Technical Appendix: Heat Kernel, Fermions, and the Sign of Induced Gravity (Sign Conventions Fixed; Laplacian/lichnerowicz Hinge Made Explicit)

We derive the local contribution proportional to the scalar curvature R in the Euclidean one-loop effective action obtained by integrating out matter fields on a curved background. Using a Schwinger cutoff epsilon = Lambda^{-2} and the Seeley-DeWitt coefficient a1, we extract the quadratically divergent term multiplying the integral of sqrt(g) R. We fix a single Euclidean convention for the Einstein-Hilbert action, state an explicit Laplacian convention, and write the Lichnerowicz/Weitzenbock identity in a sign-robust form so that the fermionic contribution is unambiguous. We provide a unified bookkeeping coefficient A1_eff such that W_R^total = -(A1_eff / 32 pi^2) Lambda^2 integral sqrt(g) R, and hence an induced Newton constant G_ind via comparison with the Euclidean Einstein-Hilbert action. We include the minimal gauge+ghost package in background Feynman gauge, a species table, and a reproducible symbolic check of every entry.