Vacuum gravity from entropy: Stability, spectra, and exact waves

We analyze the vacuum dynamics of Gravity from Entropy, including its algebraically constrained G-field formulation. Evaluating the curvature traces over zero-, one-, and two-form sectors, we show that the complete Minkowski Hessian is exactly that of the quadratic-gravity action A R+B RμνRμν, with A=3β/P⁴ and B=5β²/(2P⁴). For diagonalizable curvature blocks, the same action reduces to a sum over eigenvalue logarithms and reproduces these coefficients exactly. A strict diagonal-curvature restriction on the perturbations is instead only a reduced subsector and excludes non-diagonalizable type-N wave curvatures. Linearizing the G-field equations and subsequently imposing the algebraic vacuum constraint reproduces the same reduced metric equation and covariant Minkowski Hessian. The spectrum contains the massless graviton, a scalar with m₀²=3/(5β), and an opposite-residue spin-2 branch with m₂²=-6/(5β)=-2m₀². For the foundational choice $β>0$, conventional Einstein normalization therefore implies a tachyonic spin-2 instability. We also show that every four-dimensional Ricci-flat metric solves the local bulk equations through quadratic curvature order, while square-zero Ricci-flat pp-waves are exact local vacuum solutions of the analytic metric-only logarithmic branch. On the isolated massless transverse-traceless eigenspace, the quadratic translation current has the standard general-relativistic normalization.