Conditioning and solver work are separate claims

The condition-number theorem concerns the ideal exact local and coarse solves and the spectrum of the preconditioned interface operators. It does not bound elapsed time, communication, memory, or total arithmetic work. For example, under a dense direct-solver model with coarse dimension $n_c$, a coarse factorization uses $O(n_c^3)$ arithmetic and $O(n_c^2)$ storage, while a dense local factorization for subdomain dimensions $n_i$ uses $O({\sum }_i{n}_i^3)$ arithmetic before parallel scheduling. Therefore a growing primal space can make the coarse solve dominate even while the condition number remains polylogarithmic. An inexact local or coarse solve requires its own spectral-equivalence and cost hypotheses.