The IPCC Math Deniers Every headline climate claim is proven false by simple mathematics applied to the data the IPCC itself certifies. These are not disagreements. They are lies. By Jonathan Cohler Mathematics is absolute. A theorem, once proved, cannot be falsified by new data. You accept it, you find a false premise, or you find a broken step. Nothing else works. Galileo called mathematics the language in which God wrote the universe. Denying a theorem is not disagreement. It is lying, and it is illiteracy. The United Nations climate reports include some real measurements almost no one disputes: thermometer readings, carbon isotope records running back five centuries, ice-core samples, radiocarbon from mid-century bomb tests. The reports accept these inputs. Simple mathematics applied to those same inputs proves the IPCC's headline conclusions false, and policies on the scale of trillions of dollars are built on those false conclusions anyway. When the arithmetic is pointed out, the arithmetic is ignored. Every one of the six public claims examined below (the warming target, the planetary energy imbalance, the carbon budget, the fossil-fuel fingerprint, the long atmospheric lifetime of CO₂, and the extreme-event attribution ratios) is proven false by junior-high-school math on the data the IPCC itself certifies. That is math denial, and in plain language it is lying. "Global Temperature" Is Not Real A thermometer measures the air touching the bulb. It does not measure the Earth. Temperature is what physicists call an intensive quantity: it belongs to a specific place and does not add. Two rooms at seventy degrees are not a single room at a hundred and forty. Pour a cup of coffee at ninety degrees into another cup at ninety degrees and you have two cups of coffee at ninety degrees. The number reported each year as the global temperature is an area-weighted average of readings from a few thousand surface weather stations. Do the math. The Earth's surface is about 510 million square kilometres. Each thermometer physically samples the air immediately around its bulb, a volume on the order of one cubic metre. A few thousand thermometer bulbs times one square metre of sampled area each is on the order of one part in a hundred billion of the Earth's surface. The stations are also distributed nothing like uniformly: dense across the United States and Europe, sparse to absent across the oceans (70 percent of the planet), central Africa, interior Asia, and the polar regions. From that vanishingly small and geographically lopsided sample, a single global number is announced each year to the tenth of a degree. Change the weighting recipe (by area, by mass, by solar flux, by land versus ocean, at the surface or at two metres) and the number changes with it. Thermodynamics supplies no equation that picks a correct weighting, because none of them yields a temperature. The Physical Tether Theorem proves it formally: an area-weighted mean of intensive readings has no physical referent. It is a statistic, not a measurement. The public claims proven false. "The planet has warmed 1.1°C since pre-industrial times." The 1.5°C warming limit of the Paris Agreement. The 2°C Paris target. "The hottest year on record." Every "degrees of warming" headline. Each one puts degrees on a number that is not a temperature. Every treaty, mandate, and target in international climate law uses that same false number as its central reference. The number is published. The units are published. Both are false. "Ocean Heat Content" Cannot Be Measured The reports claim the planet is gaining heat at roughly 0.7 watts per square metre of Earth's surface, with more than 90 percent of that supposedly stored in the ocean. The number comes from the Argo float program, about four thousand instruments profiling the upper 2000 metres. The whole calculation fails in exactly the same way the global-temperature calculation fails, and for exactly the same reason. Ocean heat content is built from temperature anomalies. Temperature is intensive. The Argo processing takes sparse point readings along each float's ascent, assigns them to the position where the float later surfaces (an offset of up to tens of kilometres), subtracts a climatological reference that is itself a statistical construct over 2004–2018 rather than any actual water, multiplies the resulting "anomaly" by seawater density and specific heat, and integrates over grid cells. Multiplying a non-physical anomaly by physical constants does not produce a physical energy. It produces a number with the units of energy stapled to it. The same thermodynamic theorem that rules out a global temperature rules out this scalar. There is no ocean heat content measurement here. There is a statistical construct carrying units of joules. Do the sampling math. The ocean holds about 1.3 billion cubic kilometres of water. Four thousand floats sampling a thin trail during each ten-day ascent cover a direct volume fraction on the order of one part in a billion per year. The deep ocean below 2000 metres, which is roughly half the ocean's volume and thermal energy, is essentially unsampled. The polar oceans are essentially unsampled. The CERES satellite flux data, which carries absolute uncertainty of 3 to 5 watts per square metre on its own, is then adjusted to match the Argo-derived number, so the two values "agree" by construction rather than independent confirmation. A full accounting of the real uncertainties puts the error bar at at least ±1 watt per square metre, bigger than the 0.7 central estimate itself. The claim is statistically indistinguishable from zero even before the thermodynamic objection. The public claims proven false. "Earth's energy imbalance has been quantified." "The oceans have absorbed 90 percent of excess heat." "Ocean heat content reaches a new record." These are advertised as the central empirical pillar of the greenhouse-warming case: the physical signature everything else is supposed to rest on. There is no physical signature here. There is a non-physical construct with the units of a signature attached to it. Ocean acidification is published the same way. Acidity is local. No ocean average exists in nature. One is computed anyway, and fisheries policy follows from it. The Carbon Ledger Proves Nothing Atmospheric CO₂ rises when inflow exceeds outflow. Split the inflow into human emissions and everything else. Rearrange. The ledger balances. It balances if humans drove the rise. It balances if the oceans did, or the soils, or any combination. The equation is true by construction. The textbook name is an accounting identity. A theorem of causal inference, straightforward enough to teach in an afternoon, proves that any proposition true purely by definition and conservation carries no information about cause. Zero information. Not weak evidence. None. The IPCC presents the balanced carbon ledger as evidence that fossil fuels caused the rise. It cannot be. The ledger would balance whatever the cause. The related slogan that about half of human emissions remain in the atmosphere is one divided quotient of two long sums. The ratio has fluctuated between 18 and 77 percent in individual years. The 44 percent figure cited as settled science is a smoothed average, not a measurement of how much human carbon is still airborne. The atmosphere turns over its carbon on roughly a four-year timescale. The stock from fifty years ago is almost entirely gone. The public claims proven false. "Fossil fuels caused the CO₂ rise." "The remaining carbon budget is X gigatonnes." "Net zero by 2050." Every carbon tax. Every cap-and-trade market. Every corporate net-zero pledge. Every ESG filing. Every one of them cites the balanced ledger as proof that fossil fuels caused the rise. Simple math proves the ledger cannot prove that. The Isotopes Prove the Opposite Fossil fuels carry a specific isotopic fingerprint. Carbon from coal, oil, and gas has a δ¹³C value near −28 per mil. Carbon leaving the oceans runs near −8. Measure the isotopic signature of the CO₂ actually being added to the atmosphere, and elementary two-source mixing gives the fossil-fuel fraction directly. One line of algebra. Three independent records converge on the same answer. The modern Keeling-plot intercept places the added carbon at about −13 per mil. Caribbean sclerosponges, which record ocean chemistry in their skeletons, give the same figure back five hundred years. The Law Dome ice core, the record anthropogenic-attribution proponents themselves cite, gives it a third time. Three systems. Five centuries. One number. Plug −13 into the mixing equation and the fossil-fuel fraction comes out near zero. Over those five centuries, human emissions rose from a trickle to about ten billion tonnes a year. Had fossil carbon been the dominant addition, the signature would have grown steadily lighter as the fossil share rose. It did not. The public claims proven false. "The isotopic signature proves the CO₂ rise is anthropogenic." "Humans are responsible for essentially all of the observed warming since 1850." This is the IPCC's own smoking gun. It is held up in every assessment as the clinching evidence of fossil-fuel attribution. Simple math on the records the IPCC itself relies on proves the opposite. The reports accept the data. The arithmetic the data forces, they ignore. The Thousand-Year CO₂ Lifetime Is a Fiction The nuclear bomb tests of the 1950s and 60s roughly doubled the atmospheric ratio of radiocarbon (¹⁴C) to ordinary carbon. After the Partial Test Ban Treaty of 1963 shut the source off, that excess has relaxed back toward the pre-bomb background along a single exponential curve with a time constant of 18.4 years. Fifty-four years of global measurements. R² = 99.998 percent. One line. A single exponential has exactly one time constant. A sum of exponentials with distinct time constants is a different function and cannot be made to look like a single exponential. This is proved by elementary linear algebra: the Vandermonde determinant does not vanish, and Cauchy-Schwarz forbids the equality. There is no fitting trick that evades it. The IPCC's impulse-response function for CO₂, called the Bern IRF and used to compute every Global Warming Potential the panel publishes, is exactly a sum of distinct exponentials plus a residual fraction that supposedly never decays at all. That residual has drifted from 13.7 percent in the 1995 assessment to 21.7 percent in the current one, which is already a tell: a function tracking a real physical process does not change its asymptote every ten years. The ¹⁴C anomaly and the bulk CO₂ pulse share the same removal operator. An operator-equivalence theorem carries the single-exponential result from the isotopic decay over to the mass response for CO₂ itself. What the data proves about the shape of the one, it proves about the shape of the other. The Bern IRF is mathematically incompatible with the measurement. Not approximately. Exactly. The Bern IRF is also not fit to any measurement in the first place: it is fit to the arithmetic mean of an ensemble of climate models tuned to the same thermodynamically meaningless global-temperature index. It is a model of an average of models, and the one direct empirical test of the atmospheric CO₂ impulse response rules it out. The public claims proven false. "CO₂ emitted today will warm the planet for centuries, even millennia." The entire structure of Global Warming Potentials (GWP₁₀₀, GWP₂₀) used to compare methane, nitrous oxide, and CO₂ in treaties and markets. The Social Cost of Carbon that drives federal cost-benefit analysis. The intergenerational-ethics argument for aggressive mitigation now, in which today's emissions supposedly burden children ten generations hence. Every one of these numbers depends on the long tail the bomb data proves does not exist. A small energy imbalance, below the detection threshold of the satellite instruments, is enough to drive the observed CO₂ rise through warming's ordinary effect on seawater solubility and soil respiration. No special mechanism is needed. No tuning is needed. Extreme-Event Attribution Is Made Up When a report declares a heat wave 150 times more likely because of climate change, the comparison is to a hypothetical Earth on which the industrial revolution never happened. That Earth was not observed. It cannot be observed. No duplicate planet exists on which to tally outcomes. A probability is a count of what did happen divided by a count of what could have happened. On a planet we do not have, nothing happened. The denominator is zero. Dividing by zero does not give a small number or an uncertain number; it gives nothing at all. The "150 times" is not a measurement. It is a number someone wrote down. Two attribution studies of the 2021 Pacific Northwest heat wave came in at "at least 150 times" and "at least 8 times." The intervals do not overlap. A count of a real quantity does not disagree with itself by a factor of twenty. When you divide by zero you can get any answer you like, and in practice researchers do. The public claims proven false. "Climate change made this hurricane X times stronger." "Scientists link this wildfire season to warming." "The flood could not have happened without greenhouse emissions." The entire industry of extreme-event attribution. The Loss and Damage fund established at COP27, which transfers payments from developed to developing nations on the basis of climate-attributed disasters. The insurance-industry models now writing climate risk into property premiums. The litigation underway against energy companies, in which attribution ratios are offered to juries as damages evidence. Every one of these downstream uses is a made-up number dressed as a measurement. The Lies at the Center of Climate Policy The individual measurements the IPCC uses are not in dispute. Thermometers work. Floats work. Isotope ratios are measured to astonishing precision. Smokestacks emit CO₂. What is in dispute is the mathematics layered on top, and the public claims that mathematics is supposed to support. A theorem proves an area-weighted mean of local temperatures is not a global temperature. The 1.5°C target refers to a number that is not a temperature. The same theorem kills ocean heat content: temperature anomalies averaged across the non-equilibrium ocean are not an energy, and the Argo-derived "0.7 W/m²" is a construct with units of joules attached, not a measurement. A theorem proves the balanced carbon ledger has no causal content. Net zero cannot be derived from it. Simple arithmetic on the isotope records puts the fossil share of the net CO₂ addition near zero. The attribution claim runs backwards from the data. Vandermonde and Cauchy-Schwarz prove the single-exponential bomb-¹⁴C decay is not a sum of distinct exponentials, and operator-equivalence carries that result from the ¹⁴C anomaly to the bulk CO₂ pulse. The Bern IRF, and the thousand-year persistence story that depends on it, describes a different function from the one the data draws. Attribution ratios divide by zero. The headline multipliers are made up. Every major policy-relevant claim the IPCC and the media put before the public is one of these constructions. Every one is proven false by junior-high-school math on the data the IPCC itself certifies. When math proves you wrong, you have three choices: accept it, show which premise is false, or show which step is broken. The IPCC does none of them. It publishes the measurements. It publishes the conclusions. It refuses to notice that the measurements kill the conclusions. Published statements that are untrue are called lies. There is no other word for it. The IPCC and the media are telling the public lies, and the lies are the public case for climate policy. Galileo called mathematics the language in which God wrote the universe. The IPCC refuses to read it. Elementary arithmetic has already disproved every headline the panel publishes. Every policy that cites those headlines is enforcing a proven lie. Trillions of dollars. International treaties. Carbon markets. Net-zero mandates. Loss-and-damage transfers. Courtroom damages. All of it is extracted and imposed on the strength of claims the data has killed.

Why we KNOW fossil fuels played a completely negligible role in the rise of atmospheric CO₂ in plain English 1. Carbon has a fingerprint. Carbon atoms come in two stable kinds. One is a little heavier than the other. Every source of carbon on Earth has its own mix of heavy and light atoms, and scientists measure that mix as a single number. Think of it as a fingerprint. Coal, oil, and gas have a fingerprint of minus 28. Carbon dioxide bubbling out of the ocean has a fingerprint of minus 8. Volcanoes are minus 5. Dissolving limestone is 0. A lower number means lighter carbon. These fingerprints are measured facts, and the IPCC uses the same numbers. 2. You can read the fingerprint of whatever is being added to the air. When carbon is added to the atmosphere, the atmosphere's fingerprint drifts toward the fingerprint of what was added. In 1958 Charles Keeling showed a simple way to read off the fingerprint of the added carbon directly from the air measurements. It is a graph. Plot the air's fingerprint against the amount of CO₂ in the air in a certain way and you get a straight line. Where that line meets the edge of the graph is the fingerprint of the net addition. This is not a computer model. It is bookkeeping. You count the heavy atoms, you count the light atoms, and the answer follows. Nothing about oceans, forests, or weather goes into it. 3. The fingerprint of the added carbon is minus 13. Koutsoyiannis (2024) did this with the four Scripps monitoring stations in Alaska, California, Hawaii, and the South Pole, using records from about 1978 to today. Every station gives the same answer: minus 13.2, give or take 0.2. These are the same records the IPCC relies on. Two completely separate measurements say the same thing. A Caribbean sponge that grows in yearly layers, like tree rings, records ocean carbon back to about the year 1500. Its fingerprint is about minus 13 for all five centuries, with no drift. Air trapped in Antarctic ice at Law Dome, analyzed by Köhler et al. (2006), also gives minus 13. Three different materials, three different laboratories, five hundred years, one number. 4. Simple mixing math sets the range from the modern data alone. Every carbon atom added to the air came either from fossil-fuel combustion or from some non-fossil-fuel source: the ocean, volcanoes, rock weathering, plants and soil. The fingerprint of the mix tells you how much came from each. It works like mixing paint. If you mix black and white and get a very light gray, you know there was very little black. Fossil-fuel carbon is the black, at minus 28. The ocean is the white, at minus 8. The measured mix is minus 13, much closer to the ocean than to fossil fuel. The modern record from 1978 to today does not by itself pin down the exact fingerprint of the non-fossil-fuel carbon. It could be anywhere from minus 8 (pure ocean) down to minus 13 (matching the mix exactly). Run the mixing math across that whole range and the fossil-fuel share comes out somewhere between zero and 26 percent. That is what we know from the modern data alone: not more than 26 percent, possibly zero. The exact answer comes in step 6. 5. The official claim is impossible under any assumption. The IPCC, NASA, NOAA, EPA, and Engelbeen, Hannon, and Burton (2024) say fossil-fuel combustion caused essentially all of the rise. That is close to 100 percent. "Dominant" by any normal meaning is at least 80 percent. Run the mixing math backwards and ask what the non-fossil-fuel carbon would have to look like for each of those claims to be true: - For fossil fuel to be even a bare 50 percent, the non-fossil-fuel carbon would need a fingerprint of plus 1.6. No carbon source on Earth is that heavy. - For 80 percent, it would need a fingerprint of plus 46. That is more than 20 times heavier than the heaviest source that exists. - For 90 percent, it would need plus 120, about 60 times heavier than anything on Earth. - For 100 percent, it would need to be infinitely heavy. The official claims do not fall a little short. They require carbon sources that do not exist. 6. Five hundred years of no change gives the exact answer: essentially ZERO (0% fossil-fuel sources). The modern data left one unknown: the fingerprint of the non-fossil-fuel carbon. The sponge and ice-core records supply it. In the year 1500 there was essentially no fossil-fuel burning, so the fingerprint of the added carbon then, minus 13, came entirely from non-fossil-fuel sources. That fixes the unknown at minus 13. Since 1500, fossil-fuel burning has grown about ten thousand times over. If fossil-fuel carbon were any real part of the added carbon, the fingerprint would have slid toward minus 28 as burning grew. It did not move. It is still minus 13 today. Put minus 13 for the non-fossil-fuel carbon and minus 13.2 for the modern mix into the mixing math and fossil fuel's share of the added carbon comes out at about 1 percent, which is zero within the measurement error. The range of zero to 26 percent from step 4 collapses to a single answer: essentially zero. 7. Why this is knowledge, not opinion. The conclusion rests on three things only: - counting atoms, - the standard measured fingerprints of fossil-fuel carbon and the other sources, and - the Scripps, sponge, and ice-core records. The IPCC uses all three. Anyone who accepts all three and rejects the conclusion is refusing to do arithmetic. The only way out is to throw away one of the three, and each one is a foundation of the IPCC's own work, and nobody has done so anywhere in the peer-reviewed literature. 8. What is actually happening. Burning fossil fuel does put CO₂ into the air. That is measured at the smokestack. But a CO₂ molecule stays in the air only about four years on average before the ocean or plants take it back. So fossil-fuel CO₂ never piles up. Add up all fossil-fuel emissions since 1850 with that four-year turnover and only about 4 percent of the carbon in today's air came from fossil fuel. The growth in atmospheric CO₂ since the industrial revolution came from non-fossil-fuel sources. Fossil fuel's share of that growth is essentially zero.