2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/128133We extend Geisser and Hesselholt's result on ``bi-relative K-theory'' from discrete rings to connective ring spectra. That is, if $\mathcal A$ is a homotopy cartesian $n$-cube of ring spectra (satisfying connectivity hypotheses), then the $(n+1)$-cube induced by the cyclotomic trace $$K(\mathcal A)\to TC(\mathcal A)$$ is homotopy cartesian after profinite completion. In other words, the fiber of the profinitely completed cyclotomic trace satisfies excision.11 pagesK-Theory and Homology19D55; 55P43; 19C40text