Heat Kernel Methods and the Sign of Induced Gravity: Resolving Conventions via the Laplacian—Lichnerowicz Identity

We derive the local one-loop contribution proportional to the scalar curvature R in the Euclidean effective action obtained by integrating out matter fields on a curved background. Using a Schwinger proper-time cutoff epsilon = Lambda^(-2) and the Seeley—DeWitt coefficient a_1, we extract the quadratically divergent term multiplying int d^4x sqrt(g) R. We fix a single Euclidean convention for the Einstein—Hilbert action, state an explicit Laplacian convention, and write the Laplacian—Lichnerowicz identity in a sign-robust form so that the fermionic contribution is unambiguous. We provide a unified bookkeeping coefficient A_1^(eff), and hence an induced Newton coupling G_ind via comparison with the Euclidean Einstein—Hilbert action. We also include the minimal gauge+ghost package in background Feynman gauge, a species table, and a reproducible verification suite. v2 adds: the equivalence of the species table with the classic counting 1/G_ind = (Lambda^2/12pi)(N_0 + 2N_(1/2) - 4N_1), gauge- and scheme-dependence caveats for the vector sector, a Weyl/Majorana caveat, corrected Wick-rotation wording, and an exact-spectrum numerical verification of every a_1 entry against closed-form spectra on S^4.