Characteristic-class constructions and normalizations are owned by algebraic topology

The Stiefel-Whitney, Chern, Pontryagin and Euler classes, the Thom class, and the universal Whitney-sum and naturality identities used on this page are constructed and proved on the algebraic-topology pages stiefel-whitney-and-euler-classes-by-universal-constructions, chern-and-pontryagin-classes-by-splitting-and-complexification and leray-hirsch-thom-isomorphism-and-gysin-sequences. This page does not construct them again: it cites the exact item ids, in particular Stiefel–Whitney classes from the projective-bundle relation, Pontryagin classes by complexification, Thom class by fiberwise normalization and Naturality, stability, and mod-two reduction of Pontryagin classes, and uses the published normalization conventions: the complexification convention $p_i(E)=(-1{)}^i{c}_{2i}(E_{\mathbb{C}})$ for Pontryagin classes and the sign carried by the Euler class under orientation reversal. It never appeals to differential-form representatives. The Stiefel-Whitney and Pontryagin numbers Stiefel-Whitney numbers of a closed manifold, Pontryagin numbers of a closed oriented manifold are the published evaluations on fundamental classes; this page adds their evaluations and boundary/product computations, as well as the Pontryagin-Thom and bordism-detection arguments. No choice principle is used beyond the AC inherited from those pages, and this remark is not a proof input to any item.