K-homology from two-operator data when μ≠1

Develop a method to obtain a K-homology class for A from the pair of selfadjoint unbounded operators (the vertical operator N and the modular lift D_) when the twist parameter μ≠1, for example by identifying a suitable single selfadjoint unbounded operator whose bounded transform yields the desired K-homology class.

Background

In the μ=1 case the authors construct a single unbounded product operator N ⊗ D_0 giving a spectral triple and a K-homology class via the bounded transform.

For μ≠1, however, the geometry is encoded by a pair (N, D_) and the paper highlights the absence of a single operator capturing the geometry, which obstructs the standard bounded-transform route to K-homology.