Data jurisdiction space, sometimes called phase transposition space, is a space created by Humanoid Interfaces designed by the Data Overmind. It "overlaps" regular space, and if regular space outside can be seen, it appears to be "frozen". According to Ryoko Asakura, who claims to specialize in its creation, nothing can enter and leave once the space has been created, but Yuki Nagato, Emiri Kimidori and Sky Canopy Dominion interface Kuyou Suou have demonstrated the ability to break into and out of the space (in Nagato's case, she was granted permission by the Data Overmind).
If the space is damaged (which can occur due to something breaking in or out, or simply through manipulating it to attack someone) then a Data Overmind interface is required to repair it afterward. The space need not be repaired by the interface which created it.
When Asakura created a space the first time, the lighting immediately dimmed (going from sunset-colored to gray), and one wall of the classroom was covered in computer-generated geometric shapes. The second time it appeared to "freeze time", as the train warning sound suddenly stopped, only to recur when the space was dissolved.
Data jurisdiction appears confusing to humans, sometimes seemingly made of geometric shapes (described in the novels), but also appearing as flickering, swirling wisps (seen in the background when Asakura disintegrates the classroom walls in the anime) of various colors, or as various "projections" (often bizarre in nature, such as miniature stacked cars or magnified protozoa) on where the original walls of the classroom were present. These backgrounds would shift unpredictably, perhaps in response to actions taken by interfaces. At least some of the backgrounds (such as grass and flowers appearing on the "floor" were real, as petals could fall off the flowers).
When Nagato was seriously injured, the background stopped shifting, appearing as stars moving in circular patterns. When Nagato began disintegrating Asakura, the background shifted to ochre sand dunes.
The space appears to enhance the abilities of humanoid interfaces. During a battle with Kyon (and later Nagato) Asakura destroyed the walls of a classroom, forming it into large chunks (whose volume would be greater than that of the destroyed walls) with stacked desks. At least some of her attacks appeared to "consume" these materials; Asakura converted desks into spikes. When Nagato removed one of these spikes from her body, it immediately transformed into a desk.
Data jurisdiction first appears in the light novel: The Melancholy of Haruhi Suzumiya, when Ryoko Asakura attempts to kill Kyon in a classroom to see how Haruhi reacts. Asakura used it to hide what she was doing, but Nagato was able to determine what was happening and break into it.
A similar space appeared in Mystérique Sign, created by a more primitive data organism, with some aspects similar to closed space. An ochre desert motif appeared when facing the primitive. Kyon would refer to this as a "return" to the ochre sand dunes in the novel.