Three Dimensions from One Reading A space that keeps the record of a completed recognition has three dimensions

The recognition ledger of [1] has binary states, changes by one posting per tick, and records each act as a balanced pair. It selects three dimensions for space under a linking hypothesis that the ledger axioms do not decide. This paper says what the hypothesis is and why it has one dimension. A ledger state is determined by its cells, so a motion that posts nothing changes nothing. A completed act and the record that registers it are two closed passes on the state graph; drawn in space they are two loops. The linking hypothesis is one sentence: the posted placement of those two loops is a ledger state distinct from their separation, that is, space keeps the record of the act. The sentence names no dimension. Two disjoint circles have a placement that no motion separates in three dimensions and in no other. So a space that keeps the record of a recognition in the placement of its two loops has three dimensions; in every other dimension the loops come apart with no post, and their placement remembers nothing. Said of space rather than of the act, the same sentence is the second axiom of recognition geometry [3]: a space is a set of configurations some recognizer tells apart. The placements of two loops are such a space in three dimensions and in no other. A world of another dimension may have a ledger; it has no space of loop placements.