A Spectral Criterion for Emergent Gravity

Abstract

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 Källen-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.

arXiv comment67 pages, 0 figures0.All scientific content, derivations, and conclusions are solely the work of the authors