Abstract
For more than a century Maxwell’s demon has been asked where the cost lies. What this paper counts is not the answer but where the cost was placed: five people put it in five different places. No new mathematical theorem and no new law is claimed.
First, one bit costs $\mathbf{2.870979\times 10^{-21}}$ J at $300$ K, that is $\mathbf{0.017919}$ eV. A $100$ W device working at that bound could erase $\mathbf{3.4831\times 10^{22}}$ bits per second(Section 3).
Second, the Szilard engine returns exactly that much from one measurement. The isothermal work is $\mathbf{2.870978885079\times 10^{-21}}$ J, differing from $k_BT\ln 2$ by $6.5\times 10^{-35}$ J(Section 4).
Third, this is the core. Irreversible gates do not all lose the same amount. AND, OR and NAND lose $\mathbf{1.188722}$ bits while XOR loses $\mathbf{1.000000}$ and erasure loses $\mathbf{1.000000}$(Section 5).
Fourth, gates that can be made reversible lose nothing. NOT, copying, Toffoli and Fredkin all lose $\mathbf{0.000000}$ bits, and the last two are permutations of $8$ states that return to the identity when applied twice(Section 6).
Fifth, the cost of a reversible AND is not erasure but room. Built from a Toffoli gate it loses $\mathbf{0}$ bits, and a third wire is required(Section 7).
Sixth, the books balance. Over $N$ cycles the work extracted and the cost of erasing the record cancel exactly, for every $N$(Section 8).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed: Maxwell’s thought experiment, the Szilard one-bit engine, Brillouin’s cost of observation, Landauer’s erasure bound, Bennett’s resolution, and the Toffoli and Fredkin gates are all standard. No measured value is cited: every number is computed from a definition or obtained by exhaustive enumeration. The debate is not declared settled: generalisations of the second law that include measurement are still being studied after Bennett, and this paper only counts the structure of the debate. No theory of feedback control is built: Section 4 integrates the isothermal work and nothing more. It is not claimed that the second law is violated, nor that it is not. What is shown is the fact that the cost was placed in different places by different people. Nothing is said about real computing hardware: Sections 5 and 6 enumerate truth tables. Relation to earlier papers: Paper 131 split $k_BT\ln 2$ into three parts and wrote that an irreversible gate does not necessarily pay one bit. This paper puts a number on that (AND pays $1.188722$). Paper 88 separated the coarse-graining entropy from the horizon entropy. What this paper treats is a third one, the entropy of a record. Paper 126 split the axiomatisation of Caratheodory into three stages. This paper views the same second law from the side of information. Paper 4 treated the resource bound on exhaustive search: Section 3 is where that bound comes from. Paper 143 showed that minimum entropy production is a fenced theorem rather than a principle. This paper likewise opens something that has been called a principle. What is added is a single table of where five people placed the cost, an exhaustive count of what irreversible gates lose, and a numerical demonstration that the cost of reversibility is room rather than erasure.
1. Notation and placement
Setting: divide a box of gas with a partition and place a small being that lets only the fast molecules through one way. A temperature difference appears at no apparent cost, and work can be drawn from it. The second law appears to be violated. Maxwell described this in a letter in $1867$.
For over a century the question has been posed as: the cost must be somewhere. What this paper counts is not the answer but where the cost was placed.
The debate is not declared settled here, and it is not claimed either that the second law is violated or that it is not.
2. Five people, five different places
| who | when | where the cost was said to lie |
|---|---|---|
| Maxwell | $1867$ | **nowhere** (hence the paradox) |
| Szilard | $1929$ | **measurement**, $k_BT\ln 2$ to learn one bit |
| Brillouin | $1951$ | **the physical act of looking**, a photon is needed |
| Landauer | $1961$ | **erasure**, $k_BT\ln 2$ to erase one bit |
| Bennett | $1982$ | **erasure only**, measurement can in principle be free |
Observation 2.1 All five are trying to save the second law. What differs is only which operation the cost was assigned to: measurement, the act of looking, or erasure.
Observation 2.2 Here is the separator. The phrase the demon’s cost does not name an operation. Until one is named, no one is right or wrong: and once one is named, it is settled which of the five is being discussed.
Observation 2.3 Bennett’s resolution worked because it separated logical from physical reversibility. Measurement can be made logically reversible; erasure cannot. The cost therefore falls on erasure and not measurement.
3. The price of one bit
Check 3.1: compute $k_BT\ln 2$ at $300$ K.
| quantity | value |
|---|---|
| $k_BT\ln 2$ | $\mathbf{2.870979\times 10^{-21}}$ J |
| the same in electron volts | $\mathbf{0.017919}$ eV |
| bits a $100$ W device could erase at the bound | $\mathbf{3.4831\times 10^{22}}$ per second |
Observation 3.2 The number is small; it is the size of thermal motion at room temperature. It is the value underneath the resource bound of Paper 4.
Observation 3.3 Paper 131 split this expression into three parts. $k_B$ converts units, $\ln 2$ chooses the unit of information, and the physics is only in $T$. This paper stands on that and counts how many bits are paid.
4. The Szilard engine: work from one measurement
Setting: put one molecule in a box, insert a partition, measure which side it is on, and expand isothermally from that side. The work is the integral of pressure over volume.
Check 4.1: integrate the isothermal expansion from $V$ to $2V$ over $2000001$ points.
| measured | value |
|---|---|
| isothermal work | $\mathbf{2.870978885079\times 10^{-21}}$ J |
| $k_BT\ln 2$ | $2.870978885079\times 10^{-21}$ J |
| difference | $6.5\times 10^{-35}$ J |
Observation 4.2 The work extracted equals the price of one bit exactly. Szilard read this as a cost attached to measurement. It balances what was extracted, so the reading is natural.
Observation 4.3 Yet the same number balances the cost of erasure as well. That two numbers agree does not decide which operation carries the cost: a balance alone fixes no address.
5. The core: irreversible gates do not all lose the same amount
Setting: assume uniform inputs and call the amount lost by a gate the information of the input minus that of the output. Enumerate the truth tables exhaustively.
| gate | input | information of output | lost | injective |
|---|---|---|---|---|
| AND | $2$ bits | $0.811278$ | $\mathbf{1.188722}$ | no |
| OR | $2$ bits | $0.811278$ | $\mathbf{1.188722}$ | no |
| NAND | $2$ bits | $0.811278$ | $\mathbf{1.188722}$ | no |
| XOR | $2$ bits | $1.000000$ | $\mathbf{1.000000}$ | no |
| erasure | $1$ bit | $0.000000$ | $\mathbf{1.000000}$ | no |
| NOT | $1$ bit | $1.000000$ | $\mathbf{0.000000}$ | yes |
| copying | $1$ bit | $1.000000$ | $\mathbf{0.000000}$ | yes |
Observation 5.1 An irreversible gate pays one bit is not correct. AND loses $1.188722$ bits while XOR loses exactly $1.000000$: both are irreversible gates with two inputs and one output, and the amounts differ.
Observation 5.2 The reason is the bias of the output. The output of AND is zero three times and one once, carrying only $0.811278$ bits. The output of XOR is balanced and carries exactly one bit. What is lost is fixed by the bias of the output, not of the input.
Observation 5.3 Paper 131 wrote that an irreversible gate does not necessarily pay one bit. This paper puts the number there. It pays $1.188722$ bits, not one. And only erasure and XOR pay exactly one.
6. Gates that can be made reversible lose nothing
Check 6.1: examine the Toffoli and Fredkin gates on all $8$ states.
| gate | distinct outputs | lost | returns to identity twice |
|---|---|---|---|
| Toffoli | $8 / 8$ | $\mathbf{0.000000}$ | **yes** |
| Fredkin | $8 / 8$ | $\mathbf{0.000000}$ | **yes** |
Observation 6.2 Both are permutations of eight states, and both are involutions. Applying either twice gives the identity, so no inverse needs to be built; the same gate serves.
Observation 6.3 Both are universal. Any logic circuit can be built from either alone: hence computation itself requires no erasure. That is the content of Bennett’s conclusion.
7. A reversible AND: the cost is room, not erasure
Check 7.1: build AND from a Toffoli gate. Supply a third line set to zero and send $(a,b,0)$ to $(a,b,a\wedge b)$.
| input | output |
|---|---|
| $(0,0,0)$ | $(0,0,0)$ |
| $(0,1,0)$ | $(0,1,0)$ |
| $(1,0,0)$ | $(1,0,0)$ |
| $(1,1,0)$ | $(1,1,1)$ |
Observation 7.2 Four inputs go to four different outputs; the map is injective. The information lost is $\mathbf{0.000000}$ bits: the same AND went from $1.188722$ to zero by a change of construction alone.
Observation 7.3 It is not free. A third wire is required. The cost did not vanish; it changed form: from erasure into room. And room must eventually be cleared. The cost of erasure arrives then.
Observation 7.4 So there is a sixth place for the cost: when it is paid. Reversible computation did not remove the cost; it postponed it.
8. The demon’s books
Check 8.1: over $N$ cycles, place the work extracted beside the cost of erasing the record.
| $N$ | work extracted | cost of erasure | net |
|---|---|---|---|
| $1$ | $2.870979\times 10^{-21}$ J | $2.870979\times 10^{-21}$ J | $\mathbf{0}$ |
| $10$ | $2.870979\times 10^{-20}$ J | $2.870979\times 10^{-20}$ J | $\mathbf{0}$ |
| $100$ | $2.870979\times 10^{-19}$ J | $2.870979\times 10^{-19}$ J | $\mathbf{0}$ |
| $1000$ | $2.870979\times 10^{-18}$ J | $2.870979\times 10^{-18}$ J | $\mathbf{0}$ |
Observation 8.2 The net is exactly zero for every $N$. The demon can extract work; but so long as its memory is finite, it must eventually erase. At that moment everything extracted is returned.
Observation 8.3 With an infinite memory the demon could extract work forever. What breaks in that case is not the second law but the assumption of finiteness: as in Paper 136, where an exact equality hides in a tail, one must say which assumption is doing the work before saying what was broken.
9. Separator and fences
Separator: which operation the cost is assigned to. Measurement, the physical act of looking, erasure, the reservation of room, or the clearing of it. And when it is paid.
Fences:
- The debate is not declared settled: generalisations of the second law including measurement are still being studied.
- It is not claimed that the second law is violated, nor that it is not.
- Nothing is said about real hardware: truth tables were counted.
- No theory of feedback control is built.
- None of the five is called right: only that they placed the cost differently.
10. Roll call and closing
- One bit costs $2.870979\times 10^{-21}$ J (Check 3.1): $0.017919$ eV.
- The Szilard engine returns exactly that (Check 4.1): a difference of $6.5\times 10^{-35}$ J.
- AND loses $1.188722$ bits and XOR exactly $1.000000$ (Section 5).
- The separator is the bias of the output (Observation 5.2): $0.811278$ against $1.000000$.
- Toffoli and Fredkin lose $0$ bits (Check 6.1): and both are involutions.
- A reversible AND loses $0$ bits and needs a third wire (Section 7).
- The net is exactly zero for every $N$ (Check 8.1).
Closing: the demon’s cost was not one place. Maxwell placed none, Szilard placed it on measurement, Brillouin on the act of looking, Landauer on erasure, and Bennett on erasure alone. All five were trying to save the second law, and what differed was only which operation they named. And once numbers are put in, an irreversible gate pays one bit turns out to be wrong as well: AND pays $1.188722$ bits, and only erasure and XOR pay exactly one. A sixth place then appeared: reversible computation did not remove the cost but turned it into room and postponed it. It is paid later.
Revision Record
Second edition (September 2026): no claim, theorem or check has been altered from the first edition. In the English references, the lines citing earlier papers of this corpus still carried their Japanese titles; they now carry the English titles. Nothing of substance has changed, and nothing has been withdrawn.
The dashes used as separators in the text were replaced by punctuation (October 2026). No claim, number or reference was changed.
References
- Maxwell, J. C. (1871). Theory of Heat. London: Longmans, Green.
- Szilard, L. (1929). Uber die Entropieverminderung in einem thermodynamischen System bei Eingriffen intelligenter Wesen. Zeitschrift fur Physik, 53.
- Brillouin, L. (1951). Maxwell’s demon cannot operate: information and entropy I. Journal of Applied Physics, 22.
- Landauer, R. (1961). Irreversibility and heat generation in the computing process. IBM Journal of Research and Development, 5.
- Bennett, C. H. (1982). The thermodynamics of computation: a review. International Journal of Theoretical Physics, 21.
- Leff, H. S., Rex, A. F. (2003). Maxwell’s Demon 2. Bristol: Institute of Physics Publishing.
- Yamagishi, Y. (2026). The Physical Resource Bound on Brute-Force Search (3rd ed.) [Paper 4]. Zenodo. 10.5281/zenodo.21404131
- Yamagishi, Y. (2026). Two Entropies [Paper 88]. Zenodo. 10.5281/zenodo.21747809
- Yamagishi, Y. (2026). The Existence of an Integrating Factor Does Not Give You a Temperature [Paper 126]. Zenodo. 10.5281/zenodo.22079437
- Yamagishi, Y. (2026). Of $k_BT\ln2$, Only the $T$ Is Physics [Paper 131]. Zenodo. 10.5281/zenodo.22080240
- Yamagishi, Y. (2026). The Equality Is Exact, and the Average Is Held by the Tail [Paper 136]. Zenodo. 10.5281/zenodo.22081230
- Yamagishi, Y. (2026). Minimum Entropy Production Is Not a Principle but a Theorem with a Fence [Paper 143]. Zenodo. 10.5281/zenodo.22082367
Reference
Yamagishi, Y. (2026). Whose Cost Is the Demon’s? Five People Put It in Five Places [Paper 248]. YOI: https://www.yamagishiyuuki.com/yoi/248
DOI: 10.5281/zenodo.22160169
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 248. This paper is a preprint and has not been peer reviewed.