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 [19, Sec. 5]. 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 8φ7 summation, Rogers' 6φ5 summation, and theta interpolation, but not Bailey's nonterminating 8φ7 summation, which is recovered as a consequence. We also record two quadratic one-family examples and discuss a multi-kernel outlook.
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