  We define the rigidity of a Feynman integral to be the smallest dimension
over which it is non-polylogarithmic. We argue that massless Feynman integrals
in four dimensions have a rigidity bounded by 2(L-1) at L loops, and we show
that this bound may be saturated for integrals that we call marginal: those
with (L+1)D/2 propagators in (even) D dimensions. We show that marginal Feynman
integrals in D dimensions generically involve Calabi-Yau geometries, and we
give examples of finite four-dimensional Feynman integrals in massless $\phi^4$
theory that saturate our predicted bound in rigidity at all loop orders.
