The Surveyor: A Monitoring Water-Filling Theorem for Conservation Portfolios

Series S2 (Monitoring & Certification) of the Viridis Canon. A steward who monitors many ecosystems under one budget has lacked a first-principles rule for splitting finite monitoring power across heterogeneous sites; effort is allocated by convention. Building on the D-Score Measurement Closure Theorem and fusing it with water-filling resource allocation, this paper proves the Monitoring Water-Filling Theorem (MWT): the steward's deadweight functional is convex, so a unique optimum exists, and it is a generalized water-filling solution governed by a single broadcast monitoring shadow price λmon. Four structural laws follow: (i) optimal effort scales as P* ∝ D−1/3; (ii) no site is optimally left dark; (iii) uniform monitoring is wasteful by a power-mean factor ρ ≥ 1; (iv) a critical certification budget Pcrit marks where added budget stops buying physical certifiability. This is the eleventh self-application of the Intelligence Bound (the ‘Surveyor’). Scope of verification. The Lean 4 proofs (Aristotle-verified, zero sorry) certify the mathematical structure, not empirical magnitudes. Working paper; not peer-reviewed.