Abstract
The absolute temperature scale is taken to be one thing. What this paper shows is that the first one laid down was logarithmic, that absolute zero was not a finite point on it, and that unattainability there was a definition rather than a theorem. No new theorem or law is claimed.
First, the demand fixes the shape of the scale——requiring that a unit of heat falling one degree yield the same work at every temperature gives a logarithmic scale with $A=\mathbf{320.554545}$ for ice at $0$ and steam at $100$ (Section 2).
Second, on this scale absolute temperatures form a geometric progression——one degree down multiplies $T$ by $\mathbf{0.9968852670}$, and a factor of ten in $T$ is $\mathbf{738.1041}$ degrees (Section 2).
Third, and this is the core. The demand is in fact satisfied——the work is $\mathbf{0.0031147330}$ at $T=1000$ and at $T=3$ alike, with a spread of $\mathbf{0.000\times10^{0}}$ (Section 3).
Fourth, the present scale does not satisfy it——one kelvin down yields $\mathbf{0.3333333333}$ at $T=3$ K and $\mathbf{0.0010000000}$ at $T=1000$ K, the ratio between $T=30$ and $T=300$ being exactly $\mathbf{10.000000}$ (Section 3).
Fifth, absolute zero has a different standing——on the logarithmic scale $T=100$ K is $\mathbf{-322.1095}$ degrees and $1$ K is $\mathbf{-1798.3178}$, so zero lies at minus infinity (Section 4).
Sixth, the separator is what was chosen to be held constant——hold the work constant and the scale is logarithmic; equate ratios of heat with ratios of temperature and the present scale appears (Section 5).
Keywords: absolute temperature, Kelvin temperature scale, logarithmic scale, absolute zero, third law of thermodynamics.
Scope (what is not claimed)
No new theorem or law is claimed——Carnot’s theory, Thomson’s two scales and the third law of thermodynamics are all standard. No historical detail is asserted——it was confirmed against the primary source that Thomson’s 1848 paper required a unit of heat descending from $T$ to $T-1$ to give the same mechanical effect whatever $T$; nothing further about intent or context is claimed. The caloric theory is not judged——the 1848 framework treats heat as conserved, and this paper only translates its demand into present thermodynamic language to obtain numbers. The third law is not proved——only the difference in standing of unattainability is described. No scale is proposed for use——it is not suggested that a logarithmic scale should be adopted. The choice of fixed points is not discussed——ice at $0$ and steam at $100$ were taken for exposition. Relation to earlier papers: Paper 403 showed that the order of temperature is not the order of hotness——the two scales here agree on order and differ in spacing, so agreement on order does not fix a scale. Paper 379 showed that “twice as hot” has three answers, only one giving a ratio of two——on a logarithmic scale “twice” means something else again, a fourth answer standing in the prehistory. Paper 509 showed that “unattainable” has four senses——this adds, on the thermodynamic side, an earlier one that is unattainable by definition. Paper 95 separated convention from fact——which scale to adopt is convention, while the Carnot efficiency $1-T'/T$ is fact. Paper 300 showed that sharing a root is decidable——the two scales are separate roots under the criterion of moving a variable. What is added is building the scale from the demand to obtain $A=\mathbf{320.554545}$, confirming the demand at four temperatures, measuring the present scale’s inverse dependence as a factor of $\mathbf{10.000000}$, showing zero at minus infinity through the constant step of $\mathbf{738.1041}$ degrees, and placing the separator on what is held constant.
1. Notation and the Question
Setting: a Carnot engine taking heat $Q$ from $T$ and rejecting it at $T'$ yields work $W=Q (1-T'/T)$.
The demand of 1848: Thomson chose his scale $\theta$ so that a unit of heat descending from $\theta$ to $\theta-1$ would give the same work whatever the value of $\theta$.
Observation 1.1 this is a demand on a scale, not a claim about nature. Which demand to impose is a choice——in the terms of Paper 95, a scale lies on the side of convention.
Observation 1.2 a different demand gives a different shape. The present scale demands that ratios of heat equal ratios of temperature——two scales can agree on order and differ entirely in spacing.
Question: what scale follows from the demand of 1848?
2. Building the Scale from the Demand
Computation 2.1: the $\theta$ satisfying the demand is logarithmic in absolute temperature. Fixing ice at $0$ and steam at $100$,
$$\theta(T)=A\,\ln\frac{T}{273.15},\qquad A=\frac{100}{\ln(373.15/273.15)}$$gives $A=\mathbf{320.554545}$.
| absolute temperature (K) | the 1848 scale (degrees) |
|---|---|
| $373.15$ | $\mathbf{100.0000}$ |
| $273.15$ | $\mathbf{0.0000}$ |
| $100.00$ | $\mathbf{-322.1095}$ |
| $10.00$ | $\mathbf{-1060.2136}$ |
| $1.00$ | $\mathbf{-1798.3178}$ |
Observation 2.2 the fixed points return. Ice at $\mathbf{0.0000}$, steam at $\mathbf{100.0000}$——a check on the construction, not a claim.
Observation 2.3 geometric becomes uniform. Each factor of ten in $T$ is $\mathbf{738.1041}$ degrees——$100$ K to $10$ K and $10$ K to $1$ K span the same interval.
Observation 2.4 so absolute zero is not on the scale. Each division of $T$ by ten drops the same amount, without end——zero lies at minus infinity and cannot be written as a finite reading.
3. Is the Demand Actually Met
Computation 3.1: one degree down on this scale multiplies $T$ by $\exp(-1/A)=\mathbf{0.9968852670}$. Measuring the work $1-T'/T$ for a unit of heat at several $T$.
| $T$ (K) | $T'$ (K) | work |
|---|---|---|
| $1000.00$ | $996.8853$ | $\mathbf{0.0031147330}$ |
| $300.00$ | $299.0656$ | $\mathbf{0.0031147330}$ |
| $30.00$ | $29.9066$ | $\mathbf{0.0031147330}$ |
| $3.00$ | $2.9907$ | $\mathbf{0.0031147330}$ |
| spread | — | $\mathbf{0.000\times10^{0}}$ |
Observation 3.2 this is the core. Moving $T$ over a factor of more than three hundred leaves the work unchanged——the demand is met, so the scale follows correctly from it.
Computation 3.3: doing the same on the present scale, where one kelvin down yields $1/T$.
| $T$ (K) | work for one kelvin down |
|---|---|
| $1000.00$ | $\mathbf{0.0010000000}$ |
| $300.00$ | $\mathbf{0.0033333333}$ |
| $30.00$ | $\mathbf{0.0333333333}$ |
| $3.00$ | $\mathbf{0.3333333333}$ |
Observation 3.4 the far side of the boundary. The ratio between $T=30$ and $T=300$ is exactly $\mathbf{10.000000}$——the present scale does not meet the demand of 1848, so the two are different objects.
Observation 3.5 the gap opens at low temperature. At $T=3$ K a single kelvin turns a third of the heat into work——a degree weighs more the colder it is, and the logarithmic scale is an attempt to even that weight.
4. The Standing of Unattainability
Observation 4.1 on the logarithmic scale absolute zero is unattainable by definition. Its reading is minus infinity——one need not even say that finitely many operations cannot cover an infinite interval. There is no point there.
Observation 4.2 on the present scale unattainability is a theorem. Zero kelvin is a finite point, and the third law says it cannot be reached——one is a property of the scale, the other a claim about physics.
Observation 4.3 the census of Paper 509 was missing a predecessor. That paper found four senses of “unattainable”——unattainability by definition is a separate root from all of them, since changing the scale removes it.
Observation 4.4 order is unchanged. A logarithm is monotone, so both scales agree on which of two bodies is hotter——as Paper 403 showed, agreement on order does not fix a scale.
5. The Separator
Observation 5.1 the separator is what was chosen to be held constant. Hold the work constant and the scale is logarithmic; equate ratios of heat with ratios of temperature and the present scale appears——one physics, two scales.
Observation 5.2 separate roots under the criterion of Paper 300. Move $T$ and the work holds still on one scale and varies as $1/T$ on the other——the same quantity behaves differently, so the roots are not shared.
Observation 5.3 a fourth answer for Paper 379. That paper found three answers to “twice as hot”——on the logarithmic scale “twice” means a ratio of $\exp(\theta/A)$ in absolute temperature, unlike any of them.
Observation 5.4 which is right cannot be asked. A scale is a convention——but choosing one fixes the consequences, and whether unattainability is a definition or a theorem is settled after the choice.
Observation 5.5 what is not said. Neither historical detail nor the caloric theory is judged——the demand was confirmed against the primary source, and nothing beyond it is claimed.
6. Roll Call and Closing
- $A=\mathbf{320.554545}$ (Computation 2.1): from ice at $0$, steam at $100$.
- the fixed points return (Observation 2.2): a check on the construction.
- $\mathbf{738.1041}$ degrees per decade (Observation 2.3): geometric becomes uniform.
- $\mathbf{-1798.3178}$ at $1$ K (Computation 2.1): zero lies beyond.
- $\mathbf{0.9968852670}$ per degree (Computation 3.1): always the same ratio.
- work constant at $\mathbf{0.0031147330}$ (Computation 3.1): the core; spread $\mathbf{0}$.
- $\mathbf{0.3333333333}$ at $T=3$ K now (Computation 3.3): inverse in $T$.
- ratio $\mathbf{10.000000}$ (Observation 3.4): the far side.
- different standing of unattainability (Observation 4.1): definition or theorem.
- order unchanged (Observation 4.4): a scale is not fixed by order.
- the separator is what is held constant (Observation 5.1): work, or ratio.
Closing: The absolute temperature scale is taken to be one thing. The first one laid down was logarithmic——the demand that a unit of heat falling one degree yield the same work at every temperature gives a logarithmic scale with $A=\mathbf{320.554545}$ for ice at $0$ and steam at $100$. The fixed points return as $\mathbf{0.0000}$ and $\mathbf{100.0000}$, checking the construction. Geometric steps become uniform ones: a factor of ten in $T$ is $\mathbf{738.1041}$ degrees, whether from $100$ K to $10$ K or from $10$ K to $1$ K. So absolute zero is not on the scale at all, with $T=100$ K at $\mathbf{-322.1095}$ and $1$ K at $\mathbf{-1798.3178}$, falling without end. The demand is met——one degree down multiplies $T$ by $\mathbf{0.9968852670}$, and the work is $\mathbf{0.0031147330}$ at $T=1000$ K and at $T=3$ K alike, with a spread of $\mathbf{0.000\times10^{0}}$. Moving $T$ by more than a factor of three hundred changes nothing, so the scale follows correctly from the demand. The present scale does not meet it: one kelvin down yields $\mathbf{0.3333333333}$ at $T=3$ K and $\mathbf{0.0010000000}$ at $T=1000$ K, with a ratio between $T=30$ and $T=300$ of exactly $\mathbf{10.000000}$. A degree weighs more the colder it is, and the logarithmic scale evens that weight. Unattainability has a different standing there——on the logarithmic scale zero reads minus infinity and is unattainable by definition; one need not even invoke finitely many operations, since there is no point there. On the present scale zero kelvin is a finite point and unattainability is the third law, a claim about physics. Paper 509 found four senses of the word, and unattainability by definition is a separate root from all of them, since changing the scale removes it. A logarithm is monotone, so both scales agree on which body is hotter, and as Paper 403 showed, order does not fix a scale. The separator is what was chosen to be held constant——hold the work and the scale is logarithmic; equate ratios and the present scale appears, from one and the same physics. Under the criterion of Paper 300, moving $T$ leaves the work still on one scale and varying as $1/T$ on the other, so the roots are not shared. Paper 379 found three answers to “twice as hot”; the logarithmic scale supplies a fourth. Which is right cannot be asked, since a scale is a convention, but the choice fixes whether unattainability is a definition or a theorem. To be honest——no historical detail is asserted beyond the demand itself, which was confirmed against the primary source. The caloric theory is not judged; the 1848 framework was translated into present thermodynamic language to obtain numbers. The third law was not proved, no scale is recommended for use, and the fixed points were chosen for exposition. What can be said is that a demand fixes the shape of a scale, and that on the resulting scale absolute zero is unattainable by definition, and no further.
Revision Record
2nd Edition (October 2026): The Japanese title of Paper 509 in the Japanese references was brought into line with its corrected title. No claim or number was changed.
References
- Carnot, S. (1824). Reflexions sur la puissance motrice du feu. Bachelier.
- Chang, H. (2004). Inventing Temperature: Measurement and Scientific Progress. Oxford University Press.
- Thomson, W. (1848). On an Absolute Thermometric Scale Founded on Carnot’s Theory of the Motive Power of Heat, and Calculated from Regnault’s Observations. Philosophical Magazine.
- Thomson, W. (1854). Thermo-electric Currents. Transactions of the Royal Society of Edinburgh.
- Yamagishi, Y. (2026). Convention and Fact [Paper 95]. Zenodo. 10.5281/zenodo.21758621
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). “Twice as Hot” Has Three Answers, and Only One Gives a Ratio of $\mathbf{2}$ [Paper 379]. Zenodo. 10.5281/zenodo.22698802
- Yamagishi, Y. (2026). The Order of Temperature Is Not the Order of Hotness [Paper 403]. Zenodo. 10.5281/zenodo.22699405
- Yamagishi, Y. (2026). “Unattainable” Has Four Senses in This Corpus, and Two of Them Sit in One Field [Paper 509]. Zenodo. 10.5281/zenodo.22861975
Reference
Yamagishi, Y. (2026). The First Absolute Scale Was Logarithmic, and There Absolute Zero Was Unattainable by Definition [Paper 534]. YOI: https://www.yamagishiyuuki.com/yoi/534
DOI: 10.5281/zenodo.23014117
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 534. This paper is a preprint and has not been peer reviewed.