Protected Finite Spectra for Induced Gravity: Complete Monotonicity, Optimal Mass Placement, and Threshold-Resolved Dynamics

We formulate finite induced-gravity matter spectra as an exact moment-cancellation and spectral-design problem. For \(p+1\) distinct positive squared masses, the weights annihilating the first \(p\) moments are the barycentric coefficients of a divided difference. Their protected proper-time kernel has a fixed sign, and its normalized quotient is completely monotone with a positive B-spline representation. This implies resolvent product identities, log-convexity, and Hankel-positivity constraints. For three masses we derive an exact induction-time law and a controlled all-orders derivative expansion. Under a fixed endpoint and absolute-weight budget, the induced Einstein stiffness is strictly concave in the intermediate squared mass and uniquely maximized at the identric mean. A conformal light-sector condition selects a dyadic spectrum within the specified model. We retain individual thresholds in the curvature-squared effective action and obtain a conditional region with nonnegative formal scalar and spin-two pole masses; the massive spin-two residue remains negative. The pole criterion depends on subtraction conventions and finite local counterterms, so it is not a claim of a ghost-free ultraviolet completion. Numerical and exact-algebraic checks distinguish identities from model-dependent physical interpretations.