In 1979 the US National Academy of Sciences asked the MIT atmospheric physicist Jule Charney to convene a panel and answer one question: how much would the planet warm at equilibrium if humans doubled the atmospheric CO₂ concentration? Five climate modelers, the primitive coupled atmosphere-ocean models of the day, thirty pages of report — the answer was 1.5–4.5°C, central estimate near 3°C. Forty-five years and three orders of magnitude more compute later, the IPCC AR6 (2021) range is 2.5–4°C with a best estimate of 3°C. The bound has tightened slightly; the central number has not moved.
Equilibrium climate sensitivity (ECS) is the long-term surface warming following a sustained doubling of pre-industrial CO₂, from 280 to 560 ppm. The radiative forcing from CO₂ alone is about 4 W/m² and produces only ~1.1°C of warming if everything else is held constant. The residual uncertainty lives in feedbacks: water vapor amplifies (warmer air holds more water, water itself is a greenhouse gas), surface albedo amplifies (melting ice exposes darker ocean and land), and clouds — the deepest unknown — can either amplify or damp depending on whether high or low cloud cover responds more strongly to warming. Slow Earth-system feedbacks (permafrost carbon, ice-sheet response) operate on longer timescales. ECS is not directly observable because the planet is out of equilibrium. It is estimated from three triangulating lines: instrumental-period warming combined with energy-balance accounting, paleoclimate constraints from past warm and cold periods (Pliocene, Last Glacial Maximum), and process-level emergent constraints from comparing model output to observation. Sherwood et al. (2020) is the canonical recent synthesis: a likely range of 2.6–3.9°C, the basis for AR6's narrowing. The policy implication is direct. A 3°C ECS implies a finite carbon budget for any temperature target, and that budget is what national pledges and international agreements ultimately negotiate over. Tail-risk concerns enter at the high end: AR6's very likely range is 2–5°C, but the residual probability above 5°C — values implying civilization-threatening warming on plausible emissions trajectories — is small and non-zero. Weitzman's 2009 dismal theorem argued that this fat tail dominates rational policy regardless of best-estimate cost-benefit analysis.
Three fronts. Kilometer-scale cloud-resolving global models (DYAMOND, ECMWF's Destination Earth) are becoming feasible on the largest supercomputers and should tighten the cloud-feedback range that dominates current uncertainty. Paleoclimate refinements — especially of the Pliocene, ~3 Mya, ~3°C warmer than pre-industrial at ~400 ppm CO₂, the closest past analogue to where humans are pushing the system — are tightening the upper end. Direct observation of how Earth's energy balance has responded to the past decade's warming is slowly adding constraints. The most-studied number in climate science may finally narrow further — or widen, if clouds turn out badly.