Composite Universal Constants Combining 2–5 Known Constants Reveal Latent Connections Between Disparate Physical Regimes and the Role of Dimensionless Constants in Systems of Units

Abstract

We introduce a new method of dimensional analysis based on complete systems of units, such as the metric and Planck systems, in which fundamental dimensionless constants arise naturally. In fact, it is the reformulated Planck system that communicates its dimensionless constants to the metric or any other system. The method reveals additional complex dynamical scales and physical effects beyond those amenable to conventional dimensional analysis. We formulate our strategy in simple settings involving pairs of seemingly unrelated constants, and then we extend the analysis to more complicated cases involving combinations of three to five well-known universal constants. In constructions involving several unrelated constants, the method captures increasingly complex effects and places two or more disparate physics areas into a single framework connecting them by never-before-seen combinations of fundamental dimensionless constants, such as the fine-structure constant and the gravitational coupling constant. Thus, this method provides a pathway to blending descriptions of two or more fundamental interactions that have so far eluded a consistent theoretical formulation.

1. Introduction

1.1. Impetus from Previous Work

This work builds on a puzzling, yet innovative numerical relation discovered between totally unrelated universal constants in metric-system (SI) units [1], viz.

$$\begin{array}{ccc}{10}^{-6}{k}_{\mathrm{B}}& =& 1.3806\dots \times {10}^{-29}{\mathrm{J}\mathrm{K}}^{-1}\hfill \\ e\sqrt{G_{★}}& =& 1.3806\dots \times {10}^{-29}{{\mathrm{C}}^2\mathrm{kg}}^{-1}\hfill \end{array},$$

(1)

where

$$G_{★}=4\pi {\epsilon }_{0}G,$$

(2)

and the various universal constants involved in these equalities are listed for convenience in

Table 1. Universal physical constants *.

Symbol	Name	Definition	SI Value	SI Unit	SDs
Reformulated Planck System (RPS)
e	Elementary charge		$1.602176634\times {10}^{-19}$	$\mathrm{C}$	Exact
$m_{\mathrm{e}}$	Electron mass		$9.1093837139\left(28\right)\times {10}^{-31}$	$\mathrm{kg}$	11
$k_{\mathrm{B}}$	Boltzmann constant		$1.380649\times {10}^{-23}$	$\mathrm{J}{\mathrm{K}}^{-1}$	Exact
${\epsilon }_0$	Vacuum permittivity		$8.8541878188\left(14\right)\times {10}^{-12}$	$\mathrm{F}{\mathrm{m}}^{-1}$	11
${\mu }_0$	Vacuum permeability		$1.25663706127\left(20\right)\times {10}^{-6}$	$\mathrm{H}{\mathrm{m}}^{-1}$	12
$N_{\mathrm{A}}$	Avogadro number		$6.02214076\times {10}^{23}$	—	Exact
$f_{\mathrm{A}}$	Avogadro factor		$10.05530213\dots$	—	Exact
Composite Universal Constants
	Vacuum-related
K	Coulomb constant	${\displaystyle K=1/\left(4\pi {\epsilon }_0\right)}$	$8.9875517862\left(14\right)\times {10}^9$	$\mathrm{N}{\mathrm{m}}^2{\mathrm{C}}^{-2}$	11
c	Speed of light in vacuum	${\displaystyle c=1/\sqrt{{\mu }_0{\epsilon }_0}}$	299,792,458	$\mathrm{m}{\mathrm{s}}^{-1}$	Exact
$Z_0$	Impedance of free space	${\displaystyle Z_0={\mu }_{0}c}$	$376.730313412\left(59\right)$	$\Omega$	12
${\mathcal{R}}_{\mathrm{P}}$	Planck resistance	${\displaystyle {\mathcal{R}}_{\mathrm{P}}=Z_0/\left(4\pi \right)}$	$29.9792457960\left(47\right)$	$\Omega$	12
	Mole-related
R	Molar gas constant	${\displaystyle R=N_{\mathrm{A}}{k}_{\mathrm{B}}}$	$8.314462618\dots$	$\mathrm{J}{\mathrm{K}}^{-1}$	Exact
F	Faraday constant	${\displaystyle F=N_{\mathrm{A}}e}$	96,485.33212…	$\mathrm{C}$	Exact
${\aleph }_{\mathrm{A}}$	Reduced Avogadro number	${\displaystyle {\aleph }_{\mathrm{A}}=N_{\mathrm{A}}/f_{\mathrm{A}}}$	$5.989020203\times {10}^{22}$	—	Exact
	Gravity-related
G	Gravitational constant	$\mathcal{N}\left(G\right)=\mathcal{N}\left(K{({10}^{-6}{k}_{\mathrm{B}}/e)}^2\right)$	$6.674015081\left(1\right)\times {10}^{-11}$	${\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$	10
$M_{\mathrm{P}}$	Planck mass	$M_{\mathrm{P}}={\aleph }_{\mathrm{A}}{m}_{\mathrm{e}}$	$5.455628310\left(2\right)\times {10}^{-8}$	$\mathrm{kg}$	10
$L_{\mathrm{P}}$	Planck length	$L_{\mathrm{P}}=G{M}_{\mathrm{P}}/c^2$	$4.051264068\times {10}^{-35}$	$\mathrm{m}$	10
h	Planck constant	$h=G{M}_{\mathrm{P}}^2/c$	$6.62607015\times {10}^{-34}$	$\mathrm{J}{\mathrm{Hz}}^{-1}$	Exact
$G_{★}$	Gravoelectric G	${\displaystyle G_{★}=4\pi {\epsilon }_{0}G}$	$7.425843255\times {10}^{-21}$	${\mathrm{C}}^2{\mathrm{kg}}^{-2}$	Exact
$G_B$	Gravomagnetic G	${\displaystyle G_B=G{\mu }_0/\left(4\pi \right)}$	$6.674015081\times {10}^{-18}$	${\mathrm{m}}^4{\mathrm{s}}^{-2}{\mathrm{C}}^{-2}$	10
	MOND-related
$a_0$	MOND critical acceleration	$\mathcal{N}\left(a_0\right)=\mathcal{N}\left(4\pi {\epsilon }_0\right)$	$1.1126500562\times {10}^{-10}$	$\mathrm{m}{\mathrm{s}}^{-2}$	11
${\mathcal{A}}_0$	MOND universal constant	$a_{0}G=\mathcal{N}\left(G_{★}\right)$	$7.425843255\times {10}^{-21}$	${\mathrm{m}}^4{\mathrm{kg}}^{-1}{\mathrm{s}}^{-4}$	Exact
	Assorted combinations
$R_{\infty }$	Rydberg constant	${\displaystyle R_{\infty }=m_{\mathrm{e}}{e}^4/\left(8{\epsilon }_0^2{h}^{3}c\right)}$	10,973,731.568157(12)	${\mathrm{m}}^{-1}$	14
$\sigma$	Stefan–Boltzmann constant	${\displaystyle \sigma =2{\pi }^5{k}_{\mathrm{B}}^4/\left(15h^3{c}^2\right)}$	$5.670374419\dots \times {10}^{-8}$	${\mathrm{Wm}}^{-2}{\mathrm{K}}^{-4}$	Exact
${\Phi }_0$	Magnetic-flux quantum	${\displaystyle {\Phi }_0=h/\left(2e\right)}$	$2.067833848\dots \times {10}^{-15}$	$\mathrm{Wb}$	Exact
${\mathcal{R}}_{\mathrm{K}}$	von Klitzing constant	${\displaystyle {\mathcal{R}}_{\mathrm{K}}=h/e^2}$	$2.581280745\dots \times {10}^4$	$\Omega$	Exact
$r_{\mathrm{c}}$	Electron Compton radius	${\displaystyle r_{\mathrm{c}}=h/\left(m_{\mathrm{e}}c\right)}$	$2.426310235\dots \times {10}^{-12}$	$\mathrm{m}$	Exact
	Stoney units
$M_{\mathrm{S}}$	Stoney mass	$\sqrt{e^2{\left(G_{★}\right)}^{-1}}$	$1.859248778\times {10}^{-9}$	$\mathrm{kg}$	Exact
$L_{\mathrm{S}}$	Stoney length	$\left({\displaystyle \frac{{\mu }_0}{4\pi }}\right)\sqrt{e^2{G}_{★}}$	$1.380649000\times {10}^{-36}$	$\mathrm{m}$	10
	RPS coupling constants
$\alpha$	Fine-structure constant	${\displaystyle \alpha =K{e}^2/\left(hc\right)}$	$1.1614097321\left(2\right)\times {10}^{-3}$	—	11
${\alpha }_{\mathrm{w}}$	Weak coupling constant	${\displaystyle {\alpha }_{\mathrm{w}}=\sqrt{\alpha }}$	$3.407946203\times {10}^{-2}$	—	10
${\alpha }_{\mathrm{s}}$	Strong coupling constant	${\displaystyle {\alpha }_{\mathrm{s}}={\left(f_{\mathrm{A}}\right)}^2\alpha }$	$1.174290938\times {10}^{-1}$	—	10
${\alpha }_{\mathrm{g}}$	Gravitational coupling constant	${\displaystyle {\alpha }_{\mathrm{g}}={\left({\aleph }_{\mathrm{A}}\right)}^{-2}}$	$2.787972231\times {10}^{-46}$	—	10

* The dimensionless function $\mathcal{N}\left(x\right)$ denotes the SI magnitude of x apart from units.

. For the gravitational constant G, we used here the value determined recently in Ref. [1].

Evidently, the numerical equality in Equation (1) reveals the existence of a physical relation in which the factor of ${10}^{-6}$ represents a universal constant $C_4$ with dimensions of ${\left[{\epsilon }_{0}G\right]}^{1/2}{[k_{\mathrm{B}}/e]}^{-1}$, in which each pair of constants represents a well-known composite universal constant [1,2]. Thus, we obtain from Equation (1) the dimensionally balanced equation

$$e\sqrt{G_{★}}=C_4{k}_{\mathrm{B}},$$

(3)

where

$$C_4={10}^{-6}{\mathrm{F}\mathrm{K}\mathrm{kg}}^{-1},$$

(4)

in SI units. Here, symbol $C_4$ denotes that this is a composite constant obtained from an amalgamation of four well-known universal constants, viz. $\{G,K,e,k_{\mathrm{B}}\}$ (Section 4.3), among many such constants examined in this work (Table 1).

Equation (4) is full of surprises:

(a)

The magnitude is a precise power of ten; the leading factor is 1 to a precision of at least five significant digits (SDs) when the CODATA value of G is used [1].

(b)

Adopting the SI magnitude of $C_4$ as ${10}^{-6}$ exactly, as in Equations (1) and (3), allows for a determination of the Newtonian gravitational constant G with a precision of 10 SDs (see Table 2 in Ref. [1] and the G-entry in Table 1 below).

(c)

This is only the second instance of a constant possessing a pure power-of-ten SI magnitude; hitherto, the only other example was the magnetic permeability of the vacuum ${\mu }_0/\left(4\pi \right)$, whose SI magnitude is ${10}^{-7}$ to 10 SDs (see Section 2.2.1).

(d)

The SI units of $C_4$ reveal an unexpected combination of dimensions drawn from three seemingly unrelated areas of physics: capacitance [$\mathcal{C}$], temperature [$\Theta$], and mass [M]. Although certain pairs of these quantities appear together in both theoretical and experimental contexts (e.g., kinetic theory of gases or capacitors), we are unaware of any physical effect in which all three quantities appear simultaneously.

The appearance of three disparate SI units in $C_4$ (Equation (4)) may also be viewed from a broader theoretical perspective. Since $\mathcal{C}$, $\Theta$, and M belong to distinct physical domains, a constant combining all three would naturally be expected to arise in a framework capable of relating these realms in a deeper physical level. In that sense, $C_4$ may be regarded as suggestive of a physical effect whose full interpretation would result from a more complete unified theory, were such a theory presently available.

1.2. Universal Constants Utilized in This Work

It certainly seems that we are facing complications in both the magnitude and the dimensions of constant $C_4$ (Equation (4)). Although we are not presently equipped with an inclusive theory to provide an explanation of capacitance related to both temperature and mass, we can still leverage such unusual properties to obtain new empirical results concerning interesting combinations of several universal constants.

The well-known universal physical constants considered in this work are summarized in Table 1. We have searched for multiple independent combinations of 2–5 constants with the following characteristics: (i) each combined constant (or its reciprocal) has a power-of-ten magnitude precise to about five SDs or more; and (ii) the units attached to these constants relate disparate branches of physics that have not been previously studied in unison.

The strategy employed in our search is first demonstrated in three simple cases, each utilizing only two fundamental constants from those listed in Table 1—we have not been able to find a fourth such example. Then, we proceed to analyze in detail 17 additional composite constants that all have nearly pure power-of-ten SI magnitudes (see

Table 2. Summary of new composite constants.

Symbol	Definition	SI Value	SI Unit	Section	Related Themes in Science
		Section 2.  Two Combined Constants
$A_2$	$K/c^2$	${10}^{-7}$	${\mathrm{kg}\mathrm{m}\mathrm{C}}^{-2}$	Section 2.2.1	Biot–Savart law for charges
$B_2$	$m_{\mathrm{e}}{R}_{\infty }$	${10}^{-23}$	${\mathrm{kg}\mathrm{m}}^{-1}$	Section 2.2.2	Atomic unit of linear mass density
$C_2$	$a_0/\left(4\pi {\epsilon }_0\right)$	${10}^0=1$	${\mathrm{m}}^2{\mathrm{s}}^{-2}{\mathrm{F}}^{-1}$	Section 2.2.3	Capacitors, supercapacitors
		Section 3.  Three Combined Constants
$A_3$	$e^2{m}_{\mathrm{e}}{\Phi }_0^2$	${10}^{-97}$	${{\mathrm{kg}}^3{\mathrm{m}}^4\mathrm{s}}^{-2}$	Section 3.1	Electron spin and mass
$B_3$	$G{\Phi }_0/k_{\mathrm{B}}$	${10}^{-2}$	${\mathrm{m}}^3\mathrm{K}{\mathrm{kg}}^{-1}{\mathrm{s}}^{-1}{\mathrm{C}}^{-1}$	Section 3.2	Neutron stars
$C_3$	$e^2{\Phi }_0^2/R_{\infty }$	${10}^{-74}$	${\mathrm{kg}}^2{\mathrm{m}}^5{\mathrm{s}}^{-2}$	Section 3.3	Electron spin and Compton radius
$D_3$	$k_{\mathrm{B}}{m}_{\mathrm{e}}/{\mu }_0$	${10}^{-47}$	$\mathrm{N}{\mathrm{C}}^2{\mathrm{K}}^{-1}$	Section 3.4	Fermi temperature, Fermi gas
		Section 4.  Four Combined Constants
$A_4$	$c^2{k}_{\mathrm{B}}^2/\left(G{e}^2\right)$	${10}^{19}$	$\mathrm{kg}{\mathrm{V}}^2{\mathrm{m}}^{-1}{\mathrm{K}}^{-2}$	Section 4.1	Fisher information
$B_4$	$e{h}^2/\left(G^2{\mu }_0^2\right)$	${10}^{-53}$	${\mathrm{kg}}^4\mathrm{C}{\Omega }^{-2}$	Section 4.2	Mass–charge scaling $Q\propto 1/M^4$
$C_4$	$e\sqrt{G_{★}}/k_{\mathrm{B}}$	${10}^{-6}$	$\mathrm{F}\mathrm{K}{\mathrm{kg}}^{-1}$	Section 4.3	Larmor radius, plasma magnetic rigidity
$D_4$	$\sigma {\left(h{R}_{\infty }\right)}^2/c$	${10}^{-68}$	${\mathrm{kg}}^3\mathrm{m}{\mathrm{s}}^{-4}{\mathrm{K}}^{-4}$	Section 4.4	Nambu–Goto strings, Regge trajectories
$E_4$	${(FK{\Phi }_0/\sigma )}^2$	${10}^{15}$	${\left(\mathrm{kg}{\mathrm{m}}^5{\mathrm{K}}^4{\mathrm{C}}^{-2}\right)}^2$	Section 4.5	Plasma current sheets, MHD turbulence,
					magnetic reconnection, solar wind
$F_4$	$h^2\sigma /\left({\Phi }_0{R}_{\infty }^2\right)$	${10}^{-73}$	${\mathrm{J}}^2\mathrm{C}{\mathrm{K}}^{-4}$	Section 4.6	Superconducting magnetic fields
$G_4$	$h^{2}G{R}_{\infty }/{\sigma }^2$	${10}^{-55}$	${\mathrm{m}}^6{\mathrm{s}}^2{\mathrm{K}}^8{\mathrm{kg}}^{-1}$	Section 4.7	Orbital motion in $-k/r$ potentials
$H_4$	$h{\Phi }_0^2{\sigma }^2/m_{\mathrm{e}}$	${10}^{-47}$	${\mathrm{kg}}^4{\mathrm{m}}^6{\mathrm{s}}^{-9}{\mathrm{C}}^{-2}{\mathrm{K}}^{-8}$	Section 4.8	Wakefields, accelerators
$I_4$	$k_{\mathrm{B}}^{2}R\sigma /c^2$	${10}^{-69}$	${\mathrm{kg}}^4{\mathrm{m}}^4{\mathrm{s}}^{-7}{\mathrm{K}}^{-7}$	Section 4.9	Photon ensemble, lasers
		Section 5.  Five Combined Constants
$A_5$	$G{[ce/\left(k_{\mathrm{B}}K\right)]}^2$	${10}^{-5}$	${\mathrm{K}}^2{\mathrm{s}}^4{\mathrm{C}}^6{\mathrm{kg}}^{-5}{\mathrm{m}}^{-5}$	Section 5.1	Johnson–Nyquist thermal noise, inductors
$B_5$	$G^2{m}_{\mathrm{e}}{\Phi }_0^2/\left({\mu }_0{k}_{\mathrm{B}}\right)$	${10}^{-51}$	${\mathrm{m}}^7\mathrm{K}{\mathrm{kg}}^{-1}{\mathrm{s}}^{-4}$	Section 5.2	Entropy density, sound energy, aeroacoustics
$C_5$	$G^2{c}^2{m}_{\mathrm{e}}\sigma /{\Phi }_0$	${10}^{-26}$	${\mathrm{m}}^6\mathrm{C}{\mathrm{kg}}^{-1}{\mathrm{s}}^{-8}{\mathrm{K}}^{-4}$	Section 5.3	Zitterbewegung (electron trembling motion)
$D_5$	$G^{2}K{m}_{\mathrm{e}}\sigma /{\Phi }_0$	${10}^{-33}$	${\mathrm{m}}^7{\mathrm{C}}^{-1}{\mathrm{s}}^{-8}{\mathrm{K}}^{-4}$	Section 5.4	Zitterbewegung magnetic field, QED

for a summary).

1.3. Outline

The remainder of the paper is organized as follows:

• In Section 2, we summarize the steps of the search method, and we demonstrate the strategy in simple terms using pairs of seemingly unrelated well-known universal constants.
• In Section 3, Section 4 and Section 5, we present several more complicated cases involving various combinations of three to five unrelated constants, respectively, that point toward new physical scalings and effects.
• In Section 6, we briefly discuss our results and summarize our conclusions concerning pure power-of-ten composite constants.
• In Appendix A, we analyze four additional important pairs of unrelated constants whose SI magnitudes are not pure powers of ten. They show that the same method can be naturally extended to analyze any composite constants in physics irrespective of their magnitudes.

2. Strategy and Simple Demonstrations with Pairs of Fundamental Constants

2.1. Methodology

We search Table 1 for combinations of 2–5 constants with SI magnitudes (or their reciprocals) of $1.0000\dots \times {10}^N$, where N is an integer. We expect the N values to be linear combinations of $-7$ and $-6$ because the only known precise powers of ten (${10}^{-7}$ and ${10}^{-6}$) occur in the calculations of ${\mu }_0/\left(4\pi \right)$ and $\mathcal{N}\left(G\right)$, respectively, where $\mathcal{N}\left(G\right)$ denotes the numerical SI magnitude of the Newtonian gravitational constant G (Table 1). In the process, we avoid using the constant pairs ${\mu }_0{\epsilon }_0$ and $R/F$ that can easily produce equivalent relations when substituted for $1/c^2$ and $k_{\mathrm{B}}/e$ [2], respectively.

We also avoid using the reduced Avogadro number ${\aleph }_{\mathrm{A}}$ [1], which is not a well-known unit at present. We prefer instead to work with the familiar man-made (subjective) Avogadro number $N_{\mathrm{A}}$, which has also been retained as a fundamental unit in the reformulated Planck system (RPS) of units shown at the top of Table 1. We note that Planck’s h is a derived constant in the RPS, and Dirac’s ℏ is not at all used in any composite constant (see Ref. [1] for details).

The Stefan–Boltzmann constant $\sigma$ is the only constant in which ℏ was retained to maintain its well-known SI magnitude; this ℏ was replaced by $h/\left(2\pi \right)$ in the definition of $\sigma$ in Table 1. On the other hand, the definitions of the Planck units, the Compton radius, and the dimensionless coupling constants were given in terms of Planck’s h, replacing thus Dirac’s ℏ and restoring the correct geometries in these definitions.

This reversion to the original Planck system of units is at the core of the new RPS [1]. Despite its adoption and consistent implementation, the constants ℏ and ${\aleph }_{\mathrm{A}}$ do occasionally appear in the results, but only in strict 3D and 2D geometries, respectively, where they really belong. The spin of the electron is still $\hslash /2$, and ensembles of particles are naturally described by ${\aleph }_{\mathrm{A}}$, the number preferred by nature, instead of the familiar man-made number $N_{\mathrm{A}}$.

The SI units of the new constants are generally expected to be unfamiliar combinations of vacuum, thermal, mechanical, electromagnetic (EM), and quantum mechanical units. It is precisely these unit combinations that may hold new physics or scalings, and we have two distinct ways of analyzing their properties:

• Part 1. We use dimensional analysis to uncover relations between disparate or seemingly unrelated physical dimensions. This procedure is not straightforward because the dimensional reductions performed are not unique [3]. From prior experience, we always try to limit the number of fundamental physical quantities involved in a new composite constant to as few as possible: a single reduced quantity produces a constant value, whereas two or three quantities describe scaling relations.
• Part 2. We recast the dimensional forms of the combined constants in terms of Planck units. This procedure is straightforward and results in relations that necessarily involve some inherent geometric factors and, more importantly, some of the dimensionless coupling constants listed at the bottom of Table 1. Such combinations of disparate couplings have never before been seen in physics.

The appearance of fundamental physical quantities in Part 1 allows us to determine previously unknown effects or scales (or some known threshold values) that describe interactions stemming from disparate areas of physics. On the other hand, the appearance of dimensionless coupling constants in Part 2 solves a long-standing problem in physics (see, e.g., Refs. [4,5]): it shows that, at the very least, the fine-structure constant (FSC) and the gravitational coupling constant must be included in systems of units other than the RPS, which are otherwise incomplete.1^,2 In our methodology, all unit transformations between the Planck/RPS system and the SI system are carried out by well-known (though little appreciated [4]) dimensionless constants.

All combinations considered below are drawn from subsets of the dimensionful constants listed in Table 1. In Section 2.2, we illustrate the above methodology with three simple examples involving pairs of well-known universal constants. In Section 3, Section 4 and Section 5, we subsequently analyze more complex cases involving three to five constants, respectively. All composite constants under the present investigation are summarized in Table 2 for convenience.

2.2. Three Simple Combinations of 2 Universal Constants

2.2.1. Subset  $A2:=\{K,c\}$

We consider the constant $A_2$ such that

$$A_2={\displaystyle \frac{K}{c^2}}={10}^{-7}{\mathrm{kg}\mathrm{m}\mathrm{C}}^{-2}.$$

(5)

In this exceptional case that combines two vacuum constants, it is known that $A_2={\mu }_0/\left(4\pi \right)$ [6,7].

Part 1.—The SI units in Equation (5) imply the dimensionally exact relation $ML=A_2{Q}^2$. Imagining a particle of mass M and charge Q orbiting in a magnetic field B with azimuthal speed $\mathit{v}$ at radius L, we use the centripetal force $F=M{\mathit{v}}^2/L$ to eliminate M and the Lorentz force $F=Q\mathit{v}B$ to eliminate F, and we find that

$$B=A_2{\displaystyle \frac{Q\mathit{v}}{L^2}},$$

(6)

which is the scalar analog of the well-known nonrelativistic equation (a variant of the Biot–Savart law)

$$\mathit{B}={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{Q\left(\mathit{v}\times \widehat{\mathit{r}}\right)}{r^2}},$$

(7)

where the radial unit vector is $\widehat{\mathit{r}}=\mathit{r}/r$.

Part 2.—Next, we replace the units in Equation (5) with the corresponding RPS quantities, viz. ${\mathrm{kg}\mathrm{m}\mathrm{C}}^{-2}\to M_{\mathrm{P}}{L}_{\mathrm{P}}/Q_{\mathrm{P}}^2$, and we find that

$$A_2={\displaystyle \frac{M_{\mathrm{P}}{L}_{\mathrm{P}}}{Q_{\mathrm{P}}^2}}.$$

(8)

In this simple example, the combination of the three Planck units carries both the correct magnitude and the correct SI units of $A_2={\mu }_0/\left(4\pi \right)$.3

In more complicated cases analyzed below, the RPS units do not carry the correct magnitude, which must then be adjusted by introducing well-known dimensionless constants, revealing thus the presence of additional physics in such composite constants.

2.2.2. Subset  $B2:=\{m_{\mathrm{e}},R_{\infty }\}$

We consider the constant $B_2$ such that

$$B_2=m_{\mathrm{e}}{R}_{\infty }={10}^{-23}{\mathrm{kg}\mathrm{m}}^{-1}.$$

(9)

The nature of $B_2$ is spectroscopic since the Rydberg constant is involved, but it is doubtful that it has ever been used in atomic physics because $B_2$ represents a linear mass density. Compared to the corresponding Schwarzschild-type4 linear density, $M/R_{\mathrm{S}}\sim c^2/G\sim {10}^{27}\mathrm{kg}{\mathrm{m}}^{-1}$, the magnitude of $B_2$ is tiny, which seems to be appropriate for spectroscopy and electronic transitions.

Part 1.—The SI units in Equation (9) imply the dimensionally exact relation $B_2=M/L$, which describes a unit of measurement (not a physical law). Thus, we proceed to compare $B_2$ to the Planck/RPS unit of linear mass density $c^2/G$; we write

$$B_2=\kappa \left({\displaystyle \frac{c^2}{G}}\right),$$

(10)

where the dimensionless scale factor $\kappa$ takes the value

$$\kappa =7.425843255\times {10}^{-51}.$$

(11)

This scale factor hides a surprise: the leading coefficient $7.425\dots$ originally appeared in the magnitude of $G_{★}$ (Ref. [1] and Table 1) as a complicated composite constant. Thus, $\kappa$ can be rewritten as $\kappa ={10}^{-30}\mathcal{N}\left(G_{★}\right)$, and using the G-entry in Table 1, as

$$\kappa ={10}^{-42}\mathcal{N}{\left(k_{\mathrm{B}}/e\right)}^2.$$

(12)

Remarkably, factor $\kappa$ carries entropy and charge information, which then ties into the magnitude of $B_2$, viz.

$$\mathcal{N}\left(B_2\right)={10}^{-42}{\displaystyle \frac{\mathcal{N}{\left(k_{\mathrm{B}}c\right)}^2}{\mathcal{N}\left(e^{2}G\right)}}.$$

(13)

Therefore, the apparently simple density magnitude $\mathcal{N}\left(B_2\right)$ is determined by the vacuum and entropy, gravity, and EM properties; but the additional ingredients that separate it from the RPS linear mass density $c^2/G$ are entropy and electric charge that are brought in by factor $\kappa$ (Equation (12)).

The magnitude of the fraction in Equation (13) is $|A_4|={10}^{19}$ to 10 SDs; thus it holds that $\mathcal{N}\left(B_2\right)={10}^{-42}\mathcal{N}\left(A_4\right)$. The subset of the four constants $A4:=\{G,c,e,k_{\mathrm{B}}\}$ will be analyzed in detail in Section 4.1 below.

Part 2.—Next, we replace the units in Equation (9) with the corresponding RPS quantities, viz. ${\mathrm{kg}\mathrm{m}}^{-1}\to M_{\mathrm{P}}/L_{\mathrm{P}}$, and we find that

$$B_2=\kappa \left({\displaystyle \frac{M_{\mathrm{P}}}{L_{\mathrm{P}}}}\right),$$

(14)

which is equivalent to Equation (10) above. Thus, when the starting equation (here, Equation (9)) represents a unit of measurement (rather than a physical law), the analyses in Parts 1 and 2 produce the same result.

Probably the most important conclusion drawn from the above analysis of Equation (9) is that the Planck/RPS unit of linear mass density ${\mu }_{\mathrm{P}}=c^2/G$ is not appropriate for atomic or subatomic scales, and that the appropriate spectroscopic scale $B_2$ seems to be influenced by two additional constants, e and $k_{\mathrm{B}}$, coming from disparate areas of physics (EM theory and thermodynamics, respectively).

Scale Factor $\mathsf{\kappa }$.—The scale factor $\kappa$ in Equation (14) admits another interpretation that must rely on known dimensionless constants (and not on dimensionful constants stripped of their units, as in Equation (12)). This is necessary because the ratio $M_{\mathrm{P}}/L_{\mathrm{P}}$ already carries the units of the dimensionful constant $B_2$, and it also shows clearly why well-known dimensionless constants must be included in systems of units (see discussion in Section 2.1).

In our case, the RPS already includes the various coupling constants listed in Table 1 as derived units. Thus, we express the unitless factor $\kappa$ (Equation (11)) in terms of geometric and coupling constants as follows:

Combining Equations (9) and (10), we find that $\kappa =G{m}_{\mathrm{e}}{R}_{\infty }/c^2$, and substituting the definition of $R_{\infty }$ from Table 1, this expression takes the equivalent forms

$$\kappa =2{\pi }^2{\left({\displaystyle \frac{K{e}^2}{hc}}\right)}^2\left({\displaystyle \frac{G{m}_{\mathrm{e}}^2}{hc}}\right)=2{\pi }^2{\alpha }^2{\alpha }_{\mathrm{g}}.$$

(15)

The geometric factor of $2{\pi }^2$ appears because $R_{\infty }$ is defined in terms of ${\epsilon }_0$, whereas Planck/SI systems use the unit $4\pi {\epsilon }_0=1/K$ [6,7,8,9]. For a reduced Rydberg constant ${\tilde{R}}_{\infty }=m_{\mathrm{e}}{e}^4/\left[{\left(4\pi {\epsilon }_0\right)}^2{h}^{3}c\right]$, the corresponding equation for the reduced $\tilde{\kappa }=G{m}_{\mathrm{e}}{\tilde{R}}_{\infty }/c^2$ results in the simple expression

$$\tilde{\kappa }={\alpha }^2{\alpha }_{\mathrm{g}}.$$

In either case, we note that such a remarkable combination of dimensionless coupling constants has not been heretofore seen in physics.

Compton Radius ${\mathit{r}}_{\mathrm{c}}$.—One may speculate further that there must be other dimensionless constants expressed in terms of only one or the other coupling constant. The FSC appears prominently in Section 3, Section 4 and Section 5, so here we visit only the much simpler case of the gravitational coupling constant:

We define the Compton radius of the electron in the RPS as

$$r_{\mathrm{c}}\equiv {\displaystyle \frac{h}{m_{\mathrm{e}}c}},$$

(16)

and we use it to obtain yet another characteristic linear mass density, viz. $m_{\mathrm{e}}/r_{\mathrm{c}}$, which can be easily cast to the form

$${\displaystyle \frac{m_{\mathrm{e}}}{r_{\mathrm{c}}}}={\alpha }_{\mathrm{g}}\left({\displaystyle \frac{c^2}{G}}\right).$$

(17)

This equation is directly comparable to Equation (10) for $B_2$, with ${\alpha }_{\mathrm{g}}$ providing the scale here in place of $\kappa$. Apparently an atomic-scale constant, the linear density $m_{\mathrm{e}}/r_{\mathrm{c}}$ is nevertheless much larger than $B_2$ (Equation (9)) by a factor of $3.7544\times {10}^4$.

2.2.3. Subset  $C2:=\{{\epsilon }_0,a_0\}$

We consider the constant $C_2$ such that

$$C_2={\displaystyle \frac{a_0}{4\pi {\epsilon }_0}}=1{\mathrm{m}}^2{\mathrm{s}}^{-2}{\mathrm{F}}^{-1},$$

(18)

where $a_0$ is the critical MOND acceleration (Table 1) [10,11] and F represents the unit of Farad.5

Part 1.—The SI units in Equation (18) imply that ${\mathrm{m}}^2{\mathrm{s}}^{-2}{\mathrm{F}}^{-1}\to {\mathrm{V}}^2{\mathrm{kg}}^{-1}$, where V represents the unit of Volt. Thus, $C_2=1{\mathrm{V}}^2{\mathrm{kg}}^{-1}$, establishing thus the dimensional relation [1]

$$C_2={\displaystyle \frac{{\mathcal{V}}^2}{M}},$$

(19)

where $\mathcal{V}$ represents voltage.

This combination of quantities is well-known, though rarely used. It occurs in the energetics of capacitors in vacuum, where the specific energy ${\mathcal{E}}_M=E/M$ stored in a capacitor by a potential difference $\mathcal{V}$ is

$${\mathcal{E}}_M={\displaystyle \frac{\mathcal{C}}{2}}\left({\displaystyle \frac{{\mathcal{V}}^2}{M}}\right),$$

(20)

where $\mathcal{C}=Q/\mathcal{V}$ is the capacitance. Recasting this equation in terms of the physically relevant energy density ${\mathit{u}}_0=\rho {\mathcal{E}}_M$ (where $\rho$ is the volume density of the material) and using Equation (19), we find that

$${\mathit{u}}_0=\left({\displaystyle \frac{C_2}{2}}\right)\rho \mathcal{C}.$$

(21)

Apparently, constant $C_2$ specifies a special constant energy density, since $\rho$ and $\mathcal{C}$ are also constants fixed at the design stage of the capacitor.

As an example, we turn to industrial supercapacitors (EDLC; Ref. [12]). Typical EDLC specifications are $\rho =0.7\mathrm{g}{\mathrm{cm}}^{-3}$, $M=30\mathrm{g}$, and $\mathcal{C}=100\mathrm{F}$, for a nominal operating value of ${\mathit{u}}_{\mathrm{op}}=8.5\mathrm{J}{\mathrm{cm}}^{-3}$ at ${\mathcal{V}}_{\mathrm{op}}=2.7\mathrm{V}$. According to these values, the special energy density (21) is ${\mathit{u}}_0=0.035\mathrm{J}{\mathrm{cm}}^{-3}$, only 0.4% of ${\mathit{u}}_{\mathrm{op}}$. This corresponds to a low operating voltage of ${\mathcal{V}}_0=0.173\mathrm{V}$, a value that can also be obtained from Equation (19), viz. $\mathcal{N}\left({\mathcal{V}}_0\right)=\mathcal{N}\left(\sqrt{M}\right)$ in SI units.

Part 2.—Next, we replace the units in Equation (18) with the corresponding RPS quantities, viz. ${\mathrm{m}}^2{\mathrm{s}}^{-2}{\mathrm{F}}^{-1}\to L_{\mathrm{P}}^2/\left(T_{\mathrm{P}}^2{\mathcal{C}}_{\mathrm{P}}\right)\to c^2/{\mathcal{C}}_{\mathrm{P}}$, and we find that

$$C_2={\displaystyle \frac{a_0}{a_{\mathrm{P}}}}\left({\displaystyle \frac{c^2}{{\mathcal{C}}_{\mathrm{P}}}}\right),$$

(22)

where $a_{\mathrm{P}}$ is the Planck acceleration and ${\mathcal{C}}_{\mathrm{P}}$ is the Planck capacitance in the RPS. The leading dimensionless factor has the value $a_0/a_{\mathrm{P}}=5.015425001\times {10}^{-62}$. We note that gravity is implicitly present in Equation (22), as the Planck capacitance is defined by ${\mathcal{C}}_{\mathrm{P}}=G_{★}{M}_{\mathrm{P}}/c^2$ in the RPS [1].

3. Four Cases Each Involving Three Fundamental Constants

3.1. Subset  $A3:=\{e,m_{\mathrm{e}},{\Phi }_0\}$

We consider the constant $A_3$ such that

$$A_3=e^2{m}_{\mathrm{e}}{\Phi }_0^2={10}^{-97}{{\mathrm{kg}}^3{\mathrm{m}}^4\mathrm{s}}^{-2}.$$

(23)

There are two points that need to be made about the quantum of magnetic flux ${\Phi }_0=h/\left(2e\right)$: (a) the factor of 2 was inserted to reflect the Cooper pairs of electrons [13] that are responsible for superconductivity [14]; and (b) the magnetic flux represented by ${\Phi }_0$ is missing a geometric factor of $2\pi$ because both h and e are intrinsically 3D quantities, whereas flux through a surface is strictly a 2D quantity [1]. This latter issue is clearly resolved below by the reductions themselves, where the factor of $2\pi$ appears explicitly. So, there is no need to modify the definition of ${\Phi }_0$.

Part 1.—The SI units in Equation (23) imply the dimensionally exact relation $A_3=M{J}^2$, where J represents angular momentum. For an electron with spin $s=\pm 1/2$ and $J_z=sh/\left(2\pi \right)$, we then find that

$$A_3={\left(2\pi \right)}^2{m}_{\mathrm{e}}{J}_z^2.$$

(24)

The geometric factor of ${\left(2\pi \right)}^2$ could be thought as normalization for the term ${\Phi }_0^2$ embedded in $A_3$, where it actually imprints ${\Phi }_0$ by $2\pi$, the signature of 2D surface geometry. Then, ${\Phi }_0/\left(2\pi \right)$ becomes formally a 2D magnetic flux. Furthermore, by combining Equations (23) and (24), we recover the well-known result that $J_z=\pm \hslash /2$ for electronic spins. Here, Dirac’s constant is defined, as usual, by $\hslash =h/\left(2\pi \right)$, but it is not used in the definitions of RPS units or dimensionless coupling constants.

Part 2.—Next, we replace the units in Equation (23) with the corresponding RPS quantities, viz. ${{\mathrm{kg}}^3{\mathrm{m}}^4\mathrm{s}}^{-2}\to M_{\mathrm{P}}^3{L}_{\mathrm{P}}^4/T_{\mathrm{P}}^2$, and we find that

$$A_3={\displaystyle \frac{1}{4{\aleph }_{\mathrm{A}}}}\left({\displaystyle \frac{M_{\mathrm{P}}^3{L}_{\mathrm{P}}^4}{T_{\mathrm{P}}^2}}\right).$$

(25)

The unitless scale factor introduces the reciprocal of the reduced Avogadro number, which effectively represents the gravitational coupling constant, since ${\alpha }_{\mathrm{g}}={\left({\aleph }_{\mathrm{A}}\right)}^{-2}$ (Table 1). The factor of 1/4 is traced back to the Cooper-pair term of $4e^2$ in the flux quantum squared ${\Phi }_0^2$.

Reduced Avogadro Number ${\aleph }_{\mathrm{A}}$.—The appearance of ${\aleph }_{\mathrm{A}}$ (as a dimensionless constant that relates Planck/RPS units to SI units in Equation (25)) indicates that this is nature’s choice to describe a certain number of particles (contained in approximately 0.1 moles of any substance [1]). This should be contrasted to the man-made (subjective) Avogadro number $N_{\mathrm{A}}$, which appears below in the RPS transformations of composite constants only when it is included a priori in the definitions of such constants.

3.2. Subset  $B3:=\{G,k_{\mathrm{B}},{\Phi }_0\}$

We consider the constant $B_3$ such that

$$B_3={\displaystyle \frac{G{\Phi }_0}{k_{\mathrm{B}}}}={10}^{-2}{\mathrm{m}}^3\mathrm{K}{\mathrm{kg}}^{-1}{\mathrm{s}}^{-1}{\mathrm{C}}^{-1},$$

(26)

an unusual combination of thermal, EM, and mechanical units. The Coooper charge in ${\Phi }_0=h/\left(2e\right)$ cannot be eliminated, and this explains the presence of the unit of Coulomb. On the other hand, Planck’s constant embedded in ${\Phi }_0$, defined by $h=G{M}_{\mathrm{P}}^2/c$, is effectively a mechanical unit in the RPS (see Table 1 and Ref. [1]).

Part 1.—The SI units in Equation (26) imply the dimensional relation $B_3=L^3\Theta /\left(MTQ\right)$. Recalling that $L^3$ appears prominently in the definition of the magnetic dipole moment

$${\mu }_B\equiv \left({\displaystyle \frac{2\pi }{{\mu }_0}}\right)B{L}^3,$$

(27)

and after some manipulations ($F=Q\mathit{v}B,\mathit{v}=L/T$), Equation (26) takes the surprising form

$$B_3{M}^2=\left({\displaystyle \frac{{\mu }_0}{2\pi }}\right){\mu }_B\Theta .$$

(28)

This equation relates temperature to mass and magnetic dipole moment.6 It could turn out to be relevant to some classes of neutron stars, although so far their surface temperatures have not been linked to both properties [16,17,18].

Part 2.—Next, we replace the units in Equation (26) with the corresponding RPS quantities, viz. ${\mathrm{m}}^3\mathrm{K}{\mathrm{kg}}^{-1}{\mathrm{s}}^{-1}{\mathrm{C}}^{-1}\to L_{\mathrm{P}}^3{\Theta }_{\mathrm{P}}/\left(M_{\mathrm{P}}{T}_{\mathrm{P}}{Q}_{\mathrm{P}}\right)$, and we find that

$$B_3={\displaystyle \frac{1}{2{\alpha }_{\mathrm{w}}}}\left({\displaystyle \frac{L_{\mathrm{P}}^3{\Theta }_{\mathrm{P}}}{M_{\mathrm{P}}{T}_{\mathrm{P}}{Q}_{\mathrm{P}}}}\right),$$

(29)

where we have introduced the weak coupling constant ${\alpha }_{\mathrm{w}}=\sqrt{\alpha }$ (Table 1), rather than the square root of the FSC. Once again, the factor of 1/2 is traced back to the Cooper-pair charge of $2e$ in the flux quantum ${\Phi }_0$ [14].

3.3. Subset  $C3:=\{e,R_{\infty },{\Phi }_0\}$

We consider the constant $C_3$ such that

$$C_3={\displaystyle \frac{e^2{\Phi }_0^2}{R_{\infty }}}={10}^{-74}{\mathrm{kg}}^2{\mathrm{m}}^5{\mathrm{s}}^{-2},$$

(30)

where the various constants are listed in Table 1.

We have encountered the product $e^2{\Phi }_0^2$ in Section 3.1 as well. Because it reduces to simply ${(h/2)}^2$, this combination naturally points to the electron’s spin, and this is how we will proceed below to interpret this constant.7

Part 1.—The SI units in Equation (30) imply the dimensional relation $C_3=M^2{L}^5/T^2$. Similarly to the reductions of $A_3$ in Section 3.1, this relation takes the form

$$C_3={\left(2\pi \right)}^2{R}_{\infty }^{-1}{J}_z^2,$$

(31)

where $J_z=\pm \hslash /2$ is the electron’s spin angular momentum.

Constant $C_3$ can also be expressed in terms of the Compton radius $r_{\mathrm{c}}$ (Equation (16)). Using the relation

$$R_{\infty }^{-1}={\displaystyle \frac{r_{\mathrm{c}}}{2{\pi }^2{\alpha }^2}},$$

(32)

Equation (31) becomes

$$C_3=\left({\displaystyle \frac{2}{{\alpha }^2}}\right)r_{\mathrm{c}}{J}_z^2.$$

(33)

The residual factor of 2 can be traced to the factor of 1/2 in the Rydberg energy [6,7]. At a deeper level, this factor originates from the virial theorem for the Coulomb-bound electron in the ground state of the hydrogen atom [20].

Part 2.—Next, we replace the units in Equation (30) with the corresponding RPS quantities, viz. ${\mathrm{kg}}^2{\mathrm{m}}^5{\mathrm{s}}^{-2}\to M_{\mathrm{P}}^2{L}_{\mathrm{P}}^5/T_{\mathrm{P}}^2$, and we find that

$$C_3={\displaystyle \frac{{\aleph }_{\mathrm{A}}}{8{\pi }^2{\alpha }^2}}\left({\displaystyle \frac{M_{\mathrm{P}}^2{L}_{\mathrm{P}}^5}{T_{\mathrm{P}}^2}}\right).$$

(34)

Here again, the unitless scale factor introduces the reduced Avogadro number, which effectively represents the gravitational coupling constant since ${\alpha }_{\mathrm{g}}={\left({\aleph }_{\mathrm{A}}\right)}^{-2}$ (Table 1). Furthermore, the factor of $8{\pi }^2$ ($=4\times 2{\pi }^2$) results from the 1/2 traced back to the Cooper-pair charge of $2e$ in ${\Phi }_0$ [14] and the $2{\pi }^2$ traced to the definition of $R_{\infty }$, where ${\epsilon }_0$ has been introduced without the usual $4\pi$ geometric factor (Table 1).

Linear Mass Density ${\mathit{B}}_{\mathbf{2}}$.—Eliminating the spin angular momentum $J_z$ between Equations (24) and (31), we find that

$${\displaystyle \frac{A_3}{C_3}}=m_{\mathrm{e}}{R}_{\infty }\equiv B_2.$$

(35)

Thus, the emergence of the linear mass density constant $B_2$ (Equation (9)) is tied directly to the elimination of the dynamical component (the spin angular momentum) from the two combined equations.

3.4. Subset  $D3:=\{k_{\mathrm{B}},m_{\mathrm{e}},{\mu }_0\}$

We consider the constant $D_3$ such that

$$D_3={\displaystyle \frac{k_{\mathrm{B}}{m}_{\mathrm{e}}}{{\mu }_0}}={10}^{-47}\mathrm{N}{\mathrm{C}}^2{\mathrm{K}}^{-1},$$

(36)

where N represents the unit of force.

It is important to notice the markedly uncommon absence of the 3D geometric imprint of $1/\left(4\pi \right)$ from the magnetic permeability. This is bound to generate $4\pi$ geometric terms in the analysis that follows.

Part 1.—Equation (36) implies the dimensional relation

$$D_3=F\left({\displaystyle \frac{Q^2}{\Theta }}\right),$$

(37)

where F is force, Q is charge, and $\Theta$ is temperature. The term in the parentheses points to a degenerate Fermi gas [21]. It appears in the definition of the plasma coupling parameter $\Gamma \equiv K(Q^2/\Theta )/\left(a{k}_{\mathrm{B}}\right)$ [22,23], where a is the typical interparticle spacing (or Wigner–Seitz radius) given for all phases of matter by

$$a={\left({\displaystyle \frac{3}{4\pi n}}\right)}^{1/3},$$

(38)

where $n=N/V$ is the number density of charged particles, and the volume per particle is defined as $V/N=4\pi a^3/3$.

Using $\Gamma$, we can write constant $D_3$ as $D_3=\left(k_{\mathrm{B}}\Gamma /K\right)Fa$, and assuming that the force F between particles is Coulombic, we find that $\Theta \propto 1/a^2$, which implies that

$$\Theta \propto n^{2/3}.$$

(39)

The same scaling can also be obtained from Equation (37) for $F\propto 1/a^2$ and $Q=\mathrm{constant}$.

Thus, temperature $\Theta$ exhibits fermionic scaling (like the Fermi temperature ${\Theta }_{\mathrm{F}}\propto n^{2/3}$) up to a numerical prefactor that does not include h or ℏ. This component arises from Coulombic interactions between neighboring charges, with the $n^{2/3}$ scaling resulting from the inverse-square dependence on the Wigner–Seitz radius. Therefore, it reflects a purely EM origin, characteristic of dense classical plasmas [22] and ionic liquids [23].

Part 2.—Next, we replace the units in Equation (36) with the corresponding RPS quantities, viz. $\mathrm{N}{\mathrm{C}}^2{\mathrm{K}}^{-1}\to F_{\mathrm{P}}{Q}_{\mathrm{P}}^2/{\Theta }_{\mathrm{P}}$, and we find that

$$D_3={\displaystyle \frac{\sqrt{{\alpha }_{\mathrm{g}}}}{4\pi }}\left({\displaystyle \frac{F_{\mathrm{P}}{Q}_{\mathrm{P}}^2}{{\Theta }_{\mathrm{P}}}}\right).$$

(40)

The factor of $4\pi$ appears here to compensate for the missing 3D geometric tag of ${\mu }_0$ in Equation (36). On the other hand, $\sqrt{{\alpha }_{\mathrm{g}}}=1/{\aleph }_{\mathrm{A}}$, which introduces the reduced Avogadro number ${\aleph }_{\mathrm{A}}$ (Table 1) in Equation (40), effectively a number of particles. That was indicated by the analysis in Part 1, because the number density of charged particles is a controlling property in degenerate Fermi gases, classical plasmas, and ionic liquids.

An important point to note here is that it is ${\aleph }_{\mathrm{A}}$ that appears in Equation (40), not the well-known Avogadro number $N_{\mathrm{A}}$. This, once again, highlights the physical significance of ${\aleph }_{\mathrm{A}}$ [1] against the importance of the arbitrary man-made constant $N_{\mathrm{A}}$, which is nevertheless retained in the RPS for continuity and historical reasons (see the RPS section in Table 1).

4. Nine Cases Each Involving Four Fundamental Constants

4.1. Subset  $A4:=\{G,c,e,k_{\mathrm{B}}\}$

We consider the constant $A_4$ such that8

$$A_4={\displaystyle \frac{c^2{k}_{\mathrm{B}}^2}{G{e}^2}}={10}^{19}\mathrm{kg}{\mathrm{V}}^2{\mathrm{m}}^{-1}{\mathrm{K}}^{-2},$$

(41)

where V represents the unit of Volt.

Part 1.—The SI units in Equation (41) are equivalent to the composite units $\mathrm{Pa}{\mathrm{Wb}}^2{\mathrm{K}}^{-2}$, where Pa and Wb represent Pascal and Weber, respectively. These units imply the dimensional relation

$$A_4=\mathit{u}{\left({\displaystyle \frac{{\Phi }_B}{\Theta }}\right)}^2,$$

(42)

where $\mathit{u}$ is energy density (or pressure) and ${\Phi }_B$ is magnetic flux. This relation can be rewritten in terms of a special form of the Fisher information $\mathcal{I}(\Theta )$ [24]. For an observable power-law response ${\Phi }_B(\Theta )\propto {\Theta }^n$, the sensitivity is $\mathcal{S}=\partial {\Phi }_B/\partial \Theta =n({\Phi }_B/\Theta )$, and then $\mathcal{I}(\Theta )$ takes the form

$$\mathcal{I}(\Theta )={\left({\displaystyle \frac{\mathcal{S}}{{\sigma }_{{\Phi }_B}}}\right)}^2={\left({\displaystyle \frac{n}{{\sigma }_{{\Phi }_B}}}\right)}^2{\left({\displaystyle \frac{{\Phi }_B}{\Theta }}\right)}^2,$$

(43)

where ${\left({\sigma }_{{\Phi }_B}\right)}^2$ denotes the variance of the Gaussian noise associated with the observable flux ${\Phi }_B(\Theta )$.

Next, combining Equations (42) and (43), we find that

$${\mathcal{S}}^2=n^2\left({\displaystyle \frac{A_4}{\mathit{u}}}\right),$$

(44)

which indicates that the sensitivity of the observable flux to temperature is $\mathcal{S}\propto 1/\sqrt{\mathit{u}}$. This is an unexpected result—that the Fisher information may also be affected by the energy density of the system—and no such property has been reported to date.

If $\mathit{u}$ is assumed to be the energy density of the magnetic field B, then $\mathit{u}\propto B^2$ and $\mathcal{S}\propto 1/B$. Thus, weaker magnetic fields allow for greater sensitivity, and vice versa. Under the additional assumption that the noise level ${\sigma }_{\Phi }$ is field-independent, then the Fisher information scales as $\mathcal{I}(\Theta )\propto 1/B^2$. These conclusions are, however, heavily dependent on the initial assumption of a power-law response of ${\Phi }_B$ to temperature. This parameter space needs to be explored by experiment.

Part 2.—Next, we replace the units in Equation (41) with the corresponding RPS quantities, viz. $\mathrm{kg}{\mathrm{V}}^2{\mathrm{m}}^{-1}{\mathrm{K}}^{-2}\to M_{\mathrm{P}}{\mathcal{V}}_{\mathrm{P}}^2/\left(L_{\mathrm{P}}{\Theta }_{\mathrm{P}}^2\right)$, where ${\mathcal{V}}_{\mathrm{P}}$ is the Planck voltage, and we find that

$$A_4={\alpha }^{-1}\left({\displaystyle \frac{M_{\mathrm{P}}{\mathcal{V}}_{\mathrm{P}}^2}{L_{\mathrm{P}}{\Theta }_{\mathrm{P}}^2}}\right),$$

(45)

where [1]

$${\alpha }^{-1}=861.022576584\left(132\right).$$

(46)

Once again, we find that the composite constant under consideration is scaled to the corresponding RPS units by a single dimensionless coupling constant—the little-known RPS FSC inverse “861” that deposes the famous yet unphysical number “137” [25] which suffers from an unacceptable mixing of 2D and 3D geometric factors.

4.2. Subset  $B4:=\{G,{\mu }_0,e,h\}$

We consider the constant $B_4$ such that

$$B_4={\displaystyle \frac{e{h}^2}{G^2{\mu }_0^2}}={10}^{-53}{\mathrm{kg}}^4\mathrm{C}{\Omega }^{-2}.$$

(47)

In this case too, ${\mu }_0$ is not tagged by the $4\pi$ geometry of 3D space. But $4\pi$ prefactors do not appear in the analysis because it utilizes the impedance of free space $Z_0={\mu }_{0}c$, which is also missing its usual 3D tag. Thus, these $4\pi$ factors are subject to cancellation. A similar cancellation of 2D geometric factors was outlined in Note 6.

Part 1.—The SI units in Equation (47) imply the dimensional relation

$$M^{4}Q=Z_0^2{B}_4\equiv \mathcal{D},$$

(48)

where $Z_0$ is the impedance of free space (Table 1), and the new hybrid constant is

$$\mathcal{D}=1.419257291\times {10}^{-48}{\mathrm{kg}}^4\mathrm{C}.$$

(49)

Equation (48) is a scaling law that connects disparate physical scales:

(a)

$Q=\mathcal{D}/M^4$: For $M=M_{\mathrm{P}}$, we find that $Q=e={\alpha }_{\mathrm{w}}{Q}_{\mathrm{P}}$.—The Planck mass scale $M_{\mathrm{P}}$ produces “naturally” the elementary charge e, which is connected to the Planck charge by the weak coupling constant. Hence,

$$\mathcal{D}={\alpha }_{\mathrm{w}}{M}_{\mathrm{P}}^4{Q}_{\mathrm{P}}.$$

(50)

(b)

$M={(\mathcal{D}/Q)}^{1/4}$: For $Q=Q_{\mathrm{P}}$, we find that $M={\alpha }_{\mathrm{w}}^{1/4}{M}_{\mathrm{P}}$.—The Planck charge scale $Q_{\mathrm{P}}$ produces a mass scale M, which is a fraction of the Planck mass ($M=0.430M_{\mathrm{P}}$).

Part 2.—Next, we replace the units in Equation (47) with the corresponding RPS quantities, viz. ${\mathrm{kg}}^4\mathrm{C}{\Omega }^{-2}\to {\mathrm{M}}_{\mathrm{P}}^4{Q}_{\mathrm{P}}/Z_0^2$, and we find that

$$B_4={\alpha }_{\mathrm{w}}\left({\displaystyle \frac{M_{\mathrm{P}}^4{Q}_{\mathrm{P}}}{Z_0^2}}\right).$$

(51)

where the weak coupling constant serves again as the link between $B_4$ and the corresponding RPS units.

4.3. Subset  $C4:=\{G,K,e,k_{\mathrm{B}}\}$

We consider the constant $C_4$ introduced in Equation (3), viz.9

$$C_4={\displaystyle \frac{e\sqrt{G_{★}}}{k_{\mathrm{B}}}}={10}^{-6}\mathrm{F}\mathrm{K}{\mathrm{kg}}^{-1},$$

(52)

where $G_{★}=G/K$; thus, $G_{★}$ carries two independent constants (see also Equation (2)).10

Part 1.—The SI units in Equation (52) imply the unusual dimensional relation $C_4=\mathcal{C}\Theta /M$, where $\mathcal{C}$ represents capacitance. This relation can be rewritten dimensionally11 as

$$C_4=\Theta {\left({\displaystyle \frac{Q}{p_{\perp }}}\right)}^2,$$

(53)

where $p_{\perp }$ is assumed to represent transverse momentum. We imagine now a thermal plasma at temperature $\Theta$, permeated by a uniform magnetic field B, and a charged (Q) particle injected into this medium with transverse momentum ${\mathit{p}}_{\perp }\perp \mathit{B}$, thus executing gyrations about $\mathit{B}$-field lines. The radius of gyration is the so-called Larmor radius [19], viz.

$$R_{\mathrm{L}}={\displaystyle \frac{p_{\perp }}{\left|Q\right|B}},$$

(54)

and the magnetic rigidity [19] is

$$B{R}_{\mathrm{L}}={\displaystyle \frac{p_{\perp }}{\left|Q\right|}}.$$

(55)

On the other hand, Equation (53) predicts a magnetic rigidity that depends solely on the temperature of the plasma, viz.

$$B{R}_{\mathrm{L}}=\sqrt{{\displaystyle \frac{\Theta }{C_4}}}.$$

(56)

Experimental tests in magnetized plasmas under controlled conditions are needed to determine whether Equation (56) represents a genuine physical effect (magnetic rigidity $B{R}_{\mathrm{L}}\propto \sqrt{\Theta }$), or simply a temperature scaling relation involving the new constant $C_4={10}^{-6}\mathrm{F}\mathrm{K}{\mathrm{kg}}^{-1}$.

Part 2.—Next, we replace the units in Equation (52) with the corresponding RPS quantities, viz. $\mathrm{F}\mathrm{K}{\mathrm{kg}}^{-1}\to {\mathcal{C}}_{\mathrm{P}}{\Theta }_{\mathrm{P}}/M_{\mathrm{P}}$, and we find that

$$C_4={\alpha }_{\mathrm{w}}\left({\displaystyle \frac{{\mathcal{C}}_{\mathrm{P}}{\Theta }_{\mathrm{P}}}{M_{\mathrm{P}}}}\right),$$

(57)

where we have again introduced the weak coupling constant ${\alpha }_{\mathrm{w}}=\sqrt{\alpha }$ (Table 1), rather than the square root of the FSC.

4.4. Subset  $D4:=\{\sigma ,c,h,R_{\infty }\}$

We consider the constant $D_4$ such that

$$D_4={\displaystyle \frac{\sigma }{c}}{\left(h{R}_{\infty }\right)}^2={10}^{-68}{\mathrm{kg}}^3\mathrm{m}{\mathrm{s}}^{-4}{\mathrm{K}}^{-4},$$

(58)

where $\sigma$ represents the Stefan–Boltzmann constant (Table 1).

Part 1.—The SI units in Equation (58) imply the dimensional relation $D_4=M^{3}L/{(T\Theta )}^4$. We take the following steps:

(a)

We eliminate the temperature by using the Stefan–Boltzmann law $j=\sigma {\Theta }^4$, where $j$ is the radiant exitance with dimensions of [power][area]⁻¹, so that $j$ can, in turn, be replaced by

$$E/\left(L^{2}T\right)\to M/T^3,$$

where E represents energy.

(b)

By reducing the mechanical quantities, we obtain the relation $D_4/\sigma =M^2\mathit{v}$, where $\mathit{v}$ is velocity.

(c)

Finally, we introduce the dimensional angular momentum $J=ML\mathit{v}$ and linear mass density $\mu =M/L$, and we find the interesting relation

$$\mu J={\displaystyle \frac{D_4}{\sigma }}.$$

(59)

The unusual combination $\mu J$ occurs in string theory [26,27], particularly in studies of Regge trajectories. For a rotating open Nambu–Goto (NG) relativistic string—which supports the standard Regge-type relation $J\propto E^2$—with energy per unit length $\mu c^2$ and hence string tension $T_0=\mu c^2$, the angular momentum satisfies $J=E^2/\left(2\pi c{T}_0\right)$ or, equivalently, $J=E^2/\left(2\pi \mu c^3\right)$, from which we obtain the relation

$$\mu J={\displaystyle \frac{E^2}{2\pi c^3}}.$$

(60)

Thus, the constant $D_4/\sigma$ in Equation (59) effectively singles out a radiative/atomic energy scale in the NG framework, viz.

$$E=\sqrt{2\pi }\left(hc{R}_{\infty }\right)\simeq 34.1044\mathrm{eV},$$

(61)

where the term in the parentheses is the standard Rydberg energy ($13.6057$ eV in CODATA Refs. [6,7]) and the factor $\sqrt{2\pi }\simeq 2.5$.12 This is a surprising atomic connection because the rotating NG string is often used as an idealized model of meson-like hadronic states, where it reproduces Regge-type trajectories [27].

Part 2.—Next, we replace the units in Equation (58) with the corresponding RPS quantities, viz. ${\mathrm{kg}}^3\mathrm{m}{\mathrm{s}}^{-4}{\mathrm{K}}^{-4}\to M_{\mathrm{P}}^3{L}_{\mathrm{P}}/\left(T_{\mathrm{P}}^4{\Theta }_{\mathrm{P}}^4\right)$, and we find that

$$D_4={\displaystyle \frac{8{\pi }^9}{15}}\left({\alpha }^4{\alpha }_{\mathrm{g}}\right)\left({\displaystyle \frac{M_{\mathrm{P}}^3{L}_{\mathrm{P}}}{T_{\mathrm{P}}^4{\Theta }_{\mathrm{P}}^4}}\right).$$

(62)

In this equation, the composite factor $(8{\pi }^9{\alpha }^4/15)$ is traced directly to the geometric factors in Equation (58) introduced by the definitions of $\sigma$ and $R_{\infty }$: the Stefan–Boltzmann constant contributes $2{\pi }^5/15$ (Table 1), while the Rydberg constant contributes $2{\pi }^2{\alpha }^2$ (see also Equation (15)).

4.5. Subset  $E4:=\{F,K,{\Phi }_0,\sigma \}$

We consider the constant $E_4$ such that

$$E_4={\left(FK{\Phi }_0/\sigma \right)}^2={10}^{15}{\left(\mathrm{kg}{\mathrm{m}}^5{\mathrm{K}}^4{\mathrm{C}}^{-2}\right)}^2,$$

(63)

where F represents the Faraday constant and K represents the Coulomb constant (Table 1). Because of the overall square in this constant, it is reasonable to consider $\sqrt{E_4}$ below.

Part 1.—The SI units in Equation (63) imply the dimensional relation $\sqrt{E_4}=M{L}^5{\Theta }^4/Q^2$. We take the following steps:

(a)

We eliminate temperature by using the Stefan–Boltzmann law $j=\sigma {\Theta }^4$, where $j$ is the radiant exitance, so that $j$ can then be replaced by $E/\left(L^{2}T\right)$, where $E=M{\mathit{v}}^2$ represents energy.

(b)

Next, we notice that the dimensional combination ${\left[Q\right]}^2{\left[M\right]}^{-1}=\left[G_{★}\right]\left[M\right]$, which allows us to replace $M/Q^2=1/\left(G_{★}M\right)$.

(c)

From the above substitutions, we deduce the relation

$$\begin{array}{ccc}{L}^2{\mathit{v}}^3& =& \sigma G_{★}\sqrt{E_4}\hfill \\ & =& {\displaystyle \frac{1}{2}}Gh{N}_{\mathrm{A}}\hfill \\ & \equiv & C_{\mathrm{A}}\hfill \end{array}.$$

(64)

Although Equation (64) seems to promote a simple fluid-mechanics scaling, it should not be applicable to unmagnetized fluids because the magnetic flux quantum was introduced in the definition of $E_4$. Instead, this scaling offers a compelling framework for understanding energy transfer in both magnetohydrodynamic (MHD) and kinetic plasma turbulence. We thus outline three potential applications of Equation (64), comparing and contrasting its physical consequences with established classical theories.

1.

Velocity Scaling: In classical fluid turbulence [29], velocity fluctuations scale as $\mathit{v}\propto L^{1/3}$, implying that energy smoothly decreases as the cascade moves to smaller scales. In contrast, the above scaling implies that $\mathit{v}\propto L^{-2/3}$. This inverse relationship dictates that velocity fluctuations increase at smaller length scales, and this is a signature of energy transfer to highly localized, highly energetic MHD structures in plasma environments.

2.

Constant Power and Current Sheet Discontinuities: In Electron MHD or incompressible MHD frameworks, applicable to the sub-ion range [30,31,32], we calculate the kinematic power $\mathcal{P}$ processed by a single localized eddy or plasma filament of length scale L, scale-independent density $\rho$, and typical turbulent velocity $\mathit{v}$. We find the scaling relation

$$\mathcal{P}=\rho L^2{\mathit{v}}^3=\rho C_{\mathrm{A}}=\mathrm{const}.$$

(65)

Thus, according to Equation (64), the power flowing into smaller structures remains scale-invariant. As a result, instead of a smooth Kolmogorov-type energy decay, equal amounts of power $\mathcal{P}$ are funneled into continuously shrinking volumes. Consequently, this framework accurately describes the dynamics of sub-ion range MHD turbulence, magnetic reconnection sites, and the localized powerful explosions that generate sharp discontinuities in thin current sheets within the plasma.

3.

Kinetic and Magnetic Spectra: Applying such a constant power postulate to Fourier space yields two distinct energy spectra without relying on any constant-dissipation assumptions:

• For the 1D kinetic energy spectrum with wavenumbers $k=1/L$ and localized velocity fluctuations $\mathit{v}\left(k\right)\propto k^{2/3}$, the model predicts a positive-slope spectrum of

  $$E_{\mathrm{kin}}\left(k\right)\propto {\mathit{v}}^2\propto k^{4/3},$$

  representing an intense spatial intermittency and energy concentration at small scales.
• On the other hand, treating the sub-ion range—where only electrons are current carriers and, from Ampére’s law, $\delta B\left(k\right)\propto \mathit{v}/k\propto k^{-1/3}$—yields a 1D magnetic-energy spectrum of

  $$E_B\left(k\right)\propto {\left(\delta B\right)}^2/k\propto k^{-5/3}.$$
  This spectrum is highly consistent with established results in the gyrokinetic literature [33], Electron-MHD simulations [31], and direct spacecraft observations of the solar wind [32].

Part 2.—Next, we replace the units in Equation (63) with the corresponding RPS quantities, viz. $\mathrm{kg}{\mathrm{m}}^5{\mathrm{K}}^4{\mathrm{C}}^{-2}\to M_{\mathrm{P}}{L}_{\mathrm{P}}^5{\Theta }_{\mathrm{P}}^4/Q_{\mathrm{P}}^2$, and we find that

$$\sqrt{E_4}={\displaystyle \frac{15}{4{\pi }^5}}{N}_{\mathrm{A}}\left({\displaystyle \frac{M_{\mathrm{P}}{L}_{\mathrm{P}}^5{\Theta }_{\mathrm{P}}^4}{Q_{\mathrm{P}}^2}}\right),$$

(66)

where $N_{\mathrm{A}}$ is the usual Avogadro number, which appears because it was introduced by Faraday’s constant in the definition (63) of $E_4$. The leading numerical factor can be traced to the numerical factors included in the definitions of ${\Phi }_0$ and $\sigma$ (Table 1). We also note the absence of unitless coupling constants in Equation (66). This is because the constant $C_{\mathrm{A}}$ defined by Equation (64) does not contain the necessary ingredients (c, e, or $m_{\mathrm{e}}$); the two constants G and h are simply not enough.

4.6. Subset  $F4:=\{h,\sigma ,{\Phi }_0,R_{\infty }\}$

We consider the constant $F_4$ such that

$$F_4={\displaystyle \frac{h^2\sigma }{{\Phi }_0{R}_{\infty }^2}}={10}^{-73}{\mathrm{J}}^2\mathrm{C}{\mathrm{K}}^{-4},$$

(67)

where J represents the unit of Joule.

Part 1.—The SI units in Equation (67) imply the dimensional relation $F_4=E^{2}Q/{\Theta }^4$. We manipulate this equation using the Stefan–Boltzmann law $j=\sigma {\Theta }^4$, radiant exitance $j=M/T^3$, energy $E=M{\mathit{v}}^2$ and $E=h/T$, current $I=Q/T$, magnetic moment $\mu =I{L}^2$, and action $h=\mu BT$; and we finally find an equation (not a relation) for the magnitude of the magnetic field B, viz.

$$B={\displaystyle \frac{h^2\sigma }{F_4}}={\Phi }_0{R}_{\infty }^2=0.249\mathrm{T}.$$

(68)

Note that this magnetic-field threshold concerns two electrons (the Cooper pair in ${\Phi }_0$; Table 1), so this threshold could be relevant to superconductivity (i.e., Josephson current magnetic suppression [34,35,36] and the Shubnikov–de Haas effect in degenerate semiconductors [37,38,39,40]).

Part 2.—Next, we replace the units in Equation (67) with the corresponding RPS quantities, viz. ${\mathrm{J}}^2\mathrm{C}{\mathrm{K}}^{-4}\to E_{\mathrm{P}}^2{Q}_{\mathrm{P}}/{\Theta }_{\mathrm{P}}^4$, and we find that

$$F_4={\displaystyle \frac{\pi /15}{{\alpha }_{\mathrm{g}}{\alpha }_{\mathrm{w}}^7}}\left({\displaystyle \frac{E_{\mathrm{P}}^2{Q}_{\mathrm{P}}}{{\Theta }_{\mathrm{P}}^4}}\right),$$

(69)

where the weak coupling constant ${\alpha }_{\mathrm{w}}=\sqrt{\alpha }$ (Table 1).

On the other hand, we define the Planck unit of magnetic field as

$$B_{\mathrm{P}}\equiv {\displaystyle \frac{F_{\mathrm{P}}}{c{Q}_{\mathrm{P}}}}={\displaystyle \frac{{\alpha }_{\mathrm{w}}{c}^3}{Ge}},$$

(70)

and we recast Equation (68) in RPS units as

$$B=\left(2{\pi }^4{\alpha }_{\mathrm{g}}{\alpha }_{\mathrm{w}}^7\right)B_{\mathrm{P}}.$$

(71)

It is rather surprising that the gravitational coupling constant is involved in the above equations, but it is the only dimensionless constant that can effectively drive the enormous RPS value of $B_{\mathrm{P}}\sim {10}^{53}$ T down to laboratory-scale superconducting field magnitudes ($B\sim 0.1$–1 T).

4.7. Subset  $G4:=\{h,\sigma ,G,R_{\infty }\}$

We consider the constant $G_4$ such that

$$G_4={\displaystyle \frac{h^{2}G{R}_{\infty }}{{\sigma }^2}}={10}^{-55}{\mathrm{m}}^6{\mathrm{s}}^2{\mathrm{K}}^8{\mathrm{kg}}^{-1}.$$

(72)

Part 1.—The SI units in Equation (72) imply the dimensional relation $G_4=L^6{T}^2{\Theta }^8/M$. We manipulate this equation using the Stefan–Boltzmann law $j=\sigma {\Theta }^4$, radiant exitance $j=M/T^3$, energy $E=M{\mathit{v}}^2$, and specific angular momentum $\mathsf{\ell }=L\mathit{v}$; and we finally find a relation $E(\mathsf{\ell })$, viz.

$$E(\mathsf{\ell })={\displaystyle \frac{h^{2}G{R}_{\infty }}{{\mathsf{\ell }}^2}}.$$

(73)

This equation represents an invariant of orbital motion in a central $-k/r$ potential [41], where circular orbits obey the $E\left(J\right)$ relation $\left|E\right|=m{k}^2/\left(2J^2\right)$ with total angular momentum $J=m\mathsf{\ell }$. We rewrite this classical Newtonian equation as $\left|E\right|=k^2/\left(2m{\mathsf{\ell }}^2\right)$ for direct comparison. Thus, constant $h^{2}G{R}_{\infty }$ in Equation (73) plays the same role as $k^2/\left(2m\right)$ in the central potential.

We imagine an electron orbiting in the gravitational field of another mass $M_{★}$, in which case $k=G{M}_{★}{m}_{\mathrm{e}}$, and we equate the above two constants (i.e., $k^2/\left(2m_{\mathrm{e}}\right)=h^{2}G{R}_{\infty }$) to determine a typical magnitude of the central mass $M_{★}$. Using $R_{\infty }=2{\pi }^2{\alpha }^2/r_{\mathrm{c}}$ (Equation (32)), Equation (16) for $r_{\mathrm{c}}$, and $M_{\mathrm{P}}^2=hc/G$, we find that

$$k=2\pi \alpha \left(G{M}_{\mathrm{P}}{m}_{\mathrm{e}}\right).$$

(74)

Thus, the central mass implied by the $E(\mathsf{\ell })$ relation (73) for an orbiting electron is

$$M_{★}=2\pi \alpha M_{\mathrm{P}}\simeq 4.0\times {10}^{-10}\mathrm{kg},$$

(75)

which turns out to be a small fraction of the Planck mass ($2\pi \alpha =0.73\%$). Such a mass places the orbital dynamics of Equation (73) firmly in the sub-Planckian gravitational regime—but not at subatomic scales, where Newtonian gravity is negligible.

Part 2.—Next, we replace the units in Equation (72) with the corresponding RPS quantities, viz. ${\mathrm{m}}^6{\mathrm{s}}^2{\mathrm{K}}^8{\mathrm{kg}}^{-1}\to L_{\mathrm{P}}^6{T}_{\mathrm{P}}^2{\Theta }_{\mathrm{P}}^8/M_{\mathrm{P}}$, and we find that

$$G_4={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{15}{{\pi }^4}}\right)}^2\left({\displaystyle \frac{{\alpha }^2}{{\aleph }_{\mathrm{A}}}}\right)\left({\displaystyle \frac{L_{\mathrm{P}}^6{T}_{\mathrm{P}}^2{\Theta }_{\mathrm{P}}^8}{M_{\mathrm{P}}}}\right).$$

(76)

The leading numerical factor naturally stems from the definitions of $\sigma$ and $R_{\infty }$ (Table 1). Again, the appearance of the reduced Avogadro number ${\aleph }_{\mathrm{A}}$ indicates that this is a true natural constant, unlike $N_{\mathrm{A}}$, which is a man-made constant (see Ref. [1] for more details). For the record, the natural constant ${\aleph }_{\mathrm{A}}$ corresponds to nature’s unit of $f_{\mathrm{A}}^{-1}=0.09945002021$ mol (Table 1), or, approximately, 0.1 mol.

4.8. Subset  $H4:=\{h,\sigma ,m_{\mathrm{e}},{\Phi }_0\}$

We consider the constant $H_4$ such that

$$H_4={\displaystyle \frac{h{\Phi }_0^2{\sigma }^2}{m_{\mathrm{e}}}}={10}^{-47}{\mathrm{kg}}^4{\mathrm{m}}^6{\mathrm{s}}^{-9}{\mathrm{C}}^{-2}{\mathrm{K}}^{-8}.$$

(77)

Part 1.—The SI units in Equation (77) imply the dimensional relation $H^4=M^4{L}^6/\left(T^9{Q}^2{\Theta }^8\right)$. After some straightforward manipulations, we obtain the dimensional equalities

$${\displaystyle \frac{H_4}{h{\sigma }^2}}=\left({\displaystyle \frac{T}{h}}\right){\left({\displaystyle \frac{EL}{Q}}\right)}^2=E{\left({\displaystyle \frac{L}{Q}}\right)}^2={\displaystyle \frac{E}{{\lambda }^2}},$$

(78)

where the 1D line charge density $\lambda =Q/L$ was introduced in the last step.

The scaling $E\propto {\lambda }^2$ is not unknown in physics; it is a standard result in accelerator physics, where the energy loss of a charged bunch due to the self-induced “wakefields” is quadratic in the bunch line charge density [42,43,44]. In the wakefield formalism, this follows from the convolution structure of the wake potential or, equivalently, from the quadratic spectral weighting ${|\lambda \left(\omega \right)|}^2$ of the longitudinal impedance $Z_{\Vert }\left(\omega \right)$ in the Fourier-domain representation. Here, $\lambda \left(\omega \right)$ denotes the Fourier transform of the longitudinal charge distribution $\lambda \left(z\right)$ extending in the z-direction, and $\omega$ is the angular frequency of the Fourier modes.

Casting Equation (78) in the form $E=C_0{\lambda }^2$, the proportionality constant turns out to be

$$C_0={\displaystyle \frac{H_4}{h{\sigma }^2}}={\displaystyle \frac{{\Phi }_0^2}{m_{\mathrm{e}}}}=4.6940{\mathrm{Wb}}^2{\mathrm{kg}}^{-1},$$

(79)

which is rather surprising.

In this setting, the electron mass introduces the fundamental inertial response in an accelerating bunch or beam of electrons [43,44]. Thus, $m_{\mathrm{e}}$ serves as a mechanical impedance to EM coupling. However, because the intrinsic inertia of the electron is so small, the bunch exhibits a high dynamical susceptibility to self-induced wakefields. Consequently, the scaling $C_0\propto 1/m_{\mathrm{e}}$ in Equation (79) reflects the efficiency with which kinetic energy is transduced into EM energy of the wakefield, as the electrons lack the inertial stiffness required to resist the modulation coming from the magnetic flux ${\Phi }_0$.

Part 2.—Next, we replace the units in Equation (77) with the corresponding RPS quantities, viz. ${\mathrm{kg}}^4{\mathrm{m}}^6{\mathrm{s}}^{-9}{\mathrm{C}}^{-2}{\mathrm{K}}^{-8}\to M_{\mathrm{P}}^4{L}_{\mathrm{P}}^6/\left(T_{\mathrm{P}}^9{Q}_{\mathrm{P}}^2{\Theta }_{\mathrm{P}}^8\right)$, and we find that

$$H_4={\left({\displaystyle \frac{{\pi }^5}{15}}\right)}^2\left({\displaystyle \frac{{\aleph }_{\mathrm{A}}}{\alpha }}\right)\left({\displaystyle \frac{M_{\mathrm{P}}^4{L}_{\mathrm{P}}^6}{T_{\mathrm{P}}^9{Q}_{\mathrm{P}}^2{\Theta }_{\mathrm{P}}^8}}\right).$$

(80)

The leading numerical factor naturally comes from the definitions of $\sigma$ and ${\Phi }_0$ (Table 1).

On the other hand, we define the Planck unit of magnetic flux [1] as

$${\Phi }_{\mathrm{P}}\equiv \sqrt{{\displaystyle \frac{Kh}{c}}},$$

(81)

and we recast Equation (79) in RPS units, viz.

$$C_0={\displaystyle \frac{{\aleph }_{\mathrm{A}}}{4\alpha }}\left({\displaystyle \frac{{\Phi }_{\mathrm{P}}^2}{M_{\mathrm{P}}}}\right).$$

(82)

The leading factor of 1/4 comes from the definition of ${\Phi }_0$ (Table 1). Once again, it is surprising that the reduced Avogadro number appears in this scaling in the numerator of the unitless fraction, where it clearly dominates over the FSC, just as we also saw in Equation (34). Evidently, inertial response does play a role in an accelerated bunch or beam of electrons, the least massive free particles known in this universe.

4.9. Subset  $I4:=\{c,\sigma ,k_{\mathrm{B}},R\}$

We consider the constant $I_4$ such that

$$I_4={\displaystyle \frac{k_{\mathrm{B}}^{2}R\sigma }{c^2}}={10}^{-69}{\mathrm{kg}}^4{\mathrm{m}}^4{\mathrm{s}}^{-7}{\mathrm{K}}^{-7},$$

(83)

where R is the molar gas constant (Table 1).

Part 1.—The SI units in Equation (83) imply the dimensional relation $I^4=M^4{L}^4/\left(T^7{\Theta }^7\right)$. After some straightforward manipulations, we obtain the equalities

$${\displaystyle \frac{k_{\mathrm{B}}{I}_4}{{\sigma }^2}}={\displaystyle \frac{k_{\mathrm{B}}\Theta }{T}}{\left(M{L}^2\right)}^2={\displaystyle \frac{E}{T}}{\left(M{L}^2\right)}^2={\left(ET\right)}^3.$$

(84)

Before we can interpret this scaling physically, we must analyze the composite constant on the left-hand side. Thus, we solve Equation (84) for E, we simplify the constant, we introduce the frequency $f=1/T$, and we find that

$$E={\left({\displaystyle \frac{15}{2{\pi }^5}}{N}_{\mathrm{A}}\right)}^{1/3}hf.$$

(85)

The simplified constant scales the photon energy $hf$ up to the total energy of a classical ensemble of about 24.5 million photons, a number that is not unrealistic in laser optical technology [45].

As an example, consider a continuous-wave semiconductor laser with a spectral linewidth $\Delta f=100\mathrm{MHz}$, corresponding to a coherence time ${\tau }_{\mathrm{c}}=1/\Delta f=10\mathrm{ns}$. This coherence time defines the interval over which the optical field maintains a well-defined phase.

For visible light with frequency $f=6\times {10}^{14}\mathrm{Hz}$ and a modest output power of $\mathcal{P}=1\mathrm{mW}$, the corresponding photon flux is $dN/dt=\mathcal{P}/\left(hf\right)\simeq 2.5\times {10}^{15}{\mathrm{s}}^{-1}$. Over a single coherence time, the number of photons contained in the phase-coherent portion of the field is $N={\tau }_{\mathrm{c}}(dN/dt)\simeq 25$ million. In this regime, the radiation is well described as a classical EM field with a stable phase over the coherence interval, and the photon number fluctuations are correspondingly very small ($\Delta N\simeq \sqrt{N}\simeq 5.0\times {10}^3$ by Poisson statistics, and then $\Delta N/N\simeq 0.02\%$).

Before closing, we should also interpret the power of $N_{\mathrm{A}}^{1/3}$ in Equation (85). The power of $1/3$ appears because the photon occupation number is obtained from an integral over a 3D phase space, where the number of accessible modes scales as the cube of the characteristic frequency or wavevector. Then, deriving a single characteristic energy scale corresponds to expressing the cumulative 3D phase-space occupation in terms of its associated radial wavevector scale, which necessarily introduces a cubic-root dependence on the total occupation number. On the other hand, such a radial reduction of the 3D phase space is not encoded in the definition (83) of the constant $I_4$, nor does it appear in the RPS unit transformation described by Equation (86) below.

Part 2.—Next, we replace the units in Equation (83) with the corresponding RPS quantities, viz. ${\mathrm{kg}}^4{\mathrm{m}}^4{\mathrm{s}}^{-7}{\mathrm{K}}^{-7}\to M_{\mathrm{P}}^4{L}_{\mathrm{P}}^4/\left(T_{\mathrm{P}}^7{\Theta }_{\mathrm{P}}^7\right)$, and we find that

$$I_4={\displaystyle \frac{2{\pi }^5}{15}}{N}_{\mathrm{A}}{\displaystyle \frac{{\left(M_{\mathrm{P}}{L}_{\mathrm{P}}\right)}^4}{{\left(T_{\mathrm{P}}{\Theta }_{\mathrm{P}}\right)}^7}}.$$

(86)

The leading numerical factor13 comes from the definition of the Stefan–Boltzmann constant $\sigma$, and the Avogadro number comes from the definition of the molar gas constant R (Table 1).

5. Four Cases Each Involving Five Fundamental Constants

5.1. Subset  $A5:=\{G,c,e,k_{\mathrm{B}},K\}$

We consider the constant $A_5$ such that

$$A_5=G{\left({\displaystyle \frac{ce}{k_{\mathrm{B}}K}}\right)}^2={10}^{-5}{\mathrm{K}}^2{\mathrm{s}}^4{\mathrm{C}}^6{\mathrm{kg}}^{-5}{\mathrm{m}}^{-5},$$

(87)

where $K=1/\left(4\pi {\epsilon }_0\right)$ (Table 1) and K represents the unit of Kelvin. Constant $A_5$ combines elements of gravity, electricity, the vacuum, and Boltzmann entropy.

Part 1.—The SI units in Equation (87) are equivalent to the composite units ${\mathrm{A}}^6{\mathrm{K}}^2{\mathrm{N}}^{-5}$, where A and N represent Ampere and Newton, respectively. Eliminating the unit of force from the dimensional relation $F=I^2{\mu }_0/\left(4\pi \right)$ (a combination of the Biot–Savart law and the Lorentz force), where I represents electric current, these units imply the dimensional relation

$${\displaystyle \frac{{\Theta }^2}{I^4}}=A_5{\left({\displaystyle \frac{{\mu }_0}{4\pi }}\right)}^5={10}^{-40}{\mathrm{K}}^2{\mathrm{A}}^{-4}.$$

(88)

Surprising as it may be, this scaling is a precise statement of equipartition between thermal energy per degree of freedom $E_{\mathrm{th}}={\displaystyle \frac{1}{2}}{k}_{\mathrm{B}}\Theta$ and magnetic energy $E_B={\displaystyle \frac{1}{2}}{\mathcal{L}}_{\mathrm{S}}{I}^2$ of an inductor with inductance

$${\mathcal{L}}_{\mathrm{S}}={\displaystyle \frac{{\mu }_0}{4\pi }}{L}_{\mathrm{S}}=1.380649\times {10}^{-43}\mathrm{H},$$

(89)

where $L_{\mathrm{S}}={\alpha }_{\mathrm{w}}{L}_{\mathrm{P}}$ is the Stoney length and H represents the unit of Henry. Because the vacuum permeability is a lower limit in nature and $L_{\mathrm{S}}<L_{\mathrm{P}}$, it certainly seems that the Stoney inductance ${\mathcal{L}}_{\mathrm{S}}$ is the vacuum’s lower limit of inductive resistance (or “magnetic inertia”).

Equation (88) manifests as a fundamental spatial expression of the Johnson–Nyquist thermal noise [46,47]. An ambient thermal bath, characterized by temperature fluctuations $\Theta$, continuously imparts stochastic kinetic energy to the vacuum manifold. These thermodynamic kicks drive a microscopic thermal-noise current I. Spacetime resists this spontaneous flow via its intrinsic permeability ${\mu }_0$, momentarily capturing the kinetic energy as magnetic energy.

This magnetic storage is strictly transient. As the initial thermal excitation wanes, the vacuum’s localized magnetic field collapses and discharges the stored energy back into the surrounding thermal bath. The constant in Equation (88) acts as the fundamental scaling factor that bridges thermal fluctuations and magnetic energy. It dictates that the energy absorbed by the intrinsic magnetic field is balanced by the energy released back out to the bath, thus sustaining energy equipartition at the fundamental length scale $L_{\mathrm{S}}$ [48].

Part 2.—Next, we replace the units in Equation (87) with the corresponding RPS quantities, viz. ${\mathrm{K}}^2{\mathrm{s}}^4{\mathrm{C}}^6{\mathrm{kg}}^{-5}{\mathrm{m}}^{-5}\to {\Theta }_{\mathrm{P}}^2{T}_{\mathrm{P}}^4{Q}_{\mathrm{P}}^6/\left(M_{\mathrm{P}}^5{L}_{\mathrm{P}}^5\right)$, and we find that

$$A_5=\alpha \left({\displaystyle \frac{{\Theta }_{\mathrm{P}}^2{T}_{\mathrm{P}}^4{Q}_{\mathrm{P}}^6}{M_{\mathrm{P}}^5{L}_{\mathrm{P}}^5}}\right).$$

(90)

The same method, applied also to Equation (88), gives the relation

$$A_5{\left({\displaystyle \frac{{\mu }_0}{4\pi }}\right)}^5=\alpha \left({\displaystyle \frac{{\Theta }_{\mathrm{P}}^2}{I_{\mathrm{P}}^4}}\right).$$

(91)

Combining Equations (90) and (91), we obtain an RPS identity for the vacuum permeability, viz.

$${\displaystyle \frac{{\mu }_0}{4\pi }}={\displaystyle \frac{M_{\mathrm{P}}{L}_{\mathrm{P}}}{Q_{\mathrm{P}}^2}}={\displaystyle \frac{L_{\mathrm{P}}}{G_{★}{M}_{\mathrm{P}}}},$$

(92)

where we used $Q_{\mathrm{P}}^2=G_{★}{M}_{\mathrm{P}}^2$ [1] to obtain the second leg. In this sense, ${\mu }_0$ is determined only by mechanical units and a vacuum property, since $G_{★}=4\pi {\epsilon }_{0}G$.

5.2. Subset  $B5:=\{G,m_{\mathrm{e}},{\Phi }_0,k_{\mathrm{B}},{\mu }_0\}$

We consider the constant $B_5$ such that

$$B_5={\displaystyle \frac{G^2{m}_{\mathrm{e}}{\Phi }_0^2}{{\mu }_0{k}_{\mathrm{B}}}}={10}^{-51}{\mathrm{m}}^7\mathrm{K}{\mathrm{kg}}^{-1}{\mathrm{s}}^{-4},$$

(93)

where, notably, ${\mu }_0$ alone (without the prefactor ${\displaystyle \frac{1}{4\pi }}\simeq 0.080$) determines the precise power of ten to 6 SDs. Constant $B_5$ combines elements of gravity, magnetism, the vacuum, and Boltzmann entropy.

Part 1.—The SI units in Equation (93) imply the dimensional relation

$$B_5\mathit{u}=\Theta {\mathit{v}}^6,$$

(94)

where $\mathit{u}$ represents energy density with dimensions $\left[M\right]{\left[L\right]}^{-1}{\left[T\right]}^{-2}$. On the other hand, the second law of thermodynamics effectively relates $\mathit{u}$ to entropy density $\mathit{s}$ in a locality of temperature $\Theta$, viz. $\mathit{u}=\Theta \mathit{s}$, and Equation (94) reduces to

$$B_5\mathit{s}={\mathit{v}}^6,$$

(95)

an unusual relation scaling the entropy density $\mathit{s}$ to a high power of the velocity $\mathit{v}$.

Entropy density essentially represents the information capacity of a field or a medium; thus Equation (95) dictates that such a field acts as an immense entropic reservoir. Since $|B_5|\ll 1$, a massive amount of information is required to be stored in the field to support even microscopic kinematic fluctuations.

The high power (${\mathit{v}}^6$) of the kinematic state can be viewed in at least two ways:

(a)

Since the specific kinetic energy $E/M\propto {\mathit{v}}^2$, then $\mathit{s}\propto {(E/M)}^3$, indicating that the field coupling is very sensitive to kinematic fluctuations driven by fluctuations of the magnetic field.

(b)

In Curle’s extension of aeroacoustic theory [49,50], sound energy is radiated by dipoles at the boundaries and propagates in the medium- to the far-field, where its intensity scales as ${\mathit{v}}^6$. Here, ${\mathit{v}}^6$ represents the radiation intensity of these dipole sources; thus, Equation (95) implies that the flow’s internal entropy density $\mathit{s}$ acts as the thermodynamic source term that scales with the power required to sustain and radiate these boundary-induced fluctuations.

Part 2.—Next, we replace the units in Equation (93) with the corresponding RPS quantities, viz. ${\mathrm{m}}^7\mathrm{K}{\mathrm{kg}}^{-1}{\mathrm{s}}^{-4}\to L_{\mathrm{P}}^7{\Theta }_{\mathrm{P}}{M}_{\mathrm{P}}^{-1}{T}_{\mathrm{P}}^{-4}$, and we find that

$$B_5={\left(16\pi \alpha {\aleph }_{\mathrm{A}}\right)}^{-1}\left({\displaystyle \frac{L_{\mathrm{P}}^7{\Theta }_{\mathrm{P}}}{M_{\mathrm{P}}{T}_{\mathrm{P}}^4}}\right).$$

(96)

The factor of ${\left(16\pi \right)}^{-1}$ is a combination of ${\left(4\pi \right)}^{-1}$ (the prefactor missing by design from ${\mu }_0$ in the definition (93) of $B_5$) and ${(1/2)}^2$ coming from the definition of ${\Phi }_0^2$ (Table 1).14^,15

5.3. Subset  $C5:=\{G,c,m_{\mathrm{e}},\sigma ,{\Phi }_0\}$

We consider the constant $C_5$ such that

$$C_5={\displaystyle \frac{G^2{c}^2{m}_{\mathrm{e}}\sigma }{{\Phi }_0}}={10}^{-26}{\mathrm{m}}^6\mathrm{C}{\mathrm{kg}}^{-1}{\mathrm{s}}^{-8}{\mathrm{K}}^{-4}.$$

(97)

This constant combines elements of gravity, the vacuum, thermal radiation, and quantum mechanics.

Part 1.—The SI units in Equation (97) imply a simple dimensional relation. Using in sequence the standard dimensional conversions ${\Theta }^4=j/\sigma$, $j=M/T^3$, $Q=IT$, $\rho =M/L^3$, and $\rho =1/\left(G{T}^2\right)$, we find that

$$C_5=\sigma G^{2}I,$$

(98)

where I represents electric current. The product $G^{2}I$ is unique in physics because there is no theory unifying gravity and EM theory and no relation showing that $I\propto G^{-2}$—but it cannot be avoided. A clear intermediate step, viz. $C_5/\sigma =I/\left({\rho }^2{T}^4\right)=\mathrm{const}.$, seems to associate current with mass density and time, which is not palatable. On the other hand, the scaling $\rho T^2=1/G$ of free-fall time T is ubiquitous in gravitational dynamics and in astrophysics—although it eliminates both $\rho$ and T simultaneously, resulting in the mere determination of a constant current I.

Substituting the definitions of $C_5$ (Equation (97)) and ${\Phi }_0$ (Table 1) into Equation (98), then G and $\sigma$ both cancel out, and we find that the constant current is

$$I={\displaystyle \frac{m_{\mathrm{e}}{c}^2}{{\Phi }_0}}=2e{f}_{\mathrm{c}}=39.6\mathrm{A},$$

(99)

where $f_{\mathrm{c}}=c/r_{\mathrm{c}}$ is the Compton frequency attributed to each of the electrons in the Cooper pair captured by ${\Phi }_0$. Obviously then, the internal circulatory motion of one electron is equivalent to a current of

$$I_{1\mathrm{e}}=e{f}_{\mathrm{c}}=19.8\mathrm{A}.$$

(100)

This is a well-known result in relativistic quantum physics [51,52,53,54,55,56], corresponding directly to the so-called Zitterbewegung or the “trembling motion” of the electron. In the geometric and hydrodynamic interpretations of the Dirac equation [54,56], the electron is modeled as a localized charge e circulating with speed c at the Compton radius $r_{\mathrm{c}}$. The resulting internal current for a single electron is indeed given by Equation (100).

In the above setting, $C_5$ acts as a universal repository of disparate physics, encompassing gravitational, thermal, and quantum scales within a single dimensional value. The transition from Equation (98) to (100) represents a systematic decoupling of macroscopic factors: by stripping away the contributions of gravity (G) and thermal state ($\sigma$), we have effectively removed the two large-scale or system-wide layers from this constant. What remains is a precise description of the electron’s internal state (I). This implies that the localized, high-frequency circulation of the electron is the irreducible quantum core that persists once the broader (global) gravitational and thermodynamic physics packed into $C_5$ is removed.

Part 2.—Next, we replace the units in Equation (97) with the corresponding RPS quantities, viz. ${\mathrm{m}}^6\mathrm{C}{\mathrm{kg}}^{-1}{\mathrm{s}}^{-8}{\mathrm{K}}^{-4}\to L_{\mathrm{P}}^6{Q}_{\mathrm{P}}{M}_{\mathrm{P}}^{-1}{T}_{\mathrm{P}}^{-8}{\Theta }_{\mathrm{P}}^{-4}$, and we find that

$$C_5={\displaystyle \frac{4{\pi }^5}{15}}\sqrt{{\alpha }_{\mathrm{g}}\alpha }\left({\displaystyle \frac{L_{\mathrm{P}}^6{Q}_{\mathrm{P}}}{M_{\mathrm{P}}{T}_{\mathrm{P}}^8{\Theta }_{\mathrm{P}}^4}}\right).$$

(101)

As usual, the leading factor of $4{\pi }^5/15$ is a combination of the numerical factors coming from the definitions of ${\Phi }_0$ and $\sigma$ (Table 1).

Furthermore, the equivalent current $I_{1\mathrm{e}}$ to the trembling motion of one electron (Equation (100)) takes the following form after G and $\sigma$ are removed:

$$I_{1\mathrm{e}}=\sqrt{{\alpha }_{\mathrm{g}}\alpha }\left({\displaystyle \frac{Q_{\mathrm{P}}}{T_{\mathrm{P}}}}\right)=\sqrt{{\alpha }_{\mathrm{g}}\alpha }{\mathcal{I}}_{\mathrm{P}},$$

(102)

where ${\mathcal{I}}_{\mathrm{P}}=c^2/\sqrt{G_B}$ is the RPS current [1], and the gravomagnetic constant $G_B$ is defined in Table 1.

It is important to note that, at the Planck scale, the current $I_{1\mathrm{e}}$ is determined, in part, by the gravitational coupling constant ${\alpha }_{\mathrm{g}}$ (Table 1), the fundamental universal constant of gravity that is systematically neglected in quantum physics and in QED (see also Note 15).

5.4. Subset  $D5:=\{G,K,m_{\mathrm{e}},\sigma ,{\Phi }_0\}$

We consider the constant $D_5$ such that

$$D_5={\displaystyle \frac{G^{2}K{m}_{\mathrm{e}}\sigma }{{\Phi }_0}}={10}^{-33}{\mathrm{m}}^7{\mathrm{C}}^{-1}{\mathrm{s}}^{-8}{\mathrm{K}}^{-4}.$$

(103)

Just as $C_5$ above, this constant also combines elements of gravity, the vacuum, thermal radiation, and quantum mechanics. Then, it is not a leap to expect that these two composite constants are related in a specific way.

Part 1.—Constant $D_5$ can be written in terms of $C_5$ (Equation (97)) as

$$D_5={\displaystyle \frac{{\mu }_0}{4\pi }}{C}_5,$$

(104)

or, using Equation (98), as

$$D_5=\sigma G^2\left({\displaystyle \frac{{\mu }_{0}I}{4\pi }}\right).$$

(105)

The appearance of the term (${\mu }_{0}I$) indicates that $D_5$ is connected to the magnetic field B generated by the trembling motion of the electron (Section 5.3). Using $B=\left({\mu }_{0}I\right)/\left(2r_{\mathrm{c}}\right)$ at the center of the current loop [57] of radius $r_{\mathrm{c}}$, Equation (105) produces the trembling magnetic field of a Cooper pair of electrons, viz.

$$B={\displaystyle \frac{ec}{{\epsilon }_0}}{\left({\displaystyle \frac{m_{\mathrm{e}}}{h}}\right)}^2=1.0253\times {10}^7\mathrm{T}.$$

(106)

Thus, for a single trembling electron, the magnetic field at the center of the current loop of radius $r_{\mathrm{c}}$ turns out to be

$$B_{1\mathrm{e}}={\displaystyle \frac{1}{2}}B=5.1265\times {10}^6\mathrm{T}.$$

(107)

Compared to the QED critical value of $B_{\mathrm{crit}}={\left(m_{\mathrm{e}}c\right)}^2/(e\hslash )=4.4140\times {10}^9\mathrm{T}$ (the so-called Schwinger limit [58,59,60]), the magnitude of field $B_{1\mathrm{e}}$ is, of course, much smaller. In fact, the scaling down to the trembling electron magnetic field is controlled entirely by the FSC, viz.

$$B_{1\mathrm{e}}=\alpha B_{\mathrm{crit}},$$

(108)

a fundamental result to be kept in mind—a result precise to 12 SDs (although only 5 SDs are shown here).

This remarkable scaling is revealed only in RPS units. The old ℏ-based definition of the FSC obscures the relation by introducing a geometric term that has no place in Equation (108). The same $2\pi$ geometric contamination from the ℏ-based FSC has also been detected in the first-order correction to the Landé $g_{\mathrm{s}}$-factor of the anomalous magnetic moment of the electron, which is simply $(g_{\mathrm{s}}-2)/2=\alpha$ in RPS units [1].

Part 2.—Next, we replace the units in Equation (103) with the corresponding RPS quantities, viz. ${\mathrm{m}}^7{\mathrm{C}}^{-1}{\mathrm{s}}^{-8}{\mathrm{K}}^{-4}\to L_{\mathrm{P}}^7{Q}_{\mathrm{P}}^{-1}{T}_{\mathrm{P}}^{-8}{\Theta }_{\mathrm{P}}^{-4}$, and we find that

$$D_5={\displaystyle \frac{4{\pi }^5}{15}}\sqrt{{\alpha }_{\mathrm{g}}\alpha }\left({\displaystyle \frac{L_{\mathrm{P}}^7}{Q_{\mathrm{P}}{T}_{\mathrm{P}}^8{\Theta }_{\mathrm{P}}^4}}\right).$$

(109)

Once again, the leading factor of $4{\pi }^5/15$ is a combination of the numerical factors coming from the definitions of ${\Phi }_0$ and $\sigma$ (Table 1). The dimensionless coefficient in $D_5$ is the same as that in $C_5$ (Equation (101)) because the two constants differ only by a dimensionful factor of ${\mu }_0/\left(4\pi \right)$ (see Equation (104)).

6. Discussion and Conclusions

In this work, we have systematically explored algebraic and dimensional relations constructed from small subsets of 2–5 fundamental constants (Section 2, Section 3, Section 4 and Section 5, respectively) carved out of the main set shown in Table 1. The SI values of the composite constants that we analyzed are all precise powers of ten (Table 2).16 This choice was motivated by constant $C_4$ with an SI magnitude of ${10}^{-6}$ which appeared in previous work [1] and allowed for the determination of the Newtonian gravitational constant G to 10 SDs.17

By organizing the analysis of the composite constants listed in Table 2 according to the number of the known constants involved in each synthesis (pairs to quintets), a clear pattern emerges: seemingly disparate physical domains—gravitation, electromagnetism, quantum mechanics, and thermodynamics—admit nontrivial couplings and relations never before seen or contemplated in physics or astrophysics. Thus, this method of analysis constitutes another pathway toward investigating unified descriptions of fundamental interactions.

A central outcome of this investigation is that most of these constructions reduce to scalings that resemble known physical relations or hint at structures already present in established theory. In most cases, the composite constants naturally reproduce quantities with clear physical meaning (e.g., energies, forces, magnetic fluxes, magnetic fields, electric currents, ohmic resistances, or EM characteristic scales), although some suggest less familiar but dimensionally consistent relations. This reinforces the value of dimensional analysis when applied exhaustively and systematically, not merely as a consistency check, but as a tool for uncovering latent connections between disparate physical regimes.

The present analysis was mainly restricted to composite constants with precise power-of-ten magnitudes (Table 2). This is not a limitation of the method; it is a choice reflecting our plan to expand the search beyond the only such constant heretofore known in physics, the vacuum’s magnetic permeability ${\mu }_0/\left(4\pi \right)$ tagged by 3D geometry. From a methodological standpoint, the analysis of any other composite constants is also expected to generate additional families of relations and/or scalings. To probe this end, we extend in Appendix A the analysis to four composite constants listed in Table 1 (${\mathcal{A}}_0$, $G_{★},G_{\mathrm{B}}$, and ${\mathcal{R}}_{\mathrm{K}}$), and we discuss the resulting scalings, which are related to known physics—the Tully–Fisher/Faber–Jackson relations [61,62] in galaxies, the Larmor power Formula [9] for accelerating charges, the collapse of magnetized giant molecular clouds (GMCs) [63,64] in star-forming regions, and the quantum Hall effect [65,66,67]. These four composite constants are summarized in

Table 3. Composite constants (not precise powers of ten) discussed in Appendix A.

Symbol	Definition	SI Value	SI Unit	Section	Related Themes in Science
${\mathcal{A}}_0$	$a_{0}G$	$7.425843255\times {10}^{-21}$	${\mathrm{m}}^4{\mathrm{kg}}^{-1}{\mathrm{s}}^{-4}$	Appendix A.1	Tully–Fisher/Faber–Jackson relations
$G_{★}$	$4\pi {\epsilon }_{0}G$	$7.425843255\times {10}^{-21}$	${\mathrm{C}}^2{\mathrm{kg}}^{-2}$	Appendix A.2	Larmor power formula
$G_B$	$G{\mu }_0/\left(4\pi \right)$	$6.674015081\times {10}^{-18}$	${\mathrm{m}}^4{\mathrm{s}}^{-4}{\mathrm{A}}^{-2}$	Appendix A.3	Magnetized GMCs, star formation
${\mathcal{R}}_{\mathrm{K}}$	${\displaystyle h/e^2}$	$2.581280746\times {10}^4$	$\Omega$	Appendix A.4	Quantum Hall effect

.

The invariant quantities represented by the composite constants appear far from arbitrary; they instead reflect physically significant couplings of elementary constants across disparate physical domains. We suggest that these invariants are precursors to a fundamental set of constants inherent to unified theories bridging the gaps between mechanics, quantum mechanics, gravitation, thermal physics, and EM theory. By bridging any number of these realms, such extended frameworks will inevitably expose latent symmetries and governing principles currently obscured by the partitioning of modern physics.

As Part 2 of the dimensional method used in this work, we have also systematically transformed the SI units of each composite constant to Planck/RPS units. In most cases, the RPS units must be modulated by a specific set of dimensionless prefactors. These include the dimensionless coupling constants, the reduced Avogadro number, the Avogadro number, and geometric factors involving $2\pi$ (a 2D tag) or $4\pi$ (a 3D tag) (see Table 1 for their details):

• The appearance of the $\alpha$, ${\alpha }_{\mathrm{w}}$, and ${\alpha }_{\mathrm{g}}$ couplings in various exotic combinations turns out to be very important. It shows that such constants must be included in all systems of units to provide their services during transformations between systems. Without them, unit transformations exhibit seemingly arbitrary numerical factors that cannot be interpreted physically [4]. Here, the dimensionless constants appeared naturally because they have been included in the RPS long ago [1,5].
• The reduced Avogadro number ${\aleph }_{\mathrm{A}}$ is nature’s chosen number of particles in about 0.1 moles of matter. It also appears naturally, ever since it was discovered [1] and was included in the RPS (via constant $f_{\mathrm{A}}$ in Table 1).
• The well-known Avogadro number $N_{\mathrm{A}}$ appears only if it is implicitly used in the construction of a composite constant. This occurs when the synthesis includes the Faraday constant $F=N_{\mathrm{A}}e$ or the molar gas constant $R=N_{\mathrm{A}}{k}_{\mathrm{B}}$ (see constants $E_4$ and $I_4$, respectively; Table 2).
• The geometric and numerical factors that appear in RPS unit transformations to SI units all come from the mixed numerical structures imprinted onto some of the elementary constants of Table 1 (specifically, the man-made constants $R_{\infty }$, $\sigma$, and ${\Phi }_0$, taken from CODATA Refs. [6,7]).

The same dimensionless factors also appear in transformations of composite constants between Gaussian/cgs units and Planck/RPS/cgs units (although we did not use Gaussian/cgs units in this study). Therefore, the dimensionless constants acting as numerical prefactors in unit conversions are universal units independent of systems of measurement, and this reinforces our recommendation that they must be included in all systems of units currently employed in the physical sciences.