The modular Hamiltonian KA := − log ∆A is canonical in modular theory butgenerically nonlocal in quantum field theory, with exact locality restricted to specialcases such as the Bisognano–Wichmann theorem for Rindler wedges and the Casini–Huerta–Myers result for balls in CFT vacua. We introduce an operational algebraicdiagnostic for quasi-local modular behavior based on nested collar algebras A(0) ⊂ A(δ)and define the modular leakage functional Λ(δ; t) measuring how far the outer modularflow σA(δ)t pushes inner observables out of A(0). From this we define the modularleakage exponent αT , quantifying the best power-law suppression of leakage as thecollar thickness δ → 0+ on a fixed modular-time window |t| ≤ T . Exact localitycorresponds to αT = ∞. Our main structural result is an interior–exterior obstruction:leakage controls the strength of modularly-induced couplings between the deep interiorA(0) and the exterior algebra A(δ)′. In particular, if αT > 1 (superlinear leakagesuppression), then the interior–exterior commutator witness decays superlinearly in δ,ruling out any effective modular generator whose leading interior–exterior couplingwould produce an O(δ) (or stronger) commutator signal. This converts qualitative“nonlocality of entanglement Hamiltonians” into a falsifiable algebraic constraintwithout reference to stress tensors or background manifolds.
SIKX HILTON (Tue,) studied this question.