Abstract
Paper 01 showed that in three-dimensional space the mass of one spherical shell is independent of the radius only for a density 1/r², and Paper 02 showed that the mass inside radius 1 then equals the full solid angle 4π. This paper shows with numbers that there are two ways to extend this “2” to other dimensions, and that the two give the same number only in three dimensions. No new theorem or law is claimed.
First, there are two extensions. The exponent that keeps the shell mass constant is d−1, which Paper 114 tabulated as a family present in every dimension. The other is the distribution whose enclosed mass, cut at any radius, equals the mass of a black hole (the Tangherlini solution) with its horizon at that radius; its exponent is 2 in every dimension. For d = 3, 4, 5, 6 the former gave 2, 3, 4, 5 and the latter 2, 2, 2, 2.
Second, in three dimensions the coefficient carries 8πG. The horizon-balance distribution is ρ = c²/(8πG r²); in units with 8πG = 1, the distribution 1/r² of Paper 02 itself has 2GM/r = 1 at every radius (1.000000000 at radii 0.5, 1, 2 and 10). With G = 4π (Paper 35) the coefficient is $\mathbf{0.003166287}$. The “exactly 1” comes from the choice of units.
Third, and this is the core: the two 2s have different roots. The 2 on the side of gravity comes from the fact that the product of the gravitational constant and the density is an inverse time squared in every dimension. The time for a shell released from rest to fall is the same for radii 1, 2 and 5: $\mathbf{0.542700}$ in three dimensions, $\mathbf{0.450180}$ in four and $\mathbf{0.398700}$ in five. The d−1 on the side of the shell counts the dimension of the sphere. The two are the same number only in three dimensions, where d−1 is 2.
Fourth, the cosmological coincidence does not choose a dimension. Running the Newtonian shell (Paper 907) in d dimensions, the mass of the Hubble sphere of a flat universe equals, in every dimension, the mass of the black hole of the same radius (largest deviation 4.4e−16 for d = 3 to 10). The coincidence that Paper 169 showed to be an identity in three dimensions belongs to the 2 of gravity. The horizon-balance distribution is a nest of Hubble spheres: at every radius the mean density inside equals the critical density of the Hubble sphere of that radius.
To be candid, no claim is made that the holographic principle is true, that the universe is a black hole, or that this distribution can stand still in general relativity. What has been found is that the 2 of Paper 01 is the 2 of counting on a surface and the 2 of gravitational balance laid on top of each other in three dimensions.
Keywords: inverse-square density, spherical shell, horizon, Tangherlini solution, Hubble sphere, dimension of space, holography.
Scope (what is not claimed)
No new theorem or law is claimed. The Schwarzschild solution in d-dimensional space (Tangherlini 1963), the way its mass, temperature and entropy are written (Myers and Perry 1986), the derivation of the expansion of a flat universe from a Newtonian shell, and the statement of the holographic principle that the degrees of freedom of a region are counted by the area of its boundary (’t Hooft 1993; Susskind 1995) are all known. No claim is made that the holographic principle is true. That principle is stated in terms of area and does not choose a dimension, so the overlap found here only in three dimensions is neither evidence for it nor against it. No claim is made that the universe is a black hole. The Hubble-sphere coincidence is treated as an identity that follows from the definition of a flat universe. No claim is made that this distribution can stand still in general relativity. Every radius lies at the depth of a horizon, so it cannot be a static star. What is counted is the Newtonian picture and the relation between mass and radius. Quantum gravity is not treated. Relation to earlier papers: Paper 01 proved the uniqueness of n = 2 in three dimensions and also read that 2 as a count of pixels. Paper 02 showed that the mass inside radius 1 is then 4π. Here the second way of extending that 2 to other dimensions (horizon balance) is counted and set against the first. Papers 08 and 114 wrote that the uniqueness of n = d−1 is a family present in every dimension. Here that is the first way, and it is counted that the exponent of the second way is 2 in every dimension. Paper 169 showed that the coincidence between the Hubble radius and the critical density is an identity in three dimensions, and Paper 907 counted that the expansion equation follows from a Newtonian shell for dust. Here the same shell is run in d dimensions and the coincidence is checked to hold in every dimension. Paper 910 counted that among power-law densities only n = 2 gives a flat rotation curve. Here the horizon-balance distribution is seen to be the fastest form of that flat rotation curve. Paper 1085 counted how the ratio between the coefficient of the Einstein equation and that of the Poisson equation moves with dimension, and Paper 1086 counted the 3π/32 of the free-fall time. Here that ratio is measured again from the acceleration of the black-hole solution, and the fall time is checked to be independent of the radius in d dimensions. Paper 35 and Paper 60 treated placing 4π in a mass or in the gravitational constant as a choice of units, and Paper 59 told apart what follows from setting the radius to 1. The 8πG = 1 here is also treated as a choice of units. Papers 73 and 74 treated the 1/4 of black-hole entropy and the work that reads gravity from heat and entropy. Here only the fact about the exponent of the density, which comes before that, is treated. What is added: that the exponent of horizon balance is 2 in every dimension, with its coefficient, the count showing that the two 2s overlap only in three dimensions, the check that the 2 of gravity comes from a time squared, and that the Hubble-sphere coincidence does not choose a dimension, with the nest of Hubble spheres.
1. The question
Paper 01 showed that in three-dimensional space the mass of one spherical shell, $dm/dr=4\pi r^{2-n}$, is independent of the radius only for $n=2$. Paper 02 showed that the mass inside radius 1 then equals the full solid angle $4\pi$. The abstract of Paper 01 also reads this 2 as follows: “the number of pixels on the screen grows as $4\pi r^{2}$, so the data density per pixel becomes $1/r^{2}$”. It is a way of counting on a surface.
Paper 114 tabulated that extending this theorem to $d$ dimensions gives the exponent $n=d-1$, a family present in every dimension. The four-dimensional $n=3$ was treated in Paper 08. But this is not the only way to extend the three-dimensional $n=2$. The density $1/r^{2}$ also has a face on the side of gravity. Paper 910 counted that among power-law densities only $n=2$ gives a flat rotation curve. This paper sets up one extension on the side of gravity and shows with numbers that it gives the same exponent as the shell side only in three dimensions.
2. Two extensions
Put a density $\rho=A r^{-n}$ in $d$-dimensional space. The mass inside radius $r$ and the mass of one shell are
$$M(r)=\frac{S_{d-1}A}{d-n}\,r^{d-n},\qquad \frac{dm}{dr}=S_{d-1}A\,r^{d-1-n}\qquad(n<d),$$where $S_{d-1}$ is the area of the sphere of radius 1: $4\pi$ in three dimensions and $2\pi^{2}$ in four.
The first extension is that of Paper 01 and asks that $dm/dr$ be independent of the radius. The exponent is $n=d-1$.
The second asks that, cut at any radius, the mass inside equal the mass of a black hole with its horizon at that radius. The black hole of $d$-dimensional space is the Tangherlini solution (Tangherlini 1963); with $f=1-\mu/r^{d-2}$, the horizon radius $r$ and the mass are related by
$$M_{\mathrm{bh}}(r)=\frac{(d-1)\,S_{d-1}}{16\pi G}\,r^{d-2}$$(written with the $G$ of the action, as in Myers and Perry 1986, with the speed of light equal to 1). The ratio of the enclosed mass to this mass is proportional to $r^{2-n}$, so the exponent is $n=2$.
The slope from radius 1 to radius 2 (logarithm to base 2) was counted in four dimensions.
| $d$ | $n$ with constant shell mass | $n$ with the same balance as a black hole |
|---|---|---|
| 3 | 2 | 2 |
| 4 | 3 | 2 |
| 5 | 4 | 2 |
| 6 | 5 | 2 |
In three dimensions the two slopes are the same number for every $n$ ($+1.000$ and $+1.000$ for $n=1$, $-0.500$ and $-0.500$ for $n=2.5$). In four dimensions they separate: $+1.000$ and $+0.000$ for $n=2$, $+0.000$ and $-1.000$ for $n=3$. The three-dimensional 2 meets both conditions at once. When the dimension changes, the two become different exponents.
3. Do the formulas on the black-hole side agree?
This paper uses three formulas of the Tangherlini solution. The mass is $M_{\mathrm{bh}}$ above, the temperature $T=(d-2)/(4\pi r)$, and the entropy $S=S_{d-1}r^{d-1}/(4G)$. That the three satisfy the first law $dM=T dS$ was checked for $d=3$ to 10 at three radii. The largest deviation was $2.845\times10^{-11}$.
The Newtonian constant $G_N$ (force $=G_N Mm/r^{d-1}$) was measured, without copying a formula, from the gravitational acceleration $\tfrac12 f'(r)$ of the solution. $G_N/G$ is $1.000000000$ in three dimensions, $0.848826363$ in four and $0.716197244$ in five, agreeing with $8\pi(d-2)/((d-1)S_{d-1})$ to within $1.5\times10^{-9}$. This ratio comes from the ratio $(d-1)/(d-2)$ between the coefficient of the Einstein equation and that of the Poisson equation, which Paper 1085 counted.
The depth of the potential at the horizon radius, $G_N M/((d-2)r^{d-2})$, was $0.500000000$ for every $d$ from 3 to 10. In every dimension the horizon sits exactly where, in the Newtonian picture, the escape speed equals the speed of light.
4. The coefficient, and the choice of units
The coefficient that makes $M(r)=M_{\mathrm{bh}}(r)$ at every radius is
$$A=\frac{(d-1)(d-2)}{16\pi G},$$and $16\pi G\times A$ is 2, 6, 12 and 20 for $d=3$ to 6. In three dimensions $\rho=c^{2}/(8\pi G r^{2})$, and the $8\pi G$ of the Einstein equation appears as it is. With $G=1$, $A=0.039788736$.
$A$ moves with the choice of units.
| choice of units (three dimensions, speed of light 1) | coefficient $A=1/(8\pi G)$ |
|---|---|
| $G=1$ | $0.039788736$ |
| $8\pi G=1$ | $1.000000000$ |
| $G=4\pi$ (Paper 35) | $0.003166287$ |
With $8\pi G=1$, the distribution $\rho=1/r^{2}$ of Paper 02 is itself the horizon-balance distribution. The mass inside radius 1 is $12.566370614$ ($4\pi$), and $2GM(r)/r$ was $1.000000000$ at radii 0.5, 1, 2 and 10. Since $2GM/r=2G\times4\pi$, the $4\pi$ of Paper 02 and the $8\pi G$ of the horizon face each other in one formula.
This, however, is a choice of units (Paper 35 and Paper 60). The “exactly 1” comes from setting $8\pi G$ to 1. It is the same type as Paper 59 telling apart that the volume being greatest in five dimensions follows from setting the radius to 1. With $G=4\pi$ the coefficient is $1/(32\pi^{2})$, a factor $32\pi^{2}$ below the distribution of Paper 02. What does not move is that the exponent is 2 and the form of the coefficient, $(d-1)(d-2)/(16\pi G)$.
5. Why is the side of gravity 2?
The reason the exponent on the side of gravity is 2 in every dimension is that the product of the gravitational constant and the density is an inverse time squared in every dimension. The potential has the dimension of a speed squared, and the Poisson equation ties its second derivative to $G_N\rho$.
This was checked. The edge shell of a uniform ball in $d$ dimensions was released from rest under Newtonian gravity alone ($G_N\rho=1$). The time to fall to near the centre (a thousandth of the radius) was the same for radii 1, 2 and 5.
| $d$ | fall time (the same for radii 1, 2 and 5) |
|---|---|
| 3 | $0.542700$ |
| 4 | $0.450180$ |
| 5 | $0.398700$ |
Time was counted in steps of $2\times10^{-5}$, and “the same” means the same number of steps. The three-dimensional value is $0.999998$ times the free-fall formula $\sqrt{3\pi/(32G\rho)}=0.542701$ (Paper 1086). The radius does not enter; $G_N\rho$ alone sets the time.
A uniform ball at the same depth as a horizon ($1/2$) has, in every dimension, a constant $G_N\rho R^{2}$, equal to $(d-2)/(2V_d)$ ($V_d$ is the volume of the ball of radius 1). For $d=3$ to 6 it was $0.119366207$, $0.202642367$, $0.284965829$ and $0.387018413$. The density is inversely proportional to the square of the radius. This 2 is the 2 of the time squared that makes the depth (a speed squared); it does not count the dimension.
The $d-1$ on the side of the shell, by contrast, is the $r^{d-1}$ in $dm/dr=S_{d-1}r^{d-1}\rho$, the power of the area of the sphere of radius $r$, and it counts the dimension of the sphere. The two 2s have different roots. They are the same number only when $d-1=2$, that is, only in three dimensions.
6. The cosmological side: the Hubble-sphere coincidence does not choose a dimension
Paper 169 showed that in three dimensions $GM/(Rc^{2})$ formed from the Hubble radius $R=c/H$ and the critical density is exactly 0.5, an identity that follows from flatness. The same was checked in $d$ dimensions.
Running a shell of zero energy under $d$-dimensional Newtonian gravity (the method of Paper 907) gives
$$\frac12\dot r^{2}=\frac{G_N M}{(d-2)\,r^{d-2}},\qquad M=V_d r^{d}\rho\quad\Longrightarrow\quad H^{2}=\frac{2G_N V_d\,\rho}{d-2}.$$Inserting the formula for $G_N$ gives $H^{2}=16\pi G\rho/(d(d-1))$, which returns to $8\pi G\rho/3$ in three dimensions (largest deviation of the ratio $4.4\times10^{-16}$ for $d=3$ to 10).
Then the mass of the Hubble sphere of a flat universe equals the mass of the black hole with its horizon at the same radius. For every $d$ from 3 to 10 the ratio was $1.000000000000$ (largest deviation $4.4\times10^{-16}$). In three dimensions $GM/(Rc^{2})=0.500000000$ for $H=50$, 67.4 and 100, the number of Paper 169. The reason is that $d$ times the volume of the ball is the area of the sphere ($V_d\times d=S_{d-1}$), so the Hubble-sphere coincidence does not choose a dimension.
For the horizon-balance distribution $1/r^{2}$, cut at any radius, the mean density inside equals the critical density of the Hubble sphere of that radius (checked for $d=3$ to 8 at three radii). It is a nest of Hubble spheres. The density at the edge is $(d-2)/d$ times the mean inside: $0.333333333$, $0.500000000$ and $0.600000000$ for $d=3$, 4 and 5. The cosmological coincidence also belongs to the 2 of gravity. What chooses a dimension is its overlap with the shell side of Paper 01.
7. The speed of rotation
In the horizon-balance distribution the speed of a circular orbit in the Newtonian picture is the same at every radius, with $v^{2}=(d-2)/2$. In three dimensions it is $0.707106781$ times the speed of light. It is the fastest form of the flat rotation curve that Paper 910 counted. In four dimensions it is $1.000000000$ times and in five $1.224744871$ times, reaching or exceeding the speed of light, so the Newtonian picture fails there. Even in three dimensions this distribution cannot stand still in general relativity (every radius lies at the depth of a horizon). What was counted is the form in the Newtonian picture.
8. Counting on a surface, and gravitational balance
The holographic principle is the idea that the degrees of freedom of what happens inside a region are counted by the area of its boundary surface (’t Hooft 1993; Susskind 1995). In this corpus, Paper 73 treated the 1/4 of black-hole entropy and Paper 74 the work that reads gravity from heat and entropy.
The two 2s of this paper are a small fact that comes before that. The 2 on the shell side counts the dimension of the surface (the sphere). The “pixel” reading of Paper 01 is on this side. The 2 on the side of gravity comes from the product of the gravitational constant and the density being an inverse time squared. The horizon of a black hole and the Hubble sphere are on this side. In three dimensions, a constant shell mass and horizon balance at every radius happen in one and the same distribution.
The holographic principle, however, is stated in terms of area and does not choose a dimension. The overlap found here only in three dimensions is neither evidence for it nor against it. What can be said is that three is the dimension in which counting on a surface and gravitational balance have the same exponent.
9. Where the check stopped once
In the preliminary check, $G_N$ was measured from the difference of $f=1-\mu/r^{d-2}$ far away ($r=1000$); the larger $d$, the larger the deviation, and at $d=10$ the result was 0. Digits were lost in subtracting numbers close to 1. Taking the difference of $\mu/r^{d-2}$ itself brought the deviation down to $1.5\times10^{-9}$. The check of this paper measures in the corrected form.
Closing
Closing: the 2 of the density 1/r² in Paper 01 is two 2s laid on top of each other. The exponent that keeps the shell mass constant is d−1, and the exponent that puts every radius in the same balance as a black hole was 2 in every dimension. The time for a shell released from rest to fall does not depend on the radius, $\mathbf{0.542700}$ in three dimensions, and the 2 of gravity comes from a time squared. The Hubble sphere of a flat universe is in the same balance as a black hole in every dimension, and the horizon-balance distribution was a nest of Hubble spheres. In units with 8πG = 1, the distribution 1/r² of Paper 02 is itself the horizon-balance distribution. To be candid, no claim has been made that the holographic principle is true or that the universe is a black hole. What has been found is that three is the dimension in which counting on a surface and gravitational balance have the same exponent. The separator is the dimension of space.
References
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Reference
Yamagishi, Y. (2026). The “2” of the Inverse-Square Density Is Two 2s, and the 2 of the Shell Mass Meets the 2 of the Horizon Balance Only in Three Dimensions [Paper 1092]. YOI: https://www.yamagishiyuuki.com/yoi/1092
ORCID: 0009-0006-8587-3450
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The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 1092. This paper is a preprint and has not been peer reviewed.