Spectral Residues, π-Weighted Gravity, and New Arithmetic–Geometric Invariants

This paper explores a novel extension of classical gravitational functionals, including the cosmological constant, Einstein–Hilbert action, and boundary terms, by reinterpreting them as residues of spectral zeta functions. By promoting the constant π to a smooth positive field π(x), we derive π-twisted heat-kernel coefficients that introduce new scalar invariants involving (∇ln π)², □ln π, and their couplings to curvature.