Here is a number that sums up the problem. In 2025 the global average surface temperature was about 1.37 °C above the 1850–1900 level because of human activity, according to the annual "Indicators of Global Climate Change" update (reported in June 2026), and it is rising by about 0.27 °C each decade. At today's emissions, the remaining carbon budget for holding warming to 1.5 °C was estimated in the 2025 edition at about three years' worth.
A budget in years, a trend in tenths of a degree per decade: those are the outputs of a very small set of ideas. This piece builds them from the bottom: a planet as one equation, the push from CO₂, the feedbacks that triple it, the ocean's delay, and the running total called the carbon budget. It ends with how the IPCC speaks about certainty, because reading a climate report means reading that language. This is the science. What societies should do with it is a question of values and trade-offs and is argued elsewhere in this major.
Put the planet in a box. Sunlight comes in; the planet reflects some (its albedo, about 0.30) and absorbs the rest, spread over a sphere, which is a quarter of the area that faces the Sun. It must radiate the same amount out as infrared. The greenhouse gases make the planet a less efficient radiator, which we can represent with an effective emissivity ε less than 1.
Set ε = 1 (no greenhouse effect) and the planet sits at 255 K, which is −18 °C. Set ε to about 0.612 and you get 288 K, which is about 15 °C: the actual average. That single "knob" is the greenhouse effect. A forcing F, such as extra CO₂, is added on the left, and the temperature rises until the right-hand side catches up.
Extra CO₂ gives a forcing that grows with the logarithm of the concentration: F = 5.35 × ln(C ÷ C₀) W/m², where C₀ is the pre-industrial 280 ppm. Doubling gives 3.7 W/m² from this formula; the IPCC's more detailed assessment of the effective radiative forcing of a doubling is 3.93 ± 0.47 W/m². At the 2025 annual mean of 427.35 ppm (NOAA, Mauna Loa) the formula gives 2.3 W/m².
Zero-dimensional model with S = 1361 W/m² and σ = 5.670374×10−8. The forcing is added to absorbed sunlight, F = 5.35 ln(C/280); the model has no feedbacks. The curve is the radiated heat εσT⁴; the planet settles where it crosses the dashed line.
Move the CO₂ slider from 280 to 560 and the model warms by 1.1 °C. That is the "no feedback" answer, set by how steeply a planet's radiation climbs with temperature (about 3.3 W/m² for each degree). It is almost exactly what the IPCC calls the Planck response, −3.22 W/m² per °C. If that were the whole story, doubling CO₂ would warm the planet by about 1.2 °C. It isn't, for the next reason.
Warm the planet and other things change, and some of them change the energy balance again. The IPCC (AR6, Chapter 7) assesses the main ones, each as a number in W/m² per °C. A positive number amplifies warming.
| Feedback | Best estimate | Why |
|---|---|---|
| Planck (warmer = radiates more) | −3.22 | The stabiliser; always negative |
| Water vapour + lapse rate | +1.30 | Warmer air holds more vapour, itself a greenhouse gas |
| Surface albedo | +0.35 | Ice and snow melt, darker surface absorbs more |
| Clouds | +0.42 (−0.10 to +0.94) | The largest uncertainty; depends on cloud height and type |
| Net | −1.16 (likely −1.54 to −0.78) | Still negative, so the climate is stable, but the Planck response is multiplied by about 2.8 |
The equilibrium temperature rise for a doubling of CO₂ is the equilibrium climate sensitivity (ECS): ECS = forcing ÷ (−net feedback). Because the net feedback sits in the denominator, a small change in it makes a large change in ECS, and the uncertainty is lopsided. The AR6 assessment combines process understanding, the historical record, paleoclimate evidence and models into a best estimate of 3 °C, likely 2.5 to 4 °C, very likely 2 to 5 °C. This narrowed the old 1.5 to 4.5 °C range that stood for decades.
Planck fixed at −3.22 W/m² per °C; forcing for doubled CO₂ 3.93 W/m² (IPCC AR6 Chapter 7, Table 7.10 and Table 7.4). Adding the AR6 feedbacks gives ECS of about 3.4 °C; AR6 notes the process-based net feedback is slightly less negative than the value implied by the overall ECS assessment, which is why the headline figure is 3 °C. Values past the last slider setting here are not assessed.
Across the IPCC's likely range for the cloud feedback (+0.12 to +0.72) ECS moves from about 2.7 to 4.6 °C; remove the water vapour feedback and the sensitivity falls to about 1.6 °C. Cloud feedback is the largest source of uncertainty; the combined water-vapour and lapse-rate feedback is the largest single contributor to the warming. A caution from the IPCC: some CMIP6 climate models have a higher sensitivity than the assessed range, traced mainly to a stronger amplifying cloud feedback, and AR6 therefore does not simply average the models; it constrains its projections with observations.
The ECS is where the planet ends up after centuries. The warming we meet in a lifetime is less, because the ocean is still absorbing heat (see Earth Systems, 36/13). The transient climate response (TCR) is the warming at the moment CO₂ has doubled in a standard experiment where it rises 1% a year, about year 70. The IPCC's best estimate is 1.8 °C, likely 1.4 to 2.2 °C, below the ECS of 3 °C.
A simple model captures it: a thin upper layer (air, land, the top of the ocean) exchanges heat with a huge deep ocean. Try it. Choose when the CO₂ stops rising.
Two-layer energy balance: upper layer 8 W·yr/m²/K, deep ocean 100, heat exchange 0.7 W/m²/K (illustrative values near published fits), forcing 3.93 W/m² per doubling, CO₂ rising 1% a year until the chosen multiple, then held constant. Stepped at 0.1 years.
With ECS 3 °C the model warms about 2.0 °C at year 70 (the TCR; the IPCC's assessed value is 1.8 °C), but if CO₂ is merely held at that level the planet keeps warming for centuries, reaching about 2.7 °C by year 300, because the deep ocean is still catching up. Holding a concentration constant is not the same as stopping emissions. When emissions stop, the ocean and land keep pulling CO₂ out of the air. The IPCC concludes that achieving net zero CO₂ emissions, with non-CO₂ forcing declining, would be sufficient to prevent additional human-caused warming.
Here is the idea that turns the climate problem into a countdown. For CO₂, the warming is roughly proportional to the cumulative amount emitted, no matter how fast. The IPCC assesses each 1,000 gigatonnes of CO₂ to add 0.45 °C (likely range 0.27 to 0.63 °C), the TCRE, with high confidence in the near-linear relationship. It follows that to stop warming at any level you must reach net zero CO₂, and that any chosen limit comes with a fixed total, a remaining carbon budget.
| Limit (50% chance) | IPCC AR6 budget from 1 Jan 2020 |
|---|---|
| 1.5 °C | 500 Gt CO₂ (400 for a 67% chance) |
| 1.7 °C | 850 Gt CO₂ |
| 2.0 °C | 1,350 Gt CO₂ (1,150 for a 67% chance) |
Human CO₂ is about 42 Gt a year (2024: about 38.2 from fossil fuels and industry plus about 4.2 from land use; Forster et al., 2025). Those AR6 budgets were set at the start of 2020. The annual update by Forster and colleagues re-bases them each year with new data: its 2025 edition put the 1.5 °C budget at about 130 Gt CO₂ from the start of 2025 (central estimate, 50% chance), a little over three years at current emissions. A June 2026 update was reported as showing the 1.5 °C budget gone in about three years and the 1.7 °C budget in about 12; those figures come from a news report, so check the latest edition of the paper.
Budgets: IPCC AR6 SPM Table SPM.2 (from 1 Jan 2020, 50% chance) and Forster et al. 2025 (from 1 Jan 2025). Emissions: total CO₂ 2024, 42.4 Gt (38.2 fossil + 4.2 land use). The AR6 presets start at 42.4 for simplicity, not at 2020's actual emissions. A projection of arithmetic, not a forecast.
Play with the sliders and a hard fact appears. Even if emissions fell by 7% a year from the start of 2025 (faster than the roughly 5% fall of the 2020 pandemic year), the sum of all future emissions would be about 600 Gt: far more than 130. To stay inside the 130 Gt budget, emissions would have needed to drop by about a third every year from the start of 2025. That is why climate scientists talk about overshoot: crossing the level and then using net-negative emissions to come back. It is also why the budget is an estimate with wide error bars, not a cliff. The AR6 table alone carries uncertainty of about ±550 Gt for historical warming and ±220 Gt for non-CO₂ gases, so the same limit can be read as a smaller or larger total.
The measured record is where the theory meets the thermometer. Each figure below carries its date.
One point of care: a single year above 1.5 °C does not mean the Paris goal is broken. The IPCC reports global warming levels as 20-year averages, partly because one year includes natural swings such as El Niño (2023–24 was an El Niño period). Scientists do disagree about how much of the very warm 2023–24 was natural variability, how much was falling aerosol pollution (cleaner shipping fuel, falling emissions in China), and how much signals faster underlying warming; the 0.27 °C-per-decade human-induced rate is the cleaner number.
Reading the IPCC means reading a defined language. Confidence has five levels (very low, low, medium, high, very high) and reflects how much evidence there is and how well it agrees. Likelihood is a probability scale for an outcome: virtually certain is 99 to 100%, very likely 90 to 100%, likely 66 to 100%, about as likely as not 33 to 66%, unlikely 0 to 33%, very unlikely 0 to 10%, exceptionally unlikely 0 to 1%. The ranges nest: if something is very likely it is also likely. In square brackets the IPCC gives the very likely range, a 90% interval, unless stated otherwise.
Likelihood scale from the IPCC AR6 WGI Summary for Policymakers, footnote 4. "More likely than not" is greater than 50%.
Examples from AR6 show the language at work. It is unequivocal that human influence has warmed the atmosphere, ocean and land. It is virtually certain that human CO₂ is the main driver of surface-ocean acidification. ECS is very likely between 2 °C (high confidence) and 5 °C (medium confidence). On the Atlantic circulation: it is very likely to weaken this century, but there is medium confidence that it will not collapse abruptly before 2100.
That last sentence marks the edge of the field. Known with high confidence: the planet has warmed, humans are the cause, CO₂ is the main control knob, and warming scales with cumulative emissions. Genuinely uncertain: the exact sensitivity (clouds), how aerosols have masked warming, the speed of ice-sheet loss, and tipping points such as the Amazon, ice sheets and the Atlantic circulation; AR6 says low-likelihood, high-impact outcomes cannot be ruled out. Reasonable scientists disagree at those edges. The science does not say what to do; carbon taxes, nuclear power, adaptation and degrowth are choices about values, argued in Environmental Economics & Policy (36/19) and Energy Systems & Transitions (36/16).
Figures here are dated as shown and will be superseded by newer editions of the indicators and the IPCC's next report.
Climate science is an energy ledger. CO₂ pushes on it logarithmically, feedbacks multiply the response to about 3 °C per doubling with a long upper tail, the ocean delays it, and for CO₂ the warming follows the cumulative total, which is what a carbon budget counts. The IPCC says which parts are known with high confidence and which are not, and a good reader holds both.
Try this: take any climate headline with a number in it, and ask three questions: what is it measured against (which baseline years), over what period (one year or twenty), and which of the IPCC's likelihood or confidence words would fit.
Sources. IPCC AR6 WGI Summary for Policymakers (2021): warming, human-caused share, ECS, TCRE, Table SPM.2 budgets, sea level, confidence and likelihood scales, CMIP6 sensitivity note, Atlantic circulation statement. IPCC AR6 WGI Chapter 7: ERF of doubled CO₂ (3.93 ± 0.47), feedback values (Planck −3.22, water vapour + lapse rate +1.30, albedo +0.35, cloud +0.42, net −1.16, amplification 2.8), TCR 1.8 (1.4 to 2.2). Forster et al., Indicators of Global Climate Change 2024, Earth System Science Data 17 (2025): 1.52 and 1.36 °C for 2024, 0.27 °C per decade, 0.89 W/m², 130 Gt CO₂ from 2025, emissions; reported through the University of Leeds release (27 June 2025). The 2025 values (1.37 °C, about three years and about 12 years) are from a wire report of the June 2026 update (AAP, 10 June 2026), not from the paper itself. NOAA GML annual means. The two-layer model values and every interactive figure were computed in node.