Trace-power witness optimization (${\lambda }_{\max }(F{)}^d\le \mathrm{Tr}(F^d)\le D{\lambda }_{\max }(F{)}^d$)

For a positive witness acceptance operator on dimension $D$, large trace powers approximate its largest eigenvalue. With an $m$-qubit QMA quantum witness and $d=2m+2$, the usual NO instances give $\mathrm{Tr}(F^d)<2^{-d}$ and YES instances give a value above $2^{-d}$. A closed product-index expansion computes the trace by recomputation with polynomial storage, avoiding logarithms and roots in a real-arithmetic implementation.