# The Upgraded Planck System of Units That Reaches from the Known Planck Scale All the Way Down to Subatomic Scales

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction and Motivation

#### 1.1. Three Fundamental Systems of Units under Consideration

#### 1.2. Dirac’s Problematic Constant $\overline{)h}$ and the Three Widely-Used Atomic Radii

#### 1.3. Dimensionless Constants

“A special role is played by those physical quantities that are dimensionless in the SI system. We expect that such quantities are related to important physical effects. The experience of physicists confirms this.”

#### 1.4. Outline

## 2. The Building Blocks of the Upgraded Planck System

#### 2.1. Dimensional Units

#### 2.1.1. Constants Imposed by the Vacuum

#### 2.1.2. Dirac’s Constant $\overline{)h}=h\phantom{\rule{-0.166667em}{0ex}}/\phantom{\rule{-0.166667em}{0ex}}\left(2\pi \right)$

#### 2.1.3. Newton’s Gravitational Constant G

#### 2.2. Dimensionless Units

#### 2.2.1. Fine-Structure Constant ${\mathsf{\alpha}}_{\overline{)h}}$

#### 2.2.2. Gravitational Coupling Constant ${\mathsf{\alpha}}_{G}$

#### 2.2.3. Relative Strength of Gravitational Coupling ${\mathsf{\beta}}_{G}$

#### 2.3. Determining a New Atomic Mass Scale

## 3. Results within the UPS Realm

#### 3.1. Subatomic Masses

- (1)
- The mismatch between ${M}_{\mathrm{A}}$ and $\sqrt{{m}_{\mathrm{e}}{m}_{\mathrm{p}}}$ may be related to Koide’s K-constant$K=2/3$ [15], viz.$${M}_{\mathrm{A}}/\sqrt{{m}_{\mathrm{e}}{m}_{\mathrm{p}}}=0.6850\phantom{\rule{0.166667em}{0ex}},$$
- (2)
- Using the above values of first-generation quark masses and the mass of the strange quark, ${m}_{\mathrm{s}}=93.4$ MeV/${c}^{2}$ [32], we find that$$\sqrt{{m}_{\mathrm{u}}{m}_{\mathrm{s}}}/{M}_{\mathrm{A}}\simeq 0.95\phantom{\rule{0.166667em}{0ex}},$$$$\sqrt{{m}_{\mathrm{d}}{m}_{\mathrm{s}}}/\sqrt{{m}_{\mathrm{e}}{m}_{\mathrm{p}}}\simeq 0.95\phantom{\rule{0.166667em}{0ex}},$$
- (3)
- It certainly appears that there exists a ladder-type mechanism that uses G-Ms (and some scaling coefficients) to relate various particle masses (see also Table A1 in Appendix A below). Some examples (and their corresponding deviations from experiment) are:$${m}_{\mathrm{s}}=\sqrt{{m}_{\mathrm{d}}{m}_{\mathsf{\tau}}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(2.5\%)\phantom{\rule{0.166667em}{0ex}},$$$${m}_{\mathrm{s}}=\sqrt{{m}_{\mathrm{u}}{m}_{\mathrm{b}}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(1.7\%)\phantom{\rule{0.166667em}{0ex}};$$$${m}_{\mathrm{c}}=\sqrt{{m}_{\mathrm{p}}{m}_{\mathsf{\tau}}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(1.7\%)\phantom{\rule{0.166667em}{0ex}};$$$${m}_{\mathrm{c}}=\sqrt{2{m}_{\mathrm{d}}{m}_{\mathrm{t}}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(0.054\%)\phantom{\rule{0.166667em}{0ex}},$$$${m}_{\mathrm{u}}=\sqrt{2{m}_{\mathrm{d}}{m}_{\mathrm{e}}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(1.1\%)\phantom{\rule{0.166667em}{0ex}};$$$${m}_{\mathrm{p}}=\sqrt{2{m}_{\mathsf{\mu}}{m}_{\mathrm{b}}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(0.17\%)\phantom{\rule{0.166667em}{0ex}},$$$${m}_{\mathrm{b}}=\sqrt{{m}_{\mathsf{\mu}}{m}_{\mathrm{t}}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(2.1\%)\phantom{\rule{0.166667em}{0ex}};$$$${M}_{\mathrm{A}}=\sqrt{{m}_{\mathsf{\mu}}{m}_{\mathrm{u}}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(0.71\%)\phantom{\rule{0.166667em}{0ex}},$$$${M}_{\mathrm{A}}=\sqrt{\sqrt{{m}_{\mathrm{e}}{m}_{\mathsf{\mu}}}\phantom{\rule{0.166667em}{0ex}}\sqrt{{m}_{\mathrm{e}}{m}_{\mathsf{\tau}}}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(0.80\%)\phantom{\rule{0.166667em}{0ex}}.$$
- (4)
- The Higgs boson (${m}_{\mathrm{H}}=125.25$ GeV/${c}^{2}$) is certainly special, although unavoidably a part of the mass ladder. This is the only particle that is not involved in simple G-Ms with the low-mass particles. Two of its complex relations are the following:$${m}_{\mathrm{b}}=\sqrt{{m}_{\mathrm{s}}\phantom{\rule{0.166667em}{0ex}}({m}_{\mathrm{H}}/K)}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(0.21\%)\phantom{\rule{0.166667em}{0ex}},$$$$\frac{{m}_{\mathrm{H}}}{{m}_{\mathrm{b}}}=30.0\simeq \frac{{M}_{\mathrm{A}}}{{m}_{\mathrm{e}}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(2.0\%)\phantom{\rule{0.166667em}{0ex}}.$$This relation shows how the Higgs boson manages to assign mass to the much lower-mass bottom quark by using a novel mechanism not related to a G-M or Koide’s scale factor (see below).
- (5)
- The vacuum expectation value (VEV) of the Higgs field is $v=246.22$ GeV/${c}^{2}$ [29]. To within a deviation of 1.8%, we find for the compact 14 triplet H-t-v that$${m}_{\mathrm{t}}=\sqrt{{m}_{\mathrm{H}}\phantom{\rule{0.166667em}{0ex}}v\phantom{\rule{0.166667em}{0ex}}}\phantom{\rule{0.166667em}{0ex}},$$$${m}_{\mathrm{H}}=\sqrt{{m}_{\mathrm{t}}\phantom{\rule{0.166667em}{0ex}}{m}_{{\mathrm{Z}}^{0}}}\phantom{\rule{0.166667em}{0ex}}.$$Obviously, the top quark receives its mass from the Higgs mechanism, and then it participates in the G-Ms that define the masses of the other particles (see Table A1 in Appendix A). The high-mass geometric sequence Z${}^{0}$-H-t-v appears to be very compact indeed (Note 14), and its common ratio is about 1.3815. We note that W${}^{\pm}$ (mass ${m}_{{W}^{\pm}}=80.377$ GeV/${c}^{2}$) is not a member of this sequence since ${m}_{{\mathrm{Z}}^{0}}/{m}_{{\mathrm{W}}^{\pm}}={K}^{-1/4}\simeq 1.11$16. This relation provides another definition of Koide’s K in terms of the decay products of the Higgs boson (deviation 2.5%), viz.$${K}^{1/4}=\frac{{m}_{{\mathrm{W}}^{\pm}}}{{m}_{{\mathrm{Z}}^{0}}}\equiv \phantom{\rule{0.166667em}{0ex}}cos{\mathsf{\theta}}_{\mathrm{w}},$$
- (6)
- On the other hand, the G-M of ${m}_{\mathrm{H}}$ and ${m}_{{\mathrm{W}}^{\pm}}$ is 10% larger than ${m}_{{\mathrm{Z}}^{0}}$, but using empirically Koide’s constant, we find that$${m}_{{\mathrm{Z}}^{0}}=\sqrt{\left({K}^{1/2}\phantom{\rule{0.166667em}{0ex}}{m}_{\mathrm{H}}\right)\phantom{\rule{0.166667em}{0ex}}{m}_{{\mathrm{W}}^{\pm}}}\phantom{\rule{0.166667em}{0ex}},$$$${m}_{{\mathrm{W}}^{\pm}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}K\phantom{\rule{0.0pt}{0ex}}{m}_{\mathrm{H}}\phantom{\rule{0.166667em}{0ex}}.$$This relation helps us understand the important role of the exact constant $K=2/3$ [15]: K is a numerical scale factor that relates some close pairs of particle masses. Here, the Higgs field connects to Z${}^{0}$ by an inverse-mapping G-M17, viz.$${m}_{{\mathrm{Z}}^{0}}=\sqrt{{m}_{\mathrm{H}}^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}3}\phantom{\rule{0.0pt}{0ex}}\left(1/v\right)}\phantom{\rule{0.166667em}{0ex}},$$
- (7)
- Returning now to Equation (35), we see the Higgs mass is scaled by $1/K$ to participate in a G-M with ${m}_{\mathrm{s}}$ and ${m}_{\mathrm{b}}$. Although we have only a limited view of the dynamics of the Higgs mechanism in the above equations, it is apparent that this mechanism uses a set of scaling rules in the various coupling factors that appear in the Lagrangians. The origin of these scaling rules is unknown to us at this moment, but we feel confident that we have made a step in the right direction with this analysis (see Appendix A.3 for calculations of the free parameters of the Standard Model of particle physics).
- (8)
- The next and considerably more difficult step concerns the assignment of mass to the bottom quark, whose mass is much lower than the Higgs mass and the masses of its decay products. We were surprised to find yet another method being used by the Higgs mechanism for this coupling (no G-M can reach down to ${m}_{\mathrm{b}}$ because the barrier set by the Higgs VEV is not too high): the only way that we could find for this coupling was the deflation factor of 1/30, which we discuss below.

#### 3.2. The Planck Charge

#### 3.3. A New Atomic Length Scale

#### 3.4. Cosmological Scales and Some Ambivalent Superatomic Particles

## 4. Discussion

#### 4.1. Pairs of Fundamental Dimensional Units

#### 4.2. The Varied Contributions of the Vacuum

#### 4.3. Geometric-Mean Averaging and Particle-Mass Deflation in Nature

## 5. Lingering Issues, Future Prospects, and a Brief Summary

`K`. This tactic tells us that Schrödinger was not aware that he was including geometry in his constant

`K`. Dirac [7,8,9], on the other hand, believed that $\overline{)h}=\mathtt{K}$ is the true constant (not h), so we can guess that he sensed that the two constants are fundamentally different in their makeup (see Section 1.2 for more details).

## 6. Highlights

#### 6.1. Conclusions

- (1)
- Current systems of units are incomplete and incapable of describing all aspects of this universe. They do not include some of the fundamental dimensional constants, the dimensionless coupling constants, and all the restrictions installed by the vacuum itself on the material world.
- (2)
- Each force of nature must be represented in a system of units with a dimensional and a dimensionless coupling constant. If Planck’s h is dropped, then the system cannot measure quantities related to quantum phenomena. If Newton’s G is dropped, then the system does not include gravity. The vacuum also comes in with any two of its four interdependent constants $\{c,{Z}_{0},{\u03f5}_{0},{\mu}_{0}\}$, and it inserts a stamp of the 3-D geometry of space to the electric charge (the $4\pi $ term in ${e}^{2}\phantom{\rule{-0.166667em}{0ex}}/\phantom{\rule{-0.166667em}{0ex}}{\overline{)\u03f5}}_{0}$), but not to the mass.
- (3a)
- The fine-structure constant ${\mathsf{\alpha}}_{h}={\left(861.022576\right)}^{-1}$, not multiplied by $2\pi $, is the only coupling constant that must be included in absolute terms. It has been measured by experiment, and it provides the scale factor $\sqrt{{\mathsf{\alpha}}_{h}}\simeq 1/30$ used by the Higgs mechanism to deflate and couple to the bottom quark, and then to reach down to all the other lower-mass particles (Table A1 below). Furthermore, the Higgs mass is apparently related to the masses of vector bosons, quarks, and leptons by G-M averaging and Koide’s scale of 2/3 in various incarnations. The above scales should be present in the coupling constants of the various fields (see Appendix A.3).
- (3b)
- All other unitless constants must be included in relative terms because only ratios of coupling constants have physical meaning—such ratios provide relative strengths, just like the ratios of dimensional quantities do too.
- (3c)
- The modern definitions of the unitless coupling constants are incorrect because $\overline{)h}$ was used instead of Planck’s physical constant h. Dirac’s $\overline{)h}$ is a composite constant that also carries planar 2-D geometry and a descriptive unit of [rad]${}^{-1}$; the $2\pi $ term in $\overline{)h}$ has inadvertently reversed the influence of geometry on the coupling constants.
- (4)
- The vacuum is a passive entity impervious to forcing of any kind by the material world. By providing the least (but nonzero) resistance to all motions that occur in its domain, the vacuum installs upper limits on the material world (c and ${Z}_{0}$ in nearly perfect dielectrics), which must then be included in systems of units as well. These two geometry-free constants also bring the composite constants $4\pi {\u03f5}_{0}$ and ${\mu}_{0}/\left(4\pi \right)$ with them, in which the influence of 3-D geometry (the $4\pi $ term) is apparent. (Here, the vacuum’s ${\u03f5}_{0}$ and ${\mu}_{0}$ are both lower limits.) In unit combinations, such as $4\pi {\u03f5}_{0}\overline{)h}$ and ${\mu}_{0}/\left(4\pi \overline{)h}\right)$, geometry inadvertently cancels out, leaving behind unitless numerical imprints in the equations (see the three atomic radii in Section 1.2).
- (5a)
- There exists a new atomic mass scale ${M}_{\mathrm{A}}=15.0$ MeV/${c}^{2}$ that can be determined by deflating the original Planck mass ${M}_{\mathrm{p}}$ by $\sqrt{{\mathsf{\beta}}_{G}}=4.900\times {10}^{-22}$, where ${\mathsf{\beta}}_{G}$ is the relative ratio of the coupling constants of gravity and fine structure. Of course, in our expanding universe, the event took place in reverse (${M}_{\mathrm{A}}/\sqrt{{\mathsf{\beta}}_{G}}\to {M}_{\mathrm{p}}$). This inflation of scale accounted for 21.3 orders of magnitude in mass and explains how the Planck scale is connected to the atomic world. (At the same time, the atomic scale of length was deflated by the same factor to produce the tiny Planck length.)
- (5b)
- No (sub)atomic particle is found to occupy a scale value, and the measured masses in the atomic world are connected mostly by G-M averaging. By using G-M averaging, nature (a) remains impartial to designating any particle as being more significant than any other, and (b) assigns more weight to the smaller participant in the G-M, thereby assisting smaller entities in leaving their marks on the universe.
- (5c)
- We can relate characteristic atomic constants (charge e, mass ${m}_{\mathrm{e}}$, the G-M $\sqrt{{m}_{\mathrm{e}}{m}_{\mathrm{p}}}$, the Compton radius ${r}_{\mathrm{c}}$) to scale values (${q}_{\mathrm{p}},{M}_{\mathrm{p}},{M}_{\mathrm{A}},{L}_{\mathrm{A}}$, respectively), but this is not how these physical entities were created; they were created by the Higgs scalings (1/30 and 2/3) and by G-M averaging of other nearby physical entities.
- (6)
- Leptons, quarks, and bosons get their masses from the Higgs field. The boson-quark-lepton mass ladder is shown in Table A1 below. How the Higgs field acquires its mass ${m}_{\mathrm{H}}$ and its vacuum expectation value v remains a mystery; the only hints in the known masses [32] are that ${m}_{\mathrm{H}}\simeq v/2$ (to within a deviation of 1.7%) and the G-M ${m}_{\mathrm{t}}=\sqrt{{m}_{\mathrm{H}}\phantom{\rule{0.166667em}{0ex}}v\phantom{\rule{0.166667em}{0ex}}}$ (Equation (37), deviation 1.8%). The EM fine-structure constant ${\mathsf{\alpha}}_{h}$ ought to play a prominent role as well: in Appendix A.3, we find that it is likely related to the coupling constant of weak interaction ${\mathsf{\alpha}}_{\mathrm{w}}$, viz. ${\mathsf{\alpha}}_{\mathrm{w}}=\sqrt{{\mathsf{\alpha}}_{h}}\simeq 1/30$.
- (7)
- Koide’s lepton constant $K=2/3$ is one of the scaling constants used by the Higgs field and its decay products in couplings to other particles. We derived it from the first principles in Appendix A, and the same value is also applicable to the heavy quark triplet c-b-t. We also derived two additional Koide-type constants: $J=4/7$ (for the light quark triplet u-d-s) and $B=0.336$ (for the vector bosons W${}^{\pm}$-Z${}^{0}$-H). Constant B is barely 0.8% larger than the absolute minimum value of 1/3 that occurs for three equal masses.
- (8)
- In Appendix B, we pointed out four instances of a universal law that has the general form$$\left(\mathrm{a}\phantom{\rule{3.33333pt}{0ex}}\mathrm{surface}\phantom{\rule{3.33333pt}{0ex}}\mathrm{density}\right)\phantom{\rule{0.166667em}{0ex}}\propto \phantom{\rule{0.166667em}{0ex}}{\left(\mathrm{a}\phantom{\rule{3.33333pt}{0ex}}\mathrm{kinetic}\phantom{\rule{3.33333pt}{0ex}}\mathrm{scalar}\phantom{\rule{3.33333pt}{0ex}}\mathrm{quantity}\right)}^{\phantom{\rule{0.0pt}{0ex}}4}\phantom{\rule{0.166667em}{0ex}},$$

#### 6.2. Critical Questions and Answers

- (Q1)
- How does Planck mass relate to the atomic world?—The atomic mass scale ${M}_{\mathrm{A}}=15$ MeV/${c}^{2}$ inflates precisely to the Planck mass, i.e., ${M}_{\mathrm{A}}/\sqrt{{\mathsf{\beta}}_{G}}\to {M}_{\mathrm{p}}$, where ratio $\sqrt{{\mathsf{\beta}}_{G}}=4.900\times {10}^{-22}$ is a comparative dimensionless quantity (i.e., the ratio of two dimensionless constants).
- (Q2)
- What is the physical meaning of the number 137?Number $137=861/\left(2\pi \right)$, where the $2\pi $ is a geometric term carrying the descriptive unit of radian [11]; so, 137 is a composite constant, and this is the reason that we did not figure out its physical significance in the past 100 years. The actual physical constant is 861, and the scale factor $\sqrt{861}\simeq 30$ is used by the Higgs boson to assign masses dynamically to much lighter particles, starting with the bottom quark and moving on down the mass ladder (Table A1). Thus, the “deflation scale” $\sqrt{{\mathsf{\alpha}}_{h}}=1/30$ (or weak coupling constant ${\mathsf{\alpha}}_{\mathrm{w}}$) should appear in the Higgs couplings of the lower-mass particles.
- (Q3)
- What is the physical meaning of Koide’s constant?Koide’s $K=2/3$ is another scale factor used in the Higgs couplings to assign masses to lighter vector bosons (the particles W${}^{\pm}\phantom{\rule{-0.166667em}{0ex}}$ and Z${}^{0}$). Koide’s formula holds exactly for the leptons e-$\mathsf{\mu}$-$\mathsf{\tau}$ and for the heavy quarks c-b-t (corresponding proofs are given in Appendix A.2).
- (Q4)
- How does the top quark get its mass?The top quark mass is the geometric mean of the Higgs mass and the Higgs vacuum expectation value $v=246.22$ GeV/${c}^{2}$, so that ${m}_{\mathrm{t}}=\sqrt{{m}_{\mathrm{H}}v}$ to within a deviation of 1.8% from the experimentally measured ${m}_{\mathrm{t}}$ value [32].
- (Q5)
- How do Higgs vector bosons get their masses?By two different mechanisms (couplings): In the ordered compact high-mass triplet, Z${}^{0}$-H-t, the Higgs mass is the geometric mean of the Z${}^{0}$ mass and the top quark mass, viz. ${m}_{{\mathrm{Z}}^{0}}={m}_{\mathrm{H}}^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}2}/{m}_{\mathrm{t}}$. In contrast, the W${}^{\pm}$ coupling involves Koide’s scale since ${\mathrm{W}}^{\pm}=K{m}_{\mathrm{H}}$, where $K=2/3$.
- (Q6)
- How does the bottom quark get its mass?By a third coupling mechanism: The Higgs mass is scaled down by the weak coupling constant ${\mathsf{\alpha}}_{\mathrm{w}}=1/30$, so that ${m}_{\mathrm{b}}={m}_{\mathrm{H}}/30$. We have tried empirically several other scalings and G-Ms, but none of these patterns approached the experimental value of ${m}_{\mathrm{b}}$, which is much lower than ${m}_{\mathrm{H}}$ (the rest-energy gap is 121 GeV; Table A1). The assignment of mass to the bottom quark is becoming a major issue to resolve in the future by any theory that purports to describe mass assignments to lower-mass quarks.
- (Q7)
- Is Dirac’s $\overline{)h}$, rather than Planck’s h, the true universal constant?They both are, but h is a pure physical constant, whereas $\overline{)h}=h/\left(2\pi \right)$ is composite and includes also the 2-D geometric term $2\pi $ and the descriptive unit of [rad]. Due to this geometric content, a miscue was committed in the post-Planckian era when $\overline{)h}$ was adopted for the modern definitions of the fine-structure constant ${\mathsf{\alpha}}_{\overline{)h}}$ and the gravitational coupling constant ${\mathsf{\alpha}}_{G}$: the 3-D geometry of the electric field in ${\mathsf{\alpha}}_{\overline{)h}}$ was eliminated, and ${\mathsf{\alpha}}_{G}$ acquired 2-D geometry against its generic 3-D nature.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EM | Electromagnetic |

G-M | Geometric-Mean |

MOND | Modified Newtonian Dynamics |

UPS | An Upgraded Planck System Based on Electron Mass |

UPS${}^{\prime}$ | An Upgraded Planck System Based on Proton Mass |

VEV | Vacuum Expectation Value |

## Appendix A. Physical Meaning of Koide’s Lepton Constant and Similar Constants

#### Appendix A.1. Physical Interpretation

**Table A1.**Boson-Quark mass ladder in terms of the Higgs mass ${m}_{\mathrm{H}}=125.25$ GeV/${c}^{2}$ [32]. Two scales are used, Koide’s $K=2/3$ and the ${\mathsf{\alpha}}_{\mathrm{w}}=\sqrt{{\mathsf{\alpha}}_{h}}=1/30$ deflation of ${m}_{\mathrm{H}}$ down to the bottom quark mass ${m}_{\mathrm{b}}$. Lepton masses and proton mass are also shown in terms of ${m}_{\mathrm{H}}$ for a comparison of scales.

Particle | Mass-Energy | Mass Relation ${}^{\left(\mathit{a}\right)}$ | Deviation ${}^{\left(\mathit{b}\right)}$ |
---|---|---|---|

(MeV) | (%) | ||

Vector Bosons | |||

Z${}^{0}$ | $9.1188\times {10}^{4}$ | ${m}_{{\mathrm{Z}}^{0}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{K}^{3/4}\phantom{\rule{0.0pt}{0ex}}{m}_{\mathrm{H}}$ | +1.3 |

W${}^{\pm}$ | $8.0377\times {10}^{4}$ | ${m}_{{\mathrm{W}}^{\pm}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}K\phantom{\rule{0.0pt}{0ex}}{m}_{\mathrm{H}}$ | +3.9 |

Quarks | |||

top | $1.725\times {10}^{5}$ | ${m}_{\mathrm{t}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{K}^{-3/4}\phantom{\rule{0.0pt}{0ex}}{m}_{\mathrm{H}}$ | $-1.6$ |

bottom | $4.180\times {10}^{3}$ | ${m}_{\mathrm{b}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{m}_{\mathrm{H}}/30\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\mathsf{\alpha}}_{\mathrm{w}}\phantom{\rule{0.0pt}{0ex}}{m}_{\mathrm{H}}$ | $-0.12$ |

charm | $1.270\times {10}^{3}$ | ${m}_{\mathrm{c}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}2\sqrt{K/{30}^{3}\phantom{\rule{0.0pt}{0ex}}}\phantom{\rule{0.0pt}{0ex}}{m}_{\mathrm{H}}$ | $-2.0$ |

strange | 93.4 | ${m}_{\mathrm{s}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\left(K/{30}^{2}\right)\phantom{\rule{0.0pt}{0ex}}{m}_{\mathrm{H}}$ | $-0.67$ |

down | 4.67 | ${m}_{\mathrm{d}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}2\left(\phantom{\rule{-0.166667em}{0ex}}{K}^{7/4}/{30}^{3}\phantom{\rule{-0.166667em}{0ex}}\right)\phantom{\rule{0.0pt}{0ex}}{m}_{\mathrm{H}}$ | $-2.3$ |

up | 2.16 | ${m}_{\mathrm{u}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\left({K}^{2}/{30}^{3}\right)\phantom{\rule{0.0pt}{0ex}}{m}_{\mathrm{H}}$ | $-4.5$ |

Leptons & Proton | |||

electron | 0.511 | ${m}_{\mathrm{e}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\sqrt{\frac{K}{2\times {30}^{7}}}\phantom{\rule{0.0pt}{0ex}}{m}_{\mathrm{H}}$ | $-4.3$ |

muon | 105.66 | ${m}_{\mathsf{\mu}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{22}^{2}\frac{{K}^{5/2}}{\sqrt{2\times {30}^{7}}}\phantom{\rule{0.0pt}{0ex}}{m}_{\mathrm{H}}$ | $-0.45$ |

tauon | $1.777\times {10}^{3}$ | ${m}_{\mathsf{\tau}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{40}^{2}\frac{{K}^{-3/2}}{\sqrt{2\times {30}^{7}}}\phantom{\rule{0.0pt}{0ex}}{m}_{\mathrm{H}}$ | $-0.93$ |

& | |||

proton | 938.272 | ${m}_{\mathrm{p}}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}22{\left(\phantom{\rule{-0.166667em}{0ex}}\frac{2{K}^{5}}{{30}^{9}}\phantom{\rule{-0.166667em}{0ex}}\right)}^{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}1/4}\phantom{\rule{0.0pt}{0ex}}{m}_{\mathrm{H}}$ | $-0.12$ |

#### Appendix A.2. Additional Koide-Type Constants

#### Appendix A.3. Determination of Various Coupling Factors and Constants of the Standard Model

#### Appendix A.3.1. The Weinberg Angle

#### Appendix A.3.2. Vector-Boson g-Factors and Electric Charge

**Table A2.**Free parameters of the Standard Model: Comparison of empirical values with the corresponding experimental measurements. Details and references are provided in Appendix A.3.

Parameter | Equation | Determined Value | Measured Value |
---|---|---|---|

Weingberg angle (${\theta}_{\mathrm{w}}\phantom{\rule{-0.166667em}{0ex}})$ [degrees] | (A15) | 25.4 | 28.2 |

Higgs mass to VEV ratio (${m}_{\mathrm{H}}\phantom{\rule{-0.166667em}{0ex}}/v$) | (A16) | 0.54 | 0.51 |

Weak isospin coupling (g) | (A19) | 0.726 | 0.653 |

Weak hypercharge coupling (${g}^{\prime}$) | (A20) | 0.344 | 0.350 |

Electric charge (e) [natural units] | (A23) | 0.311 | 0.308 |

Weak coupling constant (${\mathsf{\alpha}}_{\mathrm{w}}$) | (42), (A24) | $0.0341$ | $0.0339$ |

Electron Yukawa coupling (${y}_{\mathrm{e}}$) | (A27) | $3.005\times {10}^{-6}$ | $2.935\times {10}^{-6}$ |

t-quark Yukawa coupling (${y}_{\mathrm{t}}$) | (A28) | 1.043 | 0.991 |

u-quark coupling factor (${g}_{\mathrm{u}}$) | (A31) | $1.267\times {10}^{-5}$ | $1.241\times {10}^{-5}$ |

d-quark coupling factor (${g}_{\mathrm{d}}$) | (A32) | $2.805\times {10}^{-5}$ | $2.682\times {10}^{-5}$ |

b-quark coupling factor (${g}_{\mathrm{b}}$) | (A34) | $2.566\times {10}^{-2}$ | $2.401\times {10}^{-2}$ |

#### Appendix A.3.3. Electron and Top-Quark Yukawa Coupling Factors

#### Appendix A.3.4. Quark g-Factors

## Appendix B. A Universal Natural Law Discovered in Widely Separated Scales

- (1)
- In quantum gravity, the energy-density shift of the Higgs field ${U}_{\mathrm{H}}$ resulting from spontaneous symmetry breaking (that prevents ultraviolet divergence) is ${U}_{\mathrm{H}}\propto {v}^{4}$, where v is the Higgs vacuum expectation value [20,55]. This relation is equivalent to$${\sigma}_{F}\phantom{\rule{0.166667em}{0ex}}\propto \phantom{\rule{0.166667em}{0ex}}{v}^{4}\phantom{\rule{0.166667em}{0ex}},$$
- (2)
- In the macroscopic realization of the Casimir effect, the same force per unit area is proportional to the fourth power of the reciprocal of distance D between parallel plates [56,57], viz.$${\sigma}_{F}\phantom{\rule{0.166667em}{0ex}}\propto \phantom{\rule{0.166667em}{0ex}}{(1/D)}^{4}\phantom{\rule{0.166667em}{0ex}}.$$The units agree in the last two relations since the VEV v has dimensions of [length]${}^{-1}$ in natural units (see Note 30 and Ref. [20]).
- (3)
- In atomic physics, the celebrated Stefan-Boltzmann law [44,45] takes the equivalent form$${\sigma}_{P}\phantom{\rule{0.166667em}{0ex}}\propto \phantom{\rule{0.166667em}{0ex}}{\Theta}^{4}\phantom{\rule{0.166667em}{0ex}},$$
- (4)
- In astrophysics, galaxies obey the relation $M\propto {V}^{4}$, where M is mass and V is rotational speed or stellar velocity dispersion in spiral [17,58,59,60] and elliptical [18,61,62] galaxies, respectively. This relation is equivalent to$${\sigma}_{I}\phantom{\rule{0.166667em}{0ex}}\propto \phantom{\rule{0.166667em}{0ex}}{V}^{4}\phantom{\rule{0.166667em}{0ex}},$$

#### Appendix B.1. Dimensional Analysis of Surface Densities

#### Appendix B.2. Physical Properties of the Various Surface Densities

- (a)
- (b)
- Density ${\sigma}_{I}$ (i.e., mass) is not regulated by inertia; mass already possesses inertia; instead, we can say that mass is force squared ${F}^{2}$ regulated by the rate of change of power $(P/T)$ or impulse squared ${\mathcal{J}}^{2}$ regulated by energy E (Equation (A46)), where E could also be viewed as the rate of change of the action integral, i.e., $(\mathcal{S}/T)$.
- (c)
- (d)
- The vacuum remains present in the ${\sigma}_{F}$ of the EM field, but it drops out from the ${\sigma}_{F}$ of the gravitational field (both distinct behaviors are shown in Equation (A43)).
- (e)
- Force surface density ${\sigma}_{F}$ (Equation (A43)) represents the conventional energy density of the force fields, whereas ${\sigma}_{P}$ (Equation (A39)) shows that vacuum inertial resistance ($I/T$) is present during the action of all forces; and this inertial resistance appears also in Equations (A41) and (A42) (${Z}_{0}^{\phantom{\rule{0.166667em}{0ex}}-1}$ and c, respectively).
- (f)
- Both sides of Equation (A40) have dimensions of Planck’s constant [h], thus $I/T\sim {q}^{2}{Z}_{0}=\left[\mathrm{action}\right]$. Higher powers of T in $I/{T}^{n}$ ($n=2,3$), i.e., higher-order derivatives, are also physically important: $I/{T}^{2}=\left[\mathrm{energy}\right]$ and $I/{T}^{3}=\left[\mathrm{power}\right]$. Equation (A45) for $I/{T}^{3}$ then implies that power stems from the third time derivative of the moment of inertia, a property that is fundamental to the emission of gravitational waves. The same relation, applied to EM waves, produces ohmic power (divide Equation (A40) by ${T}^{2}$) with dimensions of [electric current]${}^{2}\phantom{\rule{0.0pt}{0ex}}$[ohmic resistance].
- (g)
- Equations (A35)–(A38) all have the characteristic form$$[\mathrm{surface}\phantom{\rule{3.33333pt}{0ex}}\mathrm{density}\phantom{\rule{3.33333pt}{0ex}}X\phantom{\rule{-0.166667em}{0ex}}/\phantom{\rule{-0.166667em}{0ex}}A]\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\left[\mathrm{constant}\phantom{\rule{3.33333pt}{0ex}}C\right]\times {\left[\mathrm{kinetic}\phantom{\rule{3.33333pt}{0ex}}\mathrm{scalar}\phantom{\rule{3.33333pt}{0ex}}\mathrm{quantity}\phantom{\rule{3.33333pt}{0ex}}Y\right]}^{\phantom{\rule{0.0pt}{0ex}}4}\phantom{\rule{0.166667em}{0ex}},$$

## Notes

1 | Similarly, ${\overline{)\mu}}_{0}\equiv {\mu}_{0}/\left(4\pi \right)$ is the reduced vacuum permeability, and then, ${\overline{)\u03f5}}_{0}{\overline{)\mu}}_{0}=1/{c}^{2}$. The stereometric $4\pi $ terms cancel out nicely to produce the “definition” of c, which is a purely physical quantity. Furthermore, Dirac’s $2\pi $ term in $\overline{)h}$ tells us that Planck’s free photons only “see” two dimensions, no matter how they move in stereometric (3-D) space (in curves, or circles, or ellipses, etc.). We also learn that the fundamental natural constant c is produced by the vacuum itself, and it is the geometric mean of two smooth inverse Lie mappings [13] of ${\overline{)\u03f5}}_{0}$ and ${\overline{)\mu}}_{0}$ (i.e., the geometric mean of $1\phantom{\rule{-0.166667em}{0ex}}/\phantom{\rule{-0.166667em}{0ex}}{\overline{)\u03f5}}_{0}$ and $1\phantom{\rule{-0.166667em}{0ex}}/\phantom{\rule{-0.166667em}{0ex}}{\overline{)\mu}}_{0}$). |

2 | In Bohr’s model, the (nongeometric) number-parts of energies ${E}_{n}$ and radii ${r}_{n}$ are related by ${E}_{n}\propto 1/{r}_{n}=1/{n}^{2}$. Therefore, for pure numbers, we see that $\sqrt{{r}_{n}}=n$, and the coefficients of the quantized radii are essentially produced by geometric averaging, viz., ${r}_{n+1}=\sqrt{{r}_{n}{r}_{n+2}}+1$. The +1 extends the sequence back to $n=0,\phantom{\rule{0.166667em}{0ex}}{r}_{0}=0$. |

3 | Unless the calculations can be repeated successfully within another system of units in which another particle is chosen as a building block (see Note 10 below for the case at hand). |

4 | Going as far back as 1948, some fundamental equations signaled that the introduction of $2\pi $ in ${\mathsf{\alpha}}_{\overline{)h}}$ may not be appropriate, but the warning was brushed off as a mere simplification between pure numbers. The electron vertex functions and the corrections to the form factors (Ref. [20], pp. 194–196) all show the coefficient ${\mathsf{\alpha}}_{\overline{)h}}/\left(2\pi \right)$, in which the visible $\left(2\pi \right)$ term eliminates the 2-D geometry from ${\mathsf{\alpha}}_{\overline{)h}}$, and the correction to the g-factor of the electron becomes ${a}_{\mathrm{e}}\equiv (g-2)/g={\mathsf{\alpha}}_{h}=1/861.0$ (equation (6.59) in Ref. [20]). This result was first obtained by Schwinger in 1948 [21], and it was confirmed by experiments to 8 significant figures, the most accurate ones using an ingenious technique developed by Van Dyck et al. in 1987 [22]. Therefore, unknowingly, these experiments were measuring ${\mathsf{\alpha}}_{h}$ by mesuring the correction ${a}_{\mathrm{e}}$ to the g-factor of the electron. |

5 | The unit of [area] justifies the deduced geometric factor $1/\left(2\pi \right)$ in the second G-M of Equation (10). The additional factor of 2 attached to ${\u03f5}_{0}$ is a unitless imprint, but it has a geometric origin. This type of imprinting is difficult to track down in the various equations of physics when they are presented in reduced, simplified form (see also the discussion in Section 1.2 about the numerical factor of 1/4 imprinted by geometry to the Rydberg energy). |

6 | The reference unitless constant (${\mathsf{\alpha}}_{h}$ here) plays the exact same role that the standard 1-meter ruler and the standard 1-kg cylinder play in the SI system of units for length and mass, respectively. |

7 | Had we used the mass and the charge of a supermassive black hole (e.g., [30]) in Equation (15), we would have obtained a relative strength of couplings ${\mathsf{\beta}}_{G}\gg 1$ and a different system of units, which would be hard to relate to the Planck scale and even harder to use in the atomic world. |

8 | In fact, ${\mathsf{\beta}}_{G}=1$ near the Planck scale, for a particle of mass ${m}_{\u2605}={M}_{\mathrm{p}}/30=1.0\times {10}^{18}$ GeV/${c}^{2}$, where ${M}_{\mathrm{p}}=\sqrt{hc/G}$ is the original Planck mass. The deflation factor of 1/30 is also used by the Higgs boson to couple to the bottom quark (see Section 3.1 below) |

9 | We point out again that using $\overline{)h}$ in definitions (13) and (14) reverses what nature intended. It makes ${\mathsf{\alpha}}_{\overline{)h}}$ be a geometry-free value, although the geometry should have been that of ${\overline{)\u03f5}}_{0}$ coming from the electric field; and ${\mathsf{\alpha}}_{G}$ ends up with $2\pi $ radians, although it should have been geometry-free. This setup reveals that mass elements know they exist in a 3-D space, but elementary charges do not know, and they are taught accordingly by the vacuum’s insertion of ${\overline{)\u03f5}}_{0}\propto 4\pi $ and/or ${\overline{)\mu}}_{0}\propto 1\phantom{\rule{-0.166667em}{0ex}}/\phantom{\rule{-0.166667em}{0ex}}4\pi $ into the equations of the EM field. |

10 | An alternative choice, such as of two interacting protons with masses ${m}_{\mathrm{p}}$, leads to another complete system of units, say UPS${}^{\prime}$. In this case, we find that ${M}_{\mathrm{A}}^{\prime}=27.5$ GeV/${c}^{2}$ and ${\left({\mathsf{\beta}}_{G}^{\prime}\right)}^{1/2}=9.00\times {10}^{-19}$, but Equation (18) is still valid, and connects ${M}_{\mathrm{A}}^{\prime}$ with the original Planck mass ${M}_{\mathrm{p}}=\sqrt{hc/G}$. Furthermore, the scaling ${M}_{\mathrm{A}}^{\prime}/{M}_{\mathrm{A}}={m}_{\mathrm{p}}/{m}_{\mathrm{e}}$ holds precisely between the two systems of units; and the relation ${M}_{\mathrm{A}}{M}_{\mathrm{A}}^{\prime}=\left({m}_{\mathrm{e}}{m}_{\mathrm{p}}\right)/{\mathsf{\alpha}}_{h}$ is exact as well. Finally, referring to the upcoming UPS results in Section 3.1 below, the relation ${M}_{\mathrm{A}}^{\prime}=\sqrt{{m}_{\mathrm{t}}{m}_{\mathrm{b}}}$ holds to within 2.5% in the UPS${}^{\prime}$, where ${m}_{\mathrm{t}}$ and ${m}_{\mathrm{b}}$ are the masses of the top and bottom quarks, respectively; thus, ${M}_{\mathrm{A}}^{\prime}$ actively participates in the mass ladder of the UPS${}^{\prime}$, just as ${M}_{\mathrm{A}}$ does in the UPS mass ladder of Section 3.1 and Appendix A. |

11 | We knew that a rescaling of the Planck mass ${M}_{\mathrm{p}}=\sqrt{hc/G}$ by some power of ${\mathsf{\beta}}_{G}$ would produce an atomic mass. However, we did not know which power is appropriate to use. Here, we have shown that the appropriate coefficient of ${M}_{\mathrm{p}}$ is $\sqrt{{\mathsf{\beta}}_{G}}$, the G-M of ${E}_{\mathrm{grav}}$ and ${\left({E}_{\mathrm{elec}}\right)}^{-1}$ when scaling ${M}_{\mathrm{p}}$ down to lower masses. If we are scaling up, then the $-1$ exponent naturally moves on top of ${E}_{\mathrm{grav}}$ in the G-M (see Section 3.4 below). In retrospect, these two Lie-type G-M averages make sense in a “fair” universe that uses such impartial averaging to combine pairs of interacting physical quantities and units. |

12 | The realization that the vacuum also leaves unitless numerical imprints (in addition to its dimensional constants ${\overline{)\u03f5}}_{0}$, ${\overline{)\mu}}_{0}$, c, ${Z}_{0}$) is new, unexpected, and it may prove important in future work. In the future, we will have to investigate such imprints of the vacuum to the nuclear world, especially in the strong interactions and the so-called beta functions [20]. |

13 | The charm and bottom quarks have masses of ${m}_{\mathrm{c}}=1270$ MeV/${c}^{2}$ and ${m}_{\mathrm{b}}=4180$ MeV/${c}^{2}$, respectively [32]. At such high masses, something must be changing in the dynamics: for the ordered by mass triplet s-c-b, we find, to within a 1.6% accuracy, that ${m}_{\mathrm{c}}=2\sqrt{{m}_{\mathrm{s}}{m}_{\mathrm{b}}}$. We also find that the charm quark participates rather “reluctantly” in just one pure/unscaled G-M (Equation (28), referring to the compact triplet p-c-$\mathsf{\tau}$); and even that one is unusual, as it involves the proton mass ${m}_{\mathrm{p}}$. |

14 | No other available particle slots in the domain. |

15 | It will become apparent in Appendix A that the ratio 1.38 approximates ${C}_{F}=4/3$ (to within a deviation of 3.5%), where ${C}_{F}$ is the quadratic Casimir charge of the SU(3) fundamental representation of the quark potential (equation (4.45) in Ref. [33]). |

16 | |

17 | |

18 | We cannot help but wonder—if A. Sommerfeld, W. Pauli, C. Jung, R. Feynman, and many others [34] became familiar with this result, would they show the same fascination for number 861 as with 137? The particle-to-scale mass and charge ratios discussed above strongly indicate that we should turn our attention to the physics behind 861 rather than trying to find the same physics in the geometry-dependent composite ratios $137=861/\left(2\pi \right)$ and $\overline{)h}=h/\left(2\pi \right)$. |

19 | See Ref. [35] and article https://en.wikipedia.org/wiki/Planck_units in Wikipedia (accessed on 20 May 2023). |

20 | The remaining choice, the G-M of ${r}_{\mathrm{b}}$ and ${r}_{\mathrm{c}}$, would give an equivalent result, scaled by a different power of ${\mathsf{\alpha}}_{\overline{)h}}$ (${L}_{\mathrm{A}}^{\prime}={L}_{\mathrm{A}}/{\mathsf{\alpha}}_{\overline{)h}}$), such that the ${r}_{\mathrm{c}}=\sqrt{{L}_{\mathrm{A}}{L}_{\mathrm{A}}^{\prime}}$. |

21 | For comparison, the atomic rest-energies [42] of naturally occurring primordial uranium U${}_{92}^{238}$ and synthetic fermium Fm${}_{100}^{257}$ are 90% and 97% of the Higgs VEV, respectively. |

22 | Besides combining with G to produce the units of force and power in the cosmological and Planck systems, c does something else that is notable: it combines with ${\u03f5}_{0}$ or ${\mu}_{0}$ to produce a surprise unit for ohmic resistance: ${\mu}_{0}c=1/\left({\u03f5}_{0}c\right)={Z}_{0}=h/{e}^{2}$ (see Section 2.1.1). |

23 | Mass M is a special case for $n=0$ in which $M={F}^{2}/(I/{T}^{4})$ (see also Appendix B). This relation sketches the complex G-M interaction between mass and inertial change that regulates the kinematics of an object: $dp/dt=\sqrt{M\phantom{\rule{0.0pt}{0ex}}({d}^{4}I/d{t}^{4})}$, where p is momentum, or equivalently, ${d}^{4}I/d{t}^{4}=M{a}^{2}$. |

24 | In a nearly perfect dielectric, the wave impedance is $Z={Z}_{0}/(1+{\chi}_{\mathrm{e}})<{Z}_{0}$, where ${\chi}_{\mathrm{e}}>0$ is the electric susceptibility. |

25 | Note that even actual planetary orbits [46] and also theoretical orbits in the virtual Hooke potential [47] show G-M averaging in many of their properties [48,49]. The two types of elliptical orbits have fundamentally different centers and forces, but this is not enough to suppress or modify the ubiquitous geometric averaging that is so obvious in the parameters of the two sets of ellipses. |

26 | Arithmetic averaging would favor the large constant, whereas harmonic averaging would turn the tables and clearly favor the small constant. Compared to G-Ms, either one of these extreme averages treats “unfairly” one or the other participant. |

27 | |

28 | We also timidly attempted a preliminary calculation of the scaling between weak and strong interactions, as a ratio of energies ${\mathsf{\beta}}_{W}$ (Equation (53). It seems that such energy ratios/comparisons are the way to incorporate consistently the dimensionless constants into the UPS. We can then imagine a complete $\mathrm{UPS}:=\{c,{Z}_{0},G,h,{\mathsf{\alpha}}_{h},{\mathsf{\beta}}_{G},{\mathsf{\beta}}_{W},\phantom{\rule{0.166667em}{0ex}}\dots \phantom{\rule{0.166667em}{0ex}};{\overline{)\u03f5}}_{0},{\overline{)\mu}}_{0},\overline{)h}\}$ that includes geometry-free units and a set of geometry-dependent units, along with relative $\mathsf{\beta}$-ratios $\ll 1$ of unitless constants. |

29 | From (35) and (36), we get ${m}_{\mathrm{b}}=30{m}_{\mathrm{s}}/K$ (#1). From (32), (36), (38)–(40), we get ${m}_{\mathrm{b}}=30{K}^{-3/4}{m}_{\mathsf{\mu}}$ (#2). From (#1) and (#2), we get ${m}_{\mathrm{s}}={K}^{1/4}{m}_{\mathsf{\mu}}$ (#3). From (29), (#3), and (#2), we get ${m}_{\mathsf{\mu}}=30{K}^{-5/4}{m}_{\mathrm{u}}$ (#4). From (26), (#3), (#4), and (30), we get Equation (A4). Finally, from (34), (36), and (A4), we get Equation (A3). Equations (28), (31), (33) and (37) were not used. |

30 | In natural units, $\overline{)h}=c={\u03f5}_{0}=1$. By suppressing the 2-D geometry present in $\overline{)h}$ and by setting ${\u03f5}_{0}=1$ (not $4\pi {\u03f5}_{0}=1$, as in Gaussian units [36]), the 3-D geometry (the $\sqrt{4\pi}$ dividing e) imprinted on to the electric charge by the vacuum remains present in definition (A21). |

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Christodoulou, D.M.; Kazanas, D.
The Upgraded Planck System of Units That Reaches from the Known Planck Scale All the Way Down to Subatomic Scales. *Astronomy* **2023**, *2*, 235-268.
https://doi.org/10.3390/astronomy2040017

**AMA Style**

Christodoulou DM, Kazanas D.
The Upgraded Planck System of Units That Reaches from the Known Planck Scale All the Way Down to Subatomic Scales. *Astronomy*. 2023; 2(4):235-268.
https://doi.org/10.3390/astronomy2040017

**Chicago/Turabian Style**

Christodoulou, Dimitris M., and Demosthenes Kazanas.
2023. "The Upgraded Planck System of Units That Reaches from the Known Planck Scale All the Way Down to Subatomic Scales" *Astronomy* 2, no. 4: 235-268.
https://doi.org/10.3390/astronomy2040017