A Spectral Criterion for Emergent Gravity

We give a source-resolved spectral criterion for long-range tensor exchange between conserved matter sources. Exact channel elimination preserves the propagation kernel together with the source and readout maps and the direct response. Common-space estimates control the residue at the displaced critical root of a finite approximation. A positive Källén–Lehmann representation identifies this residue with a zero-invariant-mass atom even at a continuum endpoint. For a complete positive Lorentz-covariant conserved stress tensor we classify the null-shell measure and prove that its TT atomic weight vanishes. A reduced compact spectral realisation instead carries internal tensor vertices into an external two-helicity response. Conservation reduces the leading vertex of localised stationary matter to its total energy. The mixed residue then factorises into the two energies and a common propagation coefficient. Spatial resolvent estimates determine the potential and force with finite-size corrections. Normalisation through the matter vertex leaves one dimensional calibration for the overall Planck scale. The spectral residue fixes the dimensionless exchange strength. The resulting response admits an Einstein–matter effective description under the stated positivity and helicity assumptions together with universal soft coupling and local nonlinear consistency.Reading and research materialsarXiv preprintArticle and supplementary materialsJinku Guo · Northwestern Polytechnical University · ORCID 0009-0000-6600-6171.This metadata record provides a research summary and links to the work. Full texts, software and supplementary files are available at the linked repositories under their stated licences.