Luminosity is a Twilight fanfiction by Alicorn. Requires no knowledge of or affection for Twilight canon.
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.
The first several sections of Luminosity are very similar to canon in terms of the events that occur, although aspects of Bella's internal monologue differ strikingly.
A few thousand words in, the plot is unrecognizeable.