  We show that a master integrand basis exists for all planar, two-loop
amplitudes in massless four-dimensional theories which is fully stratified by
rigidity -- with each integrand being either pure and strictly polylogarithmic
or (pure and) strictly elliptic-polylogarithmic, with each of the later
involving a single elliptic curve. Such integrands can be said to have definite
rigidity.
