A Covariant Curvature-History Field Formulation for State-Dependent Infinite-Derivative Gravity
We construct a covariant auxiliary-field formulation for state-dependent infinite-derivative gravity. Null-congruence curvature-history integrals provide a natural motivation for state dependence, but they suffer from variational ambiguities and loss of smooth metric dependence in the presence of caustics and branch changes. We replace the curve-dependent history variable by a local scalar field \(C(x)\) satisfying a covariant hyperbolic equation sourced by the Kretschmann invariant. The history equation is enforced at the level of the action by a conjugate auxiliary field \(\chi(x)\). Since the dynamical nonlocality scale depends on \(C(x)\), the variation of the nonlocal form factor involves noncommuting operators; this variation is evaluated using Duhamel's formula. Treating \(C\) and \(\chi\) as independent local fields, we derive the coupled field equations and show that the stress-energy exchange between the nonlocal memory sector and the auxiliary history sector cancels on shell. The resulting local Noether identity ensures covariant conservation of the combined memory-history stress-energy tensor and compatibility with the contracted Bianchi identity.