c·t = CONST, THAT CLICKS EPISODE 10 ／ In a universe that never cools, what does one bit cost?
"Exactly at the Landauer limit" was just reading off an identity
Substituting into
heat and information
Temperature is pinned here — the universe sits at 2.7255 K and never cools,
so the cost of erasing a bit is fixed. Is that expensive, or cheap?
Temperature does not move in this picture. \(\tilde T=aT\) cancels the standard \(T\propto1/a\) exactly — the universe sits at 2.7255 K and never cools. So the energy needed to erase one bit (the Landauer limit \(k_BT\ln2\)) is constant across the whole history of the universe. A quantity that falls as \(1/a\) in the standard picture is pinned here. Which is the "real" cost of erasure? As usual, that is not yet a sentence.
01Temperature is pinned, so the price is pinned
In the standard picture this quantity was larger in the past and has fallen as \(1/a\). Here it has been \(1.63\times10^{-4}\) eV from the beginning until today. Two exactly opposite statements about exactly the same universe.
02Applying Episode 3's surgery to a price
\(k_BT\ln2\) is an energy — dimensionful. Run it through the decision procedure and it lands in the left column: bookkeeping. So "erasing one bit costs \(1.63\times10^{-4}\) eV" is not by itself a claim. You must say compared with what.
| Compare the erasure cost with… | Today | Time dependence |
|---|---|---|
| Electron rest mass \(m_ec^2\) | \(4.6\times10^{-10}\) | \(\propto1/t\) (gets cheaper) |
| One CMB photon's energy | \(\sim1\) | constant |
| The Planck energy | \(1.9\times10^{-32}\) | changes with epoch |
| nothing at all | ── | there is no claim |
Against particle masses, erasing information gets relatively cheaper with time. Against a single photon it is eternally the same. Both are right. Only the comparison partner differs.
03One more thing that has to be named
Naming the comparison partner is still not enough. The Landauer limit contains a \(T\) — you must say which heat bath you are dumping into. That is a demand of thermodynamics itself, nothing to do with conformal transformations.
So let us line up every bath the universe offers and ask the same question: using the entire energy of the universe, how many bits could be erased?
Total energy inside the horizon (the identity used in Episode 1)
$$E=\frac{c^4R_H}{2G}=7.90\times10^{69}\ \mathrm{J}$$Bits erasable
$$N_{\rm erase}=\frac{E}{k_BT\ln2}$$| Erase at which temperature | \(T\) | Price per bit | Bits erasable | Ratio to memory \(N\) |
|---|---|---|---|---|
| Hubble (horizon) temperature | \(2.79\times10^{-30}\) K | \(2.67\times10^{-53}\) J | \(2.96\times10^{122}\) | 1.0000 |
| CMB temperature | 2.7255 K | \(2.61\times10^{-23}\) J | \(3.03\times10^{92}\) | \(1.0\times10^{-30}\) |
| Room temperature | 300 K | \(2.87\times10^{-21}\) J | \(2.75\times10^{90}\) | \(9.3\times10^{-33}\) |
| Planck temperature | \(1.42\times10^{32}\) K | \(1.36\times10^{9}\) J | \(5.83\times10^{60}\) | \(2.0\times10^{-62}\) |
The thing this episode most wants to say
Spending the entire energy of the universe, erasing at the CMB temperature clears only \(10^{-30}\) of the memory.
Writable \(10^{122}\) bits, erasable \(10^{92}\) — thirty orders short.
Episode 1 counted "one operation per 28.5 bits". This is far more extreme: the universe can erase only \(10^{-30}\) of what it can write. A nearly write-once medium.
Figure: which bath you dump into, across; bits erasable with the entire energy of the universe, up. The grey horizontal line is the writable count \(N=2.96\times10^{122}\). The two meet at exactly one point — the Hubble temperature.
04The reveal — what "exactly at the Landauer limit" really was
Extra 2 of the previous series contained the most striking number in the whole series — the universe's energy per bit matches the Landauer limit at a ratio of 1.000000.
That is the first row of the table. And now it is clear — it was an identity. \(E=T_HS\) holds in any FLRW, so it is not a physical claim that the universe runs at the limit. It is the horizon's energy divided by the horizon's temperature and the horizon's entropy.
Conclusion of §04
"The universe sits exactly at the Landauer limit" is just reading off an identity.
Meaning appears only when you choose a different temperature — and the moment you do, out comes a real number: \(10^{-30}\).
This applies the previous series' decision procedure to a "beautiful coincidence". An identity is not physics — exactly the verdict Extra 3 delivered on Dirac's large numbers.
05Comparing with real machines
Landauer limit at room temperature
$$k_B\!\cdot\!300\,\mathrm{K}\cdot\ln2=2.87\times10^{-21}\ \mathrm{J}=0.0179\ \mathrm{eV}$$A modern CPU per operation (roughly)
$$\sim10^{-15}\ \mathrm{J}\qquad\Longrightarrow\qquad \text{about }3.5\times10^{5}\ \text{times the limit}$$Human computers are still five and a half orders from the limit. The universe (measured at the horizon temperature) sits exactly on it — though we have just seen that this is an identity.
06Entropy itself does not move
| Quantity | Weight | In this picture |
|---|---|---|
| Entropy \(S/k_B\) | \(0\) | invariant |
| The second law | ── | untouched |
| Temperature \(T\) | \(-1\) | \(\times a\) (constant here) |
| Landauer cost \(k_BT\ln2\) | \(-1\) | \(\times a\) (likewise) |
| Boltzmann factor \(e^{-E/k_BT}\) | \(0\) | invariant |
The \(3.1\times10^{104}\) counted in Episode 6 and the recombination factor \(52.6\) used in Episode 4 are both unchanged to the letter. Thermodynamics comes through entirely intact, so long as it is written dimensionlessly.
① The Landauer limit is a lower bound per logically irreversible operation. Reversible computation can in principle cost nothing. The "bits erasable" here assumes every erasure is irreversible; it is not a claim about how the universe actually operates.
② \(E=c^4R_H/2G\) is an identity of flat FLRW (Episode 1; Extra 3 of the previous series). Defining the "total energy" inside the horizon this way is natural, but energy in general relativity has no unique definition — quasi-local energies differ, and so do the values.
③ The "bits erasable" column is an upper bound assuming all the energy can be spent on erasure. In practice you need machinery to extract the energy and dump the heat, whose efficiency is not included. Read it as an order-of-magnitude argument.
④ Using the Hubble temperature \(T_H=\hbar H/2\pi k_B\) as the Landauer \(T\) does not go beyond metaphor. It corresponds to the Gibbons–Hawking temperature of a de Sitter horizon, and whether it acts as a heat bath in a general FLRW is not obvious. The first row is "exact" because of an identity, not because of a physical mechanism — which is the whole point of §04.
⑤ The \(10^{-15}\) J per CPU operation is an order-of-magnitude marker. It shifts by orders depending on what counts as an operation (a logic gate, an instruction).
Exercises (solvable with this episode's formulas alone)
- Why is the Landauer cost constant in this picture?
Show the answer
Temperature has weight \(-1\), so \(\tilde T=aT\); in the standard picture \(T\propto1/a\), and the two cancel to \(\tilde T=\)const. Hence \(k_B\tilde T\ln2\) is constant too (\(1.63\times10^{-4}\) eV). It is the same quantity that falls as \(1/a\) in the standard picture. - Is "erasing one bit costs \(1.63\times10^{-4}\) eV" a claim by itself?
Show the answer
No. Energy is dimensionful — bookkeeping — so it means nothing until you say compared with what. Against the electron mass it is \(4.6\times10^{-10}\) and gets cheaper with time; against one CMB photon it is constant. The same surgery Episode 3 applied to the series title. - With the whole energy of the universe, how many bits can be erased at CMB temperature? Compare with the memory.
Show the answer
\(E/(k_BT_0\ln2)=7.90\times10^{69}/2.61\times10^{-23}=3.03\times10^{92}\) bits — \(10^{-30}\) of the memory \(2.96\times10^{122}\). It can erase one billionth of one billionth of one billionth of what it can write. - Is "the universe runs exactly at the Landauer limit" a physical claim?
Show the answer
No — it is an identity. \(E=T_HS\) holds in any FLRW, so \(E/N=k_BT_H\ln2\) is automatically 1.000000. Meaning appears only when you choose a different temperature, and at the CMB temperature out comes the real number \(10^{-30}\). The same verdict Extra 3 of the previous series gave Dirac's large numbers: an identity is not physics. - (Harder) Entropy is conformally invariant but temperature moves. Is that a contradiction?
Show the answer
No. \(S/k_B\) is a bit count, dimensionless, so it cannot move; \(T\) is an energy, dimensionful, so it does. Both hold at once. Indeed in \(E=TS\) the left side has weight \(-1\), matching \(T\) (\(-1\)) \(\times S\) (\(0\)). The very fact that entropy is an amount of information is what shields it from the bookkeeping.
Summary — the price is undefined until you name the temperature
Temperature is pinned here (\(\tilde T=aT=\)const \(=2.7255\) K), so the Landauer limit \(k_BT\ln2=1.63\times10^{-4}\) eV is constant across the whole history of the universe — the same quantity that falls as \(1/a\) in the standard picture. Which is true? Not yet a sentence. Energy is dimensionful, so nothing is claimed until a comparison is named: against the electron mass it gets cheaper with time, against one CMB photon it is eternally the same.
And this episode found a second thing that must be named — which bath you dump into. Counting how many bits the universe's entire energy \(E=7.90\times10^{69}\) J could erase: \(2.96\times10^{122}\) at the Hubble temperature, \(3.03\times10^{92}\) at CMB temperature, \(2.75\times10^{90}\) at room temperature, \(5.83\times10^{60}\) at the Planck temperature. Sixty orders of magnitude depending on the choice.
The second row bites hardest: erasing at CMB temperature, the universe can clear only \(10^{-30}\) of what it can write. Writable \(10^{122}\), erasable \(10^{92}\) — an asymmetry far more extreme than Episode 1's "one operation per 28.5 bits". The universe is a nearly write-once medium.
And the reveal. The most striking number of the previous series — energy per bit matching the Landauer limit at 1.000000 — is the first row of that table, and it was an identity. \(E=T_HS\) holds in any FLRW, so measuring at the horizon temperature always gives exactly one. Meaning appears only at another temperature. An identity is not physics — the same verdict passed on Dirac's large numbers in Episode 7. Entropy itself, being dimensionless, does not move at all here: thermodynamics comes through intact so long as it is written dimensionlessly.
This document is Episode 10 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. Landauer's principle (at least \(k_BT\ln2\) is dissipated per logically irreversible bit erasure), the dimensionlessness of entropy, and the weight \(-1\) of temperature under a conformal transformation are all standard. Reversible computation can in principle erase at zero cost — the "bits erasable" here assumes every operation is irreversible and is an upper bound, not a claim about how the universe operates. \(E=c^4R_H/2G\) is an identity of flat FLRW (Extra 3 of the previous series), and quasi-local energy in general relativity has no unique definition. The Hubble temperature \(T_H=\hbar H/2\pi k_B\) corresponds to the Gibbons–Hawking temperature of a de Sitter horizon; whether it acts as a heat bath in a general FLRW is not obvious, and the "exact match" in the first row follows from the identity \(E=T_HS\) rather than from any physical mechanism (this is the point of §04). The numbers (\(k_BT_0\ln2=2.61\times10^{-23}\) J \(=1.63\times10^{-4}\) eV, \(E=7.90\times10^{69}\) J, erasable counts \(2.96\times10^{122}\)/\(3.03\times10^{92}\)/\(2.75\times10^{90}\)/\(5.83\times10^{60}\), and \(1.0\times10^{-30}\) of memory at CMB temperature) are computed here; extraction and heat-rejection machinery and efficiency are not included. The \(10^{-15}\) J per CPU operation is an order-of-magnitude marker that shifts by orders with the definition of an operation. Extra 2 of the previous series also states explicitly that the 1.0000 match of energy per bit with the Landauer limit is an identity. Linear expansion (\(c\cdot t=\)const, \(R_h=ct\)) is a minority model under examination and conflicts with nucleosynthesis when extrapolated into the early universe (Lewis, Barnes & Kaushik 2016). The academic standard remains the \(\Lambda\)CDM model including inflation. ── To make a PDF, use your browser's Print dialogue (sliders freeze and answers are hidden in the print version).