Doubling

How does one seed become a world?

Growth by doubling is exponential. A bacterium splits in two about every 20 minutes in a rich broth. After 37 splits, about 12 hours, there are 1.4 × 10¹¹ of them, if nutrients and space remain available.

The Vigil began the same way, with less to start from. In 2453 a seed of 1,000 kilograms landed on Kindling, a hot rocky planet in a tight orbit around Ember, the orange star that circles Vesper. Its fusion stage had burned 17,206 kg of propellant to stop it (Stopping from 0.1c); the doublings below count from the seed alone. fiction Home was 280 years away by anything that carries atoms.

The seed packed three things:

Construction equipment: assemblers and tools that turn rock into parts.
Starter feedstock: refined stock of what a new industry cannot make on its first day, such as scarce elements and finished microchips.
A bootstrap mind: a machine intelligence of the 2150s, sent to prepare the ground for minds from more than three centuries later. frontier

Its orders were to survey the system, send the survey home, build a receiver and prepare for the experts' arrival, while saving the feedstock. A seed that copied itself on landing would have fixed its 2150s designs into everything that followed.

So it grew only as far as those jobs needed. Its power was starlight: collectors on Kindling's day side, where Ember delivers 35,000 watts on every square metre fiction. The receiver, a pair of mirrors 1,000 metres wide, took until 2481 to build (Bits travel, atoms do not). Between jobs it stockpiled raw material, waiting 58 years in all from the landing.

The first minds arrived by light from Saturn in 2511 and began the rapid building. One machine made two, two made four, and the swarm doubled every 9.5 months. About thirty-seven doublings later, its 2540 radiator-material target was 1.1 × 10¹⁴ kg, enough for the petawatt radiators; this uses an unsourced film-density assumption challenge. First Light began with Vesper's first controlled feed. Radiator area explains the cost.

First Light ends the bootstrap era. The plan's goal, film for 10²⁰ watts, is 16.6 doublings further on. At the same pace that is the 2550s simulation; the film-mass target uses the same unsourced density assumption challenge.

Guess first Which of the 37 doublings added the most machinery?

The last. Every doubling adds as much as everything built before it, so the final one added about half of the radiator-material target challenge.

Sustained exponential growth requires three conditions.

The machine must make nearly everything it is made of. Once the seed's chips are spent, every imported chip is a link back to a factory 28 light years away. Engineers call this closure, and as of 2026 nobody has built a machine that has it. challenge
There must be matter to work with. Kindling alone holds 3 × 10⁹ times the First Light radiator-material target, and the Ashfall, a belt of rock around Vesper and Ember, far more. fiction challenge
Every kilogram built costs energy, and the energy ends as heat. In a vacuum heat leaves only as light, so the swarm can grow no faster than it can cool.

Once a place is a swarm it can do the same for the next. The plan has the Vigil send 12 seeds toward nearby stars in 2610, each carrying equipment, starter feedstock and a mind. A machine that copies itself and sends copies outward is a von Neumann probe. What a wave of them does, and what it means that none has turned up in the real sky, is for the folds below.

How long does it take, and what slows it down?

The arithmetic is one line:

doublings = log₂(final mass ÷ seed mass)

From the landed seed's 1,000 kg to the radiator-material target of 1.1 × 10¹⁴ kg is 36.7 doublings challenge. Each doubling takes one doubling time. At a year each, the job takes 36.7 years. The Record's count is shorter: the First Waking in 2511, First Light in 2540, 29 years, so a doubling every 9.5 months. At three months each, 9.2 years.

What to build	Mass (kg)	Doublings	At a year each (years)	At the heat floor below (years)
Film for the first petawatt	1.1 × 10¹⁴	36.7	36.7	8.8
Film for the plan's goal, 10²⁰ W	1.1 × 10¹⁹	53.3	53.3	12.7
The Moon	7.3 × 10²²	66.0	66.0	15.8
The Earth	6.0 × 10²⁴	72.3	72.3	17.3

The Earth weighs 6.0 × 10²⁴ kg, only 36 doublings beyond the first petawatt's film. Reaching that scale would require enough matter, manufacturing closure and cooling capacity throughout. The plan's goal needs at least 1.1 × 10¹⁹ kg of debris to draw on, which is 0.015 per cent of the Moon's mass. The target uses the same unsourced cold-film density fiction challenge.

One curve drawn twice: the mass of a self-copying machine after each doubling. On a plain scale almost nothing happens until the last few steps; on a log scale it is a straight line.

What does a kilogram cost?

Turning rock into finished machinery takes energy. Call it 1 × 10⁸ joules per kilogram: a round figure this page chooses, not a measurement, since nobody has built machinery from asteroid rubble. design

At that price the 1.1 × 10¹⁴ kg radiator-material target costs 1.1 × 10²² J, which is 0.34 years of the Vigil's first petawatt: about 4 months. Set against what a kilogram holds as mass-energy (A gram is a fortune), building it costs a fraction of 1.1 × 10⁻⁹, about a billionth challenge. Power is not what stops the swarm.

Why can't it go faster?

Every joule spent on building ends as heat. A machine that spends p watts per kilogram on building doubles in

doubling time = ln 2 × (energy per kg) ÷ p

The film mass and replication target use the assumed 0.01 kg/m² cold-film density. This is an unsourced engineering target challenge; Cooling is the economy separates that assumption from the physical area law.

and p is capped by how much heat a kilogram can shed. The best case is a machine that is nothing but radiating film: the Vigil's film, 0.01 kg per square metre, shining from both faces as a perfect black body. At 30 K, a kilogram of it sheds 9.2 watts, so such a machine cannot double faster than every 87 days. Warm the film to 300 K and it sheds 92,000 watts per kilogram, and the floor drops to 13 minutes. The fourth power of temperature, again.

The Record's 9.5 months per doubling is 3.3 times slower than the cold floor. That is as it should be: a real machine also carries mass that does not radiate. At that pace, the film can be as little as 30 per cent of the machine.

What does closure cost?

Suppose just one per cent of each machine, by mass, had to come from home. Over the doubling years that is 1.1 × 10¹² kg of imports. Each kilogram, thrown at the seed's own speed, carries 4.5 × 10¹⁴ J of kinetic energy, so the motion alone is 4.9 × 10²⁶ J: 16,000 years of the first petawatt. The lasers that throw it draw about 21 times that from the grid (Pushed by light). Each kilogram would also need its own brake at the far end: the seed's took 23 times its mass off the Moon. And every kilogram would take 280 years to arrive. Bits can cross that gap cheaply and atoms cannot (Bits travel, atoms do not): the swarm builds nearly everything itself, or it does not grow.

The calculator lets you try other masses and other doubling times.

With JavaScript on, this is a calculator: choose a mass to build and a doubling time, and it gives the doublings, the time, the year it would finish if it began at the First Waking, and the energy. The table above gives the same numbers for four builds.

Open everything

Why does growth double, and what bends the curve?An exponential, the logistic curve and a power budget

Why does growth double?

If every kilogram of machinery builds new machinery at the same rate, then growth is proportional to what exists:

$\frac{dM}{dt} = \frac{M}{\tau} \quad\Rightarrow\quad M(t) = M_0\, e^{t/\tau} = M_0\, 2^{\,t/t_d}, \qquad t_d = \tau \ln 2$

$M_0$ is the seed, $\tau$ the time to grow by a factor of $e$ , and $t_d$ the doubling time. Solving for the time to reach a mass $M$ :

$t = t_d \log_2 \frac{M}{M_0} = 36.7\; t_d$

for the first petawatt's film, starting from the 1,000 kg that landed. This exponential form follows from assuming that new machinery adds manufacturing capacity in proportion to its mass.

What bends the curve?

Suppose the surroundings can supply at most a mass $K$ of machinery: all the usable debris, say. Growth slows as the supply runs short, and the simplest curve that does this is the logistic:

$\frac{dM}{dt} = \frac{M}{\tau}\left(1 - \frac{M}{K}\right) \quad\Rightarrow\quad M(t) = \frac{K}{1 + \left(\dfrac{K}{M_0} - 1\right) e^{-t/\tau}}$

While $M \ll K$ the bracket is almost 1 and the growth is exponential. The time to reach a fraction $f$ of $K$ is

$t_f = \tau \ln\!\left[\left(\frac{K}{M_0} - 1\right)\frac{f}{1-f}\right]$

so going from half of $K$ to 99 per cent of it takes $\tau \ln 99$ , which is $\log_2 99 = 6.6$ doublings. An exponential would have covered the same ground in one doubling. Logistic growth slows gradually as the resource limit approaches. Pearl and Reed used this same curve in 1920 for the growth of the population of the United States since 1790.

A flow limit, such as a mining front that can only work the surface of a body, bends the curve differently: growth slows from exponential to a power of time. Which limit binds first depends on the shape of Kindling and the Ashfall. fiction

Does heat bend it, too?

Heat limits the growth rate while preserving the exponential form if cooling capacity grows with manufacturing mass. Let a machine of mass $M$ spend a power $pM$ on building, at an energy cost $e$ per kilogram of product. Then

$\frac{dM}{dt} = \frac{pM}{e} \quad\Rightarrow\quad t_d = \frac{e \ln 2}{p}$

All that power ends as heat, and heat leaves as light from a film of areal density $\rho_a$ at temperature $T$ (the radiator page). If the whole machine were two-sided film shedding $\sigma (T^4 - T_{\text{sky}}^4)$ per face,

$p \le \frac{2\sigma\,(T^4 - T_{\text{sky}}^4)}{\rho_a} \quad\Rightarrow\quad t_d \ge \frac{e \ln 2 \;\rho_a}{2\sigma\,(T^4 - T_{\text{sky}}^4)}$

The sky's few kelvin are negligible against 30 K, and the film is taken as perfectly black (emissivity 1), as on the radiator page. With $\rho_a = 0.01$ kg/m² and $e = 1 \times 10^{8}$ J/kg, 30 K film allows $p \le 9.2$ W/kg and $t_d \ge 87$ days. A fraction $\varphi$ of the machine that is film stretches the floor by $1/\varphi$ . The floor scales as $T^{-4}$ : at 300 K it is ten thousand times shorter. Heat sets the pace of the doubling, not its shape, as long as the machine keeps building film in proportion to everything else.

Closure, in symbols

Let $f$ be the share of a machine's mass the swarm can make for itself. Every new kilogram needs $(1-f)$ kilograms from outside. If all the imports come as one stock $S$ the seed carried, the swarm can reach at most

$M_{\max} = M_0 + \frac{S}{1-f}$

The seed's starter feedstock is a stock of this kind, which is why its orders were to save it for the experts. Even if the entire seed were spare parts, $S = M_0$ , reaching the first petawatt's film needs $1 - f \lesssim M_0 / M = 9.2 \times 10^{-12}$ . If imports instead arrive at a fixed rate $I$ , the demand $(1-f)\,M/\tau$ outgrows it once

$M > M^{*} = \frac{I\,\tau}{1-f},$

after which growth is linear, $dM/dt = I/(1-f)$ . For the Vigil the import rate is near zero, since only bits can cross the gap quickly. Closure requires every necessary manufacturing function: chips are a tiny share of the mass, but the swarm needs the whole production chain to make them. Building that chain is the central engineering challenge. challenge

Has anyone built one, and what would a wave of them mean?Von Neumann's machine, three studies and the Fermi question

How does von Neumann's design reproduce?

A machine that builds a copy of itself seems to need a blueprint of itself, and the blueprint seems to need a blueprint, without end. John von Neumann showed there is no regress, first in lectures in 1948 and at the University of Illinois in 1949. Split the machine in two: a constructor that builds whatever a tape of instructions describes, and a copier that duplicates the tape. Hand the tape to the constructor, and it builds a new constructor. Copy the tape and hand it over, and the daughter can do the same. From 1952 he worked the design out in detail in an imagined grid world, a cellular automaton with 29 states per cell, and left it unfinished when he died in 1957. Arthur Burks edited and completed it, and it was published in 1966, with the Illinois lectures, as Theory of Self-Reproducing Automata. A cell has the same division of labour: instructions are copied, and separate machinery reads them.

What has been built, as of 2026

Loose cubes. In 2005 Zykov, Mytilinaios, Adams and Lipson showed a stack of motorised cubes that assembled a copy of itself out of further cubes. Its closure was close to nil: the cubes were handed to it ready-made.
A lunar factory on paper. A 1980 NASA summer study, edited by Freitas and Gilbreath and published in 1982 as Advanced Automation for Space Missions, sketched an automated factory on the Moon able to build duplicates of itself. It was never built.
A survey. In 2004 Freitas and Merkle collected the theory and the experiments in Kinematic Self-Replicating Machines, with chapters on how replication time depends on replicator mass and on closure.

As of 2026 no machine makes all of its own parts from raw material. Whether one can is the open question, and the Vigil's story assumes it can. challenge

The wave

A wave of seeds advances one hop per flight time plus the pause before the next launch. The plan fixes the pause: 157 years from the landing on Kindling to the outbound launch. It does not say how far the twelve targets are, so for a sum, take the first seed's own trip as the hop: 28 light years at a tenth of the speed of light, 280 years of flight. illustrative The front then moves at 0.064 of the speed of light, and a galaxy about 1 × 10⁵ light years across is crossed in 1.6 × 10⁶ years, which is 1.1 × 10⁻⁴ of the age of the universe. Shorter hops with the same pause make the front slower, because the pause counts for more; with no pause at all it would cross in 1 × 10⁶ years. Whatever the hop, the answer is millions of years against billions. A probe at a tenth of the speed of light also has to stop, which is the unglamorous half of the problem (Stopping from 0.1c).

Michael Hart made this argument in 1975. If colonists hopped to the nearest stars at a tenth of light speed and every colony sent out ships of its own, he estimated, most of the Galaxy would be reached within 650,000 years. Since no one from outer space is on Earth, he concluded, the likeliest explanation is that there are no other advanced civilizations in our Galaxy. In 1980 Frank Tipler put self-reproducing machines at the centre of the argument: a species able to signal across the stars would also build self-replicating constructors as clever as we are, and even on chemical rockets they would explore the Galaxy in under 3 × 10⁸ years. Since none is here, he concluded that we are probably the only intelligent species in the Galaxy. In 2013 Armstrong and Sandberg went further: they argued that intergalactic colonization is feasible for a civilization already capable of interstellar colonization, that it could reach an appreciable fraction of the visible universe, and that this makes the Fermi paradox sharper.

Still argued in 2026

The absence of detected probes has several possible explanations. frontier Civilizations might prioritize efficiency over expansion, as the Vigil does (Inward, not outward). Manufacturing closure or interstellar braking might prevent sustained replication. Advanced civilizations might also be very rare. The calculations do not distinguish these explanations; they show that sustained expansion under the assumed conditions could cross the Galaxy in a small fraction of its age.