Eliminate companions from the geometric-backbone construction

Determine whether the pure rational-geometric Shellsort gap backbone, without the added zero-density unit companions, satisfies the same worst-case bound with exponent 1.024296451657… up to polylogarithmic factors.

Background

The paper constructs a practical Shellsort sequence from a finite tuned prefix followed by a rational-geometric backbone. To prove the sub-N^{4/3} upper bound, it augments that backbone beyond 10^{1000} with a sparse set of companion gaps of the form h_s+1. These companions provide a unit direction that removes residual congruence obstructions in the numerical-semigroup representation argument.

The completed sequence achieves a worst-case upper bound of O(N^{1.024296451657…} polylog N), matching the polynomial exponent of an external lower bound. The authors explicitly leave unresolved whether the companions are mathematically necessary or whether the uncompleted pure geometric sequence already has the same asymptotic performance.