The Manipulation-Proof Weighting Theorem: Gaming-Elasticity-Adjusted Design of Composite Biodiversity Indices

Paper supported by machine-verified Lean 4. Composite biodiversity indices that combine species, phylogenetic, and genetic diversity into a single score are increasingly used to price conservation payments and biodiversity credits. Existing design practice sets the combination weights from scientific judgments about each component's informativeness alone, holding the weights fixed once a monitoring protocol is chosen. We show this is incomplete whenever the index is used to pay a self-interested steward: components differ sharply in how cheaply their observed value can be inflated without genuine ecological improvement, and a rational steward reallocates effort toward whichever weighted channel offers the highest apparent return per dollar of gaming. We formalize this with a gaming elasticity gamma_i per channel and derive the steward's optimal injected bias in closed form. Minimizing a scale-invariant Goodhart ratio—injected bias per unit of squared informativeness—is a Rayleigh-quotient problem whose unique interior solution is the Manipulation-Proof Weighting Theorem: w_i^ast propto I_i/gamma_i, importance divided by gaming elasticity, rather than importance alone. Applied to Viridis's own D-Score defaults (w=0.5,0.3,0.2 for species, phylogenetic, and genetic diversity), and to illustrative but literature-grounded gaming-elasticity ratios, the manipulation-proof reweighting fully inverts the current ranking, cutting modeled gaming leakage by roughly three-quarters and cutting the resulting overstatement of the Intelligence Bound's deployable-capacity ceiling dI/dtle PD/(k_BTln2) by a comparable margin. All results are numerically verified (14/14); the verification gate itself caught and corrected a genuine scale-invariance error in an early draft of the governing ratio. We close with four testable predictions spanning archival registry data, a field trial design, a historical natural experiment, and a formal comparative-statics claim, and cast the result as an Intelligence-Bound self-application, the Warden.