  Kucerovsky's theorem provides a method for recognizing the interior Kasparov
product of selfadjoint unbounded cycles. In this paper we extend Kucerovsky's
theorem to the non-selfadjoint setting by replacing unbounded Kasparov modules
with Hilsum's half-closed chains. On our way we show that any half-closed chain
gives rise to a multitude of twisted selfadjoint unbounded cycles via a
localization procedure. These unbounded modular cycles allow us to provide
verifiable criteria avoiding any reference to domains of adjoints of symmetric
unbounded operators.
