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In 2003, four fragments that would appear in later lyrics of the song ("AVOID EYE CONTACT", "CHEER AT THE GALLOWS", "ROUND UP", "BURN THE WITCH") appeared in the lower left side of this artwork from the Hail To The Thief booklet:

This raises the question how developed 'Burn The Witch' was in september 2002, when Stanley did this painting in Los Angeles.
Since 2004, the menu of the Scrapbook section of radiohead.com, called 'Diet of Worms', was made up from an early draft of lyrics for 'Burn The Witch'. The fragments from the Hail To The Thief artwork are all included here, so maybe this text is from 2002 and the artwork from the L.A. sessions shows excerpts from it:



Transcript
On September 28th 2005, Thom mentioned that he had finished writing the lyrics to 'Burn The Witch' in this post from Dead Air Space:

'Burn the Witch' was one of the titles that appeared on the blackboard the band used during two weeks of rehearsals from September 19th to 30th 2005. At the end of that session, Colin posted a picture of this board on Dead Air Space on september 30th, unfortunately (and probably intentionally) in rather low resolution, so that some of it can't be deciphered:
On March 8th 2006, shortly after sessions with Mark Stent had come to an unfruitful end, another photo of a blackboard (and a perhaps slightly worried Ed) was posted on Dead Air Space. A tour had been scheduled for may and june to escape the frustration of the recent sessions and to breathe new life into the project. Rehearsals for the upcoming dates had begun recently, and this board shows which new songs were at this point considered for inclusion in the setlists. 'Burn the Witch' was one of these, but would not be played (click image for full size):
On June 24th 2006, during the second show in Berkeley, Thom teased the audience with the opening chords and gave an explanation for not playing it, that (even though ideas for an orchestral arrangement were mentioned earlier) appears to be slightly tongue-in-cheek:
On February 4th 2007, Thom posted some of the lyrics from 'Burn the Witch' on Dead Air Space. They look like a condensed version of what was seen in the 'Diet of Worms' page:

On May 14th 2008, during the show in St. Louis, a second teaser was played right before 'Videotape' as a reaction to shouts from the audience for the song:
Jonny: I also, for the first time, managed to get hold of a song before anything was on it. Cause usually it's songs are either finished or overfinished, and then the question is "Can we add strings to any of this?", and there's usually no room.

Adam: It's gilding the lily by that point.

Jonny: Usually yeah, it doesn't help. But this time, I managed to get a hold of the song 'Burn the Witch' when it was just a drum machine and a voice and nothing had have been added yet. I kept saying "let's leave it, we'll come up with something for the strings to do." And while we were doing that, we recorded strings on other things and a lot of it came out really good, I think.
Jonny: This song was one of the rare chances of getting our hands on a unfinished song so we can put strings on right at the beginning. Usually strings are a afterthought and they're decoration on the end of a song, and I've been saying for years, "wouldn't it be great to just start with strings?" And so this song was just Thom singing, and a drum machine, and nothing else. And then I wrote strings to that. So yeah, you're hearing [the London Contemporary Orchestra] strumming their violins with guitar plectrums, that's the sound of the rhythm. [...] It's only 13 players, but still it's amazing what they can do with their instruments.
Matt: The first track from 'A Moon Shaped Pool', 'Burn the Witch', it seems to be implicitly about people looking for political scapegoats.

Thom: It was originally kind of written about the tabloid witch hunts of the pedophiles a long time ago. The original impetus was reading some reports about -- I mean they still do it now, it's what they're good at -- when whatever newspaper it was started the witch hunts, started naming where these people lived and all that stuff. But then, that was a long time ago, it got shelved, and then... it seemed to have its time.
Colin: I listened to your really cool broadcast you did with Rosie Jones, and she talked about how she actually liked the first and the last songs on 'A Moon Shaped Pool', and those are two of the oldest songs. There's 'Burn the Witch', which was always like a song we had that- Thom had written this amazing chorus, and then we were trying to work out the verse, and then my brother came up with these amazing stabbing, staccato, violent strings. And then 'True Love Waits', which is this beautiful song that Thom had written a long time ago that we've sort of played or he's done in lots of different ways, you know. Whether it's an acoustic guitar on stage, like you said, the live version. Whether it's like using it with lots of loops and found sounds that Nigel did when we were making 'OK Computer', and then the one on the record is a sort of... I don't know what it is, it's like a sort of slight return to that and then... but it's a very beautiful song, so it doesn't really matter. As my brother might say, you could play it on two teaspoons in a box of matches if you've got this amazing song.
The lyrics for the song can be seen in one of eleven pieces of artwork made from several layers of tracing paper, including some early drafts beneath the final one: