External-product and Whitney-sum formulas for Thom classes

Give $\xi \times \eta$ the sum metric. The product pair $(D\xi \times D\eta ,(S\xi \times D\eta )\cup (D\xi \times S\eta ))$ is the unit pair for the maximum norm. On every nonzero fiber vector $z=(v,w)$, the formula $z\mapsto (\max \{\mathrm{\parallel }v\mathrm{\parallel },\mathrm{\parallel }w\mathrm{\parallel }\}\mathrm{/}\sqrt{\mathrm{\parallel }v{\mathrm{\parallel }}^2+\mathrm{\parallel }w{\mathrm{\parallel }}^2})z$, extended by zero, is a base-preserving homeomorphism to the sum-metric disk/sphere pair; its inverse uses the reciprocal radial ratio.

Form $u_{\xi }\times u_{\eta }$ with [F4] on the product pair and transport it across step 1.1. On the fiber over $(b,c)$ its restriction is the ordered product of the two normalized generators. The connector normalization in [F3] makes this exactly the ordered rank-$(n+m)$ generator. Thus the transported class is normalized, and [F1] identifies it with $u_{\xi \times \eta }$.

For bundles over $B$, [F2] identifies $\xi \oplus \eta$ with the pullback of $\xi \times \eta$ along $\mathrm{\Delta }:B\to B\times B$. By [F1], its Thom class is ${\mathrm{\Delta }}^{\ast }(u_{\xi }\times u_{\eta })$. The cochain definition in [F4] pulls this external product back to $u_{\xi }⌣u_{\eta }$, proving the Whitney-sum formula.

Swapping the two ordered fiber blocks crosses $n$ degree-one coordinate connectors past $m$ such connectors. The signed relative product rule in [F4] contributes $(-1)$ for each of the $nm$ crossings, so the generator and Thom class change by $(-1{)}^{nm}$. This is precisely the orientation of the block permutation.

If either base is empty the external pair and both classes are zero; ranks zero and one reduce respectively to the unit and one connector. The zero ring, zero class, identity diagonal, equal bundles, the zero vector in the radial map, maximum/sum unit boundaries, and both factor orders are all covered above. The radial ratio is locally bounded at zero and the map fixes zero. AC is used exactly through [A1] in [F1]; all product and sign formulas are finite and choice-free.