Resilience
You've watched the marble roll.
Here is the equation underneath it.
The three views on the regime-shift page each show a ball in a landscape. This is the physics of the landscape itself, and why a safe harbour can exist at all.
Scroll to descend into the valley
One number for a whole planet
Reduce Earth's entire climate to a single quantity: how far the planet's temperature sits from the Holocene average — the mild, stable baseline of the last 11,700 years. Call it ψ. Cooler than the Holocene is to the left; hotter is to the right. That single number is the horizontal axis of everything that follows.
The mathematics
ψ = (T − ⟨TH⟩) / ⟨TH⟩
The order parameter: the planet's temperature T measured against the Holocene average ⟨TH⟩, as a fraction of that average. Zero is the Holocene; positive is hotter.
A landscape with two valleys
The mathematics of a system with more than one stable state looks like a curve with two hollows. The left valley is the Holocene — the mild, stable climate of the last 11,700 years, its far cold flank the glacial world; the right, far deeper valley is Hothouse Earth. For those 11,700 years we've rested in the left valley, jostled by volcanoes and solar cycles but always rolling back. That returning is resilience — and its measure is how far we can be pushed and still return: the width of the valley, not the drop to its floor.
The mathematics
F(ψ) = F0 + aψ2 + bψ4
A Landau free energy. With a negative and b positive, the quadratic term digs two hollows and the quartic keeps them bounded — the double well of a system with two stable states.
The tilt is us
Human pressure doesn't push the ball directly — it tilts the whole landscape toward Hothouse. That pressure, H, is the combined weight of the nine planetary boundaries — seven of which are already transgressed. As their weight accumulates, the valley floor slopes: the Holocene hollow is still there, but it is shallower, and it has slid closer to the ridge. Explore the nine boundaries →
The mathematics
− γP(H)ψ
The human term. P(H) aggregates the nine planetary boundaries; coupled to ψ, it tilts the entire landscape — the further the boundaries are pushed, the steeper the slope toward the hot side. For now, read P(H) as simply H. What it really contains is the subject of the next two acts.
Without resilience, the runaway is inevitable
Let human pressure grow the way it has — steadily, year on year — and give the landscape nothing else. The Holocene valley shallows, then vanishes. The ball has nowhere to return to. It slides, irreversibly, into Hothouse. This is not pessimism; it is the paper's result for a landscape with a tilt and nothing to resist it.
A safe harbour can exist
Here is the paper's first principle, and it is not where you would expect to find it. The boundaries are not passive weights that simply add up; they respond to each other, and they respond to temperature. Written honestly, the pressure term stops being a straight line and becomes a curve — one that, under the paper's conditions, stops rising. At that level of pressure the tilt plateaus. The landscape stops being flattened, and the hollow the ball was sliding out of is still there to catch it. That plateau is the metastable Holocene-like state: the same safe harbour the Waddingtonian marble forks into on its way down. It is real, and it is not permanent — it holds only while the conditions that create it are maintained.
The mathematics
P(H) = H + gH2 + bhH3 · − c|ψ|3
Two nonlinearities, both born of the boundaries' own behaviour. In the pressure, the second- and third-order interaction terms: with g negative, dP/dH reaches zero at a particular pressure and the tilt stops growing — that stationary point is the metastable state. In the landscape, a cubic in ψ that sets the curvature there. Neither alone is enough; together, under the paper's sign conditions, they open the harbour.
And a brake to slow the slide
The second principle is friction. Any real system dissipates energy as it moves; that dissipation acts as a brake on the ball, resisting the change of state. It buys time and prevents overshoot. The Hamiltonian view already carries this as its damping term — a drag on the ball's motion, not a feature of the landscape.
The mathematics
μψ̈ = −∂F/∂ψ − κψ̇
The landscape sets the force; motion adds friction. The −κψ̇ drag dissipates energy as the state changes, slowing the slide — resilience as resistance to change.
Together, the future is bounded
Watch what has happened. Human pressure here has grown far past the point where, two acts ago, the Holocene valley vanished altogether — and the valley is still there. With a harbour to aim for and a brake to slow the approach, the tilt stays bounded and so does the trajectory. A stabilized, Holocene-like Earth stops being a hope and becomes something the mathematics permits. And the deepest point: the boundaries are not dials we merely read. Policy and stewardship can reshape the landscape itself — hold the tilt flat, add the brake.
Now you steer it
The mathematics
What this means
The mathematics does not promise a soft landing. It says something more useful: a safe harbour can exist — but only if we build it. The nine boundaries are levers, not just gauges; friction is dissipation we can choose to add, by bringing the economy inside the Earth system's limits rather than treating the planet as external to it. A Holocene-like Earth is reachable. Reaching it is a decision.
After Bertolami, O. & Nyström, M. (2026), “Setting up the physical principles of resilience in a model of the Earth System,” Earth Syst. Dynam. 17, 141–150. doi:10.5194/esd-17-141-2026.
Related: Planetary Boundaries Interactive · Tipping Interactive · The Planetary Observatory