Emergent Gravity, Matter Condensation, and Spectral Geometry in Quantum Graph Emergent Field Theory

We study whether a deliberately stripped-down graph surrogate can exhibit several geometry-facing signals at once without imposing a rigid background by hand. In the QGEFT surrogate, an annealed sparse graph carries diagonal SU (3) edge phases together with a matter-coupled U (1) sector and a complex scalar field. The system is evolved by simulated annealing under local graph and field updates, and then interrogated by observables designed to probe condensation, large-scale organization, gravity-like infrared structure, and coordinate-free spectral geometry. Across the current benchmark program, we find four reproducible features. First, a weak-coupling matter branch supports stable condensation and topological backreaction without destroying macroscopic connectivity. Second, in a larger N=2048 ensemble, the shell-averaged Ollivier-Ricci profile around the condensate becomes more consistent with a 1/r-like form, with R^20. 486, while coarse graining strengthens rather than erases the matter-curvature relation. Third, a separate finite-temperature vacuum probe exhibits a coherent but still gapped spin-2-like fluctuation channel, with correlator fit quality R^20. 960 but a nonzero extracted gap. Fourth, the intrinsic graph Laplacian of the largest connected bulk follows a clean Weyl-style counting law with dₖ₄ₘ₋2. 75 and R^20. 988 indicating near-three-dimensional manifold-like scaling in a coordinate-free observable. The narrow claim supported by the current data is not that continuum general relativity, a massless graviton, or a full continuum spacetime phase has already been derived. It is that the QGEFT surrogate can simultaneously support stable matter condensation, internally consistent infrared gravity-like diagnostics, a coherent spin-2-like channel, and manifold-like spectral scaling in its largest currently analyzed hot bulk.