We prove a nonterminating well-poised basic \(q\)-Taylor expansion with complementary remainders for a two-basis infinite-product kernel implicitly proposed by the second author in Sec. 5 of Schlosser (2008). The well-poised parameter \(c\) gives the rational \(p=0\) basis, while the elliptic nome \(p\) is a separate deformation; the infinite expansions treated here are specific to the basic case. We compute the two Taylor coefficient families and show that each one-family Taylor remainder tends to the complementary basis contribution.

The proof uses the well-poised Cooper formula, Jackson's terminating \({}_8\phi_7\) summation, Rogers' \({}_6\phi_5\) summation, and theta interpolation, but not Bailey's nonterminating \({}_8\phi_7\) summation, which is recovered as a consequence. We also record two quadratic one-family examples and discuss a multi-kernel outlook.