  We construct and study an explicit simultaneous $\mathscr{Y}$ eigenbasis of
Ion and Wu's standard representation of the $^+$stable-limit double affine
Hecke algebra for the limit Cherednik operators $\mathscr{Y}_i$. This basis
arises as a generalization of Cherednik's non-symmetric Macdonald polynomials
of type $GL_n$. We utilize links between $^+$stable-limit double affine Hecke
algebra theory of Ion and Wu and the double Dyck path algebra of Carlsson and
Mellit that arose in their proof of the Shuffle Conjecture. As a consequence,
the spectral theory for the limit Cherednik operators is understood.
