The Lost Melody Theorem for Infinite Time Blum-Shub-Smale Machines

We consider recognizability for Infinite Time Blum-Shub-Smale machines, a model of infinitary computability introduced by Koepke and Seyfferth. In particular, we show that the lost melody theorem (originally proved ITTMs Hamkins Lewis), i.e. existence non-computable, but recognizable real numbers, holds ITBMs, ITBM-recognizable numbers are hyperarithmetic both ITBM-unrecognizable appear at every level constructible hierarchy below $$L_{\omega _{1}^{\text {CK}}}$$ above $$\omega ^{\omega }$$ .