An interacting quantum system can have an unexpectedly simple pattern of energies. Instead of solving for every energy separately, sometimes we can obtain them by adding and subtracting a short list of basic energies. This is called a free-fermion spectrum. A graph records which terms in the spin Hamiltonian anticommute. Earlier work showed that excluding two kinds of induced subgraph, a claw and an even cycle, guarantees a free spectrum. This research candidate proves the converse under an important assumption: the Pauli terms must faithfully represent the graph algebra, and their couplings can vary independently. Under those conditions, the graph criterion is exact. Outside the class, couplings giving a free whole spectrum lie on an algebraic exceptional set. That does not rule out specially constrained physical models. The whole spectrum is only part of the story. A system can split into invariant blocks, each with a free spectrum. The paper gives explicit four-block formulas for periodic chains of eight and nine spins, valid at every coupling. At special couplings some blocks merge into larger coefficient sectors, and those sectors need not themselves be free. The evidence package contains written proofs, exact identities for one hundred and seven graphs, and numerical checks of a larger catalogue. A review exposed a missing counting argument. The revision repairs it by proving that every allowed parity pattern actually occurs in the stated three-mode class. More than sixteen hundred other examples remain numerical candidates. This is not a general solution for every spin chain. This is Evidence Press, Which frustration graphs give free fermions, released on the twenty-seventh of September twenty twenty-six, version one point zero candidate. It is unrefereed. The paper and evidence package are linked on this page. This AI-generated synthetic voice adds no scientific evidence.