  Shift and stretch invariance lead to the exponential-Boltzmann probability
distribution. Rotational invariance generates the Gaussian distribution.
Particular scaling relations transform the canonical exponential and Gaussian
patterns into the variety of commonly observed patterns. The scaling relations
themselves arise from the fundamental invariances of shift, stretch, and
rotation, plus a few additional invariances. Prior work described the three
fundamental invariances as a consequence of the equilibrium canonical ensemble
of statistical mechanics or the Jaynesian maximization of information entropy.
By contrast, I emphasize the primacy and sufficiency of invariance alone to
explain the commonly observed patterns. Primary invariance naturally creates
the array of commonly observed scaling relations and associated probability
patterns, whereas the classical approaches derived from statistical mechanics
or information theory require special assumptions to derive commonly observed
scales.
