Abstract
Physics formulas are full of factors of π such as 2π, 4π and (2π)³. Within the system, it has been counted that the same 2π has three distinct roots (Paper 18), that the 2 of 8π has ten origins (Paper 1085), and that things of the same number are not called the same root (Paper 300). This paper counts, in the texts of papers outside the system, what a factor of π comes from (its address). From 344 source texts (LaTeX) of arXiv papers in nuclear experiment and nuclear theory, 956 factors of π were picked up, and 300 places chosen at random were read one by one and given an address label. No new theorem or law is claimed. No paper’s name, number or text is shown; only counts are given.
First, the same 2π splits into two addresses. Of the 87 places of 2π read, 38 (43.7 per cent) were one turn of a circle or an angle (azimuthal integrals, integer multiples of 2π), and 31 (35.6 per cent) were phase and Fourier (dω/2π, 2πδ). Of the three distinct roots of Paper 18, the solid angle of a hemisphere never appeared.
Second, the same 4π splits into four addresses. The 61 places of 4π read divided into solid angle 22 (36.1 per cent), the way a coupling constant or a unit is defined (g²/4π, e²/4π, 4πα) 18 (29.5 per cent), the context of comparing the ratio of viscosity to entropy multiplied by 4π 9 (14.8 per cent), and normalization 7 (11.5 per cent).
Third, this is the core of the paper. There are factors whose form alone fixes the address and factors whose form does not. All 45 places of (2π)³ and (2π)⁴ were phase and Fourier (95 per cent range 92.1 to 100), all 10 places of 4π accompanied by the square of a length, as in 4πp² or 4πr², were solid angle, and 8 of the 9 places of πR² were a disk. Only the bare 2π and 4π do not state their address by their form.
Fourth, the address is not written in words but carried silently inside formulas. Of the 147 texts with a factor of π, 2 (1.4 per cent) said in words where a factor comes from. Factors that multiply two addresses into one number (1/(2π²)＝4π÷(2π)³ and the like) were 19 of the 300 places read (6.3 per cent). Of the 18 candidate pairs in which the same form appears with two addresses in one paper, reading found 8, and within the parts read no sentence treated the two as the same factor.
To be candid, the arXiv service stopped the download of source texts with a rate limit, so only the two nuclear fields are covered. Only the 220 characters before and after each factor were read, not the whole paper. There was one reader (an AI), and agreement with a second reader has not been measured. Whether other people’s papers are right is not judged.
Keywords: factor of pi, solid angle, Fourier transform, system of units, nuclear physics, literature census, distinct roots.
Scope (what is not claimed)
No new theorem or law is claimed. That factors of the same number can have different origins is widely known in physics. No individual paper is named. No paper’s name, title, number, authors or text is shown; only counts are given. Whether other people’s papers are right is not judged. A pair with two addresses is not called an error (within the parts read, each appeared in different formulas or in different parts of one formula). Nothing is extended beyond nuclear physics. The sample is only 344 source texts in nuclear experiment and nuclear theory, and fields are not compared. The whole papers were not read. Only the 220 characters before and after each factor were read, and what could not be decided there was put under “cannot be decided”. There was one reader. The table of labels was fixed before reading, but agreement with a second reader has not been measured. Relation to earlier papers: Paper 18 took a census of three distinct roots of 2π (the solid angle of a hemisphere, the period of a phase and the value of the odd ladder). Here, in texts outside the system, it is counted in what proportion two of them (phase and one turn of a circle) actually appear. Paper 1085 sorted the origins of the 2 of 8π into ten kinds in 43 papers of the system. Here the same census is made on a random sample of texts outside the system. Paper 1081, Paper 1083 and Paper 1088 sorted the origins of the factors of π in well-known formulas, formula by formula. Here factors picked up from texts without choosing formulas are counted. Paper 46 showed that of 16π² only 2π² (the size of the three-dimensional sphere) is geometry and the rest is the Fourier convention and the like. The 1/(2π²) counted here is 4π÷(2π)³, a distinct root of the same number. Paper 35 and Paper 60 took the m＝4π of the system to be a choice of units. The 4π of the coupling constant counted here (g²/4π) falls under the same label of “how it is defined”. Paper 300 set the criterion not to call things of the same number the same root. Added here are that the splitting of the addresses of the same 2π and 4π has been counted in texts outside the system, that factors whose form states the address, such as (2π)³ and 4πp², have been separated from factors whose form does not, such as the bare 2π and 4π, that few texts state the origin in words, and that factors multiplying two addresses into one number occur in a definite proportion.
1. The question
The $2\pi$ and $4\pi$ of physics formulas can come from different places even when they are the same number. The system has so far counted, within itself or in chosen formulas, that $2\pi$ has three distinct roots (Paper 18: the solid angle of a hemisphere, the period of a phase and the value of the odd ladder), that the 2 of $8\pi$ has ten origins (Paper 1085), and that the origins of the factors of $\pi$ in well-known formulas differ from formula to formula (Paper 1081, Paper 1083 and Paper 1088).
This paper counts, in the texts of papers outside the system, what a factor of $\pi$ comes from (below, its “address”). Factors are picked up from texts without choosing formulas, and places chosen at random are read and labelled. No paper’s name, title, number, authors or text is shown; only counts are given.
2. The sample
From the public arXiv service (OAI-PMH), the titles and abstracts of 1622131 papers in the physics set were harvested. Papers whose primary field (the first category) was nuclear experiment numbered 12621, and nuclear theory 36406. The plan was to choose source texts (LaTeX) at random with a fixed seed, 230 from each of 13 fields, 2990 in all, and to download them one at a time with at least 3 seconds between requests. After 367, however, the arXiv service returned a rate limit (429), and the download was stopped at once. Because the order of downloading had been left in the order of fields, what had been obtained was only 230 papers in nuclear experiment and 137 in nuclear theory.
| stage | papers or places |
|---|---|
| source texts downloaded | 367 |
| of which with LaTeX (experiment 212, theory 132) | 344 |
| PDF only, not found, no LaTeX | 20, 2, 1 |
| factors of $\pi$ (all picked up) | 1075 |
| of which removed as pions and the like | 119 |
| remaining factors of $\pi$ | 956 |
| texts with a factor of $\pi$ (experiment 61, theory 86) | 147 |
| places chosen at random and read | 300 |
What was picked up are pairs of a number and $\pi$ ($2\pi$, $4\pi$, $(2\pi)^3$, $4\pi p^2$, $\pi R^2$ and so on). In nuclear papers $\pi$ often denotes the pion ($\pi^0$, two-pion exchange, $\eta\to3\pi$), so those were removed without an address.
3. The table of labels and how it was read
Before reading, the table of address labels was fixed.
| label | address |
|---|---|
| solid angle, sphere | all directions $4\pi$, sphere $4\pi r^2$, momentum shell $4\pi p^2dp$, $1/(4\pi r)$ |
| circle, angle, one turn | one turn $2\pi$, azimuthal integrals, integer multiples of $2\pi$, disk $\pi R^2$ |
| phase, Fourier | $h=2\pi\hbar$, $\omega=2\pi f$, $d^3p/(2\pi)^3$, $d\omega/2\pi$, $2\pi\delta$ |
| normalization | Gaussian $\sqrt{2\pi}$, $\sqrt{4\pi}$ of spherical harmonics |
| how units or couplings are defined | $\varepsilon_0$, Gaussian units, $g^2/4\pi$, $e^2=4\pi\alpha$ |
| gravity, loop, horizon | $8\pi G$, one-loop $16\pi^2$, viscosity to entropy ratio |
| cannot be decided | undecidable from the context, two addresses multiplied together |
| remove | pions, the symbol of the stress tensor, not a factor |
The machine gave candidate labels from the adjacent symbol ($G$ of $8\pi G$, $\hbar$ of $2\pi\hbar$, the form $(2\pi)^3$) and from words within 160 characters (solid angle, loop, Fourier and so on). When reading, the machine labels were hidden and only the 220 characters before and after were read to give one label. After reading, the result was compared with the machine labels. The ranges of proportions are 95 per cent Wilson intervals (Wilson 1927).
4. Same form, different addresses
The 300 places read were divided by form (excluding 16 places where the machine cut out the form wrongly and 7 places removed).
| form | places | solid angle | circle, angle | phase | normalization | definition | other |
|---|---|---|---|---|---|---|---|
| $2\pi$ | 87 | 0 | 38 | 31 | 4 | 0 | 14 |
| $4\pi$ | 61 | 22 | 0 | 0 | 7 | 18 | 14 |
| $(2\pi)^3$, $(2\pi)^4$ | 45 | 0 | 0 | 45 | 0 | 0 | 0 |
| $4\pi p^2$, $4\pi r^2$ and the like | 10 | 10 | 0 | 0 | 0 | 0 | 0 |
| $\pi R^2$, $\pi r^2$ | 9 | 0 | 8 | 0 | 1 | 0 | 0 |
“Other” is, for $2\pi$, cannot be decided 13 and loop 1, and for $4\pi$, the viscosity to entropy ratio 9, cannot be decided 4 and loop 1.
The same $2\pi$ is one turn of a circle or an angle in 43.7 per cent (95 per cent range 33.7 to 54.1) and phase and Fourier in 35.6 per cent (26.4 to 46.1), together 79.3 per cent (69.6 to 86.5). The $2\pi$ of circle and angle appeared in $1/(2\pi p_T)$, dividing the transverse momentum distribution by the azimuth, in azimuthal integrals $\int_0^{2\pi}d\phi$ and in periodicity conditions $2\pi n$. The $2\pi$ of phase and Fourier appeared in $d\omega/2\pi$, $2\pi\delta(\omega)$, $e^{2\pi i\cdots}$ and the like.
The same $4\pi$ split four ways: solid angle 36.1 per cent (25.2 to 48.6), the way a coupling constant or a unit is defined 29.5 per cent (19.6 to 41.9), the context of comparing the viscosity to entropy ratio multiplied by $4\pi$ 14.8 per cent (8.0 to 25.7), and normalization 11.5 per cent (5.7 to 21.8). The $4\pi$ of solid angle appeared in detectors covering all directions, the volume of a uniform sphere and the Green function $1/(4\pi r)$ of a point source, and the $4\pi$ of definition in $g^2/4\pi$, $e^2/4\pi$, $4\pi\alpha$, formulas in Gaussian units and the like. All 9 places in the viscosity context multiplied the ratio $\eta/s$ by $4\pi$ to compare it with the value $1/(4\pi)$ obtained from a calculation on the gravity side (Kovtun, Son & Starinets 2005). The origin of this $4\pi$ is not decomposed in this paper.
5. Factors whose form states the address
There were factors whose form alone fixes the address. All 45 places of $(2\pi)^3$ and $(2\pi)^4$ were phase and Fourier (95 per cent range 92.1 to 100), all 10 places of $4\pi$ accompanied by the square of a length, as in $4\pi p^2$ or $4\pi r^2$, were solid angle (72.2 to 100), and 8 of the 9 places of $\pi R^2$ were a disk (56.5 to 98.0). A power of (2π) is the measure of momentum space and the square of a length is the surface of a sphere, so the form states the address.
Only the bare $2\pi$ and $4\pi$ do not state their address by their form. The same $4\pi$ counts all directions in one formula and carries the definition of a coupling constant in another. Just as Paper 35 and Paper 60 took the $m=4\pi$ of the system to be a choice of units, the $4\pi$ of $g^2/4\pi$ falls under the address of “how it is defined”. The separator is whether the form of the factor states its address or not.
6. Against the three distinct roots of Paper 18
Paper 18 counted three distinct roots of $2\pi$: the solid angle of a hemisphere (root A), the period of a phase (root B) and the value of the odd ladder $\mathrm{Vol}(S^1)$ (root C). In the 87 places of $2\pi$ read, root B (phase and Fourier) appeared 31 times, one turn of a circle or an angle, corresponding to root C, 38 times, and root A (the solid angle of a hemisphere) 0 times. The $2\pi$ of nuclear texts is carried almost entirely by two roots, B and C.
7. Candidates for places hard to tell apart
Pairs in which the same form appears with two or more addresses in one paper were output by the machine as candidates and all read. Of the 18 candidate pairs, 8 really had two or more addresses ($2\pi$ in 7, $4\pi$ in 1), and the other 10 were errors of the machine labels. There were 83 texts with $2\pi$, and in 7 of them two or more addresses lived together (because the machine left some places unlabelled, this is an underestimate). In all 8 pairs, the two addresses appeared in different formulas or in different parts of one formula (for example, in $\sum_m 2\pi\delta(x-2\pi m)$ the leading $2\pi$ is phase and the inner $2\pi m$ is one turn). Within the parts read (220 characters before and after), no sentence treated the two as the same factor.
Next, sentences that state in words where a factor comes from were output by the machine as candidates and all read (a factor of $\pi$ near the word factor, with comes from, arises, due to, convention, normalization and the like in the same sentence). Of the 12 candidate sentences (7 texts), 3 sentences (2 texts) really stated the origin of a factor of $\pi$, and the rest were form factors of pions and the like. That is 2 of the 147 texts with a factor of $\pi$ (1.4 per cent, 95 per cent range 0.4 to 4.8). The address is usually not written in words but carried silently inside formulas.
Finally, factors in which two addresses are multiplied into one number were 19 of the 300 places read (6.3 per cent, 4.1 to 9.7). The most frequent form is $1/(2\pi^2)=4\pi/(2\pi)^3$ (the solid angle of the momentum shell divided by Fourier; 7 of the 19), followed by $1/(4\pi^2)$ (3) and Fermi gas formulas such as $\mu^4/(12\pi^2)$. From the form alone the two addresses inside cannot be seen. Paper 46 showed that of $16\pi^2$ only the size of the three-dimensional sphere, $2\pi^2$, is geometry. The $2\pi^2$ of the $1/(2\pi^2)$ counted here is the ratio of $4\pi$ and $(2\pi)^3$, a distinct root of the same number (Paper 300).
8. How often the machine labels were right
Of the 195 places to which the machine gave a label, 136 (69.7 per cent, 63.0 to 75.8) agreed with the label from reading. The 105 places the machine left unlabelled were, on reading, cannot be decided 36, phase 16, circle and angle 10 and so on.
| machine label | right | 95 per cent range |
|---|---|---|
| circle, angle, one turn | 24/26 (92.3) | 75.9 to 97.9 |
| phase, Fourier | 77/100 (77.0) | 67.8 to 84.2 |
| how units or couplings are defined | 9/16 (56.2) | 33.2 to 76.9 |
| solid angle, sphere | 22/41 (53.7) | 38.7 to 67.9 |
| normalization | 3/8 (37.5) | 13.7 to 69.4 |
There were 16 places where the machine cut out the form wrongly (reading $\pi^{3/2}$ as $\pi^3$, taking the power 2 of a preceding letter as a coefficient, and so on). There were 7 places where $\pi$ was the symbol of the stress tensor or a pion label. All proportions come from the labels from reading; the machine labels were used only to produce candidates.
9. What was watched in the check
When the count was first made on abstracts only, the number of abstracts with a factor of $\pi$ came out as 0. Doubting the 0, it turned out that in the harvested abstracts $\pi$ is a single character, while only the command form was being picked up. After the fix, counting all 1622131 titles and abstracts, 5494 (0.34 per cent) had a factor of $\pi$; abstracts were not enough, so source texts were downloaded. In the self-test of the machine labels, the arrow of $4\pi\to2\pi$ was nearly read as a pion decay and the $\eta$ of the Stokes drag $6\pi\eta a$ as a pion, so sentences of my own making were added and the rules were fixed. Because the order of downloading had been left in the order of fields, the sample at the moment of the stop was biased towards nuclear physics. The check confirms that recounting the factors of $\pi$ from the texts agrees with the local candidates, and that drawing the sample again with the same seed gives the same 300 places.
Closing
Closing: reading 300 places chosen at random in 344 arXiv source texts in nuclear experiment and nuclear theory, the same 2π splits into two, one turn of a circle or an angle (43.7 per cent) and phase and Fourier (35.6 per cent), and the same 4π splits into four: solid angle, the way a coupling constant or a unit is defined, the context of the viscosity to entropy ratio, and normalization. All 45 places of (2π)³ and (2π)⁴ were Fourier and all 10 places of 4πp² or 4πr² were solid angle, so there are factors whose form states the address and factors that do not, such as the bare 2π and 4π. 2 of the 147 texts state the origin in words, and 6.3 per cent of the factors multiply two addresses together. There were 8 pairs in which the same form appears with two addresses in one paper, but within the parts read no sentence treated the two as the same factor. To be candid, the sample is only two nuclear fields, only the parts around the factors were read, and there was one reader. The separator is whether the form of the factor states its address or not.
References
- Kovtun, P. K., Son, D. T. & Starinets, A. O. (2005). Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics. Physical Review Letters.
- Wilson, E. B. (1927). Probable Inference, the Law of Succession, and Statistical Inference. Journal of the American Statistical Association.
- Yamagishi, Y. (2026). The Yamagishi Kinetic Energy Equation [Paper 18]. YOI: https://www.yamagishiyuuki.com/yoi/18; DOI: 10.5281/zenodo.21430706
- Yamagishi, Y. (2026). The Yamagishi Gravitational Constant Unpacking Theory [Paper 35]. YOI: https://www.yamagishiyuuki.com/yoi/35; DOI: 10.5281/zenodo.21430118
- Yamagishi, Y. (2026). Of $16\pi^2$, Only $\Omega_3=2\pi^2$ Is Geometry [Paper 46]. YOI: https://www.yamagishiyuuki.com/yoi/46; DOI: 10.5281/zenodo.21602422
- Yamagishi, Y. (2026). c as the Ball of Velocity Space [Paper 60]. YOI: https://www.yamagishiyuuki.com/yoi/60; DOI: 10.5281/zenodo.21598012
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. YOI: https://www.yamagishiyuuki.com/yoi/300; DOI: 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). A Hard Sphere Has Four Cross Sections, 1, 2, 4 and 8 Times $\pi a^2$, and the 4 Appears Only in Three Dimensions [Paper 1081]. YOI: https://www.yamagishiyuuki.com/yoi/1081
- Yamagishi, Y. (2026). Setting $m=4\pi$ Turns $h/m$ into $\hbar/2$, and Its 2 Has a Different Root from the 2 of the Uncertainty Bound [Paper 1083]. YOI: https://www.yamagishiyuuki.com/yoi/1083
- Yamagishi, Y. (2026). A Census of the 2 in 8π: It Has Ten Origins, and Changing the Dimension Sorts Them into Three Ways of Moving [Paper 1085]. YOI: https://www.yamagishiyuuki.com/yoi/1085
- Yamagishi, Y. (2026). The Same 4π/m, but the 4π of the Plasma Frequency Is a Solid Angle and the 4π of Bethe’s Formula Is Twice the 2π of a Ring [Paper 1088]. YOI: https://www.yamagishiyuuki.com/yoi/1088
Reference
Yamagishi, Y. (2026). In Nuclear Physics Papers 2π Splits into Two Addresses and 4π into Four, and the Factors Whose Form Alone Fixes the Address Are (2π)³ and 4πp² [Paper 1095]. YOI: https://www.yamagishiyuuki.com/yoi/1095
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 1095. This paper is a preprint and has not been peer reviewed.