Orbit Divisibility Under Octahedral Symmetry: A Combinatorial Analysis of Cube Face Colorings

We analyze the 57 orbits of the octahedral group O acting on 3-colorings of cube faces ({-1, 0, +1}^6). We prove two divisibility theorems: (A) the extended orbit size V_ext is even for all orbits with V_ext > 1, and (B) V_ext is divisible by 3 for all "trapped" orbits (those with at least one axis where both faces share the same nonzero value) with V_ext > 2. Combined, these show V_ext is divisible by 6 for all trapped orbits with V_ext > 2. The proofs use stabilizer analysis and the structure of order-3 elements in O.