Differential calculus on Hopf--Galois extension via the Durdević braiding

We introduce a class of right $H$–covariant first–order differential calculi on principal comodule algebras generated by the Durdević braiding $\sigma$ and a chosen vertical ideal. Starting from the universal calculus, a strong connection, and a right $H$–colinear splitting map, we construct $\sigma$–generated differential calculi and prove their existence for arbitrary principal comodule algebras. We show that, in this framework, universal vertical maps and connection $1$–forms descend naturally to the quotient calculus under suitable compatibility conditions. We further develop a functorial formulation of $\sigma$–generated calculi and establish a universal factorization property for the associated quotient calculi. Finally, we present explicit examples arising from quantum projective spaces and quantum lens spaces.