Green Knight spoilers
I'm finally getting around to watching Green Knight. The movie's about the triangulation of desire. Very straightforward Girard stuff.
The King recognizes his lack of peers (either among his Saxon rivals or his court) & desires the company of his sister as a neer-peer: he triangulates this through Gawain. G "externally" mediates the King's desire. G himself wants to be one of the knights, to have his desire "internally" mediated through them.
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Green Knight spoilers
So the Green Knight appears in the context of this as this figure of reciprocation and equivalence. Knick for knick, blow for blow, year after year, and so on.
Later on "The Lady" will pose an interpretation of greenness that is given without any irony from the film, so I think it's basically an endorsement of the filmmakers' perspective no the greenness of the knight.
Green, in the film, is associated with perspectives that are distanced from the mediation of desire. ...
Green Knight spoilers
The examples of greenness given in the monologue are birth, infection, and corruption: birth following from desire; infection and corruption indicating the death of desire.
The thing is, though, I think it's better to understand Green in terms of the woolly pagan perennialism that was being driven out of Britain by Christendom: the Priest could tell you about the Trinitarian mediation of desire; the druids could tell you about the smooth cycles of geronition & death.
Green Knight spoilers
Anyway, the Girardian read would highlight the importance of the Castle with the Lord & the Lady near the Green Chapel. The Lord & Lady invite G as someone to mediate their romance, & they each mediate their relationship to G through different methods: the Lord captures & releases the Fox, and the Lady releases the garter. Through this you can see that their methods for their affections depend on withholding, explaining how the couple is in an affective deadlock without G.
Green Knight spoilers
So what G is, in terms of desire, is he's this mediator who will play underdog in all of the games of Christendom because he's so desperate to avoid the equitable fate of the Green Knight.
Everything that comes after the Castle in the third act is the story of what G wants to do with this garland. It represents, again, this third way out of sustaining life by will. Specifically, the will involved here is the will to be objectified and treated as mediator to others' desire
Green Knight spoilers
Another word on the setting: you've got death as a phase of geronition (greenness) contrasted against the Christmas story, which is the commemoration of the paradoxical birth of the paschal sacrifice who will transcend death through death, the Christ.
The witchcraft of the green garter suggests a third way out: life sustained by human will until it becomes unbearable (such as through watching the deaths of others), until life must be renounced.