I've been reading through Winning Ways and was working through some examples of my own related to cooling and particles, and I managed to stump myself. If we let G = {{6 | 4} | -5}, G gives off two particles as it cools, at temperatures 1 and 5. The particle at temperature 5 is just *, but the particle at temperature 1 is

{4* | -4} - {4 | -4} = {4* | -4} + {4 | -4}

If I haven't made a mistake the canonical form for this game is

{{8* | *}, {8* | 0} | {0 | -8}, {* | -8}}

At some level this makes it pretty clear what's going on: either player can play to 0 or to * "in sente", with a threat of 8* or -8. But I'm having a hard time understanding its relationship to other known infinitesimals, aside from simple results like it being confused with 0 and * and less than ^^. I feel like there should be a way to write it in terms of tiny-8's and miny-8*'s or something, or at least some inequalities that would situate it within that hierarchy. But I haven't been able to make much headway there. Does anyone have any insight here?


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