at our wedding - they visited for Christmas -"
"I told them you weren't to know," Petunia whispered. "But it's true. I've seen things -"
The Professor rolled his eyes. "Dear, I understand that you're not familiar with the sceptical literature. You may not realise how easy it is for a trained magician to fake the seemingly impossible. Remember how I taught Harry to bend spoons? If it seemed like they could always guess what you were thinking, that's called cold reading -"
"It wasn't bending spoons -"
"What was it, then?"
Petunia bit her lip. "I can't just tell you. You'll think I'm -" She swallowed. "Listen. Michael. I wasn't - always like this -" She gestured at herself, as though to indicate her lithe form. "Lily did this. Because I - because I begged her. For years, I begged her. Lily had always been prettier than me, and I'd... been mean to her, because of that, and then she got magic, can you imagine how I felt? And I begged her to use some of that magic on me so that I could be pretty too, even if I couldn'...
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