History is usually told through particular events: the year of a great drought, an incompetent emperor, an army that changed sides. These causes are real. But in my view most of them decide only when and in what form something happens. Famine, war and the fall of dynasties recur across civilizations because behind these particular causes a small number of mathematical relations are at work. This is a judgement to be tested. If it holds, finding those relations would tell us what kind of system human society is and how it changes.
The best known of them comes from Malthus. In 1798 he pointed out that population grows at a constant rate, each generation larger than the last by a certain percentage, while food can increase only by a fixed amount each year. The food people eat can come only from the harvest of the land, so sooner or later population outruns the number the land can feed. When it does, famine, disease and war push population back down. The research of our society's founder proposes that industrial society has a relation of the same form: the return that owners of property demand grows at a constant rate, that return can be paid only out of the income actually produced each year, and income grows more slowly. The gap between them accumulates year by year until income can no longer pay the return, and the gap can be closed only by a clearing. A clearing means that the claims property owners have accumulated are cut back: sometimes by taking a little less each year over several decades, sometimes by being written off in large part in a war or a revolution.
The two relations have the same form. Two quantities growing at different rates are tied together by one account: the food people eat can come only from the harvest; the return on property can come only from income. An equation of this kind, true by definition, is what economists call an accounting identity. When the gap reaches its limit, the system must adjust. How many years the gap takes to reach the limit can be calculated from the two growth rates; once it is reached, how the adjustment happens and who bears the cost depend on the people and institutions of the time.
But this relation alone cannot explain why population in history rose and fell again and again. Written as an equation, Malthus's argument has population grow to near the number the land can feed and stay there. The American scholar Turchin examined long-run population records for Western Europe, China, northern Vietnam and elsewhere and found that population did not settle at a single number but rose and fell every two to three hundred years around a slowly rising trend. From this he concluded that agrarian society is, mathematically, a second-order dynamical system. A dynamical system is a set of quantities that change over time and affect one another. In a first-order system, too many people immediately push population down, so population settles at an equilibrium. In a second-order system, the consequences of too many people act first on other quantities. The model in Turchin's 2009 article uses the state's fiscal resources and the intensity of internal war; the number of people in the upper class comes from Goldstone's work and is not in that model. These quantities grow with population but change with a lag; by the time they begin to push population down, population may already have passed its equilibrium, so it falls back, and rise and fall follow. In his simplest model, repeated rises and falls also need random disturbances; once internal war is added as a quantity, the model rises and falls repeatedly under some parameters even without disturbances.
I want to answer three questions. What mathematical relations operate in human society? What kind of system do they make it? How does that system change? For example, after the Industrial Revolution food grew faster than population for a long time: did the old rise and fall disappear, or move to wealth and debt? And what happens after AI?
We test the answers in two ways. One is to check them against the historical records of many societies. The other is to build a simulated world called Ekumene and use it to find which quantities are truly long-run variables. A long-run variable is one that, when changed, changes the outcome hundreds of years later, for example the level at which population settles, how much each person can eat, or whether society rises and falls again and again. A quantity that only changes the year in which something happens, or the path by which it happens, is not a long-run variable. In practice we write the relations as rules and run the simulated world repeatedly from the same starting point, each time switching off or changing a single quantity while keeping everything else the same, and see whether the long-run outcome changes. By November 2026 this kind of test had been done only for agrarian society, and only a few such tests have been run; simulation of industrial society and of AI has not yet begun.
First, the historical record. The traces of Malthus's relation are clear. By the economic historian Maddison's estimate, world income per head rose by only about half between AD 1000 and 1820. Over those eight hundred years farm tools, crops and irrigation all improved, but the extra food mostly fed more people and did not make each person better off.
Turchin's second quantity has also left traces. If society's rises and falls come from a quantity that changes with a lag behind population, then when population first begins to fall that quantity is still at its peak and the pressure on society is greatest, so unrest should be concentrated in phases of falling population, not of rising population. Turchin and the Russian scholar Nefedov counted seven cycles in England, France, Rome and Russia: in phases of rising population there were on average 0.6 outbreaks of unrest per decade; in phases of falling population, 3.8. Turchin separately counted eight unified Chinese dynasties: 3.4 and 11.3 in the two phases. In every cycle there was more unrest in the falling phase.
Records can tell us how things happened; they cannot tell us whether things would have been different without some particular element. For that we need a simulated world.
Ekumene is a simulated world we have built in software. We take three regions from real geography, Mesopotamia, North China and Western Europe, and divide each into many small hexagons; each small piece of land can produce a different amount of food. The simulation moves forward year by year. Each year the program has the farmers on every piece of land till and harvest, and then gives each person a share of food; those who are fed have children as usual, those who are not have fewer, and those who go hungry for long die of starvation. Where there is a state, it first takes part of the grain to feed its rulers; the rulers also have children. Repeating these steps fifteen hundred times gives the region fifteen hundred years of history.
We need such a world because real history happened only once. We cannot make the real Mesopotamia go through another thousand years without kings and see what happens. A simulated world can do this: run the same world twice from the same starting point, changing one rule the second time and keeping everything else exactly the same. If the two runs differ hundreds of years later, for example one collapsing again and again and the other staying steady, the difference can come only from the rule that was changed. A quantity whose change alters the long-run outcome is the long-run variable we are looking for.
Each run of the program uses a long string of random numbers, deciding which year has a good harvest and which household has one more child. The string is fixed by a number called the random seed; changing the seed is like changing one's luck. A result obtained with one seed may be a coincidence of that run. So a conclusion has to be rerun with dozens of seeds to see whether it holds each time. Most of the results below used only one seed, and that is at present their greatest weakness.
We first ran a world with no state and no war, containing only farming commoners. Population grew at first, then stayed near a certain number and grew no further. The reason is diminishing returns. On the same land, when hands are few, one more person adds a good deal of food; the more people there are, the less one more person adds, until in the end it is almost nothing. When population stopped growing, each person ate on average only about 92.5 per cent of the ration the model sets as needed. An average short of the need means that some households always eat less: in the model about forty in every thousand people starve each year, exactly offsetting the newborn, so population neither rises nor falls.
What if the yield of this land rose, for example by switching to a better crop? By the formula that summarizes the model, the extra food first lets people eat better, so they have more children and fewer starve, and population rises; it keeps rising until each person again eats only about 92.5 per cent of the ration, and then growth stops. In the end there are more people, and each person is no better off than before. This is Malthus's conclusion. We have calculated this from the formula but have not yet rerun the model with a higher yield.
Next we added the state. Each year the state takes part of the harvest, so commoners have less to eat. The result is the opposite of a higher yield: fewer children are born and more people starve, so population falls; it stops falling only when people are no longer crowded on each piece of land and each person again eats what they did before. So by the formula that summarizes the model, the larger the share the state takes, the fewer people this land feeds; but the commoners who survive each eat about as much as they would without a state. How heavy the tax is decides how many people can live, not how well the living fare.
The calculation also gives a dividing line. When people are few and land is plentiful, the harvest per head within a state is about 1.4 to 1.6 times the ration needed. As noted above, for population to hold steady each person needs about 0.925 of the ration; the part of the harvest above that is the largest share the state can take while population can still be sustained. If the state takes more than that, then even when population has fallen so far that everyone has plenty of land, what remains after tax is not enough to sustain population, which keeps falling until it disappears. With the values in the model, this line lies between 34 and 42 per cent of the harvest. These two points are likewise derived from the formula; we have not rerun the model with tax rates adjusted one by one. An earlier version of the model offers supporting evidence: each time population in Mesopotamia collapsed, the share taken by the state had already risen to 48 to 53 per cent of the harvest, above this line.
At this point population in the model still only settles at one level and does not rise and fall repeatedly. To see whether Turchin's second quantity is at work, we need an on-and-off comparison. On 25 October 2026 we made this comparison on version v17 of the model. In v17 the rulers marry and have children like commoners, so their number grows and shrinks on its own; how much grain the state takes depends on how much the rulers need. The comparison changed only one thing: the number of commoners was made to set the maximum number of rulers, and rulers above that number became commoners, with every other rule unchanged. In both cases each of the three regions, Mesopotamia, North China and Western Europe, was run for fifteen hundred years, and we counted how many collapses occurred. A collapse here means that the fifty-year average of population fell by more than 30 per cent from its previous level.
Each region run for 1,500 years, counting how often "the fifty-year average of population fell by more than 30 per cent" occurred.
25 October 2026: each case run once with the same random seed, so this is a single comparison, not yet a statistic over repeated runs.
The result: with the number of rulers growing and shrinking on its own, Mesopotamia collapsed three times, North China once, and Western Europe twice; with the number set by the number of commoners, none of the three regions collapsed even once. All other rules were the same, so the difference can come only from whether the number of rulers changed on its own. In this model the number of rulers is a long-run variable: with it, population rises and falls again and again; without it, population settles at one level. This comparison used only one random seed, so it shows only what happens under that one draw of luck, and it has to be rerun with dozens of seeds.
Below is a cell-by-cell replay of the run in this comparison in which the number of rulers in Mesopotamia grew and shrank on its own.
How to read the replay. Each hexagon on the map is a piece of land. Coloured land belongs to a state, each state shown in its own colour; grey land is under no state, and the darker it is, the more people live there. The curves on the right show that year's total population, the number of rulers, and the share of the harvest taken by the state. Drag the time bar at the bottom to around year 200 and press "Play" to see the sequence described below.
From year 100 to year 200 almost every piece of land in Mesopotamia belonged to some state, and total population was between thirteen and sixteen million. The rulers numbered about seventy thousand to a hundred and ninety thousand, only about one per cent of the population, and the state took less than a tenth of the harvest.
After that the rulers kept having children. In twenty-five years they grew from about 190,000 to about 450,000. Their food came entirely from taxes, and as their number rose so did the share the state took: from 6 per cent of the harvest to 15 per cent, and 23 per cent by year 250.
The decisive period was around year 300. Between years 300 and 324 the commoners fell from about 2.91 million to about 2.32 million, while the rulers rose from about 110,000 to about 270,000. This is because, when tax grain ran short, the rulers in the model cut their other expenses first and kept their own rations, so they still ate their fill and still had children. Commoners were already starving while the number of rulers was still rising. Only in year 328 did the rulers begin to go hungry; for several years after that they made up more than a quarter of the population. On the fifty-year average, population fell by 98 per cent this time.
The model had a rule meant to protect commoners: after tax, each piece of land had to keep on average at least 0.62 of the needed ration per person. Under the model's rules, a person who eats that much for a long time just avoids starving but almost stops having children, with fertility down to about 30 per cent of normal. The rule failed to protect commoners because it was calculated on the average of the whole piece of land. Within one piece of land, households' harvests vary; an average of 0.62 left means that about half the households are left with less than 0.62. In a separate trial calculation, a piece of land left with exactly 0.62 on average loses 16.7 per cent of its adults to starvation each year. During the collapse, 10 to 16 per cent of adults in the model starved each year, consistent with the trial calculation. In the same period fewer than one in a thousand died in war each year; this collapse was almost entirely caused by hunger.
Taken together, this run, the comparison and the two calculations show that the model reproduces two textbook conclusions: living standards are set by fertility and diminishing returns; and for population to rise and fall repeatedly, a second quantity that changes with a lag behind population is needed. But the way the model reaches these conclusions differs from history in three respects, so it cannot yet serve as evidence of which quantities were long-run variables in history.
The first is how people die. In the run with only commoners, the model of that time had no disease; about forty in a thousand starved each year, and what held population down was almost entirely hunger. In history population was held down mainly by disease and by limits on fertility such as later marriage and fewer births. Later versions added everyday disease; in r28 and r29, in the stretches from peak to trough of several Mesopotamian collapses, deaths from starvation and from disease were roughly equal. How deaths divide among causes in ordinary years has not yet been checked against the historical record.
The second is the commoners' ways out. In history, when taxes were too heavy, commoners fled or resisted, the state often fell first, and population did not keep falling to zero. Commoners in the model can also migrate and rebel, but each simulation contains only one region; people cannot leave it, and rebellion hardly reduces the tax burden, so in the end they still starve.
The third is the depth of collapse and the way it ends. On 2 November we ran a comparison to test elite overproduction with two later versions of the model, r28 and r29, which already had disease. The comparison used only six runs in one region, Mesopotamia, in which seven collapses occurred; a collapse here was judged as "commoners falling to less than half within a hundred years", a different measure from the fifty-year average above, and one fixed after the results had been seen.
Seven collapses in two later versions of the model, compared with the English record.
The English figures are taken from reading notes on Turchin and Nefedov's Secular Cycles; I have not read the book itself.
Compared with the English record, the model is similar in three respects. Before the collapses, rulers made up 0.8 to 3.6 per cent of the population; in England around 1300 the figure was about 0.7 to 1.6 per cent. In all seven collapses the peak in the number of rulers came 36 to 105 years after the peak in the number of commoners, consistent in direction with the lag Turchin describes. In six of the seven, there were more rebellions in the phase of falling population than in the phase of rising population.
There are also two differences. In the model commoners fell by 88 to 98 per cent each time; in England between 1317 and 1377 the fall was about 29 to 60 per cent, and that includes deaths from the Black Death. In the model almost everyone died of hunger and everyday disease, and the state was never once overthrown; in history the main feature of this phase was elites competing for positions and civil war. In v17 the centres of Mesopotamian states were overthrown more than a thousand times, but in that version the rulers made up more than a quarter of the population at the time of collapse, which is equally unlike history.
So the next step is not to add more comparisons but first to make the ways people die, have fewer children and escape heavy taxes in the model broadly match history. We will first check three numbers against the historical record: the composition of causes of death, the share the state takes, and the share of the upper class in the population. After that, each time a rule is added, such as disease brought by famine, marriage, or the migration of foragers, we will compare the results with and without the rule over dozens of random seeds and record whether it changes the long-run outcome.
One thing has not been done. The clearest quantitative prediction of a second-order system is that the period of rise and fall is set by how fast the two sides change: the faster population grows and the faster the second quantity responds, the more frequent the rises and falls. Turchin used a human growth rate of about 2 to 3 per cent a year to derive a rise and fall every two to three hundred years. We have not yet calculated how many years apart the rises and falls in the model are, or whether that matches the model's growth rate.
Before farming, people lived by gathering and hunting, and how many people a place could feed depended on its wild fruit, game and fish. After farming appeared, the ceiling on population was set by farmland and crops; as crops were domesticated generation after generation and harvests slowly improved, the ceiling slowly rose. With the ceiling no longer fixed, we have not yet tested whether population and ceiling still form the same kind of system, or whether they produce rises and falls of their own.
Ekumene can now start from around 8750 BC, have farming appear at several places of origin, and spread outward. In two versions of 27 October, the farming frontier across the whole map advanced 0.73 to 0.93 km a year. From the dates of the earliest farming sites, archaeologists estimate that Neolithic farming spread across Europe at 0.6 to 1.3 km a year (Pinhasi et al., 2005). The two agree, but this does not yet count as a test, because the model contains a rule by which foragers turn into farmers slowly, over generations; that rule was set after we saw that farming in the model was spreading too fast. If a model has been tuned to the answer, agreeing with the answer proves nothing.
After the Industrial Revolution, food grew faster than population for a long time. By Maddison's estimate, between 1820 and 2010 world population grew from about 1 billion to about 6.9 billion, while income per head rose about tenfold. Malthus's relation is no longer the main constraint in developed countries.
The research of our society's founder holds that rise and fall did not disappear but moved to another pair of quantities. What grows at a constant rate is now the return property owners demand: interest on deposits and government bonds, rents, dividends. Owners expect this income to grow every year, and reinvest part of it once received. What supports these returns is the income actually produced each year. Take an example: a country has an income of 100 a year and property worth 500, property earns 5 per cent a year and the economy grows 2 per cent a year. If the whole return is kept and reinvested, the number of years of income that property is worth rises by about 3 per cent a year, and the share of property income in national income rises with it, from 25 per cent to 50 per cent in about 24 years. Of course this cannot go on for ever, because after paying rent and interest workers must keep enough to live on and to raise the next generation. We call this limit the ceiling on distribution.
At the ceiling, the income owners expect can no longer be obtained from workers; owners can only take less, or lose part of their property. This is the clearing mentioned at the start. There are two kinds. One takes a little less each year, for example by letting prices rise moderately while holding interest rates down; we call it a gradual clearing. The other is heavy loss in a short time, for example through war, revolution or hyperinflation; we call it a violent clearing.
Our society's founder wrote these rules as a program that runs year by year: the number of years of income that property is worth keeps rising, a clearing occurs when it reaches the ceiling on distribution, then it rises again, and so on, with a cycle of about seventy to seventy-five years. The length of the cycle depends on only two things: how much faster property grows than income each year, and how much the previous clearing reduced property. For example, if property grows 1 percentage point a year faster than income and the previous clearing halved property, the next cycle is about 70 years; if the previous clearing removed three-quarters, about 140 years. According to this research, the record of two hundred years contains two such cycles. The first ran from 1815 to 1945 and ended with the two world wars; in the second, from 1945 to today, in the thirty years after the war, countries used gradual means to have owners take a little less each year, and changed course after the 1970s. British government debt as a share of gross domestic product rose to a peak twice in these two hundred years, and came down twice.
In my view this research and Turchin give answers that differ in one respect. In this research, rise and fall come from a ceiling and a clearing: the ratio accumulates, is pushed down when it reaches the ceiling, and accumulates again. In Turchin, rise and fall come from a second quantity that changes with a lag behind population: population rises first, the second quantity follows, and by the time it pushes population down, population has already passed its equilibrium. Both structures can produce rise and fall, but different factors decide the length of the cycle: in the first, the difference in growth rates and the depth of the clearing; in the second, how fast the two sides respond. Which kind industrial society belongs to, or whether both are present, is the question this part has to answer.
Ekumene has not reached this part yet. The program was written separately; its parameters were set by its author and not fitted to any country's data. It shows that these rules can make up a consistent process, but not that the clearings in history were actually produced by them. A more convincing test is to define in advance what counts as a clearing and then count them in the records of many countries; according to our society's founder, such a test has not yet been done.
Our society's founder has proposed a quantity called the "human equivalent", by analogy with the "TNT equivalent" of nuclear weapons: how many people's worth of intelligence large language models can produce in bulk. By his worked example, the same piece of writing work done by a machine costs about one-fiftieth of human labour; my reading is that the supply of machines can grow with computing power. In agrarian society, what is constrained is land; in industrial society, in his view, what is constrained is how income is divided. If labour is no longer the constrained quantity, between which two quantities will the next rise and fall take place? Which grows at a constant rate, which supports it, and where is the ceiling? These are all still open questions, and we have not yet begun to study them.
| Relation | Historical record | Ekumene | Not yet done |
|---|---|---|---|
| Population grows at a constant rate, food more slowly; living standards stay at subsistence | AD 1000 to 1820: world income per head rose only by about half (Maddison) | With commoners only, rations settle at 0.925 of need | The conclusions on yield and tax come from the formula, not from rerunning with changed parameters; in early versions what held population down was almost entirely hunger, in the falling stretches of several r28 and r29 collapses, hunger and everyday disease were roughly equal; the composition of causes of death has not been checked against history |
| Rise and fall in agrarian society come from a second quantity with a lag behind population | Seven Western cycles (Turchin, Nefedov) and eight Chinese dynasties (Turchin): more unrest in every falling phase | v17: when the number of rulers was no longer allowed to change on its own, collapses in the three regions fell from six to none. r28, r29: seven collapses in Mesopotamia, the peak of rulers 36 to 105 years after the peak of commoners every time | Only one seed used; collapses too deep and end the wrong way; period not calculated |
| Rise and fall in industrial society come from the gap between return and income, cleared at the ceiling | Two cycles: 1815–1945, 1945 to the present | None | Define in advance what counts as a clearing and test against the records of many countries |
| Which pair of quantities AI changes | None | None | The question itself is not yet clearly framed |