Threshold Resolvent Singularities and the Infrared Structure of Linearized Gravity

We identify a sharp geometric threshold governing the infrared spectral behavior of the spatial Lichnerowicz operator on asymptotically flat three-dimensional manifolds. Let $(M,g)$ be asymptotically flat and let $L=\Delta_L$ denote the spatial Lichnerowicz operator acting on symmetric $2$-tensors. Assume $$|{\rm Riem}(x)| \lesssim r(x)^{-p} \quad \text{as } r(x)\to\infty.$$ If $p>3$, curvature is spectrally short-range: $L$ exhibits regular low-energy scattering and zero energy is not singular. At the critical decay $$|{\rm Riem}(x)| \sim r^{-3},$$ dispersion and curvature balance. Zero enters the essential spectrum, and the weighted resolvent develops a threshold singularity. For $s\in(1/2,1)$, $$\|\langle r\rangle^{-s}(L-i\varepsilon)^{-1}\langle r\rangle^{-s}\| \gtrsim \varepsilon^{-(1-s)} \quad \text{as } \varepsilon \downarrow 0 .$$ Thus, the limiting absorption principle fails at zero energy. This singularity provides a spatial spectral mechanism for the infrared sector of linearized gravity. The same inverse-cube scaling governs long-range correlations, irregular low-frequency scattering, and soft gravitational modes. Numerical simulations of a radial model and the full tensor operator confirm that $p=3$ marks a sharp transition between negligible and marginal curvature. The associated branch point at zero energy determines late-time relaxation, yielding the universal tail exponent $$t^{-(2\ell+3)},$$ a spectral consequence of nonzero ADM mass. More generally, in $d$ spatial dimensions, the critical decay $$|{\rm Riem}(x)| \sim r^{-d}$$ forms a universal boundary for curvature-coupled Laplace-type operators, encoding the infrared structure of gravity in the spectral geometry of a Cauchy slice.