Canonical forms of uptimals

Every all-small game has mean 0 and temperature 0. Therefore, the temperature theory is of no use in the study of all-small games. The main tool used in analyzing all-small games since the 1970s has been approximation by atomic weight. In the 1980s, Conway and Ryba developed (but did not publish) the uptimal theory, which is finer and more precise than the atomic weight theory. In order to study games such as clobber and push-ups, the author independently advances the theory of uptimals. In particular, the author finds the canonicals forms of all integral uptimals. The canonical forms also lead us to an algorithm for recognizing uptimals. We end with examples of rulesets with non-trivial uptimal values.