  We definitively establish that the theory of symmetric Macdonald polynomials
aligns with quantum and affine Schubert calculus using a discovery that
distinguished weak chains can be identified by chains in the strong (Bruhat)
order poset on the type-$A$ affine Weyl group. We construct two one-parameter
families of functions that respectively transition positively with
Hall-Littlewood and Macdonald's $P$-functions, and specialize to the
representatives for Schubert classes of homology and cohomology of the affine
Grassmannian. Our approach leads us to conjecture that all elements in a
defining set of 3-point genus 0 Gromov-Witten invariants for flag manifolds can
be formulated as strong covers.
