The semiclassical-gravity iterative surface-Hadamard conjecture

Let $\mathcal C$ be an initial surface, let $\omega_{\mathcal C}$ be an initial state, and let $g^{(r)}$ denote the $r$th time derivative of the metric on $\mathcal C$. Semiclassical-gravity iterative conjecture. At the $k$th stage, the $k$th derivative of the semiclassical equations on $\mathcal C$ implicitly determines a condition on $\omega_{\mathcal C}$ ensuring that the $k$th time derivative of the stress-energy tensor exists on $\mathcal C$, and also implicitly determines the $(k+3)$rd metric time derivative $g^{(k+3)}$ in terms of $\omega_{\mathcal C}$ and lower-order time derivatives. This conjecture motivates the proposed surface Hadamard condition for semiclassical gravity; the iterative determination and resulting well-posedness are not proved.