A control sequence gives the intensities of the primaries for a pixel in a display. A multi-primary display has four or more primaries, so that multiple control sequences can sometimes produce the same color. Different primaries likely consume different amounts of energy; furthermore, the energy consumption can be a complicated function. A minimal-energy control sequence for a target color produces that color with as little energy as possible. This article shows that such minimal-energy sequences take a simple geometric form when each primary’s energy function is linear. The display gamut, in CIE XY Z space, can be dissected into parallelepipeds. The originating vertex of each parallelepiped is the sum of a set of primaries at full intensity. Each edge of a parallelepiped is the translation of one primary. A color with XYZ coordinates in a certain parallelepiped is a unique linear combination of the primaries in the originating vertex, and the three edge primaries. This article proves that there exists a dissection such that these linear combinations are minimal-energy control sequences. In the generic case, this dissection is unique. An algorithm for a minimal-energy dissection is presented, along with an example.