Consistency Weight of Deterministic Feedback on No-Signaling Correlations: Exact Reduction in the 2-2-2 Scenario and the Quantum Curve

We study the consistency weight W(P;g)=Σ_{a,b} P(a,b | g_A(b), g_B(a)) that arises when a no-signaling (NS) box P is connected to a deterministic feedback g, together with its minimum Ω(P)=min_g W(P;g). For binary inputs and outputs (the 2-2-2 scenario), Ω=1−|T|/4; with m inputs, Ω is determined by the 2×2 sub-pair of inputs on which |T| is largest. For NS boxes, Ω ≥ (S_max−2)/4, and equality is attained for every S_max∈[2,4]. For quantum boxes the exact curve is Ω_Q(S_max)=1−c³, where 6c−4c³=S_max and S_max∈[2,2√2]. The quantity Ω is not a measure of nonlocality: there are local deterministic boxes with Ω=0, and S_max fixes only a lower bound on Ω. For field correlations in a local net of algebras, we show that under the Reeh–Schlieder property Ω=0 holds if and only if A₁=−A₀, B₁=−B₀, and the two measurements are perfectly correlated on the vacuum (A₀ξ₀=±B₀ξ₀). If the two regions lie in complementary wedges, and the Bisognano–Wichmann property and uniqueness of the vacuum hold, this condition is equivalent to all the measurements being c-numbers. With four or more inputs, taking the direct sum of non-overlapping sub-pairs yields a quantum box with S_glob=2√2 and Ω=0. With three inputs, where the sub-pairs necessarily overlap, the largest S_glob compatible with Ω=0 is exactly 5/2. Here S_glob is the maximum CHSH value over all 2×2 sub-pairs of inputs. For quantum boxes with three inputs we obtain the exact curve Ω_Q^(3)(S₀) in closed form on 5/2 ≤ S₀ ≤ 2√2; it starts quadratically, Ω_Q^(3)=(2/9)δ²+O(δ³) at S₀=5/2+δ, and ends at 1−(3√6+5√2)/16≈0.0988 at S₀=2√2.