Abstract
“One mole is the amount in $12$ g of carbon-$12$” was for a long time a definition. The $2019$ SI revision made it no longer a definition. This paper asks how the standing of that equals sign changed: the answer is it moved from definition to empirical law. No new mathematical theorem and no new law is claimed.
First, the direction of definition was reversed. Before, mass fixed the count; now, the count fixes the mass (Section 2).
Second, this is the core of the paper. $M(^{12}\mathrm{C})=\mathbf{11.9999999958}$ g/mol, and it is no longer exactly $12$ (Section 3).
Third, but the departure is not yet significant. The difference of $4.2\times10^{-9}$ g/mol is only $\mathbf{1.17}$ times the uncertainty of $3.6\times10^{-9}$ (Section 3).
Fourth, so it is not “it departed” but “it became able to depart”. What changed is not the value but the standing of the equals sign (Section 4).
Fifth, the same happened to four units. Kilogram, ampere, kelvin, mole; and the triple point of water is no longer exactly $273.16$ K (Section 5).
Sixth, the separator is which side was fixed. The fixed side becomes definition; the remaining side becomes measurement (Section 6).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed: the $2019$ SI revision, the defined value of the Avogadro constant, the CODATA value of $M(^{12}\mathrm{C})$, and the molar mass constant are all standard. We do not build metrology: all we use is one subtraction and one division. We do not argue the merits of the revision: why it was made and which definition is better are not treated. We look only at how the standing of an equals sign moved. We do not enter experimental detail: neither the silicon-sphere determination of the Avogadro constant nor the watt balance is treated at all. We do not assert a departure from $12$: the difference of $4.2\times10^{-9}$ is $1.17$ times the uncertainty and is not significantly different from $12$. This paper does not claim “it is no longer $12$” but states “there is no longer a guarantee that it is $12$”. We do not track CODATA updates: $11.9999999958(36)$ is a recommended value at one epoch and may be revised. The claim is the structure, not the number. We do not exaggerate the practical impact: a relative $3.5\times10^{-10}$ is invisible in everyday analytical chemistry (relative $10^{-4}$). It shows only in the world of mass standards. Relation to earlier papers: Paper 16 showed that the equals sign too has distinct roots. This paper is an instance of one equals sign changing its root within history. In $2019$ it moved from the equals sign of definition to that of empirical law. Paper 95 separated convention from fact and treated the defined value of $c$: the $N_A$ here is a defined constant of the same type as $c$, having moved to the side of the scaffolding. Paper 268 showed that the triple point being a point is not an experimental fact. This paper sets beside it that that triple point’s temperature also ceased to be a definition in the same revision. Paper 196 showed that “pressure” is four separate quantities. This paper treats one quantity whose equals sign can hold two standings. What is added is computing the departure of $M(^{12}\mathrm{C})$ from $12$ as $4.2\times10^{-9}$ g/mol, confirming that it is only $1.17$ times the uncertainty, lining up the same exchange across four units, and placing the separator at which side was fixed.
1. Notation and the Question
On $20$ May $2019$ the definitions of the SI base units were revised. For the mole the change was as follows.
| Epoch | Definition | Direction |
|---|---|---|
| Before $2019$ | the amount in $\mathbf{12}$ g of carbon-$12$ is $1$ mol | mass fixes the count |
| From $2019$ | $N_A=6.02214076\times10^{23}$ /mol exactly | the count fixes the mass |
Question: What became of the equation $M(^{12}\mathrm{C})=12$ g/mol?
Observation 1.1 Before the revision this equation was a definition. Nothing to measure, no error. It was so because it had been decided so.
Observation 1.2 Paper 16 showed that the equals sign too has distinct roots. The equals sign of a definition, of an identity, and of an empirical law are written alike but differ in standing. This paper is an instance of one equals sign changing its root.
2. The Direction of Definition Was Reversed
Observation 2.1 The chain before the revision ran thus.
| Weigh out $12$ g of carbon-$12$ (mass is the starting point) |
|---|
| $\downarrow$ the number of atoms therein is $1$ mol |
| $\downarrow$ measure that number to obtain $N_A$ (it carries error) |
Observation 2.2 The chain after the revision runs thus.
| Declare $N_A=6.02214076\times10^{23}$ (the count is the starting point, no error) |
|---|
| $\downarrow$ collect that number of carbon-$12$ atoms |
| $\downarrow$ measure their mass to obtain $M(^{12}\mathrm{C})$ (it carries error) |
Observation 2.3 The error has changed places. Before, $N_A$ carried the error and $M(^{12}\mathrm{C})$ was exact. Now $N_A$ is exact and $M(^{12}\mathrm{C})$ carries the error.
Observation 2.4 Under either system, not one physical fact has changed. A carbon atom weighs the same and so does a gram. What changed is only the choice of which to fix as convention (Paper 95).
3. The Core: $M(^{12}\mathrm{C})$ Is $11.9999999958$ g/mol
Check 3.1:
| Quantity | Value |
|---|---|
| $N_A$ (defined) | $6.02214076\times10^{23}$ /mol exactly |
| $M(^{12}\mathrm{C})$ (measured) | $\mathbf{11.9999999958}$ g/mol |
| Its uncertainty | $\pm3.6\times10^{-9}$ g/mol |
| Departure from $12$ | $\mathbf{4.2\times10^{-9}}$ g/mol |
| In relative terms | $\mathbf{3.5\times10^{-10}}$ |
Observation 3.2 This is the core of the paper. $M(^{12}\mathrm{C})$ is no longer exactly $12$: the number $12$ is now an object of measurement.
Check 3.3: But is the departure significant?
| Quantity | Value |
|---|---|
| Departure $/$ uncertainty | $\mathbf{1.1667}$ |
Observation 3.4 The departure is only $\mathbf{1.17}$ times the uncertainty. It is not significantly different from $12$: present measurements are equally consistent with its being exactly $12$.
Observation 3.5 This is the exact way to put it: not “it has departed from $12$” but “there is no longer a guarantee that it is $12$”. Whether it has departed is for future measurement to decide: previously that question could not even be posed.
Check 3.6: We look at the molar mass constant $M_u=M(^{12}\mathrm{C})/12$.
| Quantity | Value |
|---|---|
| $M_u$ | $9.999999996500\times10^{-4}$ kg/mol |
| Relative departure from $10^{-3}$ kg/mol | $3.5\times10^{-10}$ |
Observation 3.7 The everyday rule “atomic weight $\times$ g/mol gives the molar mass” tacitly uses $M_u=1$ g/mol exactly. That has become inexact by $3.5\times10^{-10}$: though almost no laboratory sees that decade.
4. Not “It Departed” but “It Became Able to Depart”
Check 4.1: We see where this difference could act.
| Setting | Relative precision | $3.5\times10^{-10}$ is |
|---|---|---|
| Everyday analytical chemistry | $10^{-4}$ | invisible |
| High-precision mass spectrometry | $10^{-6}$ | invisible |
| Mass standards, atomic clocks | $10^{-9}$ | barely visible |
Observation 4.2 In practice nothing changed. The same computations give the same answers as before. What changed is not the value but the standing of the equals sign.
Observation 4.3 So this revision is an event on the side of logic, not of measurement. No new experiment refuted $12$: $12$ was moved to a place where it could be refuted.
Observation 4.4 In the language of Paper 16, the equals sign changed its root.
| Before $2019$ | From $2019$ | |
|---|---|---|
| $M(^{12}\mathrm{C})=12$ g/mol | equals of definition | equals of empirical law ($1.17\sigma$) |
| $N_A=6.02214076\times10^{23}$ | equals of empirical law | equals of definition |
Observation 4.5 The two rows have exactly traded places. The total count of definitions and measurements is unchanged: only the choice of which to fix has been remade.
5. The Same Happened to Four Units
Check 5.1:
| Unit | Old definition | New definition (constant fixed) |
|---|---|---|
| Kilogram | the international prototype | $h=6.62607015\times10^{-34}$ J s |
| Ampere | force between two wires | $e=1.602176634\times10^{-19}$ C |
| Kelvin | triple point of water, $273.16$ K exactly | $k=1.380649\times10^{-23}$ J/K |
| Mole | $12$ g of carbon-$12$ | $N_A=6.02214076\times10^{23}$ /mol |
Observation 5.2 The right column is “exactly” throughout. The left column holds objects and phenomena; the right holds numbers: a convention is a number, not a thing.
Check 5.3: The kelvin row underwent the same exchange as the mole.
| Temperature of the triple point of water | |
|---|---|
| Before $2019$ | $273.16$ K exactly (definition) |
| From $2019$ | $273.1600$ K $\pm 0.0001$ K (measured) |
Observation 5.4 Paper 268 showed that the triple point being a point is itself not an experimental fact. Here, further, the temperature of that point is no longer a definition either: for one and the same triple point, two separate reasons for “not a fact” now overlap.
Observation 5.5 In four units, equations that had been definitions became measured values. And in all four, the departure lies inside the present uncertainty: the revision was carried out continuously; no number jumped.
6. Separator: Which Side Was Fixed
Observation 6.1 Separator. What must be moved for the standing of an equals sign to change? Which side was fixed as convention.
| The fixed side | The remaining side | |
|---|---|---|
| Standing | definition (no error) | measurement (with error) |
| Question one may ask | none | “is it really that value?” |
Observation 6.2 Of the defined side one cannot ask “is it really so?”Because it is a convention: whether the question can be posed at all is what the difference in standing consists of.
Observation 6.3 Paper 95 treated the defined value of $c$. Since $1983$, the speed of light is not an object of measurement, because the metre is defined from $c$. In $2019$, $N_A$ moved to the same side as $c$.
Observation 6.4 So the research programme of “measuring the Avogadro constant precisely” ended in $2019$. The same apparatus still runs, but the name of what it measures has changed. It now realises the kilogram.
Observation 6.5 A point to write honestly. This paper does not argue which definition is better. What is shown is the structure of an equals sign having moved, not the merits of the revision, but writing “exactly $12$” without noticing that the standing has moved is no longer accurate.
7. Roll Call and Closing
- The direction reversed (Observations 2.1--2.2): Mass fixes count $\to$ count fixes mass.
- The error changed places (Observation 2.3): From $N_A$ to $M(^{12}\mathrm{C})$.
- $M(^{12}\mathrm{C})=11.9999999958$ g/mol (Check 3.1): No longer exactly $12$.
- Departure $4.2\times10^{-9}$ g/mol (Check 3.1): Relative $3.5\times10^{-10}$.
- Departure $/$ uncertainty $=1.1667$ (Check 3.3): Not significant.
- Not “it departed” (Observation 3.5): “The guarantee is gone.”
- Invisible in practice (Check 4.1): Only at $10^{-9}$.
- Two rows trade places exactly (Observation 4.4): The totals are unchanged.
- The same in four units (Check 5.1): The right column is all numbers.
- The triple point is not exactly $273.16$ (Check 5.3): $\pm0.0001$ K.
Closing: The molar mass of carbon-$12$ is no longer exactly $12$. The $2019$ SI revision made $N_A$ a defined value and the error changed places: before, mass fixed the count; now the count fixes the mass. The measured value is $M(^{12}\mathrm{C})=\mathbf{11.9999999958}$ g/mol, a departure from $12$ of $\mathbf{4.2\times10^{-9}}$ g/mol, relative $3.5\times10^{-10}$. But it must be put exactly: the departure is only $\mathbf{1.17}$ times the uncertainty and is not significantly different from $12$. Not “it has departed from $12$,” but “there is no longer a guarantee that it is $12$.”The same exchange happened in four units: kilogram, ampere, kelvin, mole. The triple point of water is no longer exactly $273.16$ K either. One thing separates them, which side was fixed as convention. The fixed side becomes definition, and one can no longer ask “is it really that value?”The remaining side becomes measurement, and that question becomes askable. What moved in $2019$ was not a value but which equals signs that question may be put to.
Revision Record
Second edition (October 2026): the dashes used as separators in the text were replaced by punctuation; section headings containing mathematics, which had been cut off in display, were repaired; no claim, number or reference was changed.
References
- Bureau International des Poids et Mesures (2019). The International System of Units (SI), 9th edition. Sevres: BIPM.
- Tiesinga, E., Mohr, P. J., Newell, D. B. and Taylor, B. N. (2021). CODATA Recommended Values of the Fundamental Physical Constants: 2018. Reviews of Modern Physics, 93(2).
- Mills, I. M., Mohr, P. J., Quinn, T. J., Taylor, B. N. and Williams, E. R. (2006). Redefinition of the Kilogram, Ampere, Kelvin and Mole: a Proposed Approach to Implementing CIPM Recommendation 1 (CI-2005). Metrologia, 43(3).
- Andreas, B. et al. (2011). Determination of the Avogadro Constant by Counting the Atoms in a $^{28}$Si Crystal. Physical Review Letters, 106(3).
- Newell, D. B. et al. (2018). The CODATA 2017 Values of $h$, $e$, $k$, and $N_A$ for the Revision of the SI. Metrologia, 55(1).
- Yamagishi, Y. (2026). The Equals Sign Has Distinct Roots Too [Paper 16]. Zenodo. 10.5281/zenodo.21600471
- Yamagishi, Y. (2026). Convention and Fact [Paper 95]. Zenodo. 10.5281/zenodo.21758621
- Yamagishi, Y. (2026). “Pressure” Is Four Separate Quantities, and Sharing a Unit Guarantees Nothing [Paper 196]. Zenodo. 10.5281/zenodo.22107518
- Yamagishi, Y. (2026). That the Triple Point Is a Point Is Not an Experimental Fact [Paper 268]. Zenodo. 10.5281/zenodo.22169829
Reference
Yamagishi, Y. (2026). The Molar Mass of Carbon-$12$ Is No Longer Exactly $12$ [Paper 295]. YOI: https://www.yamagishiyuuki.com/yoi/295
DOI: 10.5281/zenodo.22171659
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 295. This paper is a preprint and has not been peer reviewed.