No-Signaling Boxes and Deterministic Feedback: Boxes with Ω=0 in the Tripartite Binary Scenario (Computer-Assisted)

We study the consistency weight W(P;g)=Σ_o P(o|g(o)) that arises when a tripartite binary no-signaling (NS) box P is connected to a deterministic feedback g, together with its minimum Ω(P)=min_g W(P;g). The feedbacks with W=1 on every NS box are exactly the process functions, and we confirm this for the 744 tripartite binary ones by exhaustive enumeration of all 4096 feedbacks and linear programming, so the remaining problem concerns the feedbacks with W<1 on some NS box. When each feedback is considered individually, we show that intrinsic paradoxes are vacuous, and we therefore define profile intrinsicness as the replacement notion. We then enumerate the 1224 vertices of the NS polytope with Ω=0 by applying a double-description method face by face, cross-check the enumeration with pycddlib, and classify the vertices into 15 orbits under a symmetry group of order 384. Of the 1224 vertices, 32 are local, 48 have a nonlocal bipartite marginal, and 1144 are nonlocal although all their bipartite marginals are local; we call these tripartite-only nonlocal. However, the existence of nonlocal boxes with Ω=0 does not mean that nonlocality is necessary for the paradox: there are local deterministic boxes with Ω=0, and the face F_g of every feedback g that is not a process function contains a local box. Profile intrinsicness corrects for this, and the nonlocal vertices that satisfy it are the 48 with a nonlocal bipartite marginal and, among the tripartite-only ones, the 8 pure parity boxes, which form a single orbit. Separately, the quantum non-realizability of the 13 nonlocal orbit representatives is verified at the 19-monomial NPA level with exact rational dual witnesses; this covers vertices only. As a by-product, we prove the identity p_gynin=(W₁+W₂)/8 between W for cyclic feedbacks and the success probability of the GYNIN game, together with the no-signaling bound p_gynin ≤ 1/4.