No-Signaling and Causal Evolution from Modular Flow: Operational Constraints on Emergent Dynamics

Modular automorphisms generated by Tomita–Takesaki theory are canonical butstate-dependent structures. In special QFT regimes (Bisognano–Wichmann wedgesectors, holographic entanglement wedges), modular flow coincides with geometriccausal evolution (Lorentz boosts, bulk modular flow). We define a state-independentsignaling defect ηt(A → B) that measures how much a candidate evolution αt (inparticular modular flow) fails to preserve the commutation structure between twocommuting algebras A and B. The defect is the operator-norm distance from theevolved algebra αt(B) to the commutant A′. We prove a sharp equivalence: ηt(A →B) = 0 if and only if no operational signaling channel exists from lab A to lab B underevolution αt, for all input states and all A-localized interventions. Quantitatively, ηt ≤ε implies the maximal signaling advantage is at most 2ε. In geometric regimes wheremodular flow is causal (Bisognano–Wichmann), the defect must vanish; nontrivialdefect is a hard falsification of the localization–evolution assignment. This yieldsa universal operational causality test applicable to any candidate dynamics, withimmediate consequences for emergent spacetime and holography.