We describe the index pairing between an odd K-theory class and an odd
unbounded Kasparov module by a pair of quasi-projections, supported on a
submodule obtained from a finite spectral truncation. We achieve this by pairing
the K-theory class with an asymptotic morphism determined by the unbounded
Kasparov module. We interpret the spectral localiser of Loring and Schulz-Baldes
as an instance of such an index pairing.
