A. G. Kim, P. E. Nugent, L. Wang

Quantum-gravitational spacetime foam varying in space and in time imprints, respectively, a transverse phase decoherence and an achromatic, stochastic arrival-time jitter on light from distant sources. We show that the jitter modifies the first-order temporal coherence $g^{(1)}(τ)$ and hence the observed spectrum, providing a spectroscopic probe of the temporal variation, complementary to existing imaging-channel constraints on the spatial variation. In the dynamical-foam limit, where the foam decorrelates within the light's coherence time, the spectrum splits into a coherent line core suppressed by $e^{-σ_φ^2}$, with $σ_φ^2$ the variance of the foam-induced phase jitter, and an incoherent pedestal, yielding a spectral shape-distortion bound on the foam exponent $α$. Three probes follow: the outright survival of broadband flux, the fractional width of a spectral line, and a chromatic deficit across a broadband continuum. Applied to the MeV emission lines of GRB\,221009A we obtain $α\gtrsim 0.63$, a weak, lever-arm-dominated floor. For the broadband CMB the foam induces an $e^{-σ_φ^2(ν)}$ deficit ($σ_φ^2 \propto ν^2$); a fit to the COBE/FIRAS monopole spectrum reaches $α\gtrsim 0.52$ ($95\%$ CL), and next-generation spectrometers reach only $α\gtrsim 0.54$--$0.55$ (forecast), limited by the foam--temperature degeneracy. The pedestal-independent line-width and CMB bounds are floors below the holographic value $α= 2/3$, consistent with and excluding no part of the imaging window reported for GRB\,221009A; only the conditional TeV survival bound ($α\gtrsim0.74$), which holds only for near-Planck-refresh foam ($τ_{\rm foam}\lesssim4\times10^{-28}$\,s), would exclude it --- and, with it, the window.