Lead, Don’t Lag

Koutsoyiannis, Math. Biosci. Eng. 2024

Temperature leads.
Carbon dioxide follows.

Recreated figures and tables from Demetris Koutsoyiannis’s stochastic causality paper — using the same public reconstructions and instrumental records. At every time span from 500 million years to the last 70, the data say the same thing the “CO₂ drives climate” slogan does not: T → [CO₂], not the reverse.

Walk the recordJump to Table 1
Time spans
5 eras
Causal direction
T → CO₂
Modern lag
~6–12 mo

Graphical abstract · Table 1

Characteristic lag scales with the window

Median IRF lag h½ versus analysis time scale. Longer records resolve slower biosphere and ocean mechanisms — they do not reverse who leads.

Log–log recreation of the paper’s summary figure. Each point is one row of Table 1 (causal system T → [CO₂], L = 0).

01

The slogan

Humans emit CO₂, it accumulates, the greenhouse thickens, temperature follows. Simple. The climate system is not.

02

The test

If [CO₂] caused T, the impulse response would live on negative lags of T → [CO₂] — or the reverse system would win. It does not, on any span the paper can resolve.

03

The result

Change in temperature leads; change in carbon dioxide concentration lags. Unidirectional. Proxy and instrumental. Annual to million-year.

Figure 1–2 · 541 million years

The deep record does not anoint CO₂ as the climate driver

Over the Phanerozoic the continents rearrange, solar output rises ~5%, cosmic-ray flux cycles, and the biosphere invents forests, then flowers, then us. CO₂ reconstructions swing from a few hundred to several thousand ppm — and the four temperature curves do not even agree before ~350 Ma. That is not a precision thermometer with a CO₂ knob. It is a noisy geological scrapbook.
Pangaea · ~200 Ma

Pangaea · ~200 Ma

Mesozoic supercontinent from the paper’s Figure 1 (Wikimedia, CC BY-SA 4.0). Land–sea geography is itself a climate boundary condition — independent of the modern CO₂ narrative.

Eocene · ~40 Ma

Eocene · ~40 Ma

Continents near modern positions, still no Antarctic ice sheet of today’s size. From the paper’s Figure 1 (Wikimedia, CC BY-SA 4.0).

Cm
O
S
D
C
P
Tr
J
K
Pg
Recreation of Figure 2. Temperature reconstructions: Scotese et al. 2021 (digitized GAT), Shaviv et al. 2023, Grossman & Joachimski 2022 (tropics −5 °C), Song et al. 2019. [CO₂]: Royer 2014 / Davis 2017, Foster et al. 2017 / Song et al. 2019, Berner 2008 GEOCARB. Time runs toward the present on the right, as in the paper. Diamonds on the bar: extinctions; discs: creation events.
Ma BPEventKind
485Ordovician–Silurian extinction · ~70% genus lossextinction
372Late Devonian extinction · ~35% genus lossextinction
252Permian–Triassic extinction · ~56% genus lossextinction
201Triassic–Jurassic extinction · ~43% genus lossextinction
66Cretaceous–Paleogene extinction · ~40% genus lossextinction
530Animals on land / fishcreation
430Land plants, insectscreation
360Amphibianscreation
230Dinosaurs, first mammalscreation
140Flowering plantscreation
60Primatescreation
0.3Homo sapienscreation

Figure 3 · 66 million years

After the dinosaurs: a long cooling with CO₂ in tow

Westerhold’s CENOGRID benthic δ¹⁸O stack is the best-dated deep-time climate series we have. Rae et al. compile alkenone and boron-isotope CO₂. The paper converts δ¹⁸O with the Epstein formula only for a visual scale; the causality math uses −δ¹⁸O itself, interpolated to a 100-kyr step. Resolution is better than the Phanerozoic. The story does not flip.
Pal
Eo
Olig
Mio
Recreation of Figure 3. δ¹⁸O (Westerhold et al. 2020, Table S34, ISOBENd18oLOESSsmooth) with a temperature scale, and [CO₂] from Rae et al. 2021 (alkenone 84th percentile merged with boron). Time’s arrow toward today on the right. Marks on the bar: K–Pg, PETM, Antarctic glaciation, Quaternary ice.

Figure 4 · Vostok ice core · 420 kyr

Four glacial cycles: temperature turns first

This is the figure climate presentations love to show — usually with the axes cropped and the lag unmentioned. Petit et al.’s Vostok cores carry both deuterium temperature and trapped-air CO₂. Air is younger than the enclosing ice by ~2,000 years in warm periods and ~6,000 in the coldest (Barnola). The public files already use gas age for CO₂. Even after that correction, CO₂ still lags.
Recreation of Figure 4 from the paper’s plotted Vostok series (Petit et al. 1999 / Jouzel temperature; Barnola / Petit CO₂). Sources cited: CDIAC vostok.1999.temp.dat and vostok.icecore.co2. 400 kyr on the left, present on the right.

Figure 5 · last two millennia

The hockey-stick era is the noisiest, not the cleanest

The last 2,000 years are where the political fight is, and the reconstructions show it. Moberg, Loehle & McCulloch, and Christiansen & Ljungqvist do not draw the same climate. Antarctic ice cores draw a flat CO₂ line until ~1800; stomatal densities (Kouwenberg, van Hoof) do not. Ice is a diffusion filter. Needles remember centuries the ice forgets.
Recreation of Figure 5 (upper). Loehle & McCulloch from the Ohio State tabulated file used in the paper; Moberg from NOAA WDC Paleo; Christiansen & Ljungqvist digitized from the paper’s figure. Anomalies relative to each series’ own mean.
Recreation of Figure 5 (lower). Law Dome CO₂ spline (MacFarling Meure / Etheridge, NOAA) plus the paper’s ice-core and van Hoof stomata traces. Stomata vary; the ice-core line is the one that gets printed as ‘pre-industrial 280 ppm’.

Figure 6 · instrumental era

Even the Keeling curve lags the thermometer

Systematic [CO₂] starts in 1958 at Mauna Loa and 1957 at the South Pole. Temperature reanalyses go earlier; the paper pairs ERA5 with both stations. The popular move is to overlay the two rising lines and declare proof. Once you difference out the shared trend and the seasonal cycle, temperature still leads by about half a year — and the lead is cleaner at the decadal scale than at the annual scale.
Recreation of Figure 6 (upper). ERA5 traces extracted from the paper (KNMI Climate Explorer aggregation). HadCRUT5 is shown as an independent check: the Met Office global monthly anomaly, offset to a 14.35 °C absolute so it sits on the same axis — not a claim that 14.35 °C is a sacred baseline.
Recreation of Figure 6 (lower). Mauna Loa monthly in situ (NOAA GML, through 2026) and South Pole flask. Scripps / NOAA sampling stations as in the paper. Seasonal wiggles are the land biosphere breathing — 96% of the flux — not a factory stack.

Figures 7–9 · Sections 2.6 & 3

Do not splice millennia onto months, and do not pretend Granger is physics

Two methodological points carry the skeptical load. First: mixing time scales is how political graphics are made. Second: climate is a stochastic, long-memory system. Impulse-response identification on that system is what the paper adds; differencing is how it stops two trendy lines from impersonating a mechanism.

Figure 9 · what the IRF is allowed to mean

Four shapes. Only one of them is a driver.

An impulse response g(h) says how [CO₂] reacts, h steps after a kick in T. The adverb in the paper is “potentially”: these shapes are necessary, not sufficient. The dashed mark is the mean lag μh.

Potentially causal

h = 0

g lives on h ≥ 0. Temperature now, CO₂ later. This is every row of Table 1 at L = 0.

Potentially mixed

h = 0

Mass on both sides, more on the positive. Time order still leans T first.

Hen-or-egg

h = 0

Symmetric around zero. Two exceptions in Table 1 still have positive characteristic lags.

Potentially anticausal

h = 0

g lives on h < 0. This is what [CO₂] → T would need. Figure 22 looks like this, and e collapses.

Three questions the paper actually asks

  1. Is CO₂ a climate driver, or an internal state variable of a much larger system?
  2. With ~5% of the greenhouse effect (Koutsoyiannis 2024 preprint on H₂O vs CO₂) and ~4% of annual carbon emissions, can the human term dominate the 96% biosphere term?
  3. When you stop using deterministic “the curve went up” reasoning and estimate a transfer function, which way does it point?

Models, and what they are not

GEOCARB / GEOCARBSULF are box models of weathering and burial, not weather simulators. CENOGRID is a dated δ¹⁸O splice. ERA5 is a reanalysis. Foster 2017 is a proxy ensemble with a LOESS. None of them is the causality test. The test is the IRF on the observations those products emit. Fetch status and original URLs live in Sources & methods.

  • GEOCARB III · Long-term carbon-cycle model

    Berner & Kothavala 2001, Am. J. Sci.

  • GEOCARBSULF + volcanic addendum · Coupled C–S cycle model

    Berner 2006 GCA; Berner 2008 AJS

  • GEOCARBSULF error analysis · Monte Carlo on GEOCARBSULF

    Royer, Donnadieu, Park, Kowalczyk & Goddéris 2014, Am. J. Sci.

  • Foster et al. 2017 compilation · Proxy ensemble + LOESS

    Foster, Royer & Lunt, Nat. Commun. 8:14845

  • CENOGRID · Astronomically tuned benthic splice

    Westerhold et al. 2020, Science; PANGAEA 917503 / 917717

  • Epstein δ¹⁸O–T · Visual temperature scale

    Epstein et al. 1953; as applied in Koutsoyiannis §2.3

  • ERA5 · Atmospheric reanalysis

    Hersbach et al., Copernicus CDS; paper via KNMI Climate Explorer

  • HadCRUT5 · In situ temperature analysis

    Morice et al. 2021, JGR Atmospheres; HadCRUT.5.1.0.0

  • IRF stochastic identification · Transfer-function causality

    Koutsoyiannis 2020–2024; this paper §3 and Appendix B

Figures 10–22 · the actual test

Impulse responses: g lives on the positive side of zero

Figures 11–22 in the paper are the same two-panel plot, over and over, on every pairing: estimated g(h) for several lower lags L, and explained variance as a function of L. Causal means e peaks at L = 0 and g sits on h ≥ 0. The reverse system (Figure 22) is the control. It fails.

Vostok original · 416 kyr

Table 1 · every test, one direction

The scoreboard: causal T → [CO₂] beats the reverse in every row

Explained variance of the causal system (L = 0) versus the anticausal system (L = −20). Time unit is years. O = original series, D = differenced. The two HOE (hen-or-egg) footnotes still have positive lags: temperature first, CO₂ later.
CaseO/DSpanScalel, me causale antihch½μh
PhanerozoicO491 Myr1 Myr1, 10.620.613.0 Myr1.2 Myr4.0 Myr
PhanerozoicD490 Myr1 Myr1, 10.270.192.0 Myr2.3 Myr6.4 Myr
CenozoicO66.8 Myr100 kyr1, 10.800.78400 kyr760 kyr910 kyr
Late QuaternaryO416 kyr1 kyr1, 10.840.711 kyr1 kyr5 kyr
Late QuaternaryO200.5 kyr500 yr1, 10.890.722 kyr1 kyr3 kyr
Late QuaternaryD200 kyr500 yr1, 10.220.150 mo1 kyr2 kyr
Common EraO1711 yr1 yr1, 40.490.4135 yr25 yr33 yr
Common EraO1701 yr10 yr1/10, 40.490.4035 yr26 yr33 yr
Common EraD1700 yr10 yr1/10, 40.110.0535 yr21 yr30 yr
Modern / Mauna LoaD63 yr1 yr1/12, 10.430.077 mo7 mo8 mo
Modern / Mauna LoaD54 yr10 yr1/120, 10.490.387 mo5 mo10 mo
Modern / Mauna LoaD54 yr10 yr1/120, 40.500.088 mo3.1 yr3.2 yr
Modern / South PoleD65 yr1 yr1/12, 10.300.037 mo8 mo9 mo
Modern / South PoleD56 yr10 yr1/120, 10.540.4310 mo1.2 yr10 mo
Modern / South PoleD56 yr10 yr1/120, 40.540.1710 mo3.3 yr3.3 yr

Source: Koutsoyiannis 2024, Table 1. Causal lags generally maximize explained variance. Exceptions: (a) Phanerozoic original, L = −14, h½ = 3.7 Myr, μh = 0; (b) Late Quaternary differenced, L = −1, h½ = 850 yr, μh = 1,970 yr. Both still have CO₂ lagging.

Phanerozoic

0.62 vs 0.61

Even the messiest, million-year proxies refuse [CO₂] → T.

Vostok original

0.84 vs 0.71

The textbook ice-core figure, once you stop hiding the lag.

Mauna Loa annual

0.43 vs 0.07

Modern instruments, differenced: anticausal explained variance collapses.

Sections 5–6 · what the figures are for

A complex planet, a simple slogan, and which one the data match

The paper’s closing argument is not “CO₂ does nothing.” It is that the climatic system is large, wet, alive, and old, and that the last-decade habit of treating a trace gas as the master variable is a political compression of that system — one the transfer functions refuse to endorse.

Physical mechanisms that put T first

  • Oceans. Henry’s law: warmer water holds less CO₂. Outgassing follows the heat content, not the other way around.
  • Soils and microbes. Cool periods trap carbon because degradation slows more than photosynthesis. Warm periods release it faster than plants can take it back. Bacterial and thermophilic decay are temperature-gated.
  • Respiration. Every aerobic organism exhales CO₂. Creation and extinction events — marked on Figures 2 and 3 — change that term on geological time.
  • Orbital geometry. For the late Quaternary the pacemaker is Milanković. CO₂ is a passenger with a delayed ticket.

What the narrative needs, and lacks

  • A demonstrated [CO₂] → T transfer function in observations. Not found here, not found in the 2020–2023 predecessor papers, not found when the same method is pointed at models (the models then invert reality).
  • Evidence that a ~4% anthropogenic share of annual carbon flux reversed a direction that holds from the Cambrian through 2024. The paper’s answer: perhaps — but no observational analysis has shown it. The claim lives in GCMs.
  • A greenhouse story in which water vapor, ~20× CO₂ by optical depth in Koutsoyiannis’s accounting, is a slave and CO₂ is the master. The hydrological cycle is not a footnote.
  • Permission to splice paleo averages onto monthly data. Figure 7 exists because that permission is routinely granted in public communication and almost never in the laboratory.
“No real-world, data-based, evidence has ever been provided in support of that popular speculation. Here evidence of the opposite is provided.”
Koutsoyiannis 2024, §4.3, on the claim that the modern decadal scale reverses T → [CO₂]

Data availability · sources & methods

Every series here is public. None of it is new. Some of it hid.

The paper uses no original measurements. This recreation tried the URLs named in §2, plus NOAA, PANGAEA, Met Office, and PMC mirrors when the first path failed. Failures are listed with the original link. Axis calibration of paywalled paleo panels is taken from the published plots themselves.

Fetched on first try

33

NOAA GML, HadCRUT5, Vostok NCEI, Moberg, Law Dome 2006, GEOCARB III, the paper itself.

Recovered on retry

7

CENOGRID S34, Royer 2014, Christiansen NOAA, van Hoof EuropePMC, Taylor Dome, Berner 2008.

Still paywalled

4

Shaviv 2023, Grossman 2022, Song 2019 S1, Davis 2017. Curves read from the paper’s figures.

Timed out / dead

5

Scripps Keeling, NASA GISS, CDIAC FTP (530), KNMI ERA5 dump, Etheridge 1996 NOAA path.

Models in play

Two families. Carbon-cycle boxes and proxy stacks reconstruct the deep past. Reanalyses and station files cover the instrumental era. The causality test is not a GCM — it is an impulse-response identification on those series.

GEOCARB III

Fetched

Long-term carbon-cycle model

Phanerozoic [CO₂] dashed curve in Figure 2. Weathering, degassing, and burial on a ~10 Myr step. Not a GCM.

Berner & Kothavala 2001, Am. J. Sci.

GEOCARBSULF + volcanic addendum

Recovered

Coupled C–S cycle model

Paper cites Berner 2008. 2006 Geochimica PDF is Elsevier; 2008 volcanic-weathering AJS HTML fetched. 2009 AJS O₂ addendum 404 on the Yale path.

Berner 2006 GCA; Berner 2008 AJS

Foster et al. 2017 compilation

Digitized

Proxy ensemble + LOESS

Phanerozoic [CO₂] in Figure 2. Springer supplementary xlsx 403. PMC HTML fetched; NOAA Paleo-pCO2 holds the 2012 boron file, not the 2017 LOESS product. Plotted curve digitized from the paper.

Foster, Royer & Lunt, Nat. Commun. 8:14845

CENOGRID

Recovered

Astronomically tuned benthic splice

Cenozoic −δ¹⁸O. Science Table S34 xlsx 403. PANGAEA parent ?format=html 400. Bare DOI + S34 LOESS textfile (2.3 MB) and MARUM Tables_S8_S34.xlsx recovered. Paper uses ISOBENd18oLOESSsmooth on a 100-kyr step.

Westerhold et al. 2020, Science; PANGAEA 917503 / 917717

IRF stochastic identification

Fetched

Transfer-function causality

The actual test. Table 1 is the scoreboard. Characteristic lags grow with window length because g has a heavy tail (Hurst–Kolmogorov), not because the direction flips.

Koutsoyiannis 2020–2024; this paper §3 and Appendix B

How the series were treated

No splice

Eras stay on their own clocks. The app never glues monthly Keeling onto ice-core millennia. That is the paper’s Figure 7 argument, enforced in the layout.

Figure digitization

Where a named source is paywalled or the tabulated supplement 403s, the plotted series is an axis-calibrated reading of the paper’s own figures (Koutsoyiannis 2024 Figs. 1–6). That is a recreation of what the paper used, not a new compilation.

IRF estimator (the paper)

Treat [CO₂] as the output of a linear filter driven by temperature. Estimate impulse response gⱼ. Potential causality (T → CO₂) puts g on positive lags; explained variance e peaks at computational lag L = 0; the reverse system is anticausal or weak. Characteristic lags: h_c (max cross-correlation), h½ (median of |g|), μ_h (mean of |g|).

Lag scan (this app)

Each era panel runs a transparent Pearson cross-correlation on z-scored, interpolated pairs, optionally first-differenced. Positive lag means temperature leads. The IRF gallery (Figs. 11–22) is FIR least squares with ridge, not the paper’s full §3 identifier; Table 1 numbers are transcribed from the paper.

Ice-core gas age

Vostok CO₂ is plotted on gas age (Barnola), not ice age. Enclosure lag is ~2 kyr in warm periods and ~6 kyr in the coldest. Even after that correction, CO₂ still lags deuterium temperature.

Fetch ledger

Original URL first. If it failed, a public mirror. If that failed, the paper’s figure. Status is about the file, not about the science.

The paper

  • Koutsoyiannis 2024 PDFFetched

    Stochastic assessment of temperature–CO₂ causal relationship. Math. Biosci. Eng. 21(7): 6560–6602.

    Used: all figures · Table 1 transcribed · Open access, CC BY 4.0. 1.56 MB from NTUA ITIA.

  • AIMS Press HTMLFetched

    Journal HTML of the same article.

    Used: methods text

  • ITIA supplement & SummaryLagsFetched

    Appendices plus the graphical-abstract lag figure.

    Used: Table 1, hero scatter · Supplement PDF 354 kB; SummaryLags.jpg 830 kB.

Modern [CO₂]

Modern temperature

Late Quaternary · Vostok

Common Era temperature

  • Moberg et al. 2005 NOAA WDC PaleoFetched

    nhtemp-moberg2005.txt, 1–1979 AD.

    Used: Figure 5 · Alt paleocean/ path 404. Study page 6267 is a thin HTML shell.

  • Loehle & McCulloch 2008 tabulatedFetched

    Non-tree-ring reconstruction, Ohio State file.

    Used: Figure 5 · Loehle 2007 Energy & Environment PDF: NCASI 403, Sage 403 (Cloudflare).

  • Christiansen & Ljungqvist 2012Recovered

    Climate of the Past extra-tropical NH reconstruction.

    Used: Figure 5 (paper figure + NOAA file archived) · Paper PDF fetched. Copernicus supplement PDF/ZIP 404. NOAA WDC Paleo 2012-049 recovered (144 kB). Bolin Centre page is HTML only.

Common Era [CO₂]

  • Law Dome 2000-yr (MacFarling Meure 2006 NOAA)Fetched

    law2006.txt spline used in Figure 5.

    Used: Figure 5 · lawdome-co2.txt and law1996.txt 404. Rubino 2019 CSIRO DOI resolves to a landing page, not a table.

  • van Hoof et al. 2008 stomatal CO₂Recovered

    PNAS; Table S1 in the supplement.

    Used: Figure 5 (curve digitized from the paper) · PNAS PDF/HTML 403. EuropePMC PDF recovered (414 kB). PNAS SI PDF still 403. Plotted points remain a figure reading.

  • Indermühle et al. 1999 Taylor DomeRecovered

    Holocene CO₂, NOAA Taylor Dome directory.

    Used: Figure 5 mixed ice-core series (digitized); NOAA files archived · Guessed path co2-taylor.txt 404. Directory listing revealed taylor_co2-holocene.txt, taylor_co2-latequat.txt, taylor_co2-glacial.txt — all fetched.

Cenozoic

  • Westerhold CENOGRID S34 LOESSRecovered

    PANGAEA 917717, ISOBENd18oLOESSsmooth.

    Used: Figure 3 T (paper calibration); 2.3 MB archived · doi.pangaea.de/…917503?format=html 400. Science aba6853_table_s34.xlsx 403. MARUM Tables_S8_S34.xlsx (6.5 MB) also fetched. Parent collection HTML recovered on the bare DOI.

  • Rae et al. 2021 Cenozoic CO₂Fetched

    Alkenone + boron; PANGAEA 934290.

    Used: Figure 3 [CO₂] · Annual Reviews supplement 403 (Cloudflare). p-co2.org landing fetched.

Phanerozoic [CO₂]

  • Foster, Royer & Lunt 2017Digitized

    Nat. Commun. 8:14845, last 420 Myr.

    Used: Figure 2 · Nature HTML fetched. Springer ESM xlsx/zip 403. PMC HTML fetched; PMC bin/ncomms14845-s2.xlsx 404. NOAA Paleo-pCO2 boron_isotopes_foster_2012.xlsx recovered — not the 2017 LOESS product.

  • Royer 2014 / Davis 2017 compilationRecovered

    AJS GEOCARBSULF error analysis; Climate 5(4) 76.

    Used: Figure 2 · Yale earth.geology.yale.edu/~ajs/2014/01.Royer.pdf 404. AJS attachment recovered. Davis 2017 MDPI /pdf and /htm 403 Access Denied.

  • Berner GEOCARB III (2001)Fetched

    Yale AJS PDF of the tabulated carbon-cycle model.

    Used: Figure 2 dashed · GEOCARBSULF 2006 Yale path 404. Berner 2008 volcanic AJS HTML fetched.

Phanerozoic temperature

Open-access paper under CC BY 4.0. Paleogeography in Figure 1 from Wikimedia (Pangaea 200 Ma; 40 Ma paleoglobe), CC BY-SA 4.0, as cited. This explorer is a reading of the paper, not a substitute for it.