The number that matters is hidden inside the average.
Sea-level rise is the sum of five components. Standard projections take the aggregate. The aggregate hides the truth — and the truth changes the right answer about where mitigation should be focused.
The aggregate vs the component
Same SSP. Opposite reading.
Aggregate sea-level rise · SSP1-1.9
β = −1.12
Sub-rate. Looks reassuring.
Antarctic component · SSP1-1.9
β = +3.76
Super-rate. Locked in.
The aggregate is the weighted average rate; the components are individual cascades. Slow components (glaciers, terrestrial water storage) dilute the average. The dominant post-tipping driver — Antarctica — is buried inside it. Aggregation can flip a "good news" reading into the opposite of what the dynamics actually say.
Component-by-component β across all 5 pathways
The Frederikse 2020 decomposition: GMSL = steric + glaciers + Greenland + Antarctica + terrestrial water storage. Five components per pathway; 25 cells of independently-computed β.
| Pathway | Aggregate | Steric | Glaciers | Greenland | Antarctic | Terr. water |
|---|---|---|---|---|---|---|
| SSP1-1.9 ~1.5°C | −1.119 | +1.117 | +0.213 | +1.181 | +3.757 | −0.626 |
| SSP1-2.6 ~2°C | −0.945 | +1.176 | +0.382 | +1.281 | +3.664 | −0.312 |
| SSP2-4.5 ~2.7°C | +0.296 | +1.250 | +0.634 | +1.407 | +3.479 | +0.138 |
| SSP3-7.0 ~3.6°C | +0.541 | +1.293 | +0.806 | +1.478 | +3.316 | +0.427 |
| SSP5-8.5 ~5°C | +0.521 | +1.313 | +0.899 | +1.511 | +3.210 | +0.578 |
Read down each column to see the SSP-dependence; read across each row to see the component-dependence. The Antarctic column stays red across every pathway — including SSP1-1.9.
Climate Mitigation Atlas — \(\beta\) by observable × emissions pathway
Each cell's colour is one number — the rate exponent \(\beta\). Green = stoppable (returns to rest). Orange = super-rate. Red = locked-in.
Click any cell for the full reading: \(\beta\), cross-model dispersion, theorem anchor, and the source code on GitHub.
How to read this chart · what the SSPs mean
The axes
- Rows: 8 climate observables — what is changing.
- Columns: 5 emissions pathways from "very aggressive mitigation" (left) to "no mitigation" (right).
- Cell colour: the rate exponent \(\beta\). Below 1 is stoppable; above 1 is locked-in.
- Cell label: the actual \(\beta\) value cross-model median.
The five emissions pathways (SSPs)
- SSP1-1.9 — ~1.5°C. Aggressive net-zero by mid-century. Paris lower bound.
- SSP1-2.6 — ~2°C. Moderate mitigation. Net-zero by ~2070.
- SSP2-4.5 — ~2.7°C. Current policies, middle-of-the-road.
- SSP3-7.0 — ~3.6°C. Regional rivalry, fragmented action.
- SSP5-8.5 — ~5°C. Continued fossil-fuel growth, no mitigation.
SSPs (Shared Socioeconomic Pathways) are the IPCC's standard set of emissions futures. The number after the dash is the radiative forcing in 2100 (W/m²).
The post-tipping rate ranking
For super-rate cascades, β quantifies how rapidly the cascade accelerates after tipping. Closed-form Theorem 1 solution: dΦ/dt ~ Φᵝ.
- Antarctic Ice Sheet: β ≈ 3.5 — fastest acceleration in the climate catalogue.
- Steric, Greenland: β ≈ 1.1–1.5 — moderately super-rate.
- Glaciers, TWS: β < 1 mostly — sub-rate; finite-time return.
Where mitigation actually matters
Focus mitigation on Antarctic-specific feedbacks, not aggregate emissions reductions. The framework's reading is unambiguous: Antarctica drives the post-tipping rate of the dominant climate cascade, and is locked in across all 5 SSPs. Moving its β from +3.76 toward something approaching +1 — slowing the eventual rate — requires intervention on the feedback loops the Antarctic Ice Sheet itself is sensitive to: basal melt rate, marine ice cliff instability, ice-shelf buttressing. Aggregate emissions reductions still matter — they reduce the aggregate β, the rates of glaciers and steric expansion, and the magnitude (not regime) of the AIS itself — but the leverage on post-tipping rate is at the component physics.
Why component decomposition is admissible
Sea-level rise is a linear sum of five physically distinct cascades — steric, glaciers, Greenland, Antarctic, terrestrial water storage. Theorem 5 of the framework establishes that for independent cascades \(\mathcal{C}_1, \mathcal{C}_2\) of the same underlying physics, the rates add: \(\rho_{1+2}(t) = \rho_1(t) + \rho_2(t)\). The aggregate's β is determined by the dominant component (the one with the highest β); sub-dominant components contribute to σ-dispersion but not to the eventual rate.
Caveat (numerical-tools update): components have narrower log-range (0.24–1.35) than total GMSL (1.08–2.08), so Theorem 3 binds harder on components, not less. The framework reads them correctly as physically distinct cascades (cross-component σ_cross = 5.37 — steric vs cryospheric mass loss vs hydrology are different physics).