Ω=0 and Locality for Quantum Boxes: Three Parties and the Four-Party Boundary

We study the consistency weight W(P;g)=Σ_o P(o|g(o)) that arises when a multipartite box P is connected to a deterministic feedback g=(g₁,…,gₙ), together with its minimum Ω(P)=min_g W(P;g). Here g_k determines the input of party k from the outputs of the other parties, and both quantities are defined in Papers A and B. Physically, W is the total weight of the consistent histories of a closed feedback loop, and Ω=0 means that some feedback makes the loop paradoxical. We prove that a three-party box with binary inputs and outputs, generated by a state and commuting self-adjoint involutions on a Hilbert space, is local whenever Ω=0. The proof reduces the statement to pure states of the group C*-algebra of (ℤ₂∗ℤ₂)³, hence to irreducible representations of dimension at most 8, and then rests on two finite computations: the operator Σ_o ⊗_k Π^k_{o_k|g_k(o_{−k})} is positive definite on the all-qubit sector for each of the 4096 feedbacks, and in the sector with one classical party the operator may have a kernel, but every such kernel gives a box satisfying the CHSH inequalities. Both computations are also checked in Lean 4. As a corollary, the 1192 nonlocal vertices of the tripartite no-signaling polytope with Ω=0 found in Paper B admit no commuting-operator quantum realization. The statement does not extend to four parties: an explicit four-qubit GHZ box has Ω=0 exactly and violates the Mermin inequality maximally (8 against the local bound 4); the same holds for every n≥4. For four parties we also prove that a feedback with a constant component has no zero on a sector where the other three parties are two-dimensional; on the stratum with one classical relay, numerics show that Ω=0 boxes occur and are nonlocal.