The Gaian Harmonization Theorem: A Single Thermodynamic Shadow Price Decentralizes Planetary Stewardship to the Social Optimum

Standalone thematic-series record of the Viridis Canon (Gaian Systems). The core Intelligence-Bound canon spine is unchanged (frozen at v10.0.0, record 20801185); this record links to the spine via isDerivedFrom the concept DOI 10.5281/zenodo.19317982. Gaia has no central controller. The Gaian Harmonization Theorem (GHT) asks how N uncoordinated local stewards — none observing the global planetary state — realize the centralized minimum-dissipation planetary optimum. The answer is a decentralized welfare theorem: every agent best-responding to a single shared scalar shadow price λ* reproduces the centralized social optimum exactly, with no duality gap. The price is not abstract — it is the regulated variable’s own deviation, broadcast for free by the shared environment. The model is a separable, strictly convex quadratic planetary resource-allocation program: each agent i picks an effort xi with private dissipation (ai/2)(xi−bi)² (ai > 0) under one planetary coupling constraint ∑i xi = X. Ten statements are machine-checked in Lean 4 (Aristotle, zero sorry, axioms ⊆ {propext, Classical.choice, Quot.sound}). R1 (welfare / strong duality): the price-taking best response globally minimizes each agent’s Lagrangian (ght_best_response_minimizes); a single scalar price clears the planetary constraint (ght_market_clears); marginal dissipations equalize to the common price (ght_equimarginal); the decentralized price equilibrium is the centralized minimum-dissipation optimum, no duality gap (ght_strong_duality_decentralized_eq_centralized); and a single scalar is a sufficient coordinating statistic, dimension independent of N (ght_scalar_price_minimal_sufficient). R2 (IB bandwidth ceiling): the coordination ceiling ωH = P·D/(kBT ln 2·Cλ) is strictly positive (ght_harmonization_bandwidth_from_IB) and antitone in temperature — a hotter planet tolerates a slower drift (ght_bandwidth_antitone_temp). R3 (wu-wei robustness paradox): the decentralized/centralized message-complexity ratio is ≥ 1 (ght_wuwei_robustness_ratio_ge_one) and strict for N ≥ 2 (ght_wuwei_robustness_strict): scalar-price coordination is strictly cheaper than central command — letting go makes the planet more robust. The theorem is non-vacuous via a concrete N = 2 binding instance (ght_nonvacuous). GHT is the 24th self-application of the Intelligence Bound (“the Harmonizer”), importing the canon keystone shadow price and the throughput ceiling P·D/(kBT ln 2) into the Gaian Systems area for the first time. The headline: a single thermodynamic shadow price decentralizes planetary stewardship to the social optimum, at a coordination cost that is Intelligence-Bound-limited and strictly favors light-touch (wu-wei) price signals over central command. Deferred to future runs: continuous-time primal–dual (tâtonnement) Lyapunov convergence of the price, and the information-theoretic minimality lower bound of the scalar statistic (only sufficiency is proven here). Scope. The Lean proofs certify the validity of the reasoning, not empirical magnitudes. Working paper; not peer-reviewed.