  The Witt construction describes a functor from the category of Rings to the
category of characteristic 0 rings. It is uniquely determined by a few
associativity constraints which do not depend on the types of the variables
considered, in other words, by integer polynomials. This universality allowed
Alain Connes and Caterina Consani to devise an analogue of the Witt ring for
characteristic one, an attractive endeavour since we know very little about the
arithmetic in this exotic characteristic and its corresponding field with one
element. Interestingly, they found that in characteristic one, the Witt
construction depends critically on the Shannon entropy. In the current work, we
examine this surprising occurrence, defining a Witt operad for an arbitrary
information measure and a corresponding algebra we call a thermodynamic
semiring. This object exhibits algebraically many of the familiar properties of
information measures, and we examine in particular the Tsallis and Renyi
entropy functions and applications to nonextensive thermodynamics and
multifractals. We find that the arithmetic of the thermodynamic semiring is
exactly that of a certain guessing game played using the given information
measure.
