We analyze the vacuum dynamics of gravity from entropy, including its algebraically constrained $G$-field formulation. Evaluating the curvature traces over 0-, 1-, and 2-form sectors, we show that the complete Minkowski Hessian is exactly that of the quadratic-gravity action $AR+B{R}_{\mu \nu }{R}^{\mu \nu }$, with $A=3\beta /{\ell }_{\mathrm{P}}^4$ and $B=5{\beta }^2/(2{\ell }_{\mathrm{P}}^4)$. 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 nondiagonalizable 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 $m_0^2=3/(5\beta )$, and an opposite-residue spin-2 branch with $m_2^2=-6/(5\beta )=-2m_0^2$. For the foundational choice $\beta >0$, conventional Einstein normalization yields an opposite-residue tachyonic spin-2 pole at a characteristic scale of order ${\beta }^{-1/2}$, whose physical interpretation may depend on the ultraviolet completion of the theory. 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.

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• Alternative gravity theories
• General relativity equations & solutions
• Gravitational waves