  We show how the fine structure in shift-tail equivalence, appearing in the
noncommutative geometry of Cuntz-Krieger algebras developed by the first two
authors, has an analogue in a wide range of other Cuntz-Pimsner algebras. To
illustrate this structure, and where it appears, we produce an unbounded
representative of the defining extension of the Cuntz-Pimsner algebra
constructed from a finitely generated projective bi-Hilbertian module,
extending work by the third author with Robertson and Sims. As an application,
our construction yields new spectral triples for Cuntz- and Cuntz-Krieger
algebras and for Cuntz-Pimsner algebras associated to vector bundles twisted by
equicontinuous $*$-automorphisms.
