  We carefully re-examine the issues of solving the modified Bianchi identity,
anomaly cancellations and flux quantization in the S^1/Z_2 orbifold of M-theory
using the boundary-free "upstairs" formalism, avoiding several misconceptions
present in earlier literature. While the solution for the four-form G to the
modified Bianchi identity appears to depend on an arbitrary parameter b, we
show that requiring G to be globally well-defined, i.e. invariant under small
and large gauge and local Lorentz transformations, fixes b=1. This value also
is necessary for a consistent reduction to the heterotic string in the
small-radius limit. Insisting on properly defining all fields on the circle, we
find that there is a previously unnoticed additional contribution to the
anomaly inflow from the eleven-dimensional topological term. Anomaly
cancellation then requires a quadratic relation between b and the combination
lambda^6/kappa^4 of the gauge and gravitational coupling constants lambda and
kappa. This contrasts with previous beliefs that anomaly cancellation would
give a cubic equation for b. We observe that our solution for G automatically
satisfies integer or half-integer flux quantization for the appropriate cycles.
We explicitly write out the anomaly cancelling terms of the heterotic string as
inherited from the M-theory approach. They differ from the usual ones by the
addition of a well-defined local counterterm. We also show how five-branes
enter our analysis.
