The Second Derivative, Part III: The Parts That Don't Bend

Climate · Earth System · Thresholds · Part III

The Parts That Don’t Bend

Part II measured how fast the climate is speeding up, then admitted its method was blind to the systems that matter most. This is what that blindness was hiding.

September 2026 · Part III of four · The Second Derivative series

This is Part III. Part I argued that climate change is accelerating. Part II put eleven charts behind that claim — and ended by conceding that the evidence is strong only where the Earth system integrates energy, and weak or absent where circulation and feedbacks dominate. This essay is that concession, taken seriously. It needs a different method, because for the systems in question the second derivative is not a weak diagnostic. It is the wrong one entirely.

There is a sentence at the end of Part II that I have been uneasy about ever since I wrote it: the measured second derivative does not tell you where the edges are, it tells you how quickly the distance to them is closing. It is true. It is also an evasion, because it quietly assumes we know where the edges are, and mostly we do not.

This essay is about the attempt to find out. It has a harder epistemic job than Part II, and I want to be honest about that from the first paragraph. Part II dealt in measurements with error bars. Here the central evidence is statistical inference about stability — a claim not about what a system is doing but about how well it would recover if pushed. That is a subtler thing to measure, it has a documented history of false alarms, and in the single most consequential case the methods currently contradict each other in public.

The rule this essay runs on

Every claim below is tagged as one of three things, and the tag matters more than the claim:

• Observed — a change measured directly in the instrumental record.
• Inferred — a statistical signal in observations, interpreted as loss of stability. Real data, contested interpretation.
• Projected — a model result about a threshold that has not been observed and may never be.

Most public writing about tipping points blurs these three into one voice. Almost all the disagreement in the field lives in the gap between the second and the third.

I. How you watch for a threshold you cannot see

Start with the problem. If a system sits in a stable state held together by feedbacks, its behaviour tells you almost nothing about how close it is to leaving that state. Antarctic sea ice was flat or slightly growing for thirty-seven years and then fell off a cliff. Nothing in the flat part announced the cliff.

So you cannot watch the value, and you cannot usefully watch the rate. What you can watch, in principle, is resilience: how quickly the system springs back after it gets knocked. And there is a genuinely elegant piece of mathematics that says resilience leaves a fingerprint in ordinary noisy data before anything dramatic happens.

Critical slowing down

Picture the state of a system as a ball sitting in a valley. Random weather knocks the ball around; the valley walls push it back. A deep, steep valley returns the ball quickly — small wobbles, fast recovery. Now warm the planet, and the valley gets shallower. The ball still sits in the same place, so the average looks unchanged. But it now takes longer to return after each knock, and it wanders further before it does.

Figure 1 · Why a system can look fine and be close to the edge Schematic. Both balls sit at the bottom of their valley, so both systems have the same average state. Only the shape of the valley — which you cannot observe directly — differs.

This is the whole idea. If resilience is falling, the noise around the average should get larger and more sluggish — before the state itself moves. That prediction is what makes the theory testable, and it is also where all the trouble starts.

The theory gives two concrete, measurable predictions.[1] As a system approaches a bifurcation, the variance of its fluctuations rises, and its lag-1 autocorrelation rises — meaning each observation resembles the one before it more closely, because the system is taking longer to shake off disturbances. Rising variance and rising memory, together, are the signature known as critical slowing down. Both can be computed from an ordinary time series with no knowledge of the underlying physics at all.

Figure 2 · What critical slowing down looks like in a data series Schematic. Two series with the same mean and no trend. The right-hand one has larger excursions that persist longer — higher variance and higher lag-1 autocorrelation.

Why this is attractive, and why it is dangerous. Attractive because it needs no model of the system — just a long, clean record. Dangerous for exactly the same reason: a purely statistical signature can be produced by things that have nothing to do with approaching a threshold. Section VIII is about those things.

II. The scoreboard

Here is the whole essay in one table, before the details. The column that matters is the last one.

System	What the evidence is	Status
Warm-water coral	84% of reefs hit by bleaching-level heat, 2023–25; assessed as having passed its threshold	Observed
Antarctic sea ice	Abrupt 2016 shift; summer variance doubled and season-to-season memory lengthened	Observed
Amazon rainforest	Rising lag-1 autocorrelation across >75% of the forest since the early 2000s	Inferred
Greenland ice sheet	Critical slowing down in western Greenland melt and height records	Inferred
AMOC	Statistical signals say tipping course; direct measurement says gradual weakening	Contested
West Antarctic ice sheet	Marine instability plausible; the fastest proposed mechanism has weakened on review	Projected
Permafrost	Reassessed: gradual globally, abrupt only locally — no global threshold found	Demoted

Note that the list is not uniformly alarming and not uniformly reassuring. One element has moved from “approaching” to “passed,” one has been demoted out of the category altogether, and the most consequential one is an open argument. That mix is what a live research field looks like.

III. Coral: the one that has already happened

If you want to know what a tipping point looks like from inside, it is not a graph. It is a reef.

Between January 2023 and March 2025, bleaching-level heat stress reached 84% of the world’s coral reefs, across 82 countries and territories — the fourth and by far the most severe global bleaching event on record.[2] The scale of the escalation is the part that should stop you.

Figure 3 · Four global coral bleaching events Share of the world’s reef area hit by bleaching-level heat stress in each declared global event. The monitoring agency had to add three new levels to its alert scale during the fourth.

Source: ICRI / NOAA Coral Reef Watch, fourth global bleaching event.[2] Percentages are of global reef area experiencing bleaching-level heat stress, not of reefs killed — bleached coral can recover if the heat relents.

In October 2025 the Global Tipping Points Report — 160 scientists, 87 institutions, coordinated by Tim Lenton — concluded that warm-water coral reefs have become the first Earth-system tipping element to pass its threshold.[3] The assessed threshold is about 1.2 °C of warming, with a range of 1–1.5 °C. We are at roughly 1.4 °C. The report’s judgement is that even if warming stabilises at 1.5 °C, reefs are virtually certain — over 99% probability — to tip, and that reefs at any meaningful scale return only if temperatures come back towards 1 °C.

What “crossed” does and does not mean here

This is an assessment judgement about committed loss, not a measured discontinuity in a time series. Reefs still exist; many are still alive; conservation of local stressors like overfishing and pollution can preserve fragments. What the report claims is that the large-scale system no longer has a path back at current temperatures. That is a different kind of statement from “sea level rose 10 cm,” and it rests on expert elicitation as much as on measurement.

I have put coral in the Observed row of the scoreboard anyway, because the 84% figure and the escalation in Figure 3 are hard measurements, and because no one disputes the direction. But the word “tipping point” is doing assessment work here, not arithmetic work.

IV. Antarctic sea ice: the textbook signature, in real data

Part II flagged Antarctic sea ice as a regime shift rather than an acceleration and left it there. It deserves better, because it is the one place in this entire essay where the early-warning theory of Section I can be checked against observations that have already delivered their verdict.

The history: Antarctic sea ice extent drifted slightly upward from 1979 to about 2014 — an awkward fact much used at the time by people arguing the Antarctic was fine. Then in 2016 it fell abruptly and has stayed low. In 2025 the summer minimum was 1.98 million km², effectively tied for second-lowest in the 47-year record, and the winter maximum was third-lowest.[4]

Now the interesting part. Hobbs and colleagues examined the summer record and found that the standard deviation of summer sea ice doubled, from 0.31 million km² over 1979–2006 to 0.76 million km² over 2007–22 — and that this rise in variance was accompanied by a longer season-to-season memory.[5]

Figure 4 · Both early-warning signals, in one real record Standard deviation of summer Antarctic sea ice extent, before and after the mid-2000s. The second signal — lengthening memory — cannot be drawn as a bar, but was found in the same analysis.

Why this case carries weight. Both theoretical predictions from Section I — rising variance and rising memory — showed up in the same observational record of a system that then underwent an abrupt, sustained shift. That is about as close to a validation of the method as the climate record offers.

The honest qualification is that a validated signature is not a validated mechanism. Modelling work attributes the 2016–2023 decline to sustained warming of the upper Southern Ocean combined with tropical teleconnections and an unusual atmospheric configuration in 2016.[6] Whether that constitutes a true bifurcation — a system that has moved to a new stable state and will not return — or a forced decline that happens to persist, is not settled. Ten years is not long enough to know.

V. The AMOC: where the methods contradict each other

And now the case that matters most, and the one I find hardest to write about honestly, because the specialists disagree in public and the disagreement is not a matter of degree.

The Atlantic Meridional Overturning Circulation carries warm surface water north and cold deep water south. It is the reason northwest Europe is habitable at its latitude. It has two stable states in essentially every model that can represent it, and freshwater from a melting Greenland pushes it towards the weaker one. Nobody disputes any of that. What is disputed is where it currently stands.

Figure 5 · Four methods, four incompatible answers What different lines of evidence say about when — or whether — the AMOC collapses. These are not error bars around a shared estimate. They are different claims.

Sources, in row order: Ditlevsen & Ditlevsen (2023);[7] van Westen et al. (2024);[8] RAPID observations as discussed in McCarthy et al. (2025);[9] IPCC AR6 WGI.[10] Note that even the RAPID trend is contested: some analyses hold that no statistically robust long-term trend can yet be extracted from a record that begins only in 2004.

Reading the disagreement

The statistical case rests on exactly the machinery of Section I. Ditlevsen and Ditlevsen applied rising-autocorrelation analysis to a sea-surface-temperature proxy for AMOC strength and extrapolated to a tipping time: collapse between 2025 and 2095 with 95% confidence, central estimate 2057.[7] It is a striking, specific, checkable claim, and it received enormous press.

The physics-based case is more cautious and, to my eye, more persuasive in method. Van Westen and colleagues argued that the SST proxy may not represent AMOC behaviour well enough to carry an extrapolation, and proposed instead a freshwater-transport indicator grounded in the actual overturning physics. Their conclusion is that the AMOC is on a tipping course and that the indicator reaches its minimum roughly 10 to 40 years before tipping — a warning system, deliberately not a date.[8]

The direct measurement cuts against both. The RAPID mooring array has been measuring the overturning at 26°N continuously since 2004, and shows weakening of about 1.0 [0.4–1.6] sverdrups per decade — consistent with climate-model projections, and roughly five times too slow to produce a collapse by mid-century, which would require something like 5 Sv per decade.[9] A twenty-year record cannot rule out a later acceleration. But it is the only thing here that is actually a measurement of the AMOC rather than a proxy for it.

And then the methodological objection, which applies to the first row and, by extension, to a good deal of this essay. Ben-Yami and colleagues examined whether tipping times can be extrapolated from historical data at all, and concluded that they cannot: the uncertainties compound across too many levels, the techniques optimised to predict tipping times are prone to false positives, and — a particularly uncomfortable point — historical ocean datasets show increasing variance simply because sparse early records were infilled differently, which mimics the early-warning signal exactly.[11]

A rise in variance can mean a system is losing resilience. It can also mean we got better at measuring it.

VI. The ice sheets: commitment is not collapse

Greenland and West Antarctica are the two elements whose eventual loss would reshape every coastline on Earth, and they are also where the public conversation goes wrong most reliably — because the word “tipping” gets attached to a process measured in millennia and read as though it meant next decade.

The early-warning evidence is real. Boers and Rypdal applied critical-slowing-down analysis to ice-core-derived height reconstructions and melt records from central-west Greenland and found the signature of a system losing stability.[12] The mechanism is not mysterious: the melt–elevation feedback. As the surface melts the ice sheet gets shorter, which puts its surface into warmer air, which melts it faster. That is a genuine self-reinforcing loop of the kind that can produce a threshold.

For West Antarctica the story has moved in the other direction. The most alarming proposed mechanism — marine ice cliff instability, in which exposed ice cliffs above a critical height collapse under their own weight in a self-sustaining cascade — drove some of the highest sea-level projections of the last decade. A 2024 reassessment implemented the physics in three ice-sheet models and found that Thwaites Glacier would not undergo runaway cliff retreat this century, because two stabilising effects intervene: the ice behind a collapsing cliff stretches and thins, lowering the next cliff, and glacier flow delivers replacement ice faster than the cliff retreats.[13] Thwaites is still retreating, faster, and West Antarctic collapse over centuries cannot be ruled out. But the fastest available mechanism got weaker on inspection.

Figure 6 · How long these things actually take Approximate timescale over which each transition would play out once triggered, on a logarithmic axis. Compiled from the assessment literature; these are order-of-magnitude ranges, not projections.

Why the log axis is doing moral work. Armstrong McKay et al. estimate that Greenland and West Antarctic collapse would unfold over hundreds to thousands of years, and would need a few decades above the threshold before being triggered at all.[14] A threshold crossed in the 2030s and completed in the year 3500 is still a threshold crossed — the commitment is made by people who will never see the consequence. This is the single most important thing to understand about ice-sheet tipping, and the hardest to feel.

VII. The Amazon: the clearest inferred signal

Boulton, Lenton and Boers applied the lag-1 autocorrelation method of Section I to satellite vegetation data and found that more than three-quarters of the Amazon has been losing resilience since the early 2000s, with the loss fastest in drier regions and in areas closer to human activity.[15] The mechanism is coherent: the forest generates much of its own rainfall, so deforestation and lengthening dry seasons attack the feedback that sustains it.

This is the textbook application of the method — and it has attracted the textbook objection. A 2024 analysis found that the observed resilience loss could plausibly arise from internal climate variability rather than from an approach to a threshold.[16] That does not refute the finding. It means the signal is real and the interpretation is not yet forced, which is precisely what the Inferred tag is for.

VIII. Permafrost: the one that got demoted

Permafrost has appeared on tipping-point lists for twenty years, usually as a carbon bomb: frozen soils thaw, release methane and CO&sub2;, drive more warming, thaw more soil. It is the most intuitive feedback loop in the whole catalogue.

The current assessment is that it is not a global tipping element at all. Modelling work finds that permafrost thaw is gradual at the global scale — a threshold-free response to warming — and abrupt only locally, producing many small regional transitions at different times rather than one planetary switch. A 2024 perspective found no feedback mechanism strong enough to support a global threshold.[17]

Why I put a demotion in an essay about thresholds

Because it is evidence about the field rather than about the climate. A discipline that only ever adds items to its list of catastrophes is doing advocacy. One that removes an item when the modelling stops supporting it — and permafrost was a popular item — is doing science. When the same field tells me coral has tipped, the demotion of permafrost is part of why I believe it.

The demotion is also not good news, exactly. The carbon still comes out, irreversibly, once the soil thaws; it simply comes out on a ramp rather than through a trapdoor.

IX. Why the early-warning signals might be lying

Section I sold you a beautiful piece of mathematics. This section is the invoice.

1. Some systems always show the signal

The most damaging result is also the least discussed. Jäger and Füllsack identified whole classes of systems that reliably produce rising variance and rising autocorrelation while having no critical transition anywhere in their future.[18] The signature is not specific to approaching a bifurcation. If your system happens to belong to one of those classes, the early-warning signal is a permanent false alarm, and no amount of additional data will resolve it.

2. The analyst has too many free choices

Computing an early-warning signal requires you to choose a detrending method, a sliding-window length, and a filtering bandwidth. Comparisons of methods have shown these choices materially change the result — and that warnings can be masked entirely by inter-annual atmospheric variability unless the data are aggregated or filtered in particular ways.[19] Wherever an analyst has several defensible choices and one of them yields a publishable result, the literature will drift.

3. Better instruments can imitate lost resilience

This is the objection I find genuinely hard to dismiss. Historical ocean datasets are sparse early and dense later. Filling the early gaps smooths them; the later data are not smoothed. The result is a record whose variance rises over time for reasons of measurement — producing exactly the pattern that the theory reads as approaching collapse.[11] Any AMOC early-warning result built on century-scale reconstructed sea-surface temperature has to answer this, and answering it convincingly is hard.

4. And the records are, again, too short

Detecting a change in the variance of a series needs considerably more data than detecting a change in its mean. RAPID has twenty-two years. Satellite sea ice has forty-seven. To ask of such records whether the second moment of the noise is drifting is to ask a great deal of them.

The asymmetry that makes this worse than Part II

In Part II, if the acceleration signal was noise, we would find out: the curves would fail to keep bending and the coefficient would fall towards zero. The claim was falsifiable on a decade or two of further data.

Here the structure is crueller. A threshold system gives you a flat line either way. If the warnings are false alarms, we learn that by outliving them. If they are real, we learn that by crossing the threshold. There is no observation available in advance that cleanly separates the two — which is exactly why the theory was invented, and exactly why its false-positive rate matters so much.

X. What would change my mind

Part II ended with a falsification list, and the habit is worth keeping even though it is harder to satisfy here. Recorded in advance, for a future reader to check:

• AMOC. Continued RAPID measurement through the 2030s showing weakening at or below roughly 1 Sv per decade would make the mid-century collapse estimate untenable. An observed acceleration towards 3–5 Sv per decade would do the opposite and would settle the argument in Figure 5 decisively.
• Antarctic sea ice. A sustained recovery of summer extent towards the pre-2016 range, accompanied by a fall in variance back towards 0.31 million km², would indicate an excursion rather than a regime shift.
• The method itself. A systematic retrospective — taking the early-warning signals published over the past fifteen years and scoring how many were followed by an actual transition — would tell us the false-positive rate empirically instead of theoretically. As far as I can tell, nobody has published one. It is the study I would most like to read.
• Amazon. A resilience analysis on the longer satellite record now available that shows autocorrelation stabilising or falling would substantially weaken the dieback case.
• Coral. Widespread recovery of bleached reefs during a sustained cool phase would suggest the threshold judgement was premature. The next strong La Niña is the test.

And the observation that would most strengthen the whole framework: a tipping element for which early-warning signals were published before a transition, and the transition then happened as described. Antarctic sea ice is the closest thing we have to that, and even there the signals were identified in hindsight.

XI. The same mathematics, pointing the other way

There is a symmetry in all this that the catastrophe framing hides. Self-reinforcing feedbacks and threshold behaviour are not properties of climate disasters. They are properties of complex systems, and they run in both directions.

The 2025 assessment that declared coral past its threshold also identified a set of positive tipping points that have already been crossed: solar photovoltaics and wind power globally, and electric vehicles, battery storage and heat pumps in leading markets.[3] These follow the same mathematics. Deployment lowers cost, lower cost drives deployment, and past a crossover the transition becomes self-sustaining and very hard to reverse — the S-curve, which is a tipping point seen from the happy side.

Figure 7 · A tipping point you would want to cross Schematic. The adoption curve of a technology that gets cheaper as it scales has the same shape as a climate threshold — slow, then abrupt, then irreversible — with the sign flipped.

Deliberately schematic. I have not put numbers on these axes because I did not verify adoption figures to the standard the rest of this series uses, and a curve with invented numbers on it would undo the point of the series. The claim being illustrated is structural, and it is sourced: the 2025 assessment reports these positive thresholds as crossed.[3]

XII. The bottom line, for all three parts

The argument these three essays make has a shape, and it is worth stating plainly now that it is complete.

Part I asked whether the climate is speeding up, and found that for the quantities that integrate energy the answer is yes. Part II put error bars on that and found the case strong for the energy budget, the ocean, sea level and land ice, weaker for surface temperature, and inapplicable to sea ice. Part III is about the systems that method could not see — and the honest summary is that we can measure the rate of approach far better than we can locate what is being approached.

That asymmetry is the thing I did not appreciate before writing these. The acceleration numbers in Part II are among the best-constrained quantities in observational climate science: two independent instruments, agreeing, on a growing energy imbalance. The threshold numbers in Part III are, with the single exception of coral, either statistical inferences with a documented false-positive problem or model projections with ranges spanning centuries. We know the speed much better than we know the distance to the wall.

We can measure how fast we are going far better than we can see what we are approaching.

Two conclusions follow, and they pull in opposite directions, which is why this is hard.

The first is a caution against overclaiming. “Tipping point” has become the most rhetorically abused phrase in climate communication, applied to gradual processes, to model outputs, to things that turn out on inspection to be neither abrupt nor irreversible. Permafrost was on the list for twenty years and came off it. The most alarming West Antarctic mechanism weakened under scrutiny. The AMOC headline that circulated most widely rests on a method whose own authors’ critics say cannot bear the weight. Anyone who tells you confidently when a tipping element will go is telling you something the evidence does not support.

The second is that this is not reassuring, and reading it as reassurance would be a mistake of the same kind. Coral has already gone. Antarctic sea ice shifted state within the working life of the satellites watching it, and showed both predicted warning signals while doing so. Greenland shows critical slowing down. Three-quarters of the Amazon is losing resilience. The uncertainty here is not the comfortable kind that means “probably fine.” It is the kind that means we are driving at an accelerating speed towards a wall whose distance we can only estimate to within a factor of ten — and the only variable we control is the accelerator.

Which returns us, as every one of these essays has, to the same unglamorous place. We cannot yet see the edges clearly. We can measure, precisely and with two largely independent instruments, how fast we are closing on them. That is enough information to slow down.

A postscript became Part IV. Everything above rests on a distinction the three essays kept making without ever explaining: some numbers are measurements, some are model output, and most are a mixture. A reader asked what kind of modelling actually sits underneath all this. The answer turned out to be long enough for its own essay — Part IV: The Grid and the Cloud, on where the numbers come from and how much of each one is machinery.

References

1. Scheffer, M. et al. (2009). “Early-warning signals for critical transitions.” Nature 461, 53–59. The founding statement of the variance-and-autocorrelation method. nature.com
2. ICRI / NOAA Coral Reef Watch (2025). “84% of the world’s coral reefs impacted in the most intense global coral bleaching event ever.” Fourth global bleaching event, January 2023 – March 2025. icriforum.org
3. Lenton, T. M. et al. (2025). Global Tipping Points Report 2025. University of Exeter, Potsdam Institute for Climate Impact Research, WWF and partners; 160+ scientists, 87 institutions, 23 countries. Source of the coral threshold assessment and the positive tipping points. noc.ac.uk
4. National Snow and Ice Data Center (2025). Antarctic and Arctic sea ice analyses for 2025. nsidc.org
5. Hobbs, W. et al. (2024). “Observational Evidence for a Regime Shift in Summer Antarctic Sea Ice.” Journal of Climate 37, 2263–2275. Summer standard deviation 0.31 (1979–2006) to 0.76 million km² (2007–22), with lengthened sea ice memory. ametsoc.org
6. Kusahara, K. et al. (2025). “Causes of the Abrupt and Sustained 2016–2023 Antarctic Sea-Ice Decline: A Sea Ice–Ocean Model Perspective.” Geophysical Research Letters. agupubs.onlinelibrary.wiley.com
7. Ditlevsen, P. & Ditlevsen, S. (2023). “Warning of a forthcoming collapse of the Atlantic meridional overturning circulation.” Nature Communications. Collapse estimated between 2025 and 2095 with 95% confidence, central estimate 2057. nature.com
8. van Westen, R. M., Kliphuis, M. & Dijkstra, H. A. (2024). “Physics-based early warning signal shows that AMOC is on tipping course.” Science Advances 10, eadk1189. science.org
9. McCarthy, G. D. et al. (2025). “Signal and Noise in the Atlantic Meridional Overturning Circulation at 26°N.” Geophysical Research Letters. RAPID array weakening of 1.0 [0.4–1.6] Sv per decade, 2004–2023. Note that other analyses (e.g. Worthington et al. 2021; Terhaar et al. 2025) hold that no robust long-term trend is yet extractable. agupubs.onlinelibrary.wiley.com
10. IPCC (2021). Sixth Assessment Report, Working Group I: The Physical Science Basis. AMOC collapse before 2100 assessed as very unlikely (medium confidence). ipcc.ch
11. Ben-Yami, M. et al. (2024). “Uncertainties too large to predict tipping times of major Earth system components from historical data.” Science Advances, doi:10.1126/sciadv.adl4841. science.org
12. Boers, N. & Rypdal, M. (2021). “Critical slowing down suggests that the western Greenland Ice Sheet is close to a tipping point.” PNAS 118, e2024192118. pnas.org
13. Morlighem, M. et al. (2024). “The West Antarctic Ice Sheet may not be vulnerable to marine ice cliff instability during the 21st century.” Science Advances, doi:10.1126/sciadv.ado7794. science.org
14. Armstrong McKay, D. et al. (2022). “Exceeding 1.5 °C global warming could trigger multiple climate tipping points.” Science 377, eabn7950. Sixteen tipping elements, with thresholds and timescales. science.org
15. Boulton, C. A., Lenton, T. M. & Boers, N. (2022). “Pronounced loss of Amazon rainforest resilience since the early 2000s.” Nature Climate Change 12, 271–278. nature.com
16. “Observation-inferred resilience loss of the Amazon rainforest possibly due to internal climate variability.” Earth System Dynamics 15, 913 (2024). esd.copernicus.org
17. Brovkin, V. et al. (2025), “Permafrost thaw: gradual change or climate tipping point?” Max Planck Institute for Meteorology; and Nitzbon, J. et al., perspective on permafrost uncertainties and global tipping points, One Earth (2025). mpimet.mpg.de
18. Jäger, G. & Füllsack, M. (2019). “Systematically false positives in early warning signal analysis.” PLOS ONE 14, e0211072. journals.plos.org
19. Lenton, T. M. et al. (2012). “Early warning of climate tipping points from critical slowing down: comparing methods to improve robustness.” Philosophical Transactions of the Royal Society A 370, 1185. On detrending, window length and filtering choices. royalsocietypublishing.org

Published September 2026 · Part III of The Second Derivative, a four-part series on climate acceleration · Read Part I, Part II and Part IV.

On method and tools

This article was researched and written collaboratively with Claude (Anthropic): human specification and critical review, machine research synthesis and drafting, iterative refinement through structured dialogue. The research phase involved live searches of the peer-reviewed literature on critical slowing down and tipping elements, the 2025 Global Tipping Points Report, NSIDC, ICRI/NOAA, and the AMOC observational and statistical literature on both sides of the current disagreement.

All seven figures are hand-built SVG. Figures 1, 2 and 7 are schematic and labelled as such; Figure 6 shows order-of-magnitude ranges compiled from the assessment literature rather than projections. Figures 3, 4 and 5 plot published values directly. The observed / inferred / projected tags in Section II are my own classification, applied consistently and stated as a rule before the evidence rather than after it.

Part I checked a viral explainer against the science. Part II showed the working behind its central claim. Part III is the part I was least confident writing, because the honest answer to “where are the thresholds?” is that we largely do not know — and saying so clearly matters more than sounding certain.

Authored by: Luis Matos Ferreira
Physicist & Developer