From induced curvature to a graviton: the tensor sector of the SQG substrate, the horizon bridge, and the conditions for a common light cone

The geometry companions of Spectral Quantum Gravity (SQG) established an ordered honeycomb base times the scale axis ω, a local constraint tying curvature to energy, and a positive Newton constant from the horizon entanglement of the honeycomb fermions. This fourth companion tests whether the same substrate carries a graviton, through the induced quadratic action of transverse-traceless metric perturbations.The half-circle of the phase fiber is the bridge across the horizon: the reduced correlation matrix of one side commutes with the lattice boost (proved), so the entanglement Hamiltonian is a function of it; numerically it equals λ(μ) times the boost, with λ → π for light modes. The lattice brick wall agrees with the entanglement for modes below the cutoff, and the factor between the lattice spacing and the main text's horizon cutoff is the identification ω_min ≈ 0.24–0.27 lattice spacings.For the honeycomb hopping fermions a strain moves the Dirac points of the two valleys oppositely (shift exactly 2 per unit strain, proved); the tensor response is dominated by an induced axial gauge field, whose logarithmic coefficient v κ²/24π² = 0.025 is derived and reproduced by the lattice computation, and the tensor-sector Newton constant is 30–40 times weaker than the horizon value. Kekulé order, flake superlattices, string matter and an arm along ω reduce the factor to no less than 3–11 without tuning.Fermions coupled through a frame give the two Newton constants within a factor 1.35. The graviton velocity is set at the lattice scale and depends on the regulator (1.52 or 0.97 times the matter velocity); with two regulator parameters a common light cone and the horizon G can be reached together.On the honeycomb product the tensor sector is not isotropic (factors 2–3 between modes in the base and along ω).The local lapse constraints generate the full algebra so(p, q) for every connected bipartite graph (proved) and can therefore be imposed only weakly.The two ends of the main text's global window are related by a swap symmetry of the spectrum that preserves chirality.Conclusion: SQG carries a graviton only if its fermions couple to the geometry through a frame and the base and the scale axis are treated alike at the lattice scale. The computations P361.1–P368 were registered before they were run (except the short check P367; the proofs of P369 were checked numerically afterwards); registrations, result logs and a table of the predictions met and not met are in the code archive.