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This looks a lot like the quincuncial projection applied to a cube map render, with some custom logic to subtly warp geometry in a stable way so as to make certain details pop based on distance. You can try for yourself by downloading blinky (formerly fisheye quake) [1] and experimenting with the projections. That said, it's probably not just a projected cube map. Doing so will introduce noticeable sampling artifacts, so I am guessing they also do something more to address such issues.
[1] https://github.com/shaunlebron/blinky
> which I implemented for RenderMan
Cool, you work at Pixar? I've been working on a small hobby renderer [1] for a few years, and RenderMan has been a big inspiration!
[1] https://github.com/vkoskiv/c-ray