Fat Tails and Carbon Taxes: A Spectral Resolution of the Climate Economics Debate

The central debate in climate economics — between Nordhaus-style moderate carbon pricing and Stern-style aggressive policy — is fundamentally a disagreement about the tail behavior of the damage distribution. We organize this through the Latent Number \(\rho\): heuristically, when \(\rho > 2\), thin-tail regimes admit well-posed second-moment reasoning and standard cost-benefit analysis (CBA) is not ruled out by tail divergence alone; when \(\rho \leq 2\), fat-tail effects dominate in the Pareto sense (variance diverges as \(\rho \downarrow 2\)), connecting to Weitzman's analysis of catastrophic climate risk. In a stylized proportional rescaling recorded in the companion script, policy charges scale like \(\rho/(\rho-2)\) relative to a baseline marginal-damage anchor — a device that places a thin-tail calibration (e.g.\ \(\rho \approx 5\), anchored at \\(80/ton in the numerics) and a fatter-tail calibration (e.g.\ \)\rho \approx 2.5\(, anchored at \\)150/ton) on one spectral continuum. Tipping points enter as discrete damage jumps; the machine-checked numerics include a +40\% uplift on the \\(80/ton anchor. Growth uncertainty is encoded as a strictly positive downward correction to the certainty-equivalent discount rate (without exporting a closed-form \)\sigma_g^2/(2\rho_g)$ identity to the formal layer). Seventeen real-arithmetic theorems in climate_economy_proof.py are machine-verified in Lean 4 with no user-side axioms in that file.