SPECTRAL VACUUM MECHANISM — PART XXX Emergent Gravity from Spectral Geometry:Einstein-Hilbert Action from Vacuum Hessian Heat Kernel

ABSTRACT Part XXX extends the Spectral Vacuum Mechanism (SVM) framework by deriving gravitational dynamics from spectral geometry, complementing the gauge dynamics established in Parts XXVIII–XXIX. While previous parts demonstrated that complex overlaps encode metric properties (R_ij), gauge connections (A_ij), and Yang-Mills dynamics (via coefficient a_4), Part XXX shows that the coefficient a_2 provides a gauge-invariant second-moment gravitational functional built from overlap magnitudes. Main result: The heat kernel expansion of the vacuum Hessian H yields a spectral action without requiring a Dirac operator. For the regime of small overlaps R_ij