A Planet of Daisies That Regulates Its Own Temperature

Two Kinds of Flower and One Star

The objection came first, and the model was built to answer it, which is an order of events worth holding on to. Lovelock had proposed that the Earth's atmosphere looks regulated, that its composition sits far from chemical equilibrium in a way that persists, and that the biosphere is what holds it there [1][2]. The reply from evolutionary biology was immediate and reasonable: natural selection acts on individuals, individuals cannot be selected for benefits that accrue to a planet, and a self-regulating Earth would therefore need something that looks a great deal like purpose.

Watson and Lovelock's answer was a toy [3]. Put two kinds of daisy on a bare grey world: black ones absorb more sunlight than the ground and so warm the patch they sit on, white ones reflect more and cool theirs, and both grow best at the same temperature, neither of them managing anything at all on a patch that has gone too cold or too hot. No daisy has foresight. Each grows where the temperature allows and dies at a rate fixed in advance. Then turn the star up.

The claim is that the planet's temperature barely moves.

We wanted to know whether that holds in the specific sense of a number, since the 1983 paper asserts the result qualitatively and shows it as a figure, without tabulating the luminosity interval over which regulation holds, or how much temperature drift remains inside it, or how much of either survives when the parameters change. Those quantities decide whether Daisyworld is a strong argument or a picture of one, so those are the quantities we measured.

A second thing sat on our minds throughout, and it is fair to say it now: Watson and Lovelock built this model already holding the conclusion, for the express purpose of showing the conclusion was possible, which is a legitimate scientific move and a narrow one. An existence proof tells you a thing can happen in a world with the stated properties, and tells you nothing whatever about whether the world has those properties. We tried to keep that distinction visible in every section below.

What We Ran

The club owns no laboratory and no instrument of any kind. We have a room with computers in it, so our research is computation, and this study is a program we wrote and ran, whose printed output is every number in this article. Nothing here was observed in any physical system, and the word "measured" below always means measured from our own simulated steady states, in the sense that you measure the ratio of a circle by drawing one, not in the sense that you measure a planet by looking at it.

The implementation is a single Python file using numpy for arrays and nothing else. Parameters are the published ones [3], confirmed against the original text: albedos of 0.25 for black daisies, 0.75 for white, 0.50 for bare ground; a death rate of 0.3 per unit time; a growth optimum of 22.35 °C and growth falling to zero at 4.85 and 39.85 °C; a solar constant of 917 W m⁻² at luminosity 1; and a heat transport coefficient of 2.06×10⁹ K⁴.

What the model does not have makes a longer list than what it has. The model carries no spatial dimension, hence no equator, no pole, no ocean, no season and no night; it carries no atmosphere, hence no greenhouse effect, no water vapour and no cloud; and its two species hold fixed albedos and identical growth curves, with no mutation, no evolution of colour, nothing that eats a daisy, and no competition for anything except bare ground. The heat transport coefficient is a constant that does not depend on the vegetation whose heat it is transporting, and the star brightens infinitely slowly compared with the response time of the biosphere.

Any one of those omissions sinks the result as evidence about Earth. We come back to it below.

The Equations

Three areas, and they must sum to one at every instant: \(a_b\) for black daisies, \(a_w\) for white, \(a_g\) for bare ground, with nothing else on the surface to account for. Planetary albedo is the area-weighted mean of the three surface albedos,

and the planet as a whole is a black body in radiative balance with its star,

Heat moves between patches, imperfectly. A patch of albedo \(A_i\) sits at a local temperature given by

where \(q = 0\) would be a perfectly conducting planet on which every patch sits at the planetary mean, and \(q = SL/\sigma\) would be a perfectly insulating one on which each patch is in balance with the sunlight it alone absorbs. The published value of 2.06×10⁹ K⁴ sits between the two, much nearer the conducting extreme than the insulating one, and §5 shows where it comes from.

Daisies grow into bare ground at a rate that depends on their own local temperature and die at a constant rate:

with a parabolic growth curve clipped at zero,

The third equation, for bare ground, is redundant: you could write \(a_g = 1 - a_b - a_w\), never think about it again, and lose nothing the model needs. We integrated it anyway, as \(da_g/dt = -(da_b/dt + da_w/dt)\), because a redundant equation is a free test: if the integrator and the right-hand side are both correct, the sum of the three is a conserved quantity and only floating-point rounding can move it. Across 1,162,915 fourth-order Runge–Kutta steps the worst departure of \(a_b + a_w + a_g\) from 1 was 8.882×10⁻¹⁵, which is 40 times the machine epsilon of a double-precision float and is the accumulated rounding of forty million additions. Area is conserved. Whatever else is wrong with this study, the planet is not leaking.

The whole feedback is contained in one sentence of that algebra. A black daisy raises \(A\) less than a white one does, so a planet with more black daisies has a lower albedo and a higher \(T_e\); and a higher \(T_e\) moves every local temperature up, which pushes black daisies away from their optimum and white daisies towards theirs; the flowers compete for ground, and the referee of that competition is a temperature both of them are busy changing.

Working Notes From the Club Table

The Arithmetic

Four numbers in this study can be checked with a calculator, and checking them is the fastest way to see what the model is actually doing.

The bare planet at L = 1. Albedo 0.5, so it absorbs half of 917 W m⁻². Set that equal to \(\sigma T^4\), take the fourth root, \((917 \times 0.5 / 5.67032\times10^{-8})^{1/4}\), and the planet lands at \((8.0859\times10^{9})^{1/4} = 299.870\ \mathrm{K}\), or 26.720 °C. Our integrator, run with no daisies at all, returns 299.86995 K, and the difference from the closed form is 0.000×10⁰ K at every one of the 501 luminosities. Not a near miss rounded down: the two calculations produce identical doubles.

The ceiling on life. At steady state a living population needs \(a_g \beta(T) = \gamma\), so \(a_g = \gamma/\beta\); and since growth \(\beta\) can never exceed 1, bare ground can never fall below \(\gamma = 0.30\), so daisies can never cover more than 0.70 of the surface. The sweep reaches exactly 0.7000, at L = 1.480, and the lid on the stacked chart in Figure 2 is that bound, a property of the death rate rather than of the star.

Where q comes from. Watson and Lovelock write the local temperature in a linearised form, \(T_i = T_e + q'(A - A_i)\) with \(q' = 20\) K per unit of albedo. Differentiating \(T^4\) gives the quartic version we use, with \(q = 4 T_{\mathrm{opt}}^3 q'\). Work it out: \(4 \times 295.5^3 \times 20 = 4 \times 25{,}803{,}134 \times 20 = 2.0643\times10^{9}\), and the published constant of 2.06×10⁹ is that arithmetic rounded to three figures, a difference of 0.206 per cent. We kept the published value so that the replication uses the published number. The constant is not ours to improve.

The two local temperatures. Where black and white coexist, both satisfy \(a_g\beta(T) = \gamma\) with the same \(a_g\), so \(\beta(T_b) = \beta(T_w)\), and since \(\beta\) is a parabola about \(T_{\mathrm{opt}}\) the two temperatures must straddle the optimum symmetrically. Write \(T_b = T_{\mathrm{opt}} + d\) and \(T_w = T_{\mathrm{opt}} - d\). Subtract the two local-temperature equations and \(T_e\) disappears entirely:

That cubic contains no luminosity, no albedo of the planet, and no population size. Solving it gives \(d = 4.988282517\) K, hence \(\beta = 1 - (d/h)^2 = 0.918749510\) and \(a_g = \gamma/\beta = 0.326530786\), while the integrator, told none of this, returns a bare-ground fraction of 0.326530785 across the coexistence band, a difference of 1.7×10⁻⁹, with a spread of 1.3×10⁻⁴ from one end of the band to the other. Black patches sit at 27.338283 °C, white at 17.361717 °C. Neither moves when the star brightens.

The last one is worth a moment. Between L = 0.744 and L = 1.340, a range over which the bare planet would warm by more than 35 °C, the amount of bare ground on Daisyworld holds as a fixed constant to nine decimal places, and all the star does is move ground between the black column and the white column. Regulation, written as an identity rather than as a graph.

It Regulates

Figure 1 is the result. The dashed line is the control, a planet with the same star, the same albedo for bare ground, and no life: it warms from −9.23 °C at L = 0.6 to 64.11 °C at L = 1.6, a span of 73.34 °C, and it does so almost linearly, at 71.86 °C per unit of luminosity with a worst departure from a straight line of 5.15 °C. The solid line is the same planet with daisies on it.

Inside the regulated interval the planetary temperature runs from 19.158 °C to 29.579 °C, a drift of 10.421 °C, with a mean of 22.147 °C against a growth optimum of 22.35 °C. The bare planet over the identical stretch of luminosity moves through 60.236 °C. Divide one by the other and the temperature range has been compressed 5.78-fold.

That number understates the effect, since both ends of the interval are places where regulation is in the act of failing: take the middle 90 per cent of the interval, L = 0.746 to 1.512, and the drift falls to 6.488 °C against a bare span of 53.835 °C, a suppression of 8.30. Ask instead for the stretch where the planet stays within 2 °C of the daisies' optimum, and you get L = 0.816 to 1.162, a width of 0.346, inside which the drift is 3.961 °C and the suppression is 6.65.

So the honest headline carries a range rather than a single figure: Daisyworld compresses a 60 °C forcing into 10 °C of response over 85 per cent of the luminosity range we swept, and into 4 °C over the central third. Watson and Lovelock's qualitative claim survives the arithmetic.

Figure 2 shows the mechanism as a land-use chart.

Read it left to right. Below L = 0.704 the planet is too cold and nothing lives. At L = 0.704 black daisies appear, and they appear all at once: cover goes from 0.00003 to 0.69614 in a single luminosity step of 0.002, and the temperature jumps 21.489 °C while the bare planet moves by a hundredth of a degree. Between 0.744 and 1.340 both colours are present and the bare ground fraction holds at the constant we derived above; from 1.342 to 1.556 only white daisies remain, and they are working hard, with cover peaking at exactly 0.7000 at L = 1.480, the highest the death rate permits. Then, at L = 1.558, the whole biosphere goes to zero in a single step, and the temperature jumps 32.294 °C with nothing left alive to hold it down.

Both edges are discontinuities, not slopes. The hot one is the catastrophic collapse of life under gradual forcing that Ackland, Clark and Lenton identified as a characteristic and in principle observable signature of environmental feedback [11]; the cold one is the same discontinuity running the other way. A planet that regulates does not stop regulating gently.

Figure 3 is the same run seen from underneath. The planetary albedo climbs from 0.328 to 0.675 across the regulated range, which is the entire mechanism: the biosphere is a dimmer switch made of flowers. The two local temperatures, drawn against the right-hand axis, sit frozen at 27.338 and 17.362 °C through the whole coexistence band, so the daisies are not experiencing a changing climate at all: they have arranged, without arranging anything, to experience the same climate while the star brightens by 80 per cent.

The Control

A flat line is not evidence, only a shape. The question a flat line raises is what would have to be false for the line to bend, and the only way to answer that is to break the mechanism deliberately and watch the line bend.

So we set both daisy albedos to 0.50, the same as bare ground. Nothing else changed: the daisies still grow, still compete for ground, still die at 0.3, and still carry a parabolic growth curve with the same optimum, the one thing now beyond them being any alteration of the planet's albedo.

Result: the temperature curve lies on the bare-planet control to within 2.842×10⁻¹³ K, the rounding error of a double-precision number near 300, and the temperature span over the full sweep reads 73.339 °C for the equal-albedo planet against 73.339 °C for the bare one. The line bent. And the biosphere is not absent while this happens: daisies are alive at 188 of the 501 luminosities and reach the same maximum cover of 0.7000. A large, thriving, competing biosphere with no radiative signature regulates nothing at all.

This control is the one the argument needs, and it turns Figure 1 from a picture into a claim: the daisies were not stabilising the temperature because they were alive, or because they were competing, or because the equations are nonlinear. They were stabilising it because they were coloured differently from the dirt.

How Hard You Have to Push

The published parameters are not measurements. No Daisyworld exists to measure, so the values are choices that make the point cleanly, and a replication that only confirms the choices has not learned much. We varied three of them.

The growth window. The parameter \(h\), the half-width of the temperature range in which a daisy can grow at all, is published at 17.5 K, and narrowing it makes the daisies fussier. Two things happen at once, and they pull in opposite directions. The regulated luminosity range shrinks, from 0.960 at \(h = 25\) K to 0.415 at \(h = 1\) K, while the regulation that survives gets tighter: the residual drift falls from 15.798 °C to 1.670 °C at \(h = 4\) K, and the suppression factor rises from 4.37 to 19.69. A fussy daisy that survives at all has pinned the planet very close to its own optimum, because pinning the planet was the only way left for it to survive.

The death rate. Raising \(\gamma\) does something different and worse. The regulated width falls roughly linearly, from 0.920 at \(\gamma = 0.05\) to 0.085 at \(\gamma = 0.95\), and the suppression falls with it, from 6.91 to 1.05. At \(\gamma = 0.99\) nothing survives anywhere in the sweep. The ceiling is exact and it is arithmetic: steady state requires \(a_g\beta = \gamma\), and neither \(a_g\) nor \(\beta\) can exceed 1, so at \(\gamma = 1\) no population can balance its own losses at any temperature.

Heat transport. The surprise of the study sits here. At \(q = 0\) the planet conducts perfectly, every patch sits at the planetary mean, and black and white daisies experience identical temperatures, which makes them the same organism in different paint. Regulation does not vanish. The daisies still live from L = 0.770 to 1.140, so a biosphere is still there, but the suppression factor is exactly 1.00: the drift inside the surviving range, 28.953 °C, equals the bare span over the same range. Life is present and the temperature curve is the control. Going the other way, \(q = 4\times10^{9}\) widens the regulated range to 0.945 while making the regulation sloppier, drift 15.626 °C, suppression 4.35; push to 8×10⁹ and the cold end collapses entirely, because patches are now so thermally isolated that a black daisy on a cold planet cooks itself.

The parameter scans use a coarser luminosity grid, 201 points instead of 501, which is why the published row reads 0.705 to 1.555 rather than 0.704 to 1.556, a difference of one grid step that measures the resolution of the scan rather than any disagreement.

We also hit the standing crop with a shock rather than a permanent change: hold everything at the published values, sweep to L = 1.000, then delete a fraction of the living daisies instantly and integrate back. Killing 10 per cent raises the temperature to 22.340 °C immediately, and killing 99.9 per cent raises it to 26.715 °C, which is essentially the bare planet. In every case the planet returns to 21.84125 °C, with a recovery error of −5.90×10⁻⁷ °C, because the attractor belongs to the equations rather than to the population, so the biosphere can be almost entirely destroyed and the climate comes back. The model's most impressive property is also the one with the least connection to anything real: a biosphere that recovers from 99.9 per cent mortality is a biosphere with no extinction, no genetics, and no minimum viable population.

Two Thousand Other Planets

Everything above sits at one point in parameter space, so to find out how much of the result belongs to that point we drew 2,000 random parameter sets and swept every one of them. Black albedo was drawn from \(N(0.25, 0.05^2)\) truncated to \([0.02, 0.48]\), white from \(N(0.75, 0.05^2)\) truncated to \([0.52, 0.98]\), heat transport from \(N(2.06\times10^{9}, (3\times10^{8})^2)\), death rate from \(N(0.30, 0.06^2)\), growth optimum from \(N(295.5, 4^2)\) K, and growth half-width from \(N(17.5, 3.5^2)\) K truncated to \([2, 35]\). The generator is numpy's PCG64, seeded by spawning a child stream from a SeedSequence on the master seed 20250621, so the draw is reproducible and independent of every other random number in the study.

The three summary statistics, with standard errors computed from the 2,000 runs:

The standard deviations are much larger than the standard errors, as they should be: 0.1124 in width, 4.148 °C in drift, 2.299 in suppression, which is the spread across planets and the quantity that matters. The 5th to 95th percentile range runs from 0.59 to 0.96 in regulated width, from 5.76 to 17.23 °C in drift, and from 3.16 to 9.90 in suppression. Ninety-nine point three per cent of drawn planets compress the temperature range by more than a factor of two; 70.7 per cent by more than five; 4.45 per cent by more than ten.

Not one of the 2,000 failed to regulate, which is the strongest single statement we can make in the model's favour and which is worth being precise about. It does not mean regulation is a safe bet in the real world. It means that inside the family of models defined by these equations and these distributions regulation is not a knife-edge, and that drawing a planet at random from this family gives you a functioning albedo thermostat essentially every time. We also chose the family.

Figure 5 shows the running mean settling. By 400 trials the estimate is within 0.002 of its final value and the standard error has fallen to 0.0057; the remaining 1,600 trials buy a factor of 2.3 in precision and change the central value by 0.002. We report 2,000 because it was cheap, not because it was needed.

One number in that section deserves a flag rather than a flourish. On the same 101-point luminosity grid the Monte Carlo used, the published parameters give a regulated width of 0.8500, while the mean across the 2,000 drawn planets is 0.8066 ± 0.0025. The published point therefore sits 17.3 standard errors above the mean of its own neighbourhood, which is neither an error nor a disagreement with Watson and Lovelock but a fact about where their parameters sit: a favourable choice within the family rather than a typical one. The same comparison for the residual drift puts the published point 5.41 standard errors below the mean, which is to say the published planet regulates over a wider range and more tightly than a randomly drawn relative of it, so anyone quoting Daisyworld's regulated range should know that the published number sits near the top of the distribution rather than in the middle.

The Objection We Could Not Get Rid Of

What follows is the case against everything above, put as strongly as we know how to put it, and we have not found a way round it.

Daisyworld is a machine for producing negative feedback, built by someone who wanted negative feedback: the growth curve is peaked, which guarantees that any population moving the temperature in either direction eventually punishes itself. The two species have albedos placed symmetrically either side of bare ground, which guarantees that one of them benefits from warming and the other from cooling. The heat transport coefficient is set so that local temperature differences are large enough to matter and small enough not to be fatal, and given those three choices temperature regulation is not a discovery. It follows as a theorem, and the simulation illustrates the theorem. Kirchner made this argument in a more careful form and it has never been properly answered: Daisyworld shows that regulation is possible in Daisyworld, and the leap from there to any claim about Earth is the entire disputed step [5][6].

We think this objection is mostly right, and Saunders sharpened it further by reading the parable as showing regulation arising from something other than selection at all, which is either a deep point about self-organisation or a sign that the model has been stripped of the very thing under dispute [7]. Three responses follow, in decreasing order of how much we believe them.

First, the equal-albedo control in §7 is doing real work. It shows that the regulation depends on the specific radiative coupling and not on the peaked growth curve, the competition, or the nonlinearity of the equations, all of which are still present when the control is run. The control is worth having, because "the model has feedback built into it" is a claim that can be tested rather than asserted, and the test locates the feedback precisely.

Second, the quantitative results are not implied by the construction. Nothing in the design of the model tells you that regulation holds over 85 per cent of a luminosity range, that the residual drift is 10 rather than 1 or 40 °C, that the suppression factor is 5.78, or that the collapse at the hot edge is a 32 °C discontinuity rather than a taper. Those had to be computed. Weber's analysis of homeostasis in Daisyworld makes the same point from the other direction, that the strength of the regulation depends on parameters in ways the qualitative story hides [8].

Third, and weakest: the model has proved productive as pure illustrations usually do not. It has been extended to one and two spatial dimensions, where it produces pattern formation nobody put in [13][11]; to mutating albedos, where it produces sustained oscillations [15]; to a version with explicit Darwinian competition between daisy types, where the regulation survives [9][10]. And there is now a serious proposal to take it off the computer entirely: a design for two engineered microbial strains with different pigmentation, competing under pH as the abiotic control variable rather than light or temperature, alongside further theoretical and computational case studies of the two-strain and multi-strain versions [17]. Nobody has grown it yet. The paper's own abstract is explicit that Daisyworld today "exists only as a mathematical or computational construct" and that a real biological version remains to be built. A published wet-lab demonstration, if one arrives, would be the strongest reply available to the charge that Daisyworld is unfalsifiable. A design for one is not that reply. It is a plan, and a good one, for closing exactly the gap this article is about.

What none of that does is bridge the gap: our study is a comparison between a computer program and a set of equations, conducted by people who wrote both. The only thing that could close the gap is a measurement of a real biosphere altering a real planetary variable in a stabilising direction, and this study contains no such measurement, because the club owns no planet.

A second objection we like less and cannot dismiss comes from the rocks. Tyrrell's book-length examination of Gaia concludes, from the geological record rather than from models, that Earth's history looks more like a series of destabilising excursions survived by luck than like a thermostat [16]. If that reading is right, then a model demonstrating that thermostats are possible is answering a question nobody needed answered, and we do not know how to settle it, beyond noticing that our own study cannot help.

What a Toy Can Show

It shows that a planetary thermostat does not require anything to want one. Two populations obeying local rules, competing for nothing but ground, holding a global variable steady for as long as they can and then failing catastrophically. If the charge against Gaia was that self-regulation implies foresight, the charge is answered, and it was answered in 1983. Our contribution is to put numbers on the answer: 85 per cent of a luminosity range, a suppression factor between 5.8 and 8.3 depending on how you define the interval, and a collapse that arrives in a single step.

It does not show that Earth regulates its temperature. It does not show that biospheres in general regulate anything. It cannot, because it contains no ocean, no atmosphere, no carbon cycle, no clouds, no geology, no evolution, and no time. Anyone who cites Daisyworld as evidence about the Earth system is citing an existence proof as if it were an observation, and the difference between those two things is most of what the last forty years of the argument has been about [5][6][16][12].

What we would most like a reader to take away is the shape of the thing rather than the headline. We had a claim from a 1983 paper. We rebuilt the model from the published equations, checked that it conserved area and reproduced a closed-form answer we derived independently, verified that the effect disappears when the proposed mechanism is disabled, measured the size of the effect with error bars, and then spent as much effort establishing where the result breaks as establishing that it exists. That sequence is available to anyone with a laptop and a free evening. Agreeing with a paper and checking one are different acts.

Where a different modelling choice would change the answer

Whether dead daisies can come back. The big one, and not a parameter at all. Our sweep carries the population forward from each luminosity to the next and restores any species that has fallen below 0.01 back up to 0.01, so seeds are always available. Remove the restoration and the cold edge moves from 0.704 to 0.768, costing 7.3 per cent of the width. Remove the memory entirely, so that every luminosity starts from bare ground with a sprinkling of seed, and the hot edge falls from 1.556 to 1.212: the regulated width drops from 0.852 to 0.508, a loss of 40.4 per cent of the result. The reason is physical rather than numerical. At L = 1.4 a bare planet sits at 53 °C, far above the 39.85 °C ceiling for growth, so a white daisy cannot get started there; but a white daisy that is already growing there has cooled its own patch into the comfortable range and stays. Regulation at the hot end is inherited, not achieved. Sweeping the star downward from L = 1.6 instead of upward makes the same point loudly: the regulated range becomes 0.622 to 1.212 and the two sweeps disagree by up to 33.82 °C at the same luminosity. Daisyworld has genuine hysteresis, and which branch you report is a choice.

Two species against many. With exactly two albedos, the planet has two settings. Harding showed that a community with many species spread across a range of optima both regulates over a wider band and resists perturbation better [14], and Dyke and Weaver have since argued that environmental homeostasis emerges generically in large random ecosystems without any of Daisyworld's careful construction. Our suppression factor of 5.78 is the two-species floor.

Fixed albedos against evolving ones. We froze the daisies at 0.25 and 0.75. Let albedo mutate and the picture changes in ways that are still argued about: Robertson and Robinson found that Darwinian competition tends to erode the regulation, since a daisy that is slightly more optimal for itself outcompetes one that is better for the planet [9]. Lenton and Lovelock argued that constraints on how far albedo can adapt are what preserve it [10], and Wood, Ackland and Lenton found that mutating albedo in a spatial version produces sustained oscillations rather than a steady state [15]. A fitness-based reanalysis puts the disagreement on firmer ground [4]. We did not run any of this, and it is the single most important thing our study leaves out.

Zero dimensions against a surface. The local temperature in our model is an accounting device: there is no place where a black daisy is. Put the daisies on a grid with real diffusion and the model produces spatial segregation, travelling fronts and pattern formation [13], and the collapse at the hot edge becomes a front-propagation problem rather than an instant [11]. Our discontinuity at L = 1.558 is sharp partly because there is nowhere for anything to retreat to.

The definition of regulated. We called the planet regulated wherever daisies are alive, which is the least flattering of the three definitions we considered. Defining it as "within 2 °C of the optimum" gives a width of 0.346 instead of 0.852 and a drift of 3.961 °C instead of 10.421. Both are in the output file. Any single number quoted for Daisyworld's regulated range is partly a definition.

Reproducing this

One file, one command, no packages beyond numpy.

python daisyworld.py > daisyworld-output.txt

Expected runtime is about seven minutes on a laptop; ours was 416.0 seconds with Python 3.12.3 and numpy 2.4.2. The master seed is 20250621, hard-coded at the top of the file, and the only stochastic component is the 2,000-planet Monte Carlo in section 9, which draws its stream through SeedSequence.spawn, so the whole output is deterministic. Roughly two-thirds of the runtime is that Monte Carlo; the main 501-point sweep takes 13 seconds. If your numbers differ from ours by more than the printed standard errors, something is wrong and we would like to hear about it. The interactive model runs the same sweep in your browser at the same resolution and reproduces the regulated interval, the drift and the suppression factor exactly.

References