A positive proof of the Littlewood-Richardson rule using the octahedron recurrence

We define the hive ring, which has a basis indexed by dominant weights for GL(n)(C), and structure constants given by counting hives [Knutson-Tao, "The honeycomb model of GL(n) tensor products"] (or equivalently honeycombs, or BZ patterns [Berenstein-Zelevinsky, "Involutions on Gel'fand-Tsetlin schemes... "]). We use the octahedron rule from [Robbins-Rumsey, "Determinants... "] to prove bijectively that this "ring" is indeed associative. This, and the Pieri rule, give a self-contained proof that the hive ring is isomorphic as a ring-with-basis to the representation ring of GL(n)(C). In the honeycomb interpretation, the octahedron rule becomes " scattering" of the honeycombs. This recovers some of the "crosses and wrenches" diagrams from Speyer's very recent preprint ["Perfect matchings... "], whose results we use to give a closed form for the associativity bijection.