Integral Künneth torsion polynomial ($T_{A\times B}=F_A{T}_B+T_A{F}_B+(1+t^{-1})T_A{T}_B$)

For spaces whose integral cohomology has only free and $\mathbb{Z}/2$ summands, let $F$ encode free ranks and $T$ count order-two summands by degree. An additively split integral Künneth theorem gives $F_{A\times B}=F_A{F}_B$ and the displayed rule for $T$. The $t^{-1}$ term is the cohomological degree shift of the Tor functor contribution. The formula requires this torsion type and the usual finite-type hypotheses.