Two-step chiral ring exchange forced by conjugate representations in an SU(3) honeycomb magnet

On a bipartite lattice whose two sublattices carry conjugate representations of SU(N), the cyclic advance that moves each site state one step around a plaquette is not invariant: it commutes only with the real orthogonal subgroup SO(N). The obstruction survives a change of basis exactly when the fundamental and its conjugate are inequivalent, which is to say for every N of at least three; at N = 2 a local rotation on one sublattice removes it. The smallest invariant power of the advance is the two-step rotation R = P^2. This work develops the consequences for N = 3 on the honeycomb lattice, a case in which arguments of Lieb-Schultz-Mattis type indicate that a featureless gapped ground state is not available. Labeling the six sites of a hexagon so that 1, 3, 5 lie on one sublattice and 2, 4, 6 on the other, the time-reversal-odd part of R decomposes exactly as i(R - R^dagger) = 4( chi_135 X_246 + X_135 chi_246 ), where chi is the SU(3) scalar chirality of a sublattice triangle and X is the time-reversal-even ring exchange of its partner. The identity does not depend on how the representations are assigned. Schur-Weyl duality fixes the underlying three-site identity i(P - P^dagger) = 4 chi up to a constant, and an elementary commutator fixes that constant to 4 for every N. Together with the Heisenberg exchange, unique because the bond representation is multiplicity free, these leave a one-parameter model. Exact diagonalization on honeycomb tori of 12 and 16 sites shows that the two-step term leaves the ground state an exact SU(3) singlet, while the forbidden one-step term carries it out of the singlet sector. The single-site reduced density matrix stays maximally mixed in both cases, and invariance under SO(3) alone is responsible, so a measured entropy below ln 3 tests the real subgroup and not the full color symmetry. Neither cluster is large enough to settle the strong-coupling phase, and the manuscript states what a larger calculation would require and in which symmetry sector it should be performed. All numerical results are reproducible from the companion code archive linked below.