Stability and rank cutoff under adding a trivial summand

Chern stability: by [F1] and [F3], $c({\epsilon }^r)=1$ and multiplicativity gives $c(E\oplus {\epsilon }^r)=c(E)c({\epsilon }^r)=c(E)$; the cutoff $c_i=0$ for $i>m$ is part of the definition.

Pontryagin stability: by [F2] the classes $p_i$ are unchanged when a trivial summand is adjoined, and $p_i(F)=0$ whenever $2i>n$ by the defining cutoff.

Consistency of the two parities: in the complex case the total class is unchanged, so all components are unchanged; in the real case the total Pontryagin class as a finite sum ${\sum }_i{p}_i$ is unchanged because each $p_i$ is.

Boundary cases. For $r=0$ both identities are trivial; for $E$ or $F$ trivial of rank zero, $c(0)=1$ and $p(0)=1$ by the conventions, so the identities read $c({\epsilon }^r)=1$ and $p({\epsilon }^r)=1$. The rank cutoffs are strict: they force $c_i(E)=0$ only for $i>m$ and $p_i(F)=0$ only for $2i>n$. They do not include $i=m$ or $2i=n$, where the top Chern or Pontryagin class may be nonzero. The coefficient ring $\mathbb{Z}$ is nonzero and the sums are finite. AC is used only through [A1].