Photon Dispersion and Gamma‑Ray Burst Tests of Emergent Gravity

We derive the modified dispersion relation for photons in the framework of Chrono‑Grid Dynamics (CGD), an emergent gravity model with a fundamental discrete proper time step τ (the Takt). The model predicts an energy‑dependent speed of light, v(E)≈1-ξE²/E_Pl², with ξ∼O(1). This leads to a cumulative time delay for high‑energy photons from gamma‑ray bursts (GRBs). Using data from GRB 090510, GRB 080916C, and GRB 190114C, we compute the expected delays and compare them with current observational limits from Fermi‑LAT, H.E.S.S., and MAGIC. We find that the quadratic energy dependence is consistent with all existing bounds and lies well below the current sensitivity. Future observations with the Cherenkov Telescope Array (CTA) will be able to test this prediction and potentially constrain ξ down to 10⁻³, offering a unique window into the quantum structure of spacetime.