400 milliseconds: that might be the whole of your life

Part IThe fluctuation

Chapter 1The universe never goes fully dark

"Heat death" suggests a universe that cools towards absolute zero and then simply stops. If dark energy is a cosmological constant, the ending is stranger than that. Matter and radiation thin out as space expands, but the energy density of empty space stays fixed, so it eventually dominates everything. The expansion then settles into a pure exponential, a state called de Sitter space (after Willem de Sitter, who also predicted the geodetic precession in Nº 16), with a constant expansion rate:

H_∞ = H₀√Ω_Λ ≈ 56 km/s/Mpc ≈ 1.8 × 10⁻¹⁸ s⁻¹ Using Planck 2018 values,² H₀ ≈ 67.7 km/s/Mpc and Ω_Λ ≈ 0.69. One e-fold of expansion then takes 1/H∞ ≈ 17.4 billion years.

Exponential expansion gives every observer a horizon: a sphere about 17 billion light-years away beyond which no signal sent from now on can ever reach them. It is the same cosmic event horizon as in Nº 0, which sits at 16.6 billion light-years today and creeps outwards towards c/H∞ ≈ 17.4 billion light-years. In 1977 Gibbons and Hawking showed that this horizon behaves like a black hole's: it has an entropy, and it glows with a temperature.¹

T_dS = ħH_∞ / (2πk_B) ≈ 2 × 10⁻³⁰ K The Gibbons–Hawking temperature. De Simone and colleagues quote 2.3 × 10⁻³⁰ K for the observed vacuum energy.³ This page uses k_BT_dS ≈ 3 × 10⁻⁵³ J.

That is unimaginably cold, but it is not zero, and it never goes away. Today the leftover glow of the Big Bang is 2.7 K. Once expansion is exponential it cools by a factor of e every 17.4 billion years, so it drops below the horizon's own temperature after about 69 e-folds, roughly 1.2 trillion years from now (this page's arithmetic). From then on, in any empty region, the horizon is the warmest thing there is.

Everything else winds down on its own schedule. Star formation ends after about 10¹⁴ years, stellar remnants scatter or fall into black holes, and by around 10¹⁰⁰ years even the largest black holes have evaporated.⁴ What remains is the horizon bath, and it lasts forever.

A finite box that never closes

The key property is that the region inside one horizon has a finite entropy, the Gibbons–Hawking value:

S_dS / k_B = πc⁵ / (ħGH_∞²) ≈ 3 × 10¹²² The horizon area in units of four Planck areas. For comparison, Nº 2 found a human brain holds about 6 × 10²⁶ bits of thermal entropy.

A system with finite entropy, left alone for unlimited time, keeps fluctuating. Almost every fluctuation is tiny. Very rarely, a patch of the bath fluctuates into an ordered, low-entropy arrangement, and given unlimited time every arrangement the laws allow turns up, again and again. The time to cycle through the most extreme fluctuations, the Poincaré recurrence time, is of order e^10¹²²; Page quotes e^10¹²³ years.⁵ At numbers like this the unit stops mattering: switching from years to Planck times changes the exponent by about one part in 10¹²⁰.

That is the stage for everything that follows: a bath that is almost perfectly cold, almost perfectly empty, and eternal. In such a bath, anything that is merely improbable eventually happens.

Assumptions. Chapters 1 to 8 take three things at face value: that dark energy is a true cosmological constant, that our vacuum is stable, and that empty space really does fluctuate. Chapter 10 questions the third and Chapter 11 the first two, and every one of these escape hatches matters.

Figure 1🌡️ Try it

The long cooling

If dark energy is a cosmological constant · Ch. 11 If dark energy never fades

Time after the Big Bang

Leftover Big Bang glow

Horizon glow2.2 × 10⁻³⁰ Knever changes

Space stretched since today

Which is warmer

Temperature against time, both on logarithmic scales; the time axis is broken at 10¹⁵ years to give the first trillion years room. The glow of the Big Bang follows a flat ΛCDM universe with Planck 2018 parameters. Soon after 10¹² years it drops off the bottom of the chart and keeps falling forever. The horizon's glow never changes. To the far right, beyond the axis, lies the recurrence time, of order e^10¹²² years.Both scales are squashed: each step is a jump of many powers of ten. The Big Bang's leftover glow keeps cooling forever. The horizon's glow never changes, and after about a trillion years it is the warmest thing in empty space.

Chapter 2An idea from Boltzmann's assistant

The idea that we might be a fluctuation is older than modern cosmology. In 1895 Ludwig Boltzmann was defending his statistical explanation of the second law in the pages of Nature. (His statistical argument is why heat runs from hot to cold; Nº 15 shows heat running the other way for a moment, while the entropy still rises.) Critics had asked how a law that runs one way could follow from mechanics that runs equally well backwards. He ended his letter with "an idea of my old assistant, Dr. Schuetz",⁶ aimed at a different question, why our world is so far from equilibrium at all: perhaps the universe as a whole is in equilibrium forever, and our ordered region is a rare fluctuation inside it. We see order because only a fluctuation could contain anyone to see it. It is an early version of what would later be called an anthropic argument.

The flaw was pointed out by Arthur Eddington.⁷ Small fluctuations are exponentially more common than large ones. If observers exist only by luck, the smallest lucky arrangement that produces an observer should vastly outnumber arrangements that also produce a sky full of galaxies. A fluctuating universe should be full of minimal islands of order, not worlds like ours.

Richard Feynman made the point testable in lectures published in 1963.⁸ If the order we see were a fluctuation, the most likely fluctuation would have ordered only what we have already looked at, so every new place we look should be a mess. We keep finding order in places no one has looked before. Feynman concluded that the universe is not a fluctuation, and that "the order is a memory of conditions when things started."

The problem came back in 2002. Dyson, Kleban and Susskind combined the newly discovered dark energy with the idea that a de Sitter horizon has finite entropy. A finite system that lasts forever revisits its fluctuations endlessly, so observers made by small, cheap fluctuations would vastly outnumber those coming from a fluctuation as large as our Big Bang. They argued that these assumptions lead to deep paradoxes requiring major revisions.⁹ Two years later Albrecht and Sorbo gave the minimal observers their name, Boltzmann brains, while asking whether cosmic inflation could compete with random fluctuations as an origin for our universe.¹⁰

Today the Boltzmann brain is used less as a fear than as a test. Cosmologists use it to rule out models and to choose between ways of counting observers in an infinite universe: a theory in which most observers turn out to be fluctuations is treated as broken.³

1895: Boltzmann's letter in Nature, crediting Schuetz
1931: Eddington: small fluctuations should dominate
1963: Feynman: new places should look disordered, and they don't
1977: Gibbons and Hawking: de Sitter horizons have a temperature
1998: The expansion of the universe is found to be accelerating
2002: Dyson, Kleban and Susskind: disturbing implications of a cosmological constant
2004: Albrecht and Sorbo name Boltzmann brains
2006–10: Page, Linde, De Simone and others turn the problem into a test of cosmological models
2014–17: Boddy, Carroll and Pollack question whether the vacuum fluctuates at all; Carroll argues the theories are self-undermining

Figure 2🎲 Try it

Feynman's box

Molecules

On their own side now

Chance it is fully sorted at any moment

Flukes so far

Typical wait for the next fluke

Two gases, one colour each, in a box you can turn. Wait for a fluke: once the wall is gone, the molecules mix and only rarely sort themselves again. The chance of finding all N on their own sides is 1 in 2^N, so each molecule you add doubles the wait (the wait shown is for this simulation's speeds). Feynman's test: 16 windows of 27 molecules each. A fluctuation only had to order the window you saw first, so another window just as neat would be a coincidence of 1 in 2²⁷, about 1 in 10⁸. Each new ordered window multiplies the odds for a real past by about 10⁸; each mixed one divides them by about 10 near the middle, where the gases have begun to mix, or about 75 further out.Molecules jiggle at random. With a handful, the box sometimes sorts itself by pure luck. Add more, and every extra molecule doubles the wait. In Feynman's test, look into new parts of the box: a lucky accident would only have tidied the part you had already seen.

Why "brain"? The word stands for the smallest physical system that can have the experience in question. Nothing in the argument requires neurons. Chapter 7 drops that assumption entirely.

Chapter 3The price of a fluctuation

How unlikely is a brain? Statistical mechanics gives a clean rule: in a system at temperature T, a fluctuation into some configuration is suppressed by a Boltzmann factor. For a de Sitter horizon there is a direct way to see it. The horizon obeys the first law, dS = dE/T, so when an object of energy E appears inside, the horizon loses entropy E/T. The object brings back its own internal entropy S. Together:

P ∝ exp[−(E − TS) / k_BT] = e^−E/k_BT · e^S/k_B Read the other way, 1/T = dS/dE defines temperature as a slope; for a system whose energy has a ceiling, that slope can even turn negative: Nº 17. S counts the microscopic arrangements that look the same from outside (ice is the classic example: Nº 14). De Simone et al. write the nucleation rate as e^−F/T, with F the free energy.³

The entropy term is a rounding error. Ordinary matter carries at most a few units of entropy per nucleon. A 1.4 kg brain contains about 8 × 10²⁶ nucleons, so S/k_B is at most a few times 10²⁷ (Nº 2's thermal entropy for a real brain, 5.7 × 10²⁶ bits, is about 4 × 10²⁶ in these units). The energy term is

E / k_BT_dS = mc² / k_BT_dS ≈ 3 × 10⁶⁶ per gram ≈ 4 × 10⁶⁹ for a 1.4 kg brain This page's arithmetic, with k_BT_dS ≈ 3 × 10⁻⁵³ J. If baryon number is conserved, matching antimatter must appear too, roughly doubling x; nothing below depends on that factor.

The two terms differ by 42 orders of magnitude. What a fluctuation costs is set almost entirely by its mass, and this is the most important fact on the page. Every gram removed makes a fluctuation about e^{3 × 10⁶⁶} times more likely.

What fluctuates	Mass	x in e^−x	Digits in the wait
A single protonHeavy compared with horizon quanta, which are as large as the horizon itself	1.7 × 10⁻²⁷ kg	5 × 10⁴²	2 × 10⁴²
A grain of sand	1 mg	3 × 10⁶³	1.3 × 10⁶³
A Boltzmann fragmentA guess; see Chapter 4	~0.4 kg	1.2 × 10⁶⁹	5 × 10⁶⁸
A human brain	1.4 kg	4.1 × 10⁶⁹	1.8 × 10⁶⁹
A whole human	70 kg	2.1 × 10⁷¹	9 × 10⁷⁰
The Earth	6 × 10²⁴ kg	1.8 × 10⁹⁴	8 × 10⁹³
A Boltzmann solar systemAbout one Sun's mass, borrowing De Simone et al.'s term "Boltzmann solar system"	2 × 10³⁰ kg	5.8 × 10⁹⁹	2.5 × 10⁹⁹
For scale: a low-entropy observable universePenrose's estimate of how special our initial state was. A different yardstick: no fluctuation inside one horizon can cost more than the horizon's entropy, about 3 × 10¹²²	n/a	2.3 × 10¹²³	10¹²³

Waiting times. The rate per unit volume and time is this probability times a prefactor that is irrelevant at this scale. So the expected wait for one brain-mass fluctuation inside a horizon is about e^{4 × 10⁶⁹} in any unit of time you like: a number with about 1.8 × 10⁶⁹ digits.

Brains versus universes. Roger Penrose estimated that the initial state of our observable universe was special to about one part in 10^10¹²³.¹¹ A brain is cheaper than a whole ordered universe by a factor whose exponent has more than 120 digits. That gap is the Boltzmann brain problem in a single line. In a universe that lasts forever, the observers made by the Big Bang are finite in number, while the observers made by fluctuations keep accumulating without limit.³

Figure 3💸 Try it

What would it cost to fluctuate this?

Mass

x = mc² / k_BT_dS

Chance of appearing

Digits in the expected wait

Compared with a 1.4 kg brain

With k_BT_dS = 3.06 × 10⁻⁵³ J, every gram costs 2.94 × 10⁶⁶ in the exponent. The expected wait inside one horizon is about e^x in any unit of time; written out, that number has x/ln 10 digits. One ruler: the price x is marked above; below, each object hangs at its mass (a tick every factor of 1,000 in mass). Drag along it or tap an object. Atom counts are this page's rough arithmetic: a person about 7 × 10²⁷, the Earth about 10⁵⁰, the Sun about 10⁵⁷, the Milky Way's stars and gas about 10⁶⁸, the observable universe about 10⁸⁰.Drag along the ruler or tap something, and see what it would cost to appear out of nowhere. The price is set almost entirely by mass: every gram adds about 3 × 10⁶⁶ to the power. The atom counts are rough.

Part IIThe fragment

Chapter 4The cheapest observer

Eddington's argument does not stop at the edge of the skull. If you are a fluctuation, you are most likely the cheapest fluctuation that could be having exactly this experience. Cosmologists have used this logic since Albrecht and Sorbo, and De Simone, Guth, Linde and colleagues made it precise. Their minimal Boltzmann brain has memories made of random bits that happen to match a real observer's, and it only needs to survive long enough to register one more observation, about a second. In passing, they note that having the same perceptions for one second might take far fewer than 10¹⁴ synapses, and that cutting the time to a single computation could shrink the information needed by many orders of magnitude.³

This page follows that remark to its end. Call it a Boltzmann fragment: the smallest physical system that produces exactly this moment of experience, and nothing else. A fragment does not need a body, eyes or ears, because the scene can be generated centrally, as it is in dreams. It does not need the machinery for acting, because it will never act. And it does not need the parts of you that are not taking part in this moment.

Most of your neurons may be optional

About 69 billion of your 86 billion neurons, four-fifths of the total, sit in the cerebellum.¹² In 2014, doctors in China reported a 24-year-old woman whose CT and MRI scans showed no cerebellar tissue at all.¹³ She had walked unsteadily all her life, spoke with a slur, and had mild intellectual impairment. She was also, plainly, conscious. Whatever the cerebellum does, most of the brain's neurons are not required for there to be someone home.

The cerebellum is only about a tenth of the brain's mass, roughly 150 g. Deleting it shifts the exponent by about 4 × 10⁶⁸, so a brain without a cerebellum is e^{4 × 10⁶⁸} times more likely to fluctuate than a complete one.

How small can a fragment get?

Nobody knows the minimal substrate of a conscious moment. In 2025 a large adversarial collaboration tested the two leading theories of consciousness against each other with preregistered predictions. It found information about conscious content in visual, ventrotemporal and inferior frontal cortex, and its results challenged key claims of both theories.¹⁴ An influential 2016 review argued that the brain regions tracking the specific contents of experience lie mainly in a posterior "hot zone" of the cortex.¹⁵

So any fragment mass is a guess. Suppose one moment of experience needs the posterior cortex plus the thalamic and brainstem systems that keep it awake, wired as compactly as possible: around 0.4 kg a guess. Adding the posterior share of the white matter that connects them would bring it nearer 0.6 kg, which changes nothing below but the leading digit. That fragment costs e^{−1.2 × 10⁶⁹}, and it is more likely than a whole brain by a factor of about e^{3 × 10⁶⁹}.

Memories you are not currently recalling are a different case. They are stored in the same tissue as everything else, so leaving them out saves little mass. What a fragment saves is the need for them to be right. Chapter 8 shows that, by the arithmetic of Chapter 3, getting them right costs almost nothing anyway.

Figure 4✂️ Try it

Delete your brain

Masses approximate

What's left

Chance of fluctuating it

More likely than a whole you

Approximate adult masses, rounded so the brain totals about 1.4 kg; treat them as illustrative. Each gram removed is worth 2.94 × 10⁶⁶ in the exponent. The evidence notes summarise Chapter 4.Masses are rough. Every gram you delete makes the accident about e^{3 × 10⁶⁶} times more likely, as long as someone is still home.

Chapter 5How long is now?

A fragment cannot be a single frozen arrangement of matter. On most physical theories of mind, experience is a process: signals moving, being compared, being combined. A configuration that never changes is not obviously experiencing anything. So a fragment has to run, and the question is for how long.

Timing differences of just 3 milliseconds are enough to create apparent motion, yet on one influential (and debated) model, conscious perception arrives in discrete slices that take far longer to build.¹⁶ In one experiment summarised by Herzog, Kammer and Scharnowski, a vernier (two slightly offset line segments) followed by one offset the opposite way, 60 ms in total, is seen as a single fused vernier. Magnetic stimulation of the visual cortex up to 400 ms afterwards can still change which offset is seen. The percept was not finished until roughly 400 ms later, and unconscious processing outlasted the stimulus about sevenfold.

An older illusion makes the same point more vividly. In the colour phi effect, described by Kolers and von Grünau in 1976, a red dot flashes in one place and then a green dot flashes nearby.¹⁷ People see a single dot that moves, and they report it changing colour abruptly partway along its path, before the second dot has actually appeared. The brain cannot have known the colour in advance. It must have built the experienced path after the fact and placed it earlier in time. Dennett and Kinsbourne used this to argue that there is no single moment at which an experience becomes conscious, only a brain continuously revising its account of the last fraction of a second.¹⁸

The twist. Your real brain already hands each moment a short, constructed history. The content of "now" is edited after the fact. A fragment with a pre-loaded past is only an extreme version of something your brain does every few hundred milliseconds.

The present you feel can seem to span longer than this. William James, borrowing E. R. Clay's phrase, called it the specious present;¹⁹ Ernst Pöppel proposed about 2 to 3 seconds, though a 2017 review by Peter White found little evidence for any fixed window. But its content can be carried by traces held within a single moment, just as memories are. So a fragment must run for a fraction of a second, up to about 0.4 seconds. In the two-stage model most of that window is unconscious processing, and a 2026 EEG study from Herzog's group fits that picture: a signature of unconscious processing about 200 ms after the sequence began, then one of the integrated percept at about 400 to 600 ms.¹⁶ A fragment could in principle arrive with the unconscious stage already done, so this is an upper estimate. This page uses the generous figure: four hundred milliseconds.

Figure 5👀 Try it

Your "now" is edited

Look at the cross, then press Start. Watch the dots from the corner of your eye.

Gap between the two flashes

Where did the dot change colour?

Watch a few cycles, then pick an answer.

Each dot flashes for 150 ms, and the cycle never exceeds three flashes a second. Many people see one dot sliding across that changes colour partway along. Some don't on a screen; the effect depends on timing, distance and display. Below the demo: what was actually shown, against what is typically seen.Each dot flashes for 150 ms. Many people see one dot glide across and change colour partway, before the second dot has even appeared. If you don't see it, that's normal too: it depends on your screen and how far away you sit.

Chapter 6Two ways to fluctuate a mind

So far, every fluctuation has been a thermal one, drawn out of the horizon's warm bath. Its cost depends only on energy, E/k_BT_dS. Once it forms, an object lasts as long as its own physics allows (a brain in cold, empty space would probably keep working for only seconds), and that extra time is free.

Don Page and others also considered brains that appear as brief quantum vacuum fluctuations, which do not need a thermal bath at all.⁵ Page estimates their suppression heuristically from the action, energy multiplied by duration:

P_vac ∼ exp(−E t / ħ) versus P_th ∼ exp(−E / k_BT_dS) Page calls these "brief", "medium" and "long" brains. For a 1 kg brain he estimates exponents of about 10⁴² (lasting the nanosecond light takes to cross it), 10⁵⁰ (lasting 0.1 s), and 10⁶⁹ (the thermal case).

Thermal fluctuation

Source: The horizon's heat bath
Cost: E / k_BT_dS
Depends on: Mass only
Duration: Free once formed
1.4 kg brain: e^{−4 × 10⁶⁹}

Vacuum fluctuation

Source: Quantum uncertainty in empty space
Cost: ≈ E t / ħ, on Page's rough estimate
Depends on: Mass × duration
Duration: Paid for linearly
1.4 kg, 0.4 s: e^{−4.8 × 10⁵⁰}

On this heuristic the two channels line up: a thermal fluctuation costs about as much as a vacuum fluctuation lasting t* ≈ ħ/(k_BT_dS) = 2π/H∞ ≈ 110 billion years (on Page's own convention, P ∼ e^−2Et/ħ, it is half that, about 55 billion years). A brain that appears thermally is, in effect, paying for a duration of several Hubble times, which fits Page's name for it, "long". Anything briefer than that is cheaper as a vacuum fluctuation.

In the vacuum channel, brevity is rewarded in the exponent, and this is where the fragment idea gets its teeth, if the heuristic holds:

Vacuum fluctuation	Mass	Duration	x in e^−x
A whole life as a fluctuation	1.4 kg	80 years	3 × 10⁶⁰
One moment, whole brain	1.4 kg	0.4 s	4.8 × 10⁵⁰
One moment, fragmentFragment mass is a guess	~0.4 kg	0.4 s	1.4 × 10⁵⁰

Living your whole life as a fluctuation costs about e^{−3 × 10⁶⁰}. Living only this moment, as a 0.4 kg fragment, costs about e^{−1.4 × 10⁵⁰}. The moment is more likely by a factor whose exponent is about twenty billion times the moment's own. If vacuum fluctuations like these really happen, the most likely mind with your experience is not merely small. It is brief.

Caveat. Whether vacuum fluctuations can produce real, lasting structures is disputed. Some authors argue that if they can, the problem only gets worse; others argue that a stationary vacuum produces nothing at all. Chapter 10 covers both views.

Figure 6

Mass, time and the two channels

Mass

How long it lasts

Vacuum channel (Page's rough estimate), x ≈ mc²t/ħ

Thermal channel, x = mc²/k_BT

Cheaper channel

Against a whole life as a fluctuation

Mass across, duration up, both logarithmic; drag the point, tap a labelled point, or use the sliders. Diagonal lines join equal vacuum costs, which on Page's heuristic estimate grow with mass × time. Above t* ≈ ħ/k_BT_dS ≈ 110 billion years (on Page's own convention, P ∼ e^−2Et/ħ, it is half that, about 55 billion years) a thermal fluctuation is cheaper, and its price no longer depends on how long it lasts. A whole life (1.4 kg for 80 years) costs e^{−3 × 10⁶⁰}.

Chapter 7The lightest possible mind

Everything so far assumed the fluctuation is made of neurons. Many philosophers and physicists assume instead that experience depends only on the computation being performed, not on what performs it. De Simone and colleagues adopt this view, which they call substrate independence, as a working hypothesis, while noting in a footnote that some of the authors themselves are sceptical.³ If it is right, the cheapest fragment is not a lump of brain tissue at all. It is whatever physical system can run one moment's computation for the least energy.

Two limits, both met in Nº 2 (the Bekenstein bound in its chapter on ceilings, the speed limit in the chapter after), set the floor. The Margolus–Levitin theorem says a system with average energy E above its ground state can perform at most 2E/(πħ) operations per second, counting operations as Seth Lloyd does, as flips between two states.²⁰ (A long run through distinct states goes at half that rate, the 2E/h of Nº 2; with that count the energy floor below doubles, which changes no conclusion.) The Bekenstein bound says a region of radius R and energy E holds at most 2πER/(ħc) nats of information.²¹ Add one more condition: the parts of the device must be able to signal each other within the moment, so its radius R can be at most about ct. Then for N elementary operations on I nats of information over a time t:

E ≥ πħN / (2t) and E ≥ ħI / (2πt) The first from Margolus–Levitin, the second from Bekenstein with R ≤ ct. These are the same limits behind Seth Lloyd's "ultimate laptop".²²

Feed those energies into the two channels of Chapter 6:

thermal: x ≳ max(π²N, I) / (H_∞t) vacuum: x ≳ max(πN/2, I/2π) This page's estimate. Inputs: N ~ 10¹⁵ operations in one moment (roughly every synapse updating once) a guess, on I ~ 10¹⁶ bits ≈ 7 × 10¹⁵ nats of information (Nº 2 found 6 × 10¹⁶ bits at the synapse level; De Simone et al. use 10¹⁶).

The numbers are startling. For a 0.4-second moment, the bounds allow as little as about 4 × 10⁻¹⁹ J, roughly the energy of a single blue photon, though only for a device that saturates the Bekenstein bound, which no known computing system does. To hold 10¹⁶ bits on that budget, the device must have a radius of at least about 84,000 km, around a fifth of the way to the Moon, and light takes 0.28 s to get from its centre to its edge, just inside the moment. As a thermal fluctuation it costs e^−10³⁴, about 35 orders of magnitude cheaper in the exponent than a brain.

Two stranger results follow. In the thermal channel, a slower thought is cheaper, because the energy needed falls as 1/t. Stretch the moment to 1/H∞, the far future's Hubble time of about 17 billion years, and the cost falls to about e^−10¹⁶, which matches the bound De Simone et al. found for a device that saturates the Bekenstein bound at the horizon's scale. If computation is all that matters, a thought spread across billions of light-years and billions of years would, from inside, feel exactly like this one. In the vacuum channel, the cost no longer depends on speed at all: it is roughly e to the minus the number of operations, about e^{−1.6 × 10¹⁵}.

One moment, by design	Channel	x in e^−x
Human brain	Thermal	4 × 10⁶⁹
Boltzmann fragment, 0.4 kg (guess)	Thermal	1.2 × 10⁶⁹
Boltzmann fragment, 0.4 s	Vacuum	1.4 × 10⁵⁰
Ideal moment machine, 0.4 sOne blue photon of energy, ~84,000 km in radius, if the bounds could be reached	Thermal	1.4 × 10³⁴
Ideal moment machine, ~17 billion yearsConsistent with De Simone et al.'s bound	Thermal	~10¹⁶
Ideal moment machine, any speed	Vacuum	1.6 × 10¹⁵

The cheapest designs stop being minds

There is a catch, and it is a deep one. De Simone et al. point out that a device approaching the horizon's size would be stretched by the expansion until its parts could no longer communicate, unless something held it together, and that saturating the Bekenstein bound with something able to compute may be impossible. They give a vivid example. A naive estimate suggests that a gas cloud about 200 light-years in radius, holding 10¹⁶ bits, could form from today's 2.7 K background with no suppression at all. They argue it still could not think, because thermal noise would wipe out any information it tried to store.³

So the price falls as the design gets stranger, and somewhere along the way the design stops being able to have an experience. The true minimum lies between a 1.4 kg brain and a horizon-sized whisper of radiation, and nobody knows where.

Figure 7

Design the cheapest mind

Functionalism assumed

Information it holds, I

Operations in one moment, N

How long the moment takes, t

Minimum energy

Minimum radius

Thermal channel

Vacuum channel

E_min = max(πħN/2t, ħI ln2/2πt) from the Margolus–Levitin and Bekenstein bounds, with the device's radius limited to about ct. R_min = ħcI ln2/2πE. Then x_thermal = E/k_BT_dS and x_vacuum ≈ Et/ħ (Page's rough estimate). These are this page's estimates, not published results, and they assume a device that reaches both bounds, which no known system does. The picture is to scale: the device's minimum size against familiar objects.

Chapter 8Your memories are free

In 1921 Bertrand Russell pointed out that there is no logical impossibility in the world having sprung into existence five minutes ago, complete with people who remember a past that never happened.²³ He judged the hypothesis "logically tenable, but uninteresting." The fluctuation picture gives Russell's hypothesis a mechanism, and a price.

Here is the price. A fragment holding your exact memories, rather than random ones, is a more specific arrangement. Specifying 10¹⁶ to 10¹⁷ bits costs a factor of roughly e^−10¹⁶ to e^−10¹⁷. For a fragment made of brain tissue, which already costs e^−10⁶⁹, that shifts the exponent by about one part in 10⁵². In brain tissue, accurate-seeming memories are essentially free. In the cheapest machines of Chapter 7, the vacuum-channel and Hubble-time designs priced at about e^−10¹⁵ to e^−10¹⁶, they are not: your exact memories rather than random ones make such a machine less likely by a factor of about e^{7 × 10¹⁵}, as large as its whole price or larger, though vacuum decay (Chapter 11) still wins by far. Memories that are actually true are never free: they require a real past, with a real low-entropy Big Bang behind it, which by Penrose's estimate is special to one part in 10^10¹²³. Counted this way, false memories are vastly cheaper than true ones.

Everything you could check is inside the moment

Try to prove you are not a fragment. Look at a photograph from last year; the image and your recognition of it are both part of this moment. Ask a friend; their answer arrives as part of this moment. Reread the paragraph above; the feeling of having read it before is a present trace. Even the memory of having already checked is present content. Nothing available to you comes from outside the moment, because "available to you" means "part of this moment".

The ordinary Boltzmann brain story has a weak point here, and Feynman found it. A brain that fluctuates into existence with your memories should expect its next observation to be incoherent, because nothing outside it is ordered.³ Every orderly moment you experience counts as evidence against being an ordinary Boltzmann brain.

Why the fragment survives Feynman's test. A fragment has no eyes or ears, so it never makes a next observation of the outside world; and if it formed as a brief vacuum fluctuation, it does not outlast this moment. If another orderly moment occurs, it belongs to a different system, which carries its own fabricated memory of this one. Evidence never accumulates in any single place, so the test that defeats ordinary Boltzmann brains does not touch fragments. (Try "You are a fragment" in Figure 2.)

Philosophers have a name for reasoning about which moment you are. Nick Bostrom's strong self-sampling assumption says you should reason as if your present observer-moment were a random sample from all observer-moments in its class.²⁴ Apply it to an eternal de Sitter universe and the conclusion follows at once: if almost all moments like this one are fragments, this one probably is too. This is the strongest form of the argument and the darkest point on the page. The next four chapters are about why almost no physicist accepts it.

Figure 8🕵️ Try it

Prove you're real

Pick a test. Each one comes back with a result, and every result arrives as part of the present moment.Every result comes back as part of the present moment. That's the problem.

Part IIIWays out, and the chain

Chapter 9The argument that eats itself

Lay the reasoning out step by step and something odd happens:

We observe thermodynamics, an accelerating universe, and the physics of horizons.
Together these imply an eternal de Sitter bath.
That bath produces fragments that vastly outnumber real observers with this exact experience.
So this moment is most likely a fragment.
Then the observations in step 1 are fabricated memories, and nothing supports step 2.
Back to step 1, with no reason left to believe it.

Sean Carroll argues that this is the real problem with Boltzmann brains. The issue is not that observations rule them out but that theories predicting them are cognitively unstable: they "cannot simultaneously be true and justifiably believed."²⁵ He stresses that the dangerous fluctuations are observers locally identical to ordinary ones, which is exactly what a fragment is. His conclusion is that a cosmology predicting such observers should be modified or rejected, rather than rescued by a clever choice of how to count them.

Not everyone finds that decisive. One reply is that the same self-undermining structure appears in every radical sceptical scenario, from Descartes's deceiving demon onward. A cognitively unstable theory cannot be believed, but that does not show it is false. Russell had said as much in 1921: tenable, but uninteresting. There is also a quieter difficulty underneath. Arguments of this kind depend on choosing a reference class, the set of moments you count yourself as a random member of, and there is no agreed way to choose it.²⁴

A 2025 analysis by David Wolpert, Carlo Rovelli and Jordan Scharnhorst sharpens the point. Treating the universe's entropy as a time-symmetric random process, they argue that the Boltzmann brain hypothesis, the second law and trust in our records each rest on a choice of which moment, and which entropy, to condition on, and that physics alone does not make that choice. On this view the Boltzmann brain hypothesis and the second law are "equally legitimate (or not)".²⁶ One recent proposal for that choice comes from Feraz Azhar, Alan Guth and Mohammad Hossein Namjoo (Physical Review D, June 2026): reason as if you were drawn at random from all the predicted observer-moments that share your exact subjective state. That is how this page counts. They deliberately set the measure problem aside.²⁶

Chapter 10Does empty space even fluctuate?

Everything so far assumed that the de Sitter vacuum actually fluctuates, that rare configurations really appear in it from time to time. That assumption is less secure than it sounds.

Boddy, Carroll and Pollack distinguish two things physicists call "quantum fluctuations". Vacuum fluctuations, the kind behind the Casimir effect, are features of a stationary quantum state; nothing changes over time. Boltzmann fluctuations are dynamical events in which a system really passes through a rare configuration at some moment. Boltzmann brains need the second kind.

In 2014 the same three authors argued that a de Sitter patch embedded in a theory with an infinite-dimensional space of quantum states settles into a quiescent vacuum: a stationary state in which no dynamical fluctuations happen at all.²⁷ In a later paper framed in the many-worlds interpretation, they concluded that such a state does assign a nonzero probability to finding a brain-like configuration if one looked, but that looking is itself a time-dependent process, and it never happens in a vacuum that does not change.²⁸ (In the many-worlds picture these authors use, without something that records the difference, a quantum possibility never becomes an event; Nº 10 shows that recording, decoherence, splitting a Schrödinger's cat into alive and dead branches that no longer interfere.)

The escape hatch has a condition. If each horizon patch is instead a closed system with a finite number of quantum states, as its finite entropy suggests, the same authors find that recurrences and Boltzmann fluctuations return. Which picture is right depends on quantum gravity, which nobody yet has. Other authors have examined whether brains could fluctuate into existence even from an empty, zero-temperature vacuum, and found that if they can, the problem gets worse.²⁹

Chapter 11Saved by the end of the world

The cleanest way out is for the bath not to last forever. De Simone et al. state the condition precisely: to avoid Boltzmann brain domination, in every vacuum that can produce them, brains must nucleate more slowly than that vacuum itself decays.³

Nº 0 showed that if the Standard Model is the whole story, our vacuum is metastable. Its decay is suppressed by e^−B with a bounce action B ≈ 1,910 (Nº 0's simple estimate) for the 173.1 GeV top quark used in the full calculation,³⁰ or roughly 2,500 for today's lighter world average (Nº 0's extrapolation). Compare that with the exponents in the rest of this page:

Process	x in e^−x
Standard Model vacuum decayNº 0; exact value depends on the top quark mass	≈ 1,900–2,500
Ideal moment machine, vacuum channel	1.6 × 10¹⁵
Ideal moment machine, thermal, Hubble-time slow	~10¹⁶
Boltzmann fragment, vacuum, 0.4 s	1.4 × 10⁵⁰
Human brain, thermal	4 × 10⁶⁹

Even against the cheapest mind on this page, vacuum decay wins by a factor of roughly e^10¹⁵. Prefactors cannot rescue the comparison. In their analysis, De Simone et al. treat any factor between e^−10¹⁴ and e^10¹⁴ as "roughly of order one".³ In time: Nº 0 estimated that a bubble able to reach us should take well over 10⁶⁰⁰ years to form, while the first thermal brain in a horizon volume takes a number of years with about 2 × 10⁶⁹ digits. Our patch of space is overwhelmingly likely to be swept away by a bubble of true vacuum long before it ever produces a single thermal Boltzmann brain. The most destructive process in Nº 0 turns out to be the cure for this one.

The fine print: how do you count infinity?

"Long before" is a statement about one patch. In an eternally expanding universe there are infinitely many patches, and whether fluctuations outnumber ordinary observers depends on how you compare infinite sets, which is known as the measure problem. In 2006 Don Page argued that if you weight by volume, our vacuum must decay faster than the volume of space grows to keep Boltzmann brains from dominating, which would make it more likely than not to decay within about 20 billion years.³¹ The Standard Model's decay comes nowhere close. Andrei Linde showed that some volume-weighted schemes avoid Page's conclusion,³² and Page replied with a natural-looking scheme in which the problem comes back, underlining how ambiguous ratios of infinite volumes are.³³ Under measures that follow a single observer's causal patch, or that cut off at a fixed amount of expansion, the simple rate comparison above is what matters.³

Three more conditions

This escape works only if the Standard Model vacuum really is metastable. New physics at high energies could make it stable, and then the hatch closes. Second, the whole problem assumes dark energy is a constant, and it might not be. In 2025 the DESI collaboration reported a preference for dark energy that changes over time: 3.1 standard deviations combining its data with the cosmic microwave background, and 2.8 to 4.2 once supernovae are added. Its July 2026 analysis, which adds a new Lyman-alpha measurement, puts the preference at 2.7 standard deviations for DESI with the cosmic microwave background, and 3.0 to 3.5 once supernovae are added (3.2 with the recalibrated DES sample), still short of the 5σ discovery standard; Nº 1 has the latest.³⁴ If vacuum energy fades towards zero, the horizon temperature ħH/2πk_B goes to zero and thermal brain production vanishes with it. De Simone et al. caution that even a zero-energy vacuum might still make brains as quantum fluctuations; Chapter 10 covers that debate.³ If it turns negative, the universe eventually recollapses. One published fit to the DES and DESI data, by Luu, Qiu and Tye, adds an ultralight field to the cosmological constant and finds the constant most likely negative; in their benchmark model the universe stops expanding, falls back and ends in a crunch about 20 billion years from now, at a total age of about 33 billion years.³⁵ It is one model built on a hint that has not reached discovery strength, and its timescale matches Page's 20 billion years only by coincidence. Either way there is no eternal bath. A recollapse would also do something vacuum decay cannot. A bubble ends one patch while the rest of space keeps growing, so it rescues us only under some ways of counting, such as patch by patch or with a cutoff at a fixed amount of expansion; a universe that recollapses has a finite history, so in our universe the fragments never get their endless time, whichever way you count.

The third condition comes from string theory. When Kachru, Kallosh, Linde and Trivedi proposed a way to build de Sitter vacua in string theory in 2003, they also argued that, under certain conditions, such vacua always decay before the recurrence time that troubled Dyson, Kleban and Susskind, though not necessarily before the first brains.³⁶ Some theorists go further and conjecture that quantum gravity does not allow long-lived de Sitter space at all: Bedroya and Vafa's trans-Planckian censorship conjecture would cap the lifetime of our present phase at about 2.4 trillion years.³⁶ Both rest on assumptions about quantum gravity that are actively debated, so neither is an established result. But if anything like the second holds, the eternal bath of Chapter 1 never exists.

Figure 9🏁 Try it

The race of endings

┆ Page: decay needed within about 20 billion years┆ decay needed within about 20 billion years

Doubly logarithmic: each bar's length is log₁₀(log₁₀ of the time in years), which is close to the logarithm of the number of digits in that time. Everything from today to a bubble able to reach us fits in the first three units (up to a thousand digits). The first mind comes about twelve units further out: the wait for the cheapest mind on this page has about 7 × 10¹⁴ digits. Brain-era times are waiting times inside one horizon volume. "Zoom" spreads out the first thousand digits.Each bar shows how many digits the number of years has, on a squashed scale, so that numbers this big fit on one screen. Flip the switches to see which ending wins. Zoom in to see the nearer endings spread out.

Chapter 12Should anyone worry?

No experiment can show that this moment is not a fragment, and none ever will; that is built into the hypothesis. But a hypothesis that cannot be tested is not thereby likely. Here is where the argument stands:

Way out	What it needs	Status
Cognitive instability	Only logic	Shows the conclusion cannot be rationally believed, not that it is false
Quiet vacuum	An infinite-dimensional space of quantum states	Open; depends on quantum gravity
Vacuum decay	A metastable Standard Model vacuum and a patch-based measure	Plausible on current measurements; not established
Dark energy fades or turns negative	Evolving dark energy	A live hint at about 3σ; one 2025 model ends in a crunch about 20 billion years from now; not established
Forever is forbidden	Quantum gravity that limits how long de Sitter space can last	Conjecture (swampland); actively debated
Substrate matters	Consciousness needing more than computation	Unknown; it would remove the ideal machines of Chapter 7 but not brain-tissue fragments
Measure problem	A principled way to count infinite observers	Unsolved; every conclusion here depends on it, unless the whole universe ends

Physicists treat a theory that predicts fragment-dominated observers as a theory that has gone wrong, and several features that real physics may well have (a decaying vacuum, a fading dark energy, a vacuum that never fluctuates) could each remove the problem, under the conditions set out in Chapters 10 and 11. What remains is honest and slightly vertiginous. The past cannot be proven from inside the present. It never could; Russell knew that in 1921. Physics only puts a price on the alternative, and the price turns out to be the kind of number that forces cosmologists to change their theories.

Chapter 13The chain

Suppose the worst case anyway: fragments are all there is. Something unexpected follows. In an eternal bath, every possible moment occurs, and occurs infinitely often, including every moment that would come "after" this one, carrying a memory of it. The sequence your life seems to be already exists, scattered across space and time, with no causal thread between its pieces. The question changes from "does my next moment exist?" to "is a scattered sequence of moments, linked only by what they contain, a life?"

An old idea in several costumes

In his 1966 novel October the First Is Too Late, the astrophysicist Fred Hoyle had a character picture each moment as a pigeon-hole, with consciousness like a light moving among them, and the order of moments as a matter of logic rather than time.³⁷ In The End of Time (1999), the physicist Julian Barbour argued that time does not exist at all. There are only "Nows", configurations of the whole universe, and some of them contain records that create the impression of a past; he calls them time capsules.³⁸ On that view, a brain is a time capsule.

Hans Moravec and the novelist Greg Egan pushed the idea furthest.³⁹ In Egan's Permutation City (1994), a simulated man is computed out of order and still experiences his life in sequence, and from this the novel's hero, Paul Durham, concludes that any set of states, wherever they lie, forms an experience if their contents fit together. Egan calls this the dust theory. Its core claim is that there is nothing more to causality than the correlations between states.⁴⁰

Every fork

Each moment has many possible successors, every later moment whose memories include this one. So the chain is not a line but a tree, branching at every point where more than one continuation is consistent with what came before. Derek Parfit argued that what matters in survival is psychological continuity and connectedness, not some further fact about identity, and that branching does not destroy what matters.⁴¹ In a dust universe, "you" would be the pattern of connections, followed down every branch.

Why does the next moment make sense?

Egan himself says he doesn't take the dust theory very seriously, though he has yet to hear a convincing refutation. His reason is that our world is far too orderly: it obeys far simpler and more uniform laws than it would need to merely contain observers.⁴⁰ Notice the shape of that objection. It is Feynman's test again, one level up. Of all the possible successors to this moment, the incoherent ones vastly outnumber the coherent ones. So why does each new moment keep making sense?

The physicist Markus Müller has proposed one answer.⁴² Start from the observer instead of the world: take the observer's states as fundamental, and let the chance of each next state be given by algorithmic probability, which strongly favours continuations that a short program could generate. Müller shows that an observer governed this way will typically find itself apparently embedded in an external world that evolves according to simple, computable, probabilistic laws. He argues that this dissolves puzzles like the Boltzmann brain problem, and that some basic quantum phenomena, such as Bell-inequality violation and no-signalling, might be understood as consequences. He presents it as a starting point, not a finished theory. In a follow-up published in 2026 he calls the framework algorithmic idealism and draws the conclusion directly: you should never bet on being a Boltzmann brain, however many there are.⁴² It is contested; in a 2026 preprint written as an adversarial collaboration, Müller and the philosopher Kelvin McQueen set out where a realist disagrees.⁴²

P(x_n+1 | x₁…x_n) = M(x₁…x_n+1) / M(x₁…x_n), M(x) = Σ_{p → x…} 2^−|p| Schematically: the weight of a sequence of observer states is the total weight of all programs p that output it, each counted as 2 to the minus its length. Short programs, meaning simple regularities, dominate.

This is exactly what scattered moments would need in order to add up to lives: a rule for weighting the transitions between them. In Müller's version, the rule that keeps your next moment coherent and the reason there seem to be laws of physics at all would be the same thing.

Here is where that leaves us. If fragments are real, nothing can prove more than this moment. If chains are real, they exist whether or not anything physical connects them. And if something like Müller's rule holds, the coherence of the next moment is not luck. You are holding one page. Whether the rest of the book was ever printed, this page tells a story about the others, and so far every page has agreed.

Figure 10🧵 Try it

Stitch the moments

Next moments that continue the story

Branches

Moments in the thread

Each card is one moment: a scene, plus a faint strip along its top holding its memory of the moment before. Nothing physical connects these cards. "Order by content" threads them only by matching memories to scenes. Toy numbers: a scene is 640 pixels and the memory strip fixes 10 of them. Counted equally, only 2 of the 2⁶³⁰ consistent next scenes continue the story. Weighted by the length of the shortest rule that draws them, the story-continuing ones can carry almost all of the weight (about 99.99999% in this toy; the exact share depends on how the rules are written).Each card is one moment, with a faint strip on top holding its memory of the moment before. Nothing connects them except what they remember. Toy numbers: if every consistent next moment counts the same, almost all of them are noise. If simpler rules count for more, the story almost always continues.

Chapter 14Glossary and sources

De Sitter space: A universe dominated by a cosmological constant, expanding exponentially at a constant rate. Every observer in it has a horizon.
Gibbons–Hawking temperature: The temperature of a de Sitter horizon, ħH/(2πk_B), about 2 × 10⁻³⁰ K for our universe's far future.
Boltzmann brain: An observer produced by a random fluctuation rather than by evolution after a low-entropy Big Bang.
Minimal Boltzmann brain: De Simone et al.'s version: random memories that happen to match yours, lasting just long enough for one more observation.
Boltzmann fragment: This page's term for the smallest physical system that produces exactly one moment of experience and nothing else.
Thermal and vacuum fluctuations: Fluctuations drawn from a heat bath, costing E/k_BT, versus fluctuations allowed by quantum uncertainty, which Page estimates to cost roughly Et/ħ.
Action: Energy multiplied by time, in units of ħ. In Page's rough estimate, it sets the suppression of quantum vacuum fluctuations.
Margolus–Levitin bound: The maximum rate of computation for a given energy: 2E/(πħ) operations per second (2E/h for a long run of distinct states).
Bekenstein bound: The maximum information a region of given size and energy can hold: 2πER/(ħc) nats.
Observer-moment: A single moment of one observer's experience, used as the unit of anthropic reasoning in Bostrom's strong self-sampling assumption.
Measure problem: The unsolved question of how to compare infinite numbers of observers or events in an eternal universe.
Cognitive instability: Carroll's term for theories that cannot be both true and justifiably believed.
Postdiction: The brain's construction of an experience after the events it describes, as in the colour phi effect.
Dust theory: Egan's proposal that any set of states whose contents fit together constitutes an experience, wherever and whenever they occur.
Time capsule: Barbour's term for a configuration that contains records suggesting a past.
Algorithmic probability: A probability that weights each outcome by the shortest programs that produce it, favouring simple regularities.

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The universe never quite turns off

Forever breaks the odds

Every gram has a price

Pack light

How long is "now"?

Short is cheap

Your memories come free

So what would it feel like?

Why physicists don't panic

1. The argument eats itself

2. The universe might end first

3. Dark energy might fade

4. Empty space might not jiggle at all

The chain