Tomofumi Miyashita
Miyashita Clinic, Osaka, Japan.
DOI: 10.4236/jmp.2026.1710059   PDF    HTML   XML   2 Downloads   35 Views   Citations

Abstract

This report extends our previous studies of empirical relations involving the cosmic microwave background temperature and the gravitational constant. In the preceding work, the adopted readout procedure gave a CMB temperature of 2.7256307 K, in agreement with the observed value of 2.72548 ± 0.00057 K. The same procedure also gave a gravitational value of 6.674184 × 10⁻¹¹ m³∙kg⁻¹∙s⁻², close to the reported value of 6.6743 × 10⁻¹¹ m³∙kg⁻¹∙s⁻². These two results were obtained within a common structural framework in which the minimum mass, denoted by M_min, connects microscopic mass-energy quantities with macroscopic readouts. The present study focuses on the structural meaning of this minimum-mass scale. In particular, we establish a readout-level degree correspondence in which the electromagnetic numerator is first degree in the minimum-mass readout, the gravitational rate representation is second degree, and their quotient leaves one residual minimum-mass readout degree in the dimensionless proton-side gravitational-to-electromagnetic force ratio. The degree terminology used here is a same-shell, readout-covariant structural classification; it is not an assertion that $M_{\min }^2$ is an intrinsic quantity independent of the adopted unit/readout system. The purpose is not to derive the physical origin of gravity or of electromagnetism, but to clarify the algebraic and readout structure associated with the minimum mass. The analysis also revisits Coulomb’s law and Newton’s law of universal gravitation from the viewpoint of their common inverse-square dependence on distance. Both laws contain the same spatial factor associated with the square of the separation, while their electromagnetic and gravitational numerators remain distinct. This common spatial structure is treated separately from the minimum-mass degree correspondence. The resulting correspondence shows that the difference between the second-degree gravitational structure and the first-degree electromagnetic structure is directly reflected in the first-degree minimum-mass readout remaining in the force ratio, while the deeper physical origin of the degree structure remains open.

Keywords

Cosmic Microwave Background Temperature, Gravitational Constant, Minimum Mass, Minimum-Mass Degree, Coulomb’s Law, Newton’s Law of Universal Gravitation, Inverse-Square Law, Readout Structure

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1. Introduction

The symbols used in this paper are summarized in Section 2. In previous studies, we proposed a set of empirical equations relating the cosmic microwave background (CMB) temperature to several physical quantities and evaluated them numerically [1]-[5]. In particular, Equations (1) - (3) summarize characteristic CMB-temperature relations that provide the empirical starting point for the present analysis.

$\frac{G{m}_p^2}{hc}=\frac{4.5}{2}\cdot \frac{k{T}_c}{1\text{kg}\cdot c^2}\approx 9.4\times {10}^{-40}$ (1)

$\frac{G{m}_p^2}{\frac{e^2}{4\pi {\epsilon }_0}}=\frac{4.5}{2\pi }\cdot \frac{m_e}{e}\cdot hc\approx 8.09\times {10}^{-37}$ (2)

$\left(\frac{m_e{c}^2}{e}\right)\cdot \left[\frac{e^2}{4\pi {\epsilon }_0}\right]=\pi \cdot k{T}_c\approx 1.18\times {10}^{-22}$ (3)

We also introduced an empirical expression for the fine-structure constant [6] and examined its consistency with related quantities [7] [8].

$\frac{1}{\alpha }=137.0359991=136.0113077+\frac{1}{3\cdot 13.5}+1$ (4)

$13.5\cdot 136.0113077=1836.152654=\frac{m_p}{m_e}$ (5)

For clarity, the two readout redefinitions used below are introduced here before their detailed formulation in Section 2. The first redefinition, denoted F₁, is the Ω-side normalization that restores the two electromagnetic coefficient branches to the ideal coefficients 9/2 and π. The second redefinition, denoted F₂, is the subsequent table-number/readout transformation based on the A-character; in the inherited transfer rule, $J_{\text{tab}}^{A\left(X\right)}$ is used as the corresponding numerical factor. The notation F₁F₂ denotes the readout stage obtained after sequential application of F₁ and F₂. Their detailed definitions and limitations are given in Sections 2.2 and 2.5. Small deviations from the ideal values of 9/2 and π, respectively, were previously considered as a means of reducing the resulting numerical errors [9]-[14]. In the previous formulation, the coefficient associated with the dimensional factor m²/s was denoted by kL₀ [15]-[18]. In the present report, kL₀ is no longer used as an independent quantity. Instead, s_gen is explained in Equation (10).

Before the first redefinition, the two electromagnetic coefficient branches in the modified notation are written using the pre-correction values as follows.

$\frac{3.132794486}{c}\left(\text{Wb}\right)=\frac{s_{\text{gen}}{m}_e{c}^2}{e}=1.044987758\times {10}^{-8}\left(\text{Wb}\right)$ (6)

$4.487397553\cdot c\left({\text{C}}^{-1}\right)=\frac{q_m}{s_{\text{gen}}{m}_p{c}^2}=1.345287943\times {10}^9\left({\text{C}}^{-1}\right)$ (7)

Applying the first redefinition to the von Klitzing constant returns the corresponding coefficients to the ideal values of 9/2 and π [15]-[18]. In the present notation, the unit-bearing coefficients are written separately from the speed of light. Equations (8) and (9) therefore use π_unit/c and 4.5_unit ∙ c, respectively; the coefficient labels and c are not combined into a single symbol. Equations (8) - (10) are relations in the F1 readout representation.

$\frac{{\pi }_{\text{unit}}}{c}\left(\frac{\text{V}\cdot \text{m}}{\text{m}/\text{s}}\right)=\frac{s_{\text{gen}}{m}_e{c}^2}{e}=\pi /\text{c}=\text{1}\text{.047922511}\times {10}^{-8}\left(\text{Wb}\right)$ (8)

${4.5}_{\text{unit}}\cdot c\left(\frac{\text{m}/\text{s}}{\text{A}\cdot \text{m}}\right)=\frac{q_m}{s_{\text{gen}}{m}_p{c}^2}=\text{4}\text{.5c}=\text{1}\text{.349066061}\times {10}^9\left({\text{C}}^{-1}\right)$ (9)

Using the two branch relations, the generative time (s_gen) is obtained as follows.

$s_{\text{gen}}=\sqrt{\frac{\frac{{\pi }_{\text{unit}}}{c}}{{4.5}_{\text{unit}}\cdot c}\cdot \frac{h}{\left(m_p{c}^2\right)\left(m_e{c}^2\right)}}=2.044990031\times {10}^{-14}\left(\text{s}\right)$ (10)

Equation (10) replaces the former kL₀-based representation. In the present formulation, this generative time is tied directly to the minimum-mass action relation,

$M_{\text{min}}{c}^2{s}_{\text{gen}}=h=6.62607015\times {10}^{-34}\left(\text{J}\cdot \text{s}\right)$ (11)

and the minimum-mass energy is related to the CMB thermal scale by

$M_{\min }{c}^2=\frac{2\pi }{\alpha }k{T}_c=3.24014790\times {10}^{-20}\left(\text{J}\right)$ (12)

In these previous studies, the minimum mass (M_min) mainly connected the CMB-temperature scale, microscopic mass-energy quantities, electromagnetic readouts, and the gravitational-side calculation. The present study instead examines the power, or degree, with which the minimum-mass readout appears. The numerical CMB-temperature and gravitational values themselves remain those of the preceding report [18]. What is new is the degree analysis: the electromagnetic numerator is first degree in the minimum-mass readout, the gravitational rate representation is second degree, and their quotient leaves one residual minimum-mass readout degree in the dimensionless proton-side gravitational-to-electromagnetic force ratio. The underlying empirical relations and numerical values are inherited from the preceding work, whereas the present contribution is the structural classification of their minimum-mass readout degrees and the explicit distinction between same-shell structural statements and readout-stage numerical bookkeeping. Because the numerical value of M_min expressed in kilograms belongs to the adopted MKSA readout, the degree classification is made at the same-shell, readout-covariant level rather than as an intrinsic unit-system-independent statement. At the action level, the Coulomb numerator divided by c can be written as

$\frac{e^2}{4\pi {\epsilon }_{0}c}=\frac{\alpha }{2\pi }h=\frac{\alpha }{2\pi }{\left[M_{\text{min}}\right]}_{\text{MKSA}}{c}^2{s}_{\text{gen}}=7.69558236\times {10}^{-37}\left(\text{J}\cdot \text{s}\right)$ (13)

Thus, one explicit factor of the MKSA minimum-mass readout appears in the electromagnetic action-level structure. The corresponding gravitational action-level quantity is $G{m}_p^2/c$ . For the rate-type representation used below, this gravitational quantity is divided by c². The rate coefficient is defined as

$B_{\text{rate}}=\frac{4.5\alpha }{4\pi }\cdot {\left(\frac{s_{\text{gen}}}{1\text{kg}}\right)}_{F_1{F}_2}=5.33193444\times {10}^{-17}{\left(\frac{\text{s}}{\text{kg}}\right)}_{\text{MKSA}}$ (14)

The numerical value is displayed using the MKSA (s/kg) carrier. The subscript F₁F₂ denotes the readout stage after sequential application of the first and second redefinitions. The detailed definition of the second redefinition is given later in Section 2.5. The resulting proton-side gravitational rate representation is

$\frac{G{m}_p^2}{c^3}=B_{\text{rate}}{\left[M_{\min }\right]}_{\text{MKSA}}^2=6.92997483855\times {10}^{-90}\left(\text{kg}\cdot \text{s}\right)$ (15)

Equation (15) becomes numerically accurate after the second redefinition; before that step the relation is structural but not yet aligned to the accepted readout value. Equation (13) gives the first-degree minimum-mass readout structure at the action level, Equation (14) defines the rate coefficient, and Equation (15) gives the gravitational rate representation after division by c². In Equation (15), ${\left[M_{\min }\right]}_{\text{MKSA}}^2$ is explicitly a squared MKSA readout; it is not promoted to an intrinsic unit/readout-system-independent quantity. This is why a local-unit-shell (LUS) reading is required already at Equation (15). The problem is that ${\left[M_{\min }\right]}_{\text{MKSA}}^2$ looks as if it were a unit-system-independent invariant, whereas it is an MKSA readout square and therefore depends on the adopted mass-readout scale. The formal criteria are stated in Section 3.3. The degree-two statement refers instead to the covariance of the complete relation: with the adopted character assignments, $B_{\text{rate}}{\left[M_{\min }\right]}_{\text{MKSA}}^2$ and $G{m}_p^2/c^3$ both carry A = +2, which is explained in Section 2.5. Multiplying Equation (15) by c² returns $G{m}_p^2/c$ for comparison with Equation (13). At that common action level, the shared α, c², and s_gen factors cancel in the force ratio, leaving a first-degree minimum-mass readout structure. The normalized combination $M_{\min }^2/\left(m_e\cdot m_p\right)$ can remain invariant under a common same-shell mass-scale transfer, but this is distinct from invariance of $M_{\min }^2$ itself. Thus, the present degree terminology denotes a same-shell, readout-covariant structural degree, not an SI mass dimension or an intrinsic invariant degree. Coulomb’s law and Newton’s law of universal gravitation also share the same inverse-square spatial factor while their numerators remain distinct. The physical origin of the gravitational second-degree structure remains open.

2. Symbols and Basic Relations

This section summarizes the reference quantities and structural relations required for the present degree analysis. The first-redefinition framework is retained from the previous report, while only the relations needed in the present analysis are carried forward.

2.1. Reference Constants in MKSA Units (Values Taken from Standard Reference Data)

The standard physical constants listed below follow the 2018 CODATA recommended values [19], except where otherwise stated. The value of alpha is retained from the preceding work, and q_m is defined here by q_m = h/e. The CMB comparison value is taken from Fixsen (2009) [20]. The observed values of G and T_c are included for comparison only and do not enter the algebraic chain.

G: gravitational constant = 6.6743 × 10⁻¹¹ m³∙kg⁻¹∙s⁻² (comparison only).

T_c: CMB temperature = 2.72548 ± 0.00057 K (comparison only).

k: Boltzmann constant = 1.380649 × 10⁻²³ J∙K⁻¹.

c: speed of light = 299792458 m∙s⁻¹.

h: Planck constant = 6.62607015 × 10⁻³⁴ J∙s.

ε₀: electric constant = 8.8541878128 × 10⁻¹² N∙m²∙C⁻².

μ₀: magnetic constant = 1.256637062 × 10⁻⁶ N∙A⁻².

e: elementary charge = 1.602176634 × 10⁻¹⁹ C.

q_m: magnetic-monopole charge defined by q_m = h/e = 4.13566770 × 10⁻¹⁵ Wb.

m_p: proton rest mass = 1.67262192369 × 10⁻²⁷ kg.

m_e: electron rest mass = 9.1093837015 × 10⁻³¹ kg.

R_K: von Klitzing constant = 25812.80745 Ω.

α: fine-structure constant = 1/137.035999081.

2.2. Symbols and Relations Obtained after the Redefinition Procedure

This subsection transfers the first-redefinition block of the previous report [18], with the present coefficient-c notation. The redefinition is a change of readout representation, not a change of the physical constants. Hereafter, $F_1$ and $F_2$ denote the first and second readout-redefinition functions, respectively. For any readout quantity $X$ , $X^{F_1}=F_1\left(X\right)$ and $X^{F_1{F}_2}=F_2\left(F_1\left(X\right)\right)$ . The notation $F_1{F}_2$ denotes sequential application of the two functions and does not denote multiplication.

2.2.1. Symbols Obtained after the First Redefinition (F1) Procedure

Quantities written with the subscript “new” denote numerically consistent readouts after the first redefinition, as in the previous report [18]. They are historical/internal labels for that representation, not a new current-shell grammar.

$e_{new}=e\cdot \frac{4.4873976}{4.5}=1.5976897\times {10}^{-19}\left(\text{C}\right)$ (16)

$q_{m,\text{new}}=q_m\cdot \frac{\pi }{3.1327945}=4.1472823\times {10}^{-15}\left(\text{Wb}\right)$ (17)

$h_{\text{new}}=e_{new}\cdot q_{m,\text{new}}=h\cdot \frac{4.4873976}{4.5}\cdot \frac{\pi }{3.1327945}=h=6.62607015\times {10}^{-34}\left(\text{J}\cdot \text{s}\right)$ (18)

Therefore the value of Planck’s constant remains unchanged.

$R_{K,\text{new}}=\frac{q_{m,\text{new}}}{e_{\text{new}}}=R_K\cdot \frac{4.5}{4.4873976}\cdot \frac{\pi }{3.1327945}=25957.997\left(\Omega \right)$ (19)

Equation (19) can be rewritten as

$R_{K,\text{new}}=\left({4.5}_{\text{unit}}c\right)\cdot \frac{{\pi }_{\text{unit}}}{c}\cdot \frac{m_p}{m_e}=25957.997\left(\Omega \right)$ (20)

Here 4.5_unit and π_unit are unit-identification coefficients for the 1/(A∙m) and (V∙m) readouts. They represent MKSA readout deviations, not new physical constants. In the present notation they are written separately from c as 4.5_unitc and π_unit/c. The explicit factors of c cancel in the product (4.5_unitc)(π_unit/c) = 4.5_unitπ_unit.

$Z_{0,\text{new}}=2\alpha \cdot R_{K,\text{new}}=Z_0\cdot \frac{4.5}{4.4873976}\cdot \frac{\pi }{3.1327945}=378.84931\left(\Omega \right)$ (21)

$Z_{0,\text{new}}=2\alpha \cdot \left({4.5}_{\text{unit}}c\right)\cdot \frac{{\pi }_{\text{unit}}}{c}\cdot \frac{m_p}{m_e}=378.84931\left(\Omega \right)$ (22)

${\mu }_{0,\text{new}}=\frac{Z_{0,\text{new}}}{c}={\mu }_0\cdot \frac{4.5}{4.4873976}\cdot \frac{\pi }{3.1327945}=1.2637053\times {10}^{-6}\left(\text{N}\cdot {\text{A}}^{-2}\right)$ (23)

${\epsilon }_{0,\text{new}}=\frac{1}{Z_{0,\text{new}}c}={\epsilon }_0\cdot \frac{4.4873976}{4.5}\cdot \frac{3.1327945}{\pi }=8.8046642\times {10}^{-12}\left(\text{F}\cdot {\text{m}}^{-1}\right)$ (24)

$c_{\text{new}}=\frac{1}{\sqrt{{\epsilon }_0{\mu }_0}}=c=299792458\left(\text{m}\cdot {\text{s}}^{-1}\right)$ (25)

The Compton wavelength is

$\lambda =\frac{h}{mc}$ (26)

Because h and c are fixed, the Compton relation connects the length readout to the mass readout. Under the first-redefinition map only, the mass and thermal readouts remain m_e with subscript new equals m_e, and likewise for m_p and kT_c. This is not a claim of unit-system-independent invariance.

The minimum mass is defined as

${\left[M_{\min }\right]}_{\text{MKSA}}=\frac{\frac{2\pi }{\alpha }k{T}_c}{c^2}=3.605150741\times {10}^{-37}\left(\text{kg}\right)$ (27)

2.2.2. Meaning of the First Redefinition: Ω-Side Normalization

In the first redefinition, macroscopic and microscopic quantities are related through a normalization of the unit Ω. Under this normalization the unit J∙m²/s is fixed as follows:

$\begin{array}{l}\frac{4.4873976}{3.1327945}\cdot \left[1/\left(\text{A}\cdot \text{m}\right)\right]\left[1/\left(\text{V}\cdot \text{m}\right)\right]\\ =\left(4.5/\pi \right)\cdot \left[1/\left(\text{A}\cdot \text{m}\right)\right]\left[1/\left(\text{V}\cdot \text{m}\right)\right]=1/\left(\text{J}\cdot {\text{m}}^2/\text{s}\right)\end{array}$ (28)

Redefining the macroscopic unit 1 Ω yields the following relations.

$1{\Omega }_{\text{new}}={1.005624705}^{-1}{\Omega }_{\text{MKSA}}={1.005624705}^{-1}{\left(\text{Wb}/\text{C}\right)}_{\text{MKSA}}$ (29)

$e_{\text{new}}={1.005624705}^{-\frac{1}{2}}{e}_{\text{MKSA}}$ (30)

$q{m}_{\text{new}}={1.005624705}^{+\frac{1}{2}}q{m}_{\text{MKSA}}$ (31)

${\epsilon }_0{}_{{}_{\text{new}}}={1.005624705}^{-1}{\epsilon }_0{}_{{}_{\text{MKSA}}}$ (32)

${\mu }_0{}_{{}_{\text{new}}}=1.005624705{\mu }_0{}_{{}_{\text{MKSA}}}$ (33)

$R{K}_{\text{new}}=\left(4.5/4.4873976\right)\cdot \left(\pi /3.1327945\right)R{K}_{\text{MKSA}}=1.005624705R{K}_{\text{MKSA}}$ (34)

$Z{0}_{\text{new}}=1.005624705Z{0}_{\text{MKSA}}$ (35)

In the previous report the coefficient associated with m²/s was written kL₀. In the present report that object is not an independent quantity. Its previous role is carried by s_gen through M_minc²s_gen = h.

2.3. c-Normalization and Action versus Rate

Here, the notational modification means only the rearrangement of explicit factors of c; it does not change the unit-identification coefficients or their dimensional roles. In the previous report, physical quantities were expressed in c-normalized form by introducing appropriate factors of the speed of light, so that electromagnetic, mass-energy, and action-type quantities could be compared in a common structural framework. Before the notational modification that comparison was written with the absorbed maps (e ⇒ ec, q_m ⇒ q_mc, and h ⇒ h ∙ c²). After the coefficient-c separation adopted here, those absorbed symbols are no longer required: the factors of c are written explicitly, and the former c-wrapped coefficient notation becomes identity-like (e ⇒ e, q_m ⇒ q_m, h ⇒ h). This does not mean that c-normalization itself is the identity. The underlying c-normalized relations are retained. In particular the action-level comparison of e²/(4πε₀c) with $G{m}_p^2/c$ , and the subsequent rate form $G{m}_p^2/c^3$ , remain in use.

$e\Rightarrow ec\left(\text{previous}\right),e\Rightarrow e\left(\text{present}\right)$ (36)

$q_m\Rightarrow q_{m}c\left(\text{previous}\right),q_m\Rightarrow q_m\left(\text{present}\right)$ (37)

$h\Rightarrow h\cdot c^2\left(\text{previous}\right),h\Rightarrow h\left(\text{present}\right)$ (38)

${\epsilon }_0\Rightarrow {\epsilon }_0\cdot c\left(\text{previous}\right),{\epsilon }_0\Rightarrow {\epsilon }_0\cdot c\left(\text{present}\right)$ (39)

${\mu }_0\Rightarrow {\mu }_0\cdot c\left(\text{previous}\right),{\mu }_0\Rightarrow {\mu }_0\cdot c\left(\text{present}\right)$ (40)

$\text{kg}\Rightarrow \text{kg}\cdot c^2\left(\text{previous}\right),\text{kg}\Rightarrow \text{kg}\cdot c^2\left(\text{present}\right)$ (41)

$G\Rightarrow \frac{G}{c^3}\left(\text{previous}\right),G\Rightarrow \frac{G}{c^3}\left(\text{present}\right)$ (42)

$G_N\Rightarrow \frac{G_N}{c}\left(\text{previous}\right),G_N\Rightarrow \frac{G_N}{c}\left(\text{present}\right)$ (43)

where G = G_N/(1 kg), equivalently G_N = G · 1 kg, is used as an intermediate gravitational-side readout. Thus, the kilogram appearing in the MKSA dimension of G is a denominator mass unit. The maps (36) - (43) record the notational modification of the former absorbed notation.

2.4. Minimum Mass, Generative Time, and SI

$M_{\min }{c}^2{s}_{\text{gen}}=h$ (44)

$s_{\text{gen}}=h/\left(M_{\min }{c}^2\right)$ (45)

${\left[M_{\text{min}}\right]}_{\text{MKSA}}=\frac{h}{c^2{s}_{\text{gen}}}$ (46)

When a numerical kilogram value is intended, the minimum mass is written [M_min]_MKSA. The subscript MKSA marks a numerical readout in the adopted MKSA shell, not a new physical species of M_min. The former coefficient kL₀ is historical only; its role is carried by s_gen through (44). The mass-side structural invariant of the previous report is retained in the restricted form

$\text{SI}:=M_{\min }^2/\left(m_e\cdot m_p\right)$ (47)

SI denotes the structural invariant of the previous report [18]. Under a common same-shell mass-scale transfer, SI is invariant. This invariance is not from invariance of $M_{\min }^2$ itself. The additional structural quantities used in the previous report are not introduced here unless required in the subsequent analysis.

The four connection relations of the previous report [18] are retained here as useful identities. In the present notation, their former c-absorbed forms are modified only by making the factors of c explicit; the unit-identification coefficients and their dimensional roles are unchanged. Separately, in the two relations that previously contained kL₀, kL₀ is eliminated in favor of $s_{\text{gen}}$ through $M_{\text{min}}{c}^2{s}_{\text{gen}}=h$ . The other two connection relations retain the same structural content.

$\left({4.5}_{unit}c\right)e_{new}=\frac{M_{min}}{m_p}$ (48)

$\frac{q{m}_{\text{new}}}{\frac{{\pi }_{\text{unit}}}{c}}=\frac{M_{\min }}{m_e}$ (49)

$\left(\frac{{\pi }_{\text{unit}}}{c}\right)e_{\text{new}}=m_e{c}^2{s}_{\text{gen}}$ (50)

$\frac{q{m}_{\text{new}}}{\left({4.5}_{\text{unit}}c\right)}=m_p{c}^2{s}_{\text{gen}}$ (51)

2.5. A-Character Table Used in This Analysis

The second redefinition (F2) can be done using the A-character. The A-character is a readout character. It is not automatically identified with a concrete MKSA-to-readout endpoint exponent. The A-character assignments used here are inherited from the readout framework developed in the preceding report [18]; their deeper origin is not re-derived in the present paper. R_p is defined in Section 3.1. In the preceding report [18], a table-number A(X) was also used as the exponent of the transfer factor $J_{\text{tab}}^{A\left(X\right)}$ , with J_tab = 1.00056105184. The transfer rule is not applied again within the same-shell evaluation below; quantities labeled $F_1{F}_2$ are already expressed in the post-$F_2$ readout stage. In that report A(X) is a covector that assigns a table-number weight to each readout quantity X: a positive weight multiplies the readout by $J_{\text{tab}}^{A\left(X\right)}$ , a negative weight divides it by $J_{\text{tab}}^{\left|A\left(X\right)\right|}$ , and A(X) = 0 leaves the readout unchanged. Table 1 of the present paper keeps those labels as readout characters for the degree analysis; it does not apply the transfer rule inside the same-shell evaluation.

Table 1. Lists only the A-characters used in the present degree analysis.

Quantity	A(X)	Role
c	0	c-normalized reference
h	+2	action
e	−4	elementary charge
qm	+6	magnetic-monopole charge
Mmin, me, mp	+3	Mass, electron, proton
1 kg	+3	macroscopic mass unit
sgen, s	−1	time
m	−1	distance
Ω	+10	resistance
ε0	−10	electric constant
μ0	+10	magnetic constant
G	−4	gravity
e2/(4πε0)	+2	EM numerator
$G{m}_p^2$	+2	gravitational numerator
1/r2	+2	spatial factor
Rp	0	ratio
Brate	−4	rate coefficient

The table-number factor used in the second redefinition is defined by Equation (52). Each of the two coefficient branches in Equation (34) corresponds to $J_{\text{tab}}^5$ , so their product is $J_{\text{tab}}^{10}$ = RK_new/RK_MKSA.

$J_{\text{tab}}={\left[\frac{4.5}{4.487397553}\frac{\pi }{3.132794486}\right]}^{1/10}={\left[\frac{R_{K,\text{new}}}{R_{K,\text{MKSA}}}\right]}^{1/10}=1.00056105184$ (52)

3. Methods

3.1. Definitions for B_pair and B_rate

The evaluation uses three auxiliary objects: the rate coefficient B_rate, the proton-side force ratio R_p(m_p), and the pair coefficient B_pair. They are introduced here only as definitions. Their degree reading is given in Section 4.

$B_{\text{rate}}^{F_1{F}_2}=\frac{4.5\alpha }{4\pi }\cdot \frac{s_{\text{gen}}^{F_1{F}_2}}{{\left(1\text{kg}\right)}^{F_1{F}_2}}$ (53)

$R_p\left(m_p\right)=G\cdot m_p^2/\left[e^2/\left(4\pi {\epsilon }_0\right)\right]$ (54)

$B_{\text{pair}}:=B_{\text{rate}}\left(m_e/m_p\right)$ (55)

Equation (53) is written in the $F_1{F}_2$ readout stage; therefore, no additional $J_{\text{tab}}^{-4}$ factor is applied within this same-stage expression.

J_tab is a readout-transfer factor. It appears only in a cross-system reverse display and is not part of sam_e-system R_p(m_p). Its table is not required for the sam_e-shell evaluation below. B_rate is not introduced as a new coupling constant. It is the coefficient that remains when the gravitational num_erator is written in the rate form $G\cdot m_p^2/c^3$ , whose coherent slot is s/kg. The factor s_gen/(1 kg) occupies that slot; the num_erical package 4.5α/4π is the sam_e package already used on the electromagnetic side.

3.2. Inverse-Square Laws of Coulomb and Newton

Coulomb’s law and Newton’s law of universal gravitation both have an inverse-square dependence on distance. For two protons, they are written as follows.

$F_C=\frac{\frac{e^2}{4\pi {\epsilon }_0}}{r^2}$ (56)

$F_G=\frac{G{m}_p^2}{r^2}$ (57)

Table 1 gives A(1/r²) = +2, and therefore A(r²) = −2. The two force numerators have A(e²/(4πε₀)) = +2 and $A\left(G\cdot m_p^2\right)$ = +2. Consequently, A((e²/(4πε₀))r²) = +2 − 2 = 0 and $A\left(G\cdot m_p^2{r}^2\right)$ = +2 − 2 = 0. The zero A-character means that each product is invariant under the adopted readout transformation. In both cases, the numerator character +2 is balanced by the character −2 of r². The corresponding force contains the inverse-square factor 1/r² and therefore decreases in proportion to the inverse square of the distance. This common A-character structure does not mean that the electromagnetic and gravitational interactions have the same physical origin.

3.3. Solution for the LUS Problem in R_p(m_p)

The present Methods adopt only a partial LUS resolution. The solution in this section is limited to the force ratio R_p(m_p). It is not proposed as a general solution for every LUS-dependent quantity. LUS specifies the readout frame, same-shell specifies a comparison made within one such frame, and readout covariance specifies how a complete relation transforms under the adopted character rule.

3.3.1. Explanation for the LUS Problem

Here LUS means a local unit shell: a coherent readout frame in which the numerical values of mass, time, and related quantities are expressed together. The kilogram value of the minimum mass belongs to the adopted MKSA shell. The problem addressed in the degree analysis is that ${\left[M_{\min }\right]}_{\text{MKSA}}^2$ can appear to be a unit-system-independent number, while in fact it depends on the MKSA mass-readout scale of the adopted shell. Under a common same-shell mass-scale transfer, the numerical square of the minimum mass changes, even though a same-shell identity such as M_minc²s_gen = h may still hold. Therefore, a statement about ${\left[M_{\min }\right]}_{\text{MKSA}}^2$ is a statement about a readout representation, not about a unit-system-independent object.

For R_p(m_p), however, the relevant object is the complete same-shell ratio. In the gravitational numerator $G{m}_p^2$ , G contains one inverse mass dimension and $m_p^2$ contains two mass dimensions. The numerator therefore contains one net mass unit. In the electromagnetic denominator, $e^2/4\pi {\epsilon }_0=\left(\alpha /2\pi \right)hc$ , the same single mass unit is contained in h. The apparent LUS problem results from viewing the minimum-mass expression on the gravitational side separately from the mass unit contained in h on the electromagnetic side.

3.3.2. Solution for the LUS Problem

Consider the change from MKSA to MgSA, in which only the mass unit is changed from kilograms to grams. Since the mass unit in G is in the denominator, G changes by a numerical factor of 10⁻³, whereas $m_p^2$ changes by 10⁶. Therefore, $G{m}_p^2$ changes by 10³. On the electromagnetic side, h also changes by 10³, while α and c remain unchanged. Consequently, $e^2/4\pi {\epsilon }_0=\left(\alpha /2\pi \right)hc$ changes by the same factor of 10³. These two factors cancel in R_p(m_p).

The LUS problem of R_p(m_p) is therefore solved by the cancellation of the common mass-unit factor: the net mass unit in the gravitational numerator $G{m}_p^2$ is canceled by the mass unit contained in h in the electromagnetic denominator. The same transformation can be followed explicitly in Equation (59). Under the coherent change from MKSA to MgSA, the numerical readout of M_min changes from 3.605150741 × 10⁻³⁷ kg to 3.605150741 × 10⁻³⁴ g, whereas s_gen = 2.044990031 × 10⁻¹⁴ s is unchanged. Because B_rate carries the time-per-mass slot, its numerical value changes from 5.33193444 × 10⁻¹⁷ s/kg to 5.33193444 × 10⁻²⁰ s/g. Thus, on the right-hand side of Equation (59), B_rate changes by 10⁻³ while [M_min]² changes by 10⁶, and their product changes by 10³. On the left-hand side, G changes by 10⁻³ and $m_p^2$ changes by 10⁶, so $G{m}_p^2/c^3$ also changes by 10³. Equation (59) therefore remains covariant under this coherent same-shell mass-unit rescaling.

The numerical readout of M_min changes under this transformation, but the complete dimensionless force ratio R_p(m_p) does not. The common mass-unit factor in $G{m}_p^2$ is matched by the corresponding mass-unit factor in $e^2/4\pi {\epsilon }_0=\left(\alpha /2\pi \right)hc$ , and the two factors cancel in the ratio. This same-shell example does not remove the distinction between the F₁F₂ and MKSA kg/s carriers used in Equation (63).

3.3.3. Limitation for This Solution

Same-shell, in this report, means that the compared quantities are rescaled together inside one readout frame. Readout covariance means that both sides of a displayed relation transform so as to keep the same A-character. The MKSA system is a completed reference frame, not a physically privileged unit system. For the coherent change from MKSA to MgSA considered here, R_p(m_p) is unchanged because its gravitational numerator and electromagnetic denominator acquire the same mass-unit factor. No additional J_tab factor is required in the same-system ratio.

This solution is limited to the LUS problem of R_p(m_p) and to the coherent mass-unit change from kilograms to grams examined above. It does not establish a general LUS solution for arbitrary changes of length, time, or readout structure. ${\left[M_{\min }\right]}_{\text{MKSA}}^2$ remains an MKSA readout square and is not promoted to an LUS-independent intrinsic quantity. The same-shell invariance of $M_{\min }^2/\left(m_e\cdot m_p\right)$ is not invariance of $M_{\min }^2$ itself. The physical origin of the gravitational second power and the physical owner or selector of the second redefinition remain outside the present solution.

4. Results

The results below are same-shell, readout-covariant statements. ${\left[M_{\min }\right]}_{\text{MKSA}}^2$ is not an intrinsic LUS-independent quantity.

4.1. Electromagnetic First-Degree Structure

The Coulomb numerator admits one explicit minimum-mass factor,

$\frac{e^2}{4\pi {\epsilon }_0}=\frac{\alpha }{2\pi }hc=\frac{\alpha }{2\pi }{\left[M_{\text{min}}\right]}_{\text{MKSA}}{c}^3{s}_{\text{gen}}$ (58)

This is a first-degree minimum-mass readout structure.

This means that one explicit factor of the MKSA minimum-mass readout appears in the electromagnetic numerator after the spatial factor has been set aside. It does not mean that the origin of the Coulomb numerator has been derived, and it does not promote [M_min]_MKSA itself to a unit-system-independent invariant.

4.2. Gravitational Second-Degree Structure

The second-degree statement refers only to the minimum-mass readout appearing in the complete gravitational rate representation.

After division by c³,

$\frac{G{m}_p^2}{c^3}=B_{\text{rate}}{\left[M_{\text{min}}\right]}_{\text{MKSA}}^2$ (59)

Equation (59) is a readout-covariant relation. With Table 1, A([M_min]_MKSA) = +3, A(s_gen) = −1, and A(1 kg) = +3 give A(B_rate) = −4 and $A\left(B_{\text{rate}}\cdot {\left[M_{\min }\right]}_{\text{MKSA}}^2\right)$ = +2, matching $A\left(G\cdot m_p^2/c^3\right)$ = +2. The square ${\left[M_{\min }\right]}_{\text{MKSA}}^2$ transforms under a mass-readout rescaling; the complete relation remains character-closed. This matching is not invariance of Mmin² itself.

This means that the gravitational rate representation carries a second power of the MKSA minimum-mass readout and remains character-closed at A = +2. It does not mean that ${\left[M_{\min }\right]}_{\text{MKSA}}^2$ is an intrinsic LUS-independent square, and it does not explain why gravity selects the second power.

4.3. Same-Shell Invariant Composite

The normalized composite

$\text{SI}:=\frac{M_{\min }^2}{m_e{m}_p}$ (60)

is invariant under a common same-shell mass-scale transfer. This invariance is distinct from invariance of $M_{\min }^2$ itself. This means that a normalized mass composite can survive a common same-shell mass-scale transfer. It does not mean that the un-normalized square $M_{\min }^2$ is invariant, and it does not by itself select the proton or the electron.

4.4. Gravitational-to-Electromagnetic Force Ratio for Two Protons

The ratio is

$R_p\left(m_p\right)=\frac{G{m}_p^2}{e^2/\left(4\pi {\epsilon }_0\right)}$ (61)

Substitution of Equations (58) and (59) cancels the shared package α, c³, and s_gen. What remains is a first-degree minimum-mass readout structure. Coarsely,

$R_p\left(m_p\right)~\frac{{\left[M_{\text{min}}\right]}_{\text{MKSA}}^2}{{\left[M_{\text{min}}\right]}_{\text{MKSA}}}\to {\left[M_{\text{min}}\right]}_{\text{MKSA}}$ (62)

In Equation (62), the symbol ~ means only that the gravitational side contributes two powers of the minimum-mass readout and the electromagnetic side contributes one. It does not indicate numerical approximation or numerical closeness. Equation (62) therefore represents only the structural degree counting: in forming the quotient, the second-degree gravitational structure and the first-degree electromagnetic structure leave one residual minimum-mass readout degree. At this stage, that residual degree is not identified with a particular same-shell numerical value of M_min. The corresponding numerical evaluation requires the readout-stage bookkeeping shown explicitly in Equation (63).

$R_p{\left(m_p\right)}_{mixed-stage}=\frac{9}{4}\frac{{\left[M_{\min }/s\right]}_{\text{MKSA}}}{{\left(1\text{kg}/s\right)}^{F_1{F}_2}}=\frac{9}{4}\frac{{\left[M_{\min }/s\right]}_{\text{MKSA}}}{{\left[1\text{kg}/s\right]}_{\text{MKSA}}}\cdot J_{\text{tab}}^{-4}=8.09341058\times {10}^{-37}$ (63)

Equation (63) is written as a mixed-stage reverse display. In Equation (63), the F₁F₂ label belongs to the kg/s carrier in the denominator, not to the MKSA minimum-mass readout in the numerator. The factor $J_{\text{tab}}^{-4}$ is used once in the reverse-display bookkeeping and is not an intrinsic factor of R_p(m_p). The force ratio defined in Equation (61) is dimensionless and J_tab-free when evaluated within one coherent readout shell.

Restoring c³ on the gravitational side and forming the force ratio cancels the common factors alpha, c³, and s_gen at the structural level. The coefficient 9/4 follows from the ratio of the gravitational and electromagnetic coefficients, while the degree difference 2-1 leaves one residual minimum-mass readout degree. Equation (63) gives the corresponding numerical result through the mixed-stage reverse display described above. The factor $J_{\text{tab}}^{-4}$ is used only in this bookkeeping step and is not an intrinsic or final coefficient of the same-system R_p(m_p). The complete same-system force ratio remains dimensionless and J_tab-free.

Species dependence and reverse-display factors remain outside this same-shell reading. The coefficient 9/4 follows algebraically from Equations (58) and (59).

4.5. Readout Display of R_p(m_p) and the Pair Rewrite

This subsection separates same-system relations from cross-system bookkeeping and then rewrites the gravity-side expression in a pair-normalized form. When numerator and denominator are expressed in one coherent readout system, R_p(m_p) is J_tab-free. J_tab belongs only to cross-system reverse display and is not retained in the final same-system coefficient. In a mixed-stage reverse display, a factor $J_{\text{tab}}^{-4}$ may appear in order to recover the accepted numerical witness $R_p\left(m_p\right)=8.09341\times {10}^{-37}$ . That factor is bookkeeping for readout transfer, not an intrinsic part of R_p theory. B_rate is the rate-type coefficient of (59). It carries the time-per-mass slot required by [G/c³] = s/kg through s_gen/(1 kg). Its deeper normalization remains open.

The pair rewrite

$\frac{G}{c^3}=B_{\text{pair}}\frac{M_{\text{min}}^2}{m_p{m}_e},B_{\text{pair}}:=B_{\text{rate}}\left(\frac{m_e}{m_p}\right)$ (64)

introduces no new LUS problem, because m_e/m_p is invariant under common same-shell mass scaling and $M_{\min }^2/\left(m_p\cdot m_e\right)$ is the invariant composite (60). Species dependence is not thereby removed; it is deferred. This means that the pair form merely rearranges the same same-shell content. It does not remove species dependence and does not add an LUS-independent square of M_min.

5. Discussions

The present report adopts the partial LUS resolution. What is closed is the same-shell invariant composite (60), the covariant display (59), and the same-shell degree statement (62). Equation (63) is retained only as a mixed-stage reverse display and is not counted as an exact same-shell relation. What is not closed is an LUS-independent intrinsic square of M_min, the reason why gravity selects the second power, the deeper normalization of s_gen, species dependence, and the selector of 4.5 and π. MKSA remains a reference readout frame only. Same-shell cancellation is not promoted to physical equivalence of distinct shells. A candidate correspondence between distance and a discrete readout index remains unsettled, and its origin is not established here. A simple integer quantization of m_p/M_min is not pursued. α is retained as an interaction coefficient. B_pair is a reference display only. The factor 1 kg is an external macroscopic reference used in the display of the ratio, not an intrinsic component of M_min. The present results therefore give a coherent same-shell organization, but not a completed microscopic mechanism. A complete LUS solution across distinct readout systems is not claimed. A dissipative or nonequilibrium account may be relevant to the open normalization and origin questions, but no such explanation is developed in the present report [21]-[25]. Reference [21] provides only an empirical example of relaxation in a material system. References [22] and [23] provide background on nonequilibrium work and fluctuation relations, while References [24] and [25] exemplify thermodynamic and nonequilibrium perspectives on spacetime gravitational field equations. None of References [21]-[25] enters the algebraic derivation in the present paper; these citations are included only as background for possible future work. The LUS problem is not that Equation (15) is false, but that ${\left[M_{\min }\right]}_{\text{MKSA}}^2$ can be misread as a unit-system-independent square. The present report keeps that square as an MKSA readout and does not claim a complete solution across distinct shells.

6. Conclusion

The present report organizes a same-shell, readout-covariant degree correspondence for the minimum-mass readout. Within the inherited empirical relations, the electromagnetic numerator contains one explicit minimum-mass readout degree, whereas the gravitational rate representation contains two. Their quotient therefore leaves one residual minimum-mass readout degree in the dimensionless proton-side gravitational-to-electromagnetic force ratio R_p(m_p). This degree correspondence is the principal structural result of the present analysis. The empirical relations and numerical values used here are inherited from the preceding work. The present contribution is their structural classification in terms of minimum-mass readout degree and the separation of same-shell structural statements from readout-stage numerical bookkeeping. Accordingly, the result is not an independent derivation of the proton-side force ratio, nor does it establish the physical origin of gravity or electromagnetism. The coherent MKSA-to-MgSA mass-unit change provides an explicit covariance check. Under this rescaling, the numerical readout of M_min changes, and the individual quantities in the gravitational and electromagnetic representations also change, but the complete dimensionless ratio R_p(m_p) remains unchanged because the two sides acquire the same net mass-unit factor. The squared quantity ${\left[M_{\min }\right]}_{\text{MKSA}}^2$ is therefore retained as an MKSA readout square and is not interpreted as an intrinsic, unit-system-independent quantity. The common inverse-square spatial dependence of Coulomb’s law and Newton’s law is a separate structural feature and does not imply a common physical origin of the two interactions. The present analysis also does not provide a general solution of the LUS problem for arbitrary readout transformations. In particular, the physical origin of the gravitational second-degree structure and the deeper origin of the adopted readout transformation remain open.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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