Residue Monotones for Ricci Flow and π-Weighted Spectral Geometry

This paper introduces two dimensionless spectral invariants, \I_{2n} = (Z_{n-1})^n / (Z_n)^{n-1} and \J_{2n} = Z_{n-2} Z_n / (Z_{n-1})^2, constructed from the first three heat-kernel residues of Laplace-type operators on even-dimensional manifolds. These invariants serve as residue monotones for Ricci flow and provide algebraic diagnostics for symmetric spaces. Key results include: (A) a residue-only Spectral Gauss–Bonnet law on constant-curvature space forms, \I_{2n} = c_n \chi(M) / |\Gamma| with rational cnc_nc_n ; (B) a strict-convexity isoperimetric law on product spheres S2k×S2ℓS^{2k} \times S^{2\ell}S^{2k} \times S^{2\ell} with optimizer t∗=(2ℓ−1)/(2k−1)t^* = (2\ell-1)/(2k-1)t^* = (2\ell-1)/(2k-1) ; (C) an a2a_2a_2-sensitive invariant \J_{2n} distinguishing scalar and Dirac operators with explicit extremals; and (D) monotonicity of \I_{2n} along volume-normalized Ricci flow, with equality at Einstein metrics. A π-weighted (dilaton) variant is formulated, including a π-Yamabe inequality, and a 4D spectral certificate is proposed for 3D spherical space-form limits. This work bridges spectral geometry, Ricci flow dynamics, and π-weighted gravity, offering new tools for geometric analysis.