Quantum Gravity from the Elastic Medium: Einstein Equations, UV-Finite Graviton and Singularity Regulation

We derive from the 4D deformation field U_μ of the Danieletto elastic medium three quantum gravity results, unified by the same axiom dT/dF=0: (1) the Einstein equations G_μν+Λg_μν=(8πG/c⁴)T_μν from the identification h_μν=∂_μU_ν+∂_νU_μ and Bekenstein entropy [3]; (2) the graviton as the quantum mode of U_μ with spin 2, zero mass, 2 polarizations and UV-finite propagator D̃(k)∝1/[k²(1+k²/k_max²)] with k_max=1/l_P; (3) regulation of the Schwarzschild singularity — V(r→0)→−Gm/l_P (finite) — and the quantum Big Bang bounce: H²=(8πG/3)ρ(1−ρ/ρ_max). This last result is formally identical to loop quantum cosmology (LQC) [6] and to the recent thermodynamic derivation of Alonso-Serrano, Liška and Vicente-Becerril (PLB 2023) [7]. The key difference: the Danieletto framework derives H²=(8πG/3)ρ(1−ρ/ρ_max) from the modified medium graviton propagator — not from Ashtekar variables or horizon thermodynamics — producing the same result via a more direct path and from a framework that simultaneously includes the SM and canonical quantization.