The Ambainis composition lock, at every depth: the product partition of the (d+1)-fold Ambainis iterate — the construction giving the multiplicative upper bound on the partition number — is not the leaf partition of any deterministic protocol, for every depth d. Index 0 is the base partition itself (a kernel-checked eight-rectangle monochromatic partition of the base game); index 1 is the 64-rectangle object of the depth-two frontier.