[This paper has appeared: J. Pure Appl. Algebra 212 (2008) 1039-1045, 10.1016/j.jpaa.2007.07.015 .]
By simple geometric methods, we calculate the Balmer-Witt groups of
the spheres (over a base ring with 2 invertible) up to 2-primary
torsion and determine the multiplicative structure of the total
graded Witt group. An appendix clarifies an issue concerning the
canonicity of coherent Witt groups and transfer morphisms.
The original February 13, 2006, version has been replaced by a newer version on
May 29, 2006.