The metric from energy-momentum non-conservation: Generalizing Noether and completing spectral geometry

We complete the program of spectral geometry, in the sense that we show that a manifold's shape, i.e., its metric, can be reconstructed from its resonant sound when tapped lightly, i.e., from its spectrum, -- if in addition we also record the resonances' mutual excitation pattern when the driving is strong enough to reach the nonlinear regime. Applied to spacetime, this finding yields a generalization of Noether's theorem: the specific pattern of energy-momentum non-conservation on a generic curved spacetime, encoded within the quantum field theoretic scattering matrices, is sufficient to calculate the metric. Applied to quantum gravity, this shows that the conventional dichotomy of spacetime versus matter can emerge from an underlying information-theoretic framework of only one type of quantity: abstract correlators, $G^{(n)}$, that are, a priori, merely operators on $n$ tensor factors of a Hilbert space. This is because, on one hand, if abstract higher-order correlators $G^{(n>2)}$ can be diagonalized, these correlators can be represented as local quantum field theoretic vertices on a curved spacetime whose metric $g_{\mu\nu}(x)$ can be explicitly calculated. On the other hand, at sufficiently high energies, such as the Planck scale, the $G^{(n)}$ may not be even approximately representable as correlators of a local QFT on a spacetime, indicating a regime that is mathematically controlled but transcends the concepts of spacetime and matter.