A complicated quantum model can sometimes have a simple spectrum: every energy is built from a short list of basic modes. For parafermions, those modes are multiplied by roots of unity before being added. The energies can be complex, so this is not automatically a model of an ordinary Hermitian quantum system. Earlier work identified graphs that guarantee this free spectral pattern. This research candidate asks the converse: under which assumptions are those the only graphs? Its answer covers every local order from three upwards, including even and composite orders. The Hamiltonian terms must vary independently, obey specified commutation rules, and represent the full universal algebra. The complete spectrum must fit one shared mode list with the correct multiplicities. Under these conditions, a precise graph class, called oriented indifference graphs, gives the exact dividing line. The proof finds two obstructions. A three-term fork fails unless at least one coupling vanishes. Directed cycles produce too many distinct eigenvalues to fit the required pattern. This is why checking separate symmetry sectors is not enough. Each sector can look simple while their combined spectrum fails to share one global set of modes. The paper also gives a cubic-time algorithm for deciding whether reversing suitable graph edges can reach the allowed class. That operation changes the Hamiltonian; it does not preserve its spectrum. The evidence package includes written proofs, an exact census through six Hamiltonian terms, and replayable checks from separate implementations. The publication revision clarifies the earlier theorem it uses, a phase correction at even orders, and the conditions for applying the result to physical operators. This remains an unrefereed candidate. Finite tests do not establish its universal proof, and local replay is not independent peer review. This is Evidence Press, the full-spectrum parafermion converse, released on the twenty-eighth of September twenty twenty-six, version one point zero point one candidate. The paper and evidence are linked on this page. This synthetic AI voice is an explanation, not additional scientific evidence.