§1. Introduction. Sacks  showed that every computably enumerable (c.e.) degree > 0 has a c.e. splitting. Hence, relativising, every c.e. degree has a Δ2 splitting above each proper predecessor (by ‘splitting’ we understand ‘nontrivial splitting’). Arslanov  showed that 0′ has a d.c.e. splitting above each c.e. a < 0′. On the other hand, Lachlan  proved the existence of a c.e. a < 0 which has no c.e. splitting above some proper c.e. predecessor, and Harrington  showed that one could take a = 0′. Splitting and nonsplitting techniques have had a number of consequences for definability and elementary equivalence in the degrees below 0′.
Heterogeneous splittings are best considered in the context of cupping and non-cupping. Posner and Robinson  showed that every nonzero Δ2 degree can be nontrivially cupped to 0′, and Arslanov  showed that every c.e. degree > 0 can be d.c.e. cupped to 0′ (and hence since every d.c.e., or even n-c.e., degree has a nonzero c.e. predecessor, every n-c.e. degree > 0 is d.c.e. cuppable). Cooper  and Yates (see Miller ) showed the existence of degrees noncuppable in the c.e. degrees. Moreover, the search for relative cupping results was drastically limited by Cooper , and Slaman and Steel  (see also Downey ), who showed that there is a nonzero c.e. degree a below which even Δ2 cupping of c.e. degrees fails.
We prove below what appears to be the strongest possible of such nonsplitting and noncupping results.