Chapter 2. What Carries a World?

1In May 1981 a few dozen physicists and computer scientists gathered at Endicott House, a country estate outside Boston that MIT keeps for conferences, to talk about the physics of computation. The meeting had been organized by Edward Fredkin and his colleagues at MIT together with researchers from IBM. It rested on a suspicion that the two subjects in its title might be one subject seen from two sides. Computers are physical objects, so the laws of physics limit what they can do. Physical systems change according to rules, so in some sense every physical system computes. Fredkin held the stronger version of that thought. He believed the universe itself is a computer, a vast grid of simple cells updating by simple rules, and he spent much of his career looking for the rules. Richard Feynman, who gave the keynote, was more cautious. His lecture, published the following year as “Simulating Physics with Computers,” became one of the founding documents of quantum computing.

2Feynman began with a demanding standard. He asked whether a computer could imitate physics exactly, as opposed to approximately, and he set a condition on what would count. The number of computing elements needed to imitate a physical system should grow only in proportion to the volume of space and time the system occupies. A machine that met the condition would do for each small region of the world roughly what that region does for itself. It would keep pace with nature in a fair sense, never needing more machinery for one cubic meter of air than for any other. Feynman thought classical physics might be imitated this way once space and time were broken into small enough pieces. Quantum physics was another matter. The probabilities of quantum mechanics arise from amplitudes that combine across every possible configuration of a system. A machine built of classical parts can reproduce them only by tracking a number of quantities that grows exponentially with the number of particles, so that doubling the size of the system roughly squares the size of the machine. Feynman concluded that a faithful imitator of nature would have to be built of quantum parts, a computer whose own components obey the laws it imitates.

3The idea that nature itself computes is older than the conference, going back at least to Konrad Zuse, who built the first working programmable computer in Berlin in 1941 and proposed in Rechnender Raum (1969) that space might be a computing machine running the laws of physics as its program. Whether the foundation of a world computes makes no difference to its reality. What the Endicott House meeting offers the designer is its direction of approach, since everyone in the room took nature as given and asked how a machine might keep up with it.

4For the designer, no nature yet exists for the machine to keep pace with, since the physics has not been written. Feynman asked whether a machine could keep pace with nature, and the designer asks which natures a machine can keep. Behind that question stand several more. Some physics will be cheap to carry and some ruinously expensive, while some may lie beyond any machine at all. The arithmetic of the machine may leak into the laws it carries, and the machine itself will sometimes fail. Every limit of this kind will press the designer toward savings, and each saving will land somewhere in the world. That the substrate takes nothing from a world’s reality is already settled. Which worlds it can carry is a matter of engineering.

5In the carpenter’s world, the world of the third table, a boy stands at the edge of a lake toward evening. He has spent some minutes choosing a stone, flat, smooth, and about the size of his palm, and he throws it low and hard with a flick of the wrist. It touches the water four times before it sinks. At each touch a ring opens and widens, and the rings cross one another, building small peaks where they meet and canceling where a crest falls into a trough. A soft sound carries across the water with each touch. The sun is low, and its light breaks on the moving surface into a scatter of bright points, while below the surface a few fish turn away from the disturbance and settle again. The boy laughs and begins looking along the shore for another stone.

6Every part of that moment has to be carried. The rings are waves in water, whose laws the substrate carries in the way air carries sound. Each crossing of one ring with another is therefore work the machine must do, as is every point of light and every turn of a fish. Most of that work goes into the water. The lake is large, its surface is never still, and its depths are full of slow currents and shifts of temperature that no one in the carpenter’s world will ever notice. The boy is small by comparison, yet he is the reason the moment matters, since the evening is good for him and the water is good for no one except through what it gives to the creatures in it and beside it. A first impression forms easily here, that the expensive part of the moment is the water and the valuable part is the boy. The impression holds, in a sense more exact than the one it first suggests.

7What it costs to carry the boy’s evening depends above all on how deep the world goes, meaning the finest level its physics resolves. Described at the depth of our own physics, the lake is a configuration of quantum fields. A quantum state assigns an amplitude to every possible arrangement of a system’s parts, and the number of arrangements grows exponentially with the number of parts. A few hundred interacting particles, each with only two possible states, already have more joint arrangements than there are atoms in the observable universe. A lake of modest size, holding a cubic kilometer of water, contains something like 3 × 10 ^{to the power 37} molecules, so no machine built of classical parts could ever hold its full quantum state. The lake holds that state with ease, and a machine of quantum parts could do the same, as Feynman concluded and Seth Lloyd showed in “Universal Quantum Simulators” (1996) for any physics whose parts act only on their neighbors. Such a machine needs resources that grow only modestly with the size of what it carries, and its steep price lies elsewhere, in the machinery it needs for correcting its own errors.

8Described at the grain of molecules, the same lake becomes a finite problem. Each molecule is treated as a small body moving under forces from its neighbors, and a machine advances them all together in steps of about a femtosecond, a millionth of a billionth of a second, which is the time scale on which molecules vibrate. Chemists run computations of this kind every day on systems of a few million atoms. Applied to a whole lake the method is astronomically expensive, though no longer impossible in principle. Allowing a hundred elementary operations for each molecule at each step, a deliberately low figure, the whole lake asks for about 3 × 10 ^{to the power 54} operations every second. Described at the grain of fluid flow, the lake becomes cheap. Its water is treated as a continuous medium with a velocity, a pressure, and a temperature at each point, governed by the equations of hydrodynamics on a mesh of cells, and a single modern workstation can produce a surface of rings and reflections that most observers would accept as water. The gap in cost between the second and third descriptions runs to dozens of orders of magnitude, and on classical machines the gap between the first and second has no useful measure at all.

9Feynman’s problem was to imitate a nature already fixed, so for him the depth was set by the world and the cost followed from it. The designer is placed differently, because a designed world imitates nothing. The carpenter’s world is no copy of the primordial cosmos, and the reasoning that establishes this also frees its designer from any obligation to match our cosmos level for level. The designer decides where the world’s physics bottoms out. A world can have fundamental entities far coarser than our quarks and electrons. Its water might be made of a handful of particle kinds with simple interactions and nothing beneath them, or, at the extreme, it might be a true continuum whose smallest drop is still water. Among the technical choices that crowd forward first at the empty substrate is the smallest unit it will resolve. Here that choice becomes concrete. It is a choice of depth.

10Explore The ladder of lengthsThe choice can be pictured on a ladder of lengths, shown in Figure 1, which runs from the Planck length, about 10 ^{to the power −35} meters, where present theories are expected to fail, to the edge of the observable universe, some 4 × 10 ^{to the power 26} meters away. About sixty-one powers of ten separate the two ends. The nucleus of an atom sits some twenty rungs above the bottom, the atom five rungs above the nucleus, and a molecule of water just above the atom. The boy stands about thirty-five rungs up and some twenty-six below the top, nearer by this measure to the edge of everything than to the bottom of his own matter. Each rung is a place where a world’s physics could stop, and wherever the designer stops it, everything above has to be built from what lies there.Explore The ladder of lengths

11A designer who thinks of depth only as a cost will build a poorer world than they intend. A deep physics gives chemistry room to be rich, since the variety of possible molecules grows with the variety of ways fundamental things can combine. Life needs that variety in order to arise, because every account of its origin in our own world depends on a chemistry varied enough to try uncounted combinations until some of them copy themselves. Minds need a medium in which to be built, and the world’s scientists need something to find, a sequence of levels each explaining the one above. A world too shallow to hold anyone fails at the one thing that matters, however cheaply it runs. It may be lawful, persistent, and counterfactually robust, and it will still be empty.

12Natives are made of their world’s matter, and the boy is a native. If his world goes deep enough for chemistry and life to arise on their own, then his brain is a volume of that world’s matter, about 1.2 liters of it, and at molecular grain it costs what any 1.2 liters of the world costs. By the same counting, that comes to about 4 × 10 ^{to the power 42} elementary operations every second, an amount that becomes startling once it is translated into energy. In a deep world the boy is as expensive as the water he is made of, ounce for ounce.

13Counted by function instead of matter, a mind costs far less. Estimates of what a human mind requires, if it is computed at the level of its neurons and synapses, run very roughly from 10 ^{to the power 14} to 10 ^{to the power 18} operations per second. Anders Sandberg and Nick Bostrom, in Whole Brain Emulation: A Roadmap (2008), laid out a ladder of such estimates, each rung a finer level of detail at which a brain might have to be modeled for the result to be the same mind, with costs climbing steeply from the functional rungs toward the molecular ones. At the lower rungs a mind is cheap, a trivial load beside one lake. That cheapness belongs to two kinds of being. It belongs to emulated minds, such as immigrants carried in from the primordial cosmos and lodged in bodies of the world, provided a mind computed at a functional grain is still the mind it was. It also belongs to the natives of a world that is shallow beneath the level of life, whose bodies are built from coarse units arranged to do what neurons do, with nothing finer underneath.

14These two ways of counting set the first real dilemma of design. A world shallow beneath life makes minds cheap, and it may never produce them. Life in our cosmos arose, so far as anyone can tell, from a chemistry rich enough to explore an enormous space of molecules over hundreds of millions of years, and a coarse physics may offer too little variety for anything like that to happen. Its minds would then have to be built directly and placed in the world, a choice that has to be weighed against the value of letting life emerge on its own. A deep world lets life arise from chemistry and makes every body as costly as the matter it is made of. Minds may have to be built there too, since selection reached them in our world through overproduction and early death, and the ethics of design may forbid that among any creatures that might feel.

15Commitment Depth is the first decision of…Even the molecular grain used in these counts is a simplification. Chemists who compute molecules as small classical bodies are approximating the quantum behavior of electrons. A designer could write a classical molecular physics directly, with bonding rules built into its forces, but whether such a physics could support chemistry rich enough for life to arise on its own is unknown, and it is the very question on which the dilemma turns. If life needs something like quantum chemistry, the deep horn grows costlier still, and quantum machines become its natural substrate. Neither horn is forbidden. Depth is the first decision of substance the designer makes about the machine, and nearly every later decision about the machine depends on where it is set.

16Commitment Functionalism is the book’s working hypothesis…A middle path suggests itself, and it brings out a rule that governs every saving a designer might make. The designer could build a deep world, let chemistry and life run at full depth everywhere, and then compute the brains of natives at the cheaper grain of their function, as though each skull held an emulation in place of matter. The saving would be enormous, and it would be taken directly from the subjects. It would be safe only if the designer knew the grain at which consciousness depends on the physical details beneath it. The functionalist hypothesis holds that the right organization suffices for experience, and it says nothing about which level of a brain’s organization carries the experience. Nobody knows whether a mind computed at the level of its synapses is the same mind as one carried molecule by molecule, and precaution refuses to stake a person on the answer. The designer may build a shallow world openly, with natives made of coarse matter from the start. What they may not do is build a deep world and then quietly make its people shallow, because a saving taken on a subject is a harm done to someone who never agreed to bear it.

17Commitment A risk no designer may impose…The same unknown hangs over immigrants. A mind carried in from the primordial cosmos at a functional grain is a bet on exactly the question that forbids the middle path, and the two cases differ in who makes the bet and what it is set against. Natives are given no choice and gain nothing from the saving. An immigrant can take the risk knowingly, and for the dying the alternative is the end of everything, so a risk no designer may impose on natives can be reasonable for someone who accepts it freely with death as the other choice. Consent does not settle the matter. Precaution still asks that the passage be made at the finest grain the budget allows, because the question it raises, whether the one who arrives is the one who left, is the question on which the whole value of the passage depends.

18That a designer can choose depth at all is owed to a feature of physics that Kenneth Wilson explained in the early 1970s with the renormalization group. Wilson showed how the behavior of a system at large scales depends on only a few features of what happens at small scales, while the remaining details wash out as one moves upward. Water near its critical point, where liquid and vapor become indistinguishable, shares its large-scale statistics with a simple model of magnets, though water and magnets have nothing in common at the scale of atoms. Many different floors can support the same large-scale world, so the choice of depth is free within limits. The inhabitants’ science can also succeed one level at a time without reaching the bottom, as ours did when chemistry flourished for a century before anyone understood the nucleus. Whatever depth the designer chooses, its cost must be paid, and the budget it is paid from belongs to the primordial cosmos.

19Nobody knows what the machines of a civilization with superintelligence will look like. The minds that design them may invent architectures no one living can imagine, and any forecast of their blueprints is a guess. Some claims about those machines are safe all the same, because they concern limits that physics places on any machine built in the primordial cosmos, however intelligent its builders. No mind repeals the laws it runs on. The sound policy follows from that difference. It speaks with confidence about limits and with humility about blueprints, and it marks every forecast as one.

20The hard ceilings are few and generous. Norman Margolus and Lev Levitin showed in 1998 that a physical system with a given average energy can pass through only a limited number of distinguishable states each second, a number proportional to its energy. Jacob Bekenstein showed in 1981 that a region of given size and energy can hold only a finite amount of information. Signals between the parts of any machine travel no faster than light. Lloyd combined such limits in “Ultimate Physical Limits to Computation” (2000), a portrait of a one-kilogram computer working at the edge of what physics allows, which would perform around 5 × 10 ^{to the power 50} operations each second and would be, in its working state, a ball of plasma at a billion degrees. These ceilings sit far above anything a lake requires. Carrying the lake at molecular grain takes about 3 × 10 ^{to the power 54} operations each second, and by the Margolus–Levitin bound about six tonnes of mass-energy would in principle suffice to perform them. What makes the lake expensive is the heat a real machine sheds while doing the work.

21Rolf Landauer of IBM had identified the relevant cost two decades before the Endicott House meeting. In 1961 he showed that erasing a bit of information must release a minimum amount of heat, proportional to the temperature of the surroundings, which comes at room temperature to about 3 × 10 ^{to the power −21} joules. The price attaches to forgetting. A logic gate that takes two inputs and returns one output has discarded information about its inputs, and every register overwritten and every intermediate result thrown away pays the same small toll. Ordinary computers forget constantly and in practice spend many orders of magnitude more energy per operation than Landauer’s limit. The limit itself is the part no engineering can remove from a machine built that way.

22Explore The cost of a worldTake the lake again, its 3 × 10 ^{to the power 37} molecules carried at molecular grain in steps of a femtosecond, so 10 ^{to the power 15} steps each second, with the same hundred elementary operations per molecule per step. Assume, as ordinary machines do, that each operation erases about one bit, and that the machine works at room temperature. The lake’s 3 × 10 ^{to the power 54} operations per second then release roughly 10 ^{to the power 34} watts of heat. The Sun’s entire output is about 4 × 10 ^{to the power 26} watts, so the lake alone would consume the light of tens of millions of suns. The boy’s brain at the same grain, about 4 × 10 ^{to the power 42} operations each second, would release around 10 ^{to the power 22} watts, hundreds of millions of times all the power humanity now uses. At a functional grain, by contrast, a mind computed at Landauer’s limit would need at most a few thousandths of a watt, and the boy’s own brain, like ours, runs on about twenty.Explore The cost of a world

23These figures are rates, the power to carry the lake at its own pace, which no machine yet conceived could keep. Each femtosecond of the lake’s time asks for some ten operations in sequence, so logic switching in a trillionth of a second would run the lake ten thousand times slower than it lives, however much hardware was built. No quantity of hardware passes that sequential floor, since operations that must follow one another cannot be spread across more machines. A budget is better measured in years of the world’s own time, whose outside pace the inhabitants can never feel. A forgetful machine pays Landauer’s price for every operation of those years, however slowly it runs. At that price alone, such a machine in the cold of space, about three degrees above absolute zero, could carry the lake for some forty thousand of its years on the Sun’s lifetime output, and at room temperature for about four hundred.

24The estimate holds only as long as its assumption about forgetting holds, and that assumption can be broken. Charles Bennett showed in 1973 that computation can be made logically reversible, with every step undoable, so that nothing is ever erased and Landauer’s price need not be paid on each step. A reversible machine still loses a little energy in each operation, and as Michael Frank and Karpur Shukla set out in “Quantum Foundations of Classical Reversible Computing” (2021), the loss shrinks in proportion as the operation is drawn out, so that a switch thrown ten times more slowly wastes a tenth as much. Energy and waiting then become a single currency. For a given task on a machine sized to the world, the energy multiplied by the waiting is fixed. It grows with the size of the world and with the square of the number of steps in its history, since more steps mean both more operations to pay for and less time in which to draw each of them out. A machine half the size doubles it, because each of its parts must work through twice as many operations in turn. The trade holds until leakage, or the few erasures that errors, chance, and upkeep still demand, come to dominate.

25Commitment The expense of depth lies in…Counted this way, the true expense of depth lies in growing a world. Carrying Earth’s oceans at molecular grain for the three and a half billion years between the origin of life and the first minds would cost a forgetful machine, even at Landauer’s limit in the cold of space, about a hundred trillion times the Sun’s lifetime output, the starlight of thousands of galaxies. A reversible machine could pay for the same history from one star’s output, hoarded and spent slowly, but only by waiting more than three billion billion times the present age of the universe, and only if its switching lost no more than a speculative best case that no known law forbids and none guarantees. The machine itself would be no small thing. Working on every molecule at once, with a switch about the size of a small protein for each operation, it would weigh as much as seventeen Earths, a forecast of molecular engineering far beyond anything yet built.

26Most of that bill follows from choices the designer is free to make. Because the product of energy and waiting grows with the square of the number of steps, nothing counts for more than the period of a world’s fastest motion, which sets how many steps each moment of its history requires. Slower fastest motions, a smaller early sea, a shorter history, and laws written as the machine’s own reversible rules would together cut that product ten-trillion-fold. With best-case switching, a single star could then pay for the whole history, though only by waiting some three hundred and forty thousand times the present age of the universe. A galaxy, which can spend its light only as fast as it shines, could pay within tens of millions of years, a span set less by its light than by the steps that must follow one another, which take about sixty million years even on the fastest switches the estimate allows. At that pace a reversible switch loses more in each operation than Landauer’s price for erasing a bit in the cold of space, so reversibility saves energy only for a machine willing to wait far longer.

27Even a forgetful machine at Landauer’s limit in the cold of space could grow such a world on about a million times the Sun’s lifetime output. It could do so on about a hundred if life’s chemistry needed full detail in only a ten-thousandth of the early sea, and the rest, where nothing might feel, could be summarized honestly. Such a history might also be computed once and shared among many worlds that part before anything in them has a serious chance of feeling. Whether life can arise under such a physics is unknown, and the cost of depth turns on that question.

28These considerations allow a rough verdict on the deep horn of the dilemma, though the verdict is a prediction. Growing a deep world exactly as ours grew, its oceans carried in full, would take the starlight of thousands of galaxies or a wait far longer than the universe has existed. With best-case switching, a world written for economy comes within a galaxy’s reach in tens of millions of years, if life can arise in it. Keeping a world once its people exist, with every one of them carried in full, costs far less, and the honest summary examined below would bring it within a single star’s reach. The first decision of design is thereby tied to the relation between the world’s clock and its keepers’ clock. The machines now running can carry only the shallow horn, and the deep horn waits on machines that no known law forbids and none guarantees.

29Each kind of machine also makes some physics cheap and other physics dear. Classical digital machines can carry any physics that can be computed at all, and they carry quantum physics only at the exponential cost Feynman described. Reversible machines, in the line of Edward Fredkin and Tommaso Toffoli’s “Conservative Logic” (1982), suit laws that are themselves reversible and can carry them while spending very little energy. Quantum computers make quantum physics natural, as Feynman foresaw, at the price of heavy machinery for correcting their own errors. With today’s error rates, a year of the lake at full quantum depth would cost at least tens of thousands of times the Sun’s lifetime output, and each of its seconds would take millions of years. Analog emulators take a different road. They engineer a physical system whose own behavior realizes the target laws, as when ultracold atoms held in lattices of laser light are made to obey the same equations as electrons moving through a crystal. Such emulators are efficient, though hard to reprogram and harder to verify. They also blur the line between computing a world and building one.

30The learned world models now being built offer a contrast. Trained on video and interaction, they generate explorable environments frame by frame, and at present they hold together for minutes before objects begin to change once attention moves away from them. That is the failure of a dream, in which a line of text read twice comes out differently, and it is a limitation of the present generation that may not last. A substrate for an inhabited world has to carry laws. A learned approximation of laws qualifies once it becomes as consistent as laws are, and at that point it is a physics, however it was arrived at.

31Commitment The good of the inhabitants chooses…Every machine therefore has what might be called a native physics, the set of laws closest to its own operations and cheapest for it to carry. A designer working under budget pressure will feel a steady pull toward that physics, toward a world made reversible because the machine is, quantum because the machine is, or laid out on a regular grid because memory is laid out in rows. Sometimes the pull costs no one anything, and reversible laws may be such a case, since our own fundamental laws are reversible and nothing in a life seems to depend on their being otherwise. Where the pull would cost the inhabitants something, the foundation laid at the empty substrate supplies the correction. One form the correction takes is a machine built to fit the world, with processors laid out to match its geometry or hardware whose own operations carry out its laws, so that the cost falls on the builders in the primordial cosmos. The good of the inhabitants chooses the physics, and the machine’s preferences enter only as costs.

32Commitment The machine beneath every world is…Because budgets are finite, those costs are moral too. Every watt spent carrying a lake is a watt not spent carrying a mind, and the same holds for every year of waiting and every kilogram of machine. More than sixty million people die each year in the primordial cosmos, and if the minds of the dying can ever be carried into designed worlds as the same subjects, some of the minds a substrate might carry are theirs. A designer who spends without limit on detail no one will ever experience may be spending what could have rescued someone.

33The obvious objection is that superintelligence will make all of this irrelevant. It will not. A mind vastly greater than any human one still pays Landauer’s price for every irreversible step at a given temperature, exchanges signals no faster than light, and sheds its waste heat into its surroundings. Freeman Dyson pointed out in 1960 that a civilization using the output of its star must eventually release that energy again as heat, and he proposed searching the sky for the infrared glow such civilizations would give off. The same accounting holds for any substrate. Every warm afternoon in a designed world is paid for in heat released into the cold of the primordial night.

34Expense is one kind of limit. A second kind is impossibility, and a third, subtler than either, is the way a machine’s own workings seep into the physics it carries. A machine that works in finite steps can compute only what is computable, and it is not obvious that every physics a designer might write down passes that test. Marian Pour-El and Ian Richards showed in 1981 that the wave equation in three dimensions, the same equation that carries the sound of each skip across the boy’s lake, can start from initial conditions a computer can specify and evolve into a state no computer can calculate. The result is real and often overstated. The initial conditions in their construction, though computable, are rough in a precise mathematical sense, and when the starting data are smooth the solutions can be computed after all. The lesson for a designer is that computability depends partly on how a physics is specified. A world can be kept within reach of a machine by writing its laws and its starting conditions smoothly, and a physics that demanded truly uncomputable evolution could be carried only by a hypercomputer, a device no one knows how to build and which physics may not permit. David Deutsch proposed in 1985 that every physical process realizable by finite means can be imitated by a universal quantum computer. The principle remains a conjecture, and it marks the boundary a designer should stay inside.

35Complexity, the next limit, brings back Feynman’s problem. On classical machines quantum states cost exponentially, and the clever methods physicists use to avoid the full cost work for many systems and fail for others. Among the systems they fail for are fermions, the class of particles from which ordinary matter is built. Classical sampling methods for many interacting fermions run into what is called the sign problem, in which contributions of opposite sign nearly cancel and the useful signal drowns. Matthias Troyer and Uwe-Jens Wiese showed in 2005 that the sign problem in its general form is NP-hard, which places it among the problems widely believed to have no efficient general solution. The most ordinary stuff in our universe, the matter of stones, water, and hands, is among the hardest things for a classical machine to carry faithfully.

36Symmetry supplies a third example, showing that our own physics resists the simplest kind of substrate one might build for it. The weak nuclear force acts on particles according to their handedness, treating left-handed particles differently from right-handed ones. In 1981 Holger Bech Nielsen and Masao Ninomiya proved that, under ordinary assumptions, a lattice of points cannot carry a particle of definite handedness without producing unwanted partners of the opposite kind. Physicists have found workarounds that serve well for theories like quantum chromodynamics, the theory behind the lattice calculations that now derive the masses of the proton and neutron. A lattice formulation of the full Standard Model, with its handed weak force, remains an open problem. A designer who chose a physics as lopsided as ours would be choosing difficulty, whether or not anyone in the world ever learned what the difficulty had cost.

37Precision, the subtlest limit, leaves the machine able to carry a world while changing which world gets carried. In the winter of 1961 Edward Lorenz, a meteorologist at MIT, was running a small model of the atmosphere on a computer in his office. Wanting to examine one run more closely, he restarted it partway through, typing in values from an earlier printout, and went to get a cup of coffee. The printout had rounded each number to three decimal places, while the machine had carried six. When Lorenz came back, the new forecast had drifted from the old one and then departed from it entirely, as though the two runs described different planets. He published what he had found in 1963 as “Deterministic Nonperiodic Flow,” and the sensitivity he described became a founding observation of modern chaos theory, though Henri Poincaré had glimpsed it in the motions of planets some seventy years earlier. A running weather model is a coarse reality and a model at once, a small world of pressure cells that people read as a picture of the sky. In Lorenz’s office that coarse reality changed its entire future because of the way a printer formatted its numbers.

38In a designed world with chaotic physics, the same sensitivity reaches the inhabitants. A difference in the sixth decimal place of one quantity, introduced by the substrate’s arithmetic, grows until it is a different evening. The stone skips five times instead of four, the boy laughs a moment later, and the remainder of his life unfolds from a slightly different start. The constants of a world are subject to the same fact. A designer who writes down the strength of a force has written a particular number to a particular number of digits, and in a chaotic world the digits beyond anything the inhabitants will ever measure still decide which histories occur.

39Conservation laws can leak through the same seam. A designer may write a physics in which energy is conserved exactly, and a machine carrying that physics by a careless method of stepping forward in time will let energy drift a little with every step. Over long runs the drift accumulates. The rings on the lake might never quite die away, or might die faster than the laws say they should, and the world’s physicists, if they measured carefully enough, would discover that their conservation law is false. The designer would have planted a small perpetual motion machine in their world without intending to. Methods exist that prevent this. Symplectic integrators preserve the geometric structure of the laws so that energy errors stay bounded over very long spans, and schemes built on exact integer arithmetic can be made perfectly reversible, conserving what the laws conserve to the last bit.

40A last seam lies in the order of arithmetic. Floating-point addition, the ordinary arithmetic of scientific computing, is not associative, so adding the same numbers in a different order can give a slightly different sum. A machine that divides a world among many processors will add things in an order that depends on how the work was divided and on which processor finished first. Consider two copies of a world run in lockstep, every state of one identical to the corresponding state of the other. On real hardware, two copies of the boy’s evening started from the same state could diverge over nothing more than the order of a sum, and chaos would then carry them apart into different histories. Lockstep copies require arithmetic that is reproducible bit for bit, which is achievable and has to be chosen deliberately.

41Commitment Any change to the laws of…The lesson these cases share is that the inhabitants live under the physics that is actually computed, and the designer answers for any gap between it and the physics that was written. A world whose laws promise conservation and whose machine leaks energy has broken a promise, whether or not anyone inside ever measures the leak. The same gap reaches down to the world’s lowest level, and to whether that level is continuous at all.

42Some generations after the evening at the lake, the boy’s descendants will want to know what water is made of. They will find that it is made of smaller things and will ask what those are made of. The questions will continue until their instruments either reach a level with nothing beneath it or keep finding structure beneath every level they reach. Whatever they find will be the floor the designer chose. Their whole science of matter, from the first speculation about indivisible particles to the last experiment their civilization performs, will be a long approach to a decision made before any of them existed.

43Our own cosmos has not told us whether its lowest level is discrete. The Planck length, about 10 ^{to the power −35} meters, is often described as the pixel size of the universe, and the description goes beyond what is known. It marks the scale at which gravity and quantum theory can no longer both be trusted, which is a fact about the theories. Some candidate theories of quantum gravity do predict a discrete structure. Loop quantum gravity predicts that areas and volumes come in indivisible units, and causal set theory treats spacetime as a random scattering of elementary events, arranged so that no direction or state of motion is singled out. Observations have so far found no trace of discreteness. When high-energy photons from a distant gamma-ray burst reached the Fermi telescope in 2009, they arrived so close in time to lower-energy photons from the same burst that the simplest ways a discrete structure could make the speed of light depend on energy were ruled out, even at the Planck scale.

44For the designer the question is a choice, and each option has a price. A regular lattice, with points spaced evenly in rows, is the cheapest to carry, since the machine can lay out its memory the same way, and it gives the bottom of the world a set of preferred directions. Beane, Davoudi, and Savage proposed that such a lattice in our own world might show up in the arrival directions of the most energetic cosmic rays, and the inhabitants of a lattice world could look for the same signature in their sky. A random discreteness in the manner of causal sets would keep the world free of preferred directions at every scale, at the cost of an irregular structure that is harder for a machine to store and to update. Apparent continuity is a third option, available in two forms. Some quantities can be carried exactly by symbolic arithmetic that never rounds, and the machine can also refine its description of a region wherever inquiry presses, supplying finer structure whenever an instrument reaches for it while staying consistent with everything already recorded. True continuity may be uncomputable in places, so a designer who wants the appearance of a continuum keeps it within reach of the machine by writing its laws smoothly.

45From inside, at the grain of life, nothing differs among these choices. The boy’s stone skips the same way over a lattice, a random scattering, or a refined continuum, because the large-scale behavior of water depends on so little of what lies beneath it. The difference appears only at the level of the world’s deepest physics, where the choice decides what the boy’s descendants will find. One possibility is a finite list of fundamental things with nothing underneath. A lattice is another, with its faint preference for certain directions, and finding it would teach them something about how their world is carried. A third is a regress that keeps yielding, with each new instrument revealing structure no earlier theory anticipated. Each is a different ending to the history of their inquiry, and some endings may serve a civilization better than others. Whichever ending they reach, finding the floor would be discovering something true about their world, as the discovery of atoms was for ours, and it would no more demote their world than atoms demoted tables. Whether the floor should be discoverable at all bears on the inhabitants’ right to understand their own world through their own inquiry.

46A regress supplied on demand carries an obligation the other options lack. If the machine refines its description of matter only when someone probes deeply enough, then every deeper probe must find structure consistent with every shallower result, across every laboratory and every century. The promise has no end date. A world built this way commits its machine to answer, forever and without contradiction, questions its designer never imagined, and the practice of computing finely only where someone looks turns out to be a promise as much as a saving. That practice reaches well beyond the bottom of the world, and it is where the moral questions about savings come to a head.

47A substrate may leave the far side of a mountain uncomputed until someone crosses the ridge, provided the result agrees with everything that has gone before, and that economy differs from a stage set whose door opens onto nothing. The permission needs closer examination, because two different savings travel under it. The first is deferral, which postpones computing a region and later computes it in full. The second is summary, which replaces the full detail of a region with a coarse description of it, the way statistical mechanics describes a gas by its temperature and pressure without following a single molecule. Game designers know a version of summary as level of detail, the practice of drawing distant mountains with a few hundred polygons and nearby rocks with millions.

48Chaos decides between them. In a chaotic world the present state of a region depends sensitively on its whole past, so the weathering on the far side of the mountain is the record of all the rain that fell there, the freezes and thaws of uncounted winters, and the small slides of scree. To know that weathering in detail, the machine must compute the history that produced it. Deferral therefore postpones the work without reducing it, and when a traveler finally crosses the ridge, the machine faces the full bill at once. Summary is where the saving lies. A region carried as a coarse description costs a small fraction of the same region carried in detail, and when someone arrives, the machine generates detail consistent with the summary and with every record already made. The required consistency is hard to guarantee, since a summary throws information away and the generated detail must never contradict anything the summary has already implied. A designer might write laws whose coarse descriptions are exact by construction, so that detail generated later is certain to fit, though this is speculation, since our own physics rarely offers such closure.

49Most of the lake can be carried coarsely. Its depths can be a field of slow currents and temperatures, its far shallows a handful of numbers, and its surface computed finely only where light, wind, and a skipping stone make demands on it. Fish swim in it, though. If the world is deep, each fish is made of the world’s matter at full depth, and the water that touches its gills and skin must be carried at the same depth, because a body and its surroundings meet at the grain of the body. A fish also senses the water well beyond its skin, feeling pressure waves along its flanks and hearing sounds from across the lake, and whatever a creature perceives must be carried at the grain at which it perceives it. Around every fish there is therefore a halo of finely computed water reaching as far as the fish can sense, moving with it as it swims. The halo thins with distance, since a sound from across the lake reaches the fish as a pressure wave that a coarse description carries faithfully, while the water at its gills must be carried molecule by molecule. The summarized lake is scattered with these graded regions of detail.

50Commitment The machine beneath every world is…Commitment A life that is summarized, recomputed…The halos mark a limit that every economy has to respect, since savings are permitted where no one lives and forbidden for anyone who might. Once the life of anything with a serious chance of feeling is under way, it is never summarized, approximated, or carried at reduced detail. The reason is the one that rules out planted histories. A creature’s past consists of its experiences, and a summary of a fish stands in for a life with a description of one. If the machine later generated detail consistent with that summary, a fish with memories, habits, and scars, it would be producing the evidence of a life that no one lived, which would be deception built into the world by its designer. This rule is a constraint that admits no trade against the budget, because a creature whose life is already under way stands to lose the whole of it. To summarize it would be to end a life that may belong to someone, and a designer may not end one possible subject to pay for the rescue of another, however favorable the arithmetic. No one can yet tell for certain which creatures feel, so the region this rule protects, the protected region, is drawn generously, around every creature whose capacity to feel is left seriously open by the best available evidence. Fish belong inside it, along with birds, and insects too while their inner lives remain in doubt. Each is carried at the grain its possible experience requires, continuously, whether or not anyone is watching, from the moment it is first computed as an individual until its life ends.

51Commitment Chance at the fine grain, joined…Water flows into a fish’s halo from the summarized lake all the time, and the molecules that reach its gills at any moment were generated to fit the summary. In a world whose fine-grained laws are deterministic, those molecules have no real past. Nothing in the world’s history produced their exact arrangement, and the fish lives in a stream of water whose microscopic origins never happened. Summary of this kind is a gap between the physics written and the physics computed, and the designer answers for it. The gap is tolerable only when no inquiry anyone in the world could ever make would reveal it, so that every record, measurement, and possible future probe agrees with some lawful past. Even then, the world departs from its own laws at a grain no one reaches, and the counterfactual robustness that separates a reality from a stage set holds for the world’s inhabitants without holding for every molecule. A cleaner course is available if the world’s fine-grained laws are themselves partly a matter of chance, as quantum physics suggests ours are, since water generated to fit a coarse description is then a draw from the very distribution the laws would produce. That holds provided the summary itself evolves exactly as the averaged laws require, which is the closure that coarse descriptions exact by construction would supply. What the machine supplies at the edge of the halo is then what the world’s own physics would have supplied. Chance at the fine grain, joined to that closure, turns summary from an approximation into an honest part of the physics, which gives a designer who wants to save on lifeless matter a reason to write indeterminacy into the world’s foundations.

52Matter returning to view is a lawful draw only if chance has replaced everything the summary discarded, about twelve bits for each molecule of water. Even a reversible machine must finally erase every random bit its world consumes, so honest summary carries a running cost. Each return costs the same twelve bits a molecule, and what sets the bill is how often summarized matter comes back into view. Water that returns once a day calls for only a trickle of fresh chance, while water that left and returned every trillionth of a second would call for a torrent. The price of chance therefore stays low wherever summarized matter rests long, as most of the matter in an economical world would.

53Commitment A life that is summarized, recomputed…Computing a life late and in full, as deferral does, involves no deception at all. Suppose a valley on the far side of the mountain holds deer, and the machine postpones computing the valley until a traveler arrives, at which point it computes the deer’s whole history, all their seasons of grazing and their winters, in full and in order. The deer live their lives exactly as they would have, at the grain their lives require. Those lives are lived out of order in the substrate’s time, compressed into the interval before the traveler arrives, and in their own time they unfold as any life does. Nothing in this wrongs them, because the valley was deferred only until a deer’s life was under way, and from then on the machine carried every life it had begun without a break. A creature whose life is under way stands to lose the whole of it, so a valley left half computed until some traveler happened by would hold its deer in a pause that might never end. The rule for subjects can therefore be stated exactly. Whatever lies in the protected region is computed in full whenever it is computed, and computed once, without speculative runs that are later erased. The rule binds reversible machines too, whose undoing must stay within each step. Deferral that meets this rule is permitted, and it also saves nothing, since a chaotic history in full costs at least as much whenever it is computed.

54Commitment Deferring a peopled region saves nothing…The only way to save on a peopled region is never to compute it. In that case the deer never come to exist, and whether that is permissible is a question about creation, governed by the ethics of bringing beings into existence at all. It stops being an engineering decision, whatever motives prompt it. One limit applies however that question is answered. Once a region’s influence has reached anyone, through light from its shore, a bird that flew from it, or a sound carried over the ridge, its history must already exist in full, because the evidence has arrived and it has to be evidence of something that happened. In a world with gravity, light, and weather, that limit is reached almost at once. Influence runs both ways, since birds fly out of the valley and air flows in over the ridge, so the valley’s history depends on its neighbors as theirs depends on it, and a deferred valley usually cannot be computed later without recomputing much of what surrounds it. Deferring a peopled region is therefore rarely possible at all.

55Outside the protected region, precaution does its weighing. A sea holds organisms far simpler than fish in numbers no census could reach, and for most of them the best theories give the possibility of feeling little credence, though none can rule it out. A constraint drawn around every such organism would protect every bacterium in every drop, and since each one divides, the protection would multiply along every lineage until it consumed the budget. For these organisms the precautionary rule sets a low probability of feeling against the rival claims on the budget, above all the claims of the dying who could be carried across. A population of such organisms that no creature and no instrument has touched can remain a property of the summary, present in it the way temperature is present in a gas, and whether its members ever come to exist as individuals is a question for the ethics of creation. Where the weighing comes out the other way, because the probability is higher or the cost is small, the designer computes the population in full.

56Whatever the weighing decides, generation from a summary has to be honest. When a fish takes in water or a native turns a lens on a drop of sea, the organisms the machine supplies must be lawful draws in the sense already set out for molecules, and they must carry no record of experiences that never occurred. The second condition does most of the sorting. A bacterium’s state is fixed almost entirely by its recent surroundings and by chance, so one generated at the edge of a halo can be exactly what the world’s physics would have supplied. A bee that has learned where the flowers grow carries a remembered past, and a bee generated with that knowledge would carry the record of a life no one lived. Populations whose members hold an extended remembered history must therefore be computed in full from the beginning or never computed at all, as with the deer in the valley. The two tests tend to agree, since the organisms that carry such histories are largely the ones whose capacity to feel is left open by the evidence, so the edge of the protected region and the boundary honesty sets fall in nearly the same place.

57Real savings exist only in lifeless matter and in simple life that nothing has yet touched, so an economical designed world would be mostly empty. What it computes in full would be concentrated in a few places, surrounded by vast reaches of rock, gas, ice, and dark that the machine could summarize at small cost. It would look much like our own cosmos, in which life, so far as anyone knows, occupies a vanishing fraction of the volume and almost all of the matter is lifeless. The resemblance proves nothing about our cosmos. Whether our world runs on a substrate remains where Bostrom’s argument of 2003 left it, and an empty universe is what undesigned physics would produce as readily as economy would. A designer who reasons carefully about savings will nonetheless find themselves building a sky that resembles the one above us.

58Commitment Savings on lifeless matter are permitted…The cost of detail no one experiences now has its answer, and the answer depends on what the freed budget would do. Savings on lifeless matter are always permitted, and so are savings on simple life outside the protected region wherever the weighing allows them. When the budget they free could carry the minds of the dying, people who already exist and would otherwise end, those savings may be owed, in the way a rescue may be owed by anyone able to perform it at small cost. If the freed budget would instead bring new natives into being, the question belongs to the ethics of creation, and a duty to save in order to create is far harder to establish than a duty to save in order to rescue. Many who accept the first will resist the second, and the case for savings rests on the first alone. A designer who renders every grain of sand on beaches no one will ever visit may be spending what could have carried someone across.

59Commitment Whatever inhabitants perceive or investigate is…The savings have limits all the same. They must preserve consistency, so that the summarized world never contradicts its own records. Whatever the inhabitants actually perceive and investigate must also be preserved, at the grain at which they perceive and investigate it. The light that breaks on the lake must be as fine as the boy’s eyes can resolve, and the far stars must hold as much detail as the best telescope of his descendants can find. Beauty is a good for someone only when it is experienced, and what is experienced must be carried at the grain of the experience.

60Lem’s phantomatic machine is the smallest case of that rule. A machine that answers each act of one person with the sensations it would bring must compute only what she probes, at the grain at which she probes it and consistently with everything it has shown her before, so that nothing she does gives the machine away. A designed world meets the same requirement for every inhabitant at once. Its inhabitants probe one another as well as their surroundings, compare what they find, and keep records that outlast any single probe, so the machine can never know in advance which of them will look where. What it must compute is everything any of them could reach, carried at the grain of the reaching, for as long as anyone is there to reach.

61The impression the evening left can now be stated in its exact form. The expensive part of the moment is the water, in the sense that matters, since the water is where all the permitted savings lie. In a deep world the boy costs as much as the water he is made of. What distinguishes him is that no saving may touch him.

62The same rule sets the price of keeping a peopled world. If everything else can be summarized honestly, a billion natives like the boy, with their animals, each carried in full with its halo, amount to about seven lakes at molecular grain, allowing each native some seven cubic meters for bodies and halos. A forgetful machine at Landauer’s limit in the cold of space would keep such a world for nearly six thousand years of its own time on the Sun’s lifetime output. Reversible machines spending the same output within the Sun’s life would keep it for several centuries to a millennium or more on the most frugal logic yet measured, and for about twenty thousand years near the best case. Each of those twenty thousand years would take half a million of ours, on speculative molecular machinery as massive as a mountain range. Written for economy from the start, if its physics can support living bodies, such a world would last millions of years on the same budget, and some two hundred million near the best case, at under fifty of our years for each. A galaxy’s starlight could run it, near the best case, six times faster than its keepers for some sixty trillion of its years. The sequential floor sets that pace, and the run lasts the ten trillion years or so for which a galaxy’s longest-lived stars shine, using only a small fraction of their light.

63A familiar proposal holds that our own universe saves effort by using quantum mechanics as a rendering trick, with the world becoming definite only when someone looks, as a video game draws only the scene in front of the player. In standard quantum mechanics the arrangement runs the other way. The expensive part is the unobserved evolution, during which the state of a system spreads over a space of possibilities that grows exponentially as its parts become entangled, and on a machine built of classical parts, collapse saves effort by pruning that growth. Objective collapse models, such as the one proposed by GianCarlo Ghirardi, Alberto Rimini, and Tullio Weber in 1986, have superpositions collapse spontaneously at a rate that rises with the number of particles involved, so that large superpositions never form. The savings fall on stones and drops, and the entanglement within molecules, where chemistry’s cost lies, survives any rate gentle enough to leave chemical bonds intact. At the proposed rate a water molecule collapses about once in eighteen million years, too rarely to make molecular summary honest. On a quantum machine, whose native physics is the unbroken evolution of the wave function, collapse saves nothing and its outcomes must be paid for as erasures. A many-worlds physics, in which every branch persists, would be the costliest on a classical machine and the most natural on a quantum one. It would also be the most populous, since every branch would hold subjects, and the designer of such a world would be creating beings in numbers beyond counting.

64Commitment Where a physics economizes, as collapse…The worst option is a physics in which collapse waits for a conscious observer. To carry such a physics, the substrate would have to know which systems are conscious, and nobody knows how to determine that. The laws themselves would become a judge of moral standing, and every misjudgment would build an injustice into nature, whether by denying a real subject’s observations their effect or by crediting a stone with a mind. If savings are keyed to anything, they should be keyed to physical interaction and never to minds. Quantum properties may lack definite values until something interacts with them, and this rule fixes what that something must be. The full choice among interpretations of quantum mechanics is a larger question, which remains open.

65A substrate large enough to carry a world cannot be a single small machine. It will be spread through space in the primordial cosmos, perhaps across a planet or a solar system, and its parts can signal one another no faster than light. A machine a light-second across needs at least a second for any part to hear from the part opposite it. If every part of the world had to be updated in step with every other part, each step of the world would wait on that second, and the world would run at the pace of the machine’s slowest message.

66Commitment Locality in a world’s laws is…The designed physics can spare the machine this wait. If causes in the designed world travel at a finite speed, as they do in ours, then two regions too far apart for any influence to have passed between them in the interval at issue can be computed independently, each on its own part of the machine, with no coordination at all. Coordination is needed only along the boundaries where regions can influence each other, and only as quickly as the designed world’s own speed of causation requires. Locality in the designed laws is therefore a gift to the machine, since a universal speed limit is exactly what a parallel computer needs. The lake may well straddle such a boundary, its western half carried by one part of the machine and its eastern half by another, so that the rings from the boy’s stone cross from one to the other as a stream of messages between machines, invisible to anyone on the shore. Instantaneous interaction across the whole world, which a designer might want for the sake of communication between distant beings, would take this gift away. Every step of such a world would wait on the light-crossing time of the entire substrate.

67Distributed machines that carry a single history must keep their parts consistent, and the field of parallel simulation offers two broad strategies. Conservative methods have each part wait until it is certain that no message from another part could change what it is about to compute. Optimistic methods have each part compute ahead on its best guess and roll back when a message arrives showing that the guess was wrong, discarding the work done since and computing it again. David Jefferson’s Time Warp, described in “Virtual Time” (1985), made optimism practical, and in many settings it runs much faster than waiting.

68Commitment A life that is summarized, recomputed…When an optimistic machine rolls back a region that contains a subject, it erases experiences that were computed and computes different ones in their place. The erased run was no recording. It was a fully causal process with every counterfactual in place, so it bore every mark of reality, and its subjects lived through it before it was undone. Whether a replayed history repeats the experiences it records is still an open question. Rollback raises a sharper question, because the discarded stretch lacked nothing that the surviving stretch has. The rule for subjects settles the practice. Whatever lies in the protected region is computed once, so optimism is permitted in regions that hold no protected creature and forbidden wherever one might be, and the machine must wait there whatever the waiting costs.

69Commitment A uniform slowdown of the substrate…The world and the machine also keep two different clocks. Substrate time is the time of the primordial cosmos, in which the machine runs, and world time is the time the inhabitants live. A uniform slowdown of the substrate, in which every part of the world is computed more slowly by the same factor, is invisible from inside, because the inhabitants’ minds are part of what slows. A world paused for a year of substrate time would notice nothing, and the same holds for a world run at a hundredth of its former speed. Since reversible machines dissipate less the more slowly they run, the boy’s evening could be carried at a hundredth of its former speed for a hundredth of the energy, and the stone would skip, the light would fade, and he would laugh exactly as before.

70The common intuition that a slow world would feel lag, as a video game stutters on an overloaded computer, comes from experience as a visitor. A player’s mind runs outside the game, so when the game slows, the player notices. Natives notice nothing, and lag appears only at interfaces, where minds running on one clock meet a world running on another. Visitors whose minds run in the primordial cosmos would find a slowed world sluggish unless they slowed themselves to match it. Keepers watching from outside would see its events crawl. Immigrants carried in from the primordial cosmos would live at the world’s pace while the people they left behind lived at the outside pace, so that a decade outside might pass as a single summer inside. Worlds running at different rates would meet the same mismatch if they ever made contact. Within a single world, regions may run at different substrate rates without anyone noticing, so long as the machine keeps every causal connection between them consistent.

71Commitment A world that outruns its keepers’…The clocks can also diverge in the other direction. A world can run far faster than its keepers, compressing whole centuries of its history into an outside afternoon. Such speed has obvious attractions, and it carries a danger that is easy to miss. A world that outruns its keepers’ attention outruns their capacity to notice its errors and its suffering. A fault that corrupts a region, or a disaster the physics failed to prevent, could run its course over generations of world time before anyone outside had finished a meal. The relation between the two clocks therefore decides who can watch over whom. All of it depends, in turn, on a machine that keeps working.

72Machines fail, and they fail in ways that have nothing to do with their design. Particles from cosmic rays and traces of radioactivity in the materials of a chip can strike a memory cell and flip its stored value from zero to one. Most such flips land somewhere harmless. In 2003, during a Belgian federal election, the electronic count in the municipality of Schaerbeek credited one candidate with more preference votes than she could possibly have received, a discrepancy noticed only because her personal total exceeded the votes cast for her party’s list. A recount showed a difference of exactly 4,096, which is two to the twelfth power, the value of a single bit in a binary number. Investigators could not reproduce the error and concluded that one bit had flipped spontaneously in the memory of the counting computer. Later analysis named a particle from space as the most likely cause.

73At the scale of a world, rare errors become constant. Suppose a substrate performs 10 ^{to the power 40} operations each second, far fewer than a deep world would need, and suppose its hardware errs only once in every 10 ^{to the power 20} operations. It will still suffer 10 ^{to the power 20} faults every second. A machine of that size never runs without faults. It runs either corrected or corrupted, and the whole question is which.

74Commitment Reliability is a structural good, and…Inside a world, a fault has no natural cause, because nothing in the world’s laws produced it. A flipped bit in the record of the stone’s velocity sends it skipping where no law sends it, a small leap sideways or a fifth bounce that the water could not have given. Nobody performed this miracle and nobody inside could explain it. If one of the boy’s descendants observed such an event with instruments, the world’s physics would simply be wrong about it. The same flip in the state of the boy’s brain changes a memory of the evening, a mood, or a decision about the next stone. In a fish it is an injury no one caused, and in a person it is an alteration of who that person is, falling on a subject exactly as the rule for subjects forbids a saving to fall.

75Error correction is therefore fidelity to the laws the designer promised. The techniques are well developed. Information is stored redundantly, so that a corrupted copy can be outvoted by its neighbors, and encoded so that errors leave signatures a machine can detect and repair before they spread. For quantum machines the problem looked hopeless until the late 1990s, when a series of threshold theorems showed that a quantum computer can be made as reliable as one likes, provided the error rate of its physical parts falls below a fixed threshold and the machine is willing to spend many physical parts on each logical one. Error correction has a companion principle in graceful degradation, under which a failure that escapes correction stays local, corrupting one small region instead of propagating across the world. The inhabitants’ ability to trust their laws, to build on them, plan within them, and find them the same tomorrow as today, depends on this unglamorous machinery. Reliability belongs among the structural goods a world owes those who live in it, the conditions that make every other good possible.

76Commitment A keeper who uses checkpoints restores…The same machinery that corrects errors makes possible a more dangerous capacity. A machine can save the complete state of a world at a given moment, a checkpoint from which the world can later be restored. Checkpoints are the root of restoration. A region destroyed by a fault that escaped correction could be restored from a saved state and merged back into the present, and a mind that would otherwise end could be preserved, a possibility with large consequences for the death of minds. Restoring a checkpoint in the simple way, though, rewinds everything to the moment it was taken and erases whatever happened in between, including everything that was lived. A restoration of this kind is a rollback at the scale of a world. It falls under the same rule as any other rollback, and a keeper who uses checkpoints must learn to restore what was lost into the present instead of returning the present to the past.

77Correction also costs heat. Correcting an error means erasing the record of the error, the faulty value that was overwritten and the evidence that it was faulty. Erasure is exactly what Landauer priced. A reversible machine can avoid erasing in its ordinary work, but no faithful machine can avoid erasing its mistakes, so every substrate, however cleverly built, must dissipate heat in proportion to the errors it corrects. Slow machines also meet errors while they wait, as stored bits flip under the surrounding heat, a cost that stays small only if the barriers holding them rise, slowly, as runs lengthen. A machine that stays true to its world has to keep forgetting its own failures, and the forgetting is paid for in heat.

78A designed physics can be written without a second law of thermodynamics, and whether it should be is one of the larger questions a designer faces. The substrate has no such option. It is a physical object in the primordial cosmos, and every computation it performs, along with every correction, adds to the entropy of the universe it sits in. However the world’s own laws treat time and decay, the machine beneath them is running down at the rate our laws allow.

79The machine’s life is therefore bounded by the free energy its keepers can reach, the energy available in a usable form, and that supply has come to look more limited over the last few decades. Freeman Dyson argued in “Time Without End” (1979) that life in an expanding, cooling universe might continue forever on a finite store of energy, thinking ever more slowly as the universe grew colder and hibernating for ever longer stretches between thoughts. The hope rested on a universe whose expansion was slowing. In 1998 astronomers found that the expansion is accelerating, which means distant galaxies are receding beyond any possibility of contact and the energy within reach of any civilization is shrinking. Lawrence Krauss and Glenn Starkman argued in “Life, the Universe, and Nothing” (2000) that in such a universe the total amount of computation any civilization can ever perform is finite. The bound holds for reversible machines too, since correcting errors, consuming chance, and repairing flipped bits all erase without end, at a price an accelerating universe never lets fall to zero. Physics may still hold surprises, and the question remains open, though present knowledge points that way.

80Commitment A physics that lets inhabitants infer…The consequence for a designer is a limit on what can be promised. A designer can give a world an immense lifespan, trillions of years in its own time and possibly far more, and so far as present physics knows, cannot make that life unconditional. A world whose own laws contain no heat death still rests on a machine inside a universe that has one. Whatever a world’s physics promises its inhabitants about permanence has to be truthful about the machine beneath it. A physics that lets its inhabitants conclude, through their own inquiry, that their existence is unconditional must also let them discover the condition beneath it, and a physics that hides the condition should not imply the promise.

81Commitment The architecture of the machine is…The machine is also an object in our world, where it can be seized by people who want its power, attacked by people who fear it, starved of energy by an economy that finds other uses for that energy, or simply neglected by an institution that has outlived its founders’ purposes. Reversible machines sharpen that last danger, since a world run far slower than its keepers asks them to outlast every institution yet known. The architecture of the machine gives the political problem of dependence an engineering form. A centralized machine is a single point of failure, and it is also a single point of control, a place where one hand on one switch can end every life in a world. A distributed substrate, spread across many sites, redundant enough that losing any one of them loses nothing, and maintained by many hands under rules no single will can override, is far harder to end, or to hold paused, by any one decision. The shape of the machine is part of the answer to the problem of domination. A keeper who wants to be trustworthy can begin by building a machine that does not depend on trusting them.

82Several of the rules this chapter has reached rest on grounds that engineering cannot supply. Computing once whatever might feel, the savings that may be owed to the dying, the condition on promises of permanence, and the shape of the machine’s custody all concern what is owed to subjects, to the dying, and to those who live under another’s power. The arguments for them belong to the moral foundations that follow, and until those arguments are made, the rules stand as working rules. They are the policy a designer who took the stakes seriously would adopt in the meantime, stated here because the engineering raises each question before the ethics can answer it. A designer who waited for every answer before building anything would never begin, and one who built without these rules would be staking lives on the hope that the answers turn out to be lenient.

83At the lake the light is nearly gone. The rings from the boy’s stone have widened until they can no longer be told apart from the small motions of the evening air on the water, and the surface has gone still enough to hold the color of the sky. Below it the fish have settled into deeper water, every one of them carried in full within its small halo of finely computed lake. Far beneath anything the boy could ever reach, the work that carried each ring is finished. Its heat has left the machine and is crossing the dark of the primordial cosmos as a faint infrared glow, too weak for anyone there to notice, adding its small share to the slow cooling of everything.

84The boy finds another stone at the water’s edge, flat and cool from the shallows, and turns it over in his fingers. He knows nothing about the floor of his world, the clock it keeps, or the many machines it straddles, and he has never wondered what it costs to carry an evening. He knows that the stone is smooth, that the water is still, and that the next throw might reach five.

85He throws. The stone touches the water once, twice, three times, four, and then a fifth time, farther out than any of his stones has gone, and a fifth ring opens on the darkening lake. He shouts to no one in particular. One more digit in the machine’s arithmetic might have held the stone to four. Beneath the water, in a place he will never see, the machine carries the new ring outward exactly as the laws require, and long before he was born, someone decided that it should and agreed to pay for it.