The calcichordate hypothesis holds that each separate lineage of chordate (Cephalochordates, Urochordates, Craniates) evolved from its own lineage of mitrate, and thus the echinoderms and the chordates are sister groups, with the hemichordates as an out-group.
It was formulated by British Museum paleontologist Richard Jefferies.
The carpoids Cornuta and Mitrata are grouped together in a clade called Calcichordata. Cornutes and mitrates are viewed as sister groups, and mitrates represent stem group chordates. The mitrates (and thus the chordates) are all Dexiothetes, dexiothetism being a synapomorphy for the clade.
In the evolution of the chordates, the ancestors of the chordates underwent a profound remodeling of their bauplan, becoming dexiothetetic. All chordates share a common ancestor which lost its echinoderm stereom calcite skeleton. However, later revisions of the theory had each separate lineage losing its calcitic skeleton independently, as it evolved from its own mitrate ancestor, making the chordates a paraphyletic group.