  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 $GL$. We utilize links between $^+$stable-limit double affine Hecke
algebra theory of Ion-Wu and the double Dyck path algebra of Carlsson-Mellit
that arose in their proof of the Shuffle Conjecture. As a consequence, the
spectral theory for the limit Cherednik operators is understood. The symmetric
functions comprise the zero weight space. We introduce one extra operator that
commutes with the $\mathscr{Y}_i$ action and dramatically refines the weight
spaces to now be one-dimensional. This operator, up to a change of variables,
gives an extension of Haiman's operator $\Delta'$ from $\Lambda$ to
$\mathscr{P}_{as}^{+}.$ Additionally, we develop another method to build this
weight basis using limits of trivial idempotents.
