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Luminosity is a work of fanfiction. It is set in the universe of Stephenie Meyer's novel Twilight (and its sequels and companion works). The first several sections of Luminosity are very similar to canon in terms of the events that occur, although aspects of Bella's character, and her internal monologue, differ strikingly.
A few thousand words in, the plot is unrecognizeable.
The history and character of the Twilight world are intact in Luminosity up to the point where the story begins, with one exception: Bella's a rational self-awareness-junkie with a penchant for writing down everything that crosses her mind in a notebook. She maintains many of the traits and dispositions of canon - and Luminosity is a Bella/Edward story - but she's a distinctly different character.
Luminosity was inspired in part by Harry Potter and the Methods of Rationality, in which Harry is a scientifically minded and sharp-witted boy who was brought up by Petunia and her alternate-universe professor husband.