jmp Journal of Modern Physics 2153-120X 2153-1196 Scientific Research Publishing 10.4236/jmp.2026.176030 jmp-152071 Article Physics Mathematics A Readout-Based Structural Analysis of Empirical CMB Temperature Equations Related to E = hν and E = mc² Miyashita Tomofumi 1 Miyashita Clinic, Osaka, Japan

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

 24 06 2026 06 2026 17 06 697 713 12 04 2026 21 06 2026 24 06 2026 © 2026 by the authors and Scientific Research Publishing Inc. 2026 This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( https://creativecommons.org/licenses/by/4.0/ ). https://doi.org/10.4236/jmp.2026.176030

This report extends our previous studies on empirical equations involving the cosmic microwave background (CMB) temperature. Whereas our earlier work mainly discussed numerical connections among physical quantities, the present work attempts to reformulate those relations within a structural framework. In particular, Equation (12), previously introduced as a central empirical relation, is reconsidered independently of the MKSA unit system and expressed in terms of macro-micro correspondence. This reformulation clarifies how macroscopic units and microscopic quantities can be organized into a consistent structural scheme. A recalculated CMB temperature of 2.7256307 K agrees with the observed value of 2.72548 ± 0.00057 K. Rather than introducing further numerical adjustments, we examine the internal consistency of the resulting structure and show that it can be written in a closed and coherent form. The aim is not to provide a complete microphysical theory, but to examine whether E = mc2 and E = hν can be organized within a common readout-based structural framework.

 Temperature of the Cosmic Microwave Background Minimum Mass Ratio of the Gravitational Force to the Electric Force Dimensional Analysis Fine-Structure Constant 1. Introduction

The list of symbols used in this paper is summarized in Section 2. In our 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), (2), and (3) represent characteristic relationships derived from the CMB temperature.

G m p 2 h c = 4.5 2 ⋅ k T c 1   kg ⋅ c 2 G m p 2 ( e 2 4 π ε 0 ) = 4.5 2 π ⋅ m e e ⋅ h c m e c 2 e ⋅ ( e 2 4 π ε 0 ) = π ⋅ k T c

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

1 α = 137.0359991 = 136.0113077 + 1 3 ⋅ 13.5 + 1 13.5 ⋅ 136.0113077 = 1836.152654 = m p m e

Small deviations from the ideal values of 9/2 and π, respectively, were discussed as a way to reduce the induced numerical errors [9]-[14]. The coefficient associated with the dimensional factor of length (m2/s) was denoted by kL and by kL₀ when explicitly expressed in the meter-kilogram-second-ampere (MKSA) unit system [15][16]. In this work, kL0 is treated as a structural readout coefficient associated with the m2/s-type branch, rather than as a new dynamical operator or as an independently measured transport coefficient.

k L ( m 2 s ) = 1837.94538 ( m 2 s ) MKSA ≡ k L 0 ( m 2 s ) MKSA

where the subscript “MKSA” indicates the MKSA unit system.

3.1327945 ( V ⋅ m ) = k L 0 ⋅ m e c 2 e ⋅ c 4.4873976 ( 1 A ⋅ m ) = q m ⋅ c k L 0 ⋅ m p c 2

By redefining the von Klitzing constant, these deviations can be adjusted back to the ideal values of 9/2 and π [15], leading to Equations (9) and (10).

π ( V ⋅ m ) = k L 0 ⋅ m e c 2 e n e w ⋅ c 4.5 ( 1 A ⋅ m ) = q m _ n e w ⋅ c k L 0 ⋅ m p c 2

The parameter kL₀ is then obtained from Equation (11).

k L 0 = 3.1327945 4.4873976 ⋅ h c 2 m p c 2 ⋅ m e c 2 = π u n i t 4.5 u n i t ⋅ h c 2 m p c 2 ⋅ m e c 2 = 1837.94538

This leads to the following relation.

m p c 2 ⋅ m e c 2 ⋅ 4.5 u n i t π u n i t ⋅ h c 2 = ( 2 π ( 1 ) ⋅ k T c α ) 2 = 1.04985584 E   −   39

Using this framework, a calculated CMB temperature of 2.7256307 K was obtained, in agreement with the observed value of 2.72548 ± 0.00057 K. While previous work mainly focused on numerical consistency and parameter adjustments within the MKSA unit system [15][16], the underlying structural meaning of these relations has remained unclear. In particular, the role of Equation (12) and its possible connection to more fundamental physical structure require further clarification.

In the present work, we revisit these empirical relations from the viewpoint of macro-micro correspondence. The redefinition procedure used here does not imply a physical change of fundamental constants, but is introduced only as a readout-level transformation. Rather than introducing further numerical adjustments, we examine whether the relations can be reorganized into a consistent structural form based on readout-invariant quantities and their associated mass-time structure. The physical origin of the connection coefficients is not claimed to be fully explained in this paper and remains an open question.

The main purpose of the present work is to examine whether the empirical relations discussed above can be organized within a common readout-based structural framework related to E = mc2 and E = hν. The present work is therefore intended as a structural consistency analysis rather than a complete physical theory.

The remainder of this paper is organized as follows. Section 2 summarizes the symbols used in this work. Section 3 presents the methodological framework, including c-normalization and the redefinition procedure adopted in this study. Section 4 presents the main structural results and explains their physical meaning in relation to the connection between E = mc2 and E = hν. Section 5 discusses broader implications of the present framework.

 2. Symbol List 2.1. Reference Constants in MKSA Units (These Values Were Obtained from Wikipedia)

G: Gravitational constant: 6.6743 × 10−11 (m3∙kg−1∙s−2)

(we used the compensated value of 6.68917534 × 10−11 in this study)

Tc: CMB temperature: 2.72548 ± 0.00057 (K)

(we used the compensated value of 2.725630647 K in this study)

k: Boltzmann constant: 1.380649 × 10−23 (J∙K−1)

c: Speed of light: 299792458 (m/s)

h: Planck constant: 6.62607015 × 10−34 (J s)

ε₀: Electric constant: 8.8541878128 × 10−12 (N∙m2∙C−2)

μ₀: Magnetic constant: 1.25663706212 × 10−6 (N∙A−2)

e: Electric charge of one electron: −1.602176634 × 10−19 (C)

qm: Magnetic charge of one magnetic monopole: 4.13566770 × 10−15 (Wb)

(This value is only a theoretical value: qm = h/e)

mp: Resting mass of a proton: 1.67262192369 × 10−27 (kg)

me: Resting mass of an electron: 9.1093837015 × 10−31 (kg)

Rk: von Klitzing constant: 25812.80745 (Ω)

Z0: Wave impedance in free space: 376.730313668 (Ω)

α: Fine-structure constant: 1/137.035999081

 2.2. Symbol List Established in Previous Reports

2.2.1. Symbol List Obtained after the Redefinition Process

In this subsection, symbols with the subscript “new” denote quantities expressed under the redefinition scheme adopted in this work, representing numerically consistent forms of the corresponding MKSA quantities.

e n e w = e ⋅ 4.4873976 4.5 = 1.5976897 E   −   19 ( C ) q m _ n e w = q m ⋅ π 3.1327945 = 4.1472823 E   −   15 ( Wb ) h n e w = e n e w ⋅ q m _ n e w = h ⋅ 4.4873976 4.5 ⋅ π 3.1327945 = 6.62607015 E   −   34 ( J ⋅ s ) = h

Therefore, the value of Planck’s constant is unchanged.

R k _ n e w = q m _ n e w e _ n e w = R k ⋅ 4.5 4.4873976 ⋅ π 3.1327945 = 25957.997 ( Ω )

Equation (16) can be rewritten as follows:

R k n e w = 4.5 ( 1 A ⋅ m ) ⋅ π ( V ⋅ m ) ⋅ m p m e = 25957.997 ( Ω ) Z 0 _ n e w = α ⋅ 2 h n e w e n e w 2 = 2 α ⋅ R k n e w = Z 0 ⋅ 4.5 4.4873976 ⋅ π 3.1327945 = 378.84931 ( Ω )

Equation (18) can be rewritten as follows:

Z 0 _ n e w = 4.5 ( 1 A ⋅ m ) ⋅ π ( V ⋅ m ) ⋅ 2 α ⋅ m p m e = 378.84931 ( Ω ) μ 0 _ n e w = Z 0 _ n e w c = μ 0 ⋅ 4.5 4.4873976 ⋅ π 3.1327945 = 1.2637053 E   −   06 ( N ⋅ A − 2 ) ε 0 _ n e w = 1 Z 0 _ n e w ⋅ c = ε 0 ⋅ 4.4873976 4.5 ⋅ 3.1327945 π = 8.8046642 E   −   12 ( F ⋅ m − 1 ) c _ n e w = 1 ε 0 _ n e w μ 0 _ n e w = 1 ε 0 μ 0 = c = 299792458 ( m ⋅ s − 1 )

The numerical value of the speed of light remains unchanged. In the present redefinition scheme, h and c are held fixed by construction. Next, the Compton wavelength (λ) is as follows:

λ = h m c

When the values of Planck’s constant and the speed of light are unchanged, the following values in Equations (24), (25) and (26) can be kept constant.

m e _ n e w = m e ⋅ 4.4873976 4.5 ⋅ π 3.1327945 = m e = 9.1093837 E   −   31 ( kg ) m p _ n e w = m p ⋅ 4.4873976 4.5 ⋅ π 3.1327945 = m p = 1.6726219 E   −   27 ( kg ) k T c _ n e w = k T c ⋅ 4.4873976 4.5 ⋅ π 3.1327945 = k T c = 3.7631393 E   −   23 ( J )

The definition of minimum mass is as follows:

M min = 2 π ⋅ k T c α c 2 = 3.605150741 E   −   37 ( kg )

2.2.2. Algorithm Used in the Previous Work

1) Symbol list about  4.5unit and 𝜋unit

4.5unit and 𝜋unit are introduced solely as unit-identification factors.

4.5 u n i t = 4.5 ( A − 1 ⋅ m − 1 ) π u n i t = π ( V ⋅ m )

2) Algorithm used in the previous work

e n e w ⋅ c ( A ⋅ m ) = 1 4.5 u n i t ⋅ h c 2 m p c 2 ⋅ k L 0 = 4.78975317 E   −   11 ( A ⋅ m ) q m _ n e w ⋅ c ( V ⋅ m ) = π u n i t ⋅ h c 2 m e c 2 ⋅ k L 0 = 1.24332398 E   −   06 ( V ⋅ m ) m e c 2 ⋅ k L 0 ( J ⋅ m 2 s ) = π u n i t 4.5 u n i t ⋅ h c 2 m p c 2 ⋅ k L 0 = 1.50474532 E   −   10 ( J ⋅ m 2 s ) m p c 2 ⋅ k L 0 ( J ⋅ m 2 s ) = π u n i t 4.5 u n i t ⋅ h c 2 m e c 2 ⋅ k L 0 ( J ⋅ m 2 s ) = 2.76294215 E   −   07 ( J ⋅ m 2 s ) h c 2 ( J ⋅ m 2 s ) = π u n i t 4.5 u n i t ⋅ h c 2 m p c 2 ⋅ k L 0 ⋅ h c 2 m e c 2 ⋅ k L 0 ( J ⋅ m 2 s ) = 5.95521 E   −   17 ( J ⋅ m 2 s ) 2 π ⋅ k T c α ⋅ k L 0 = π u n i t 4.5 u n i t ⋅ h c 2 m p c 2 ⋅ k L 0 ⋅ h c 2 m e c 2 ⋅ k L 0 = 5.95521 E   −   17 ( J ⋅ m 2 s ) G n e w c 3 ( s kg = m 2 / s J ) = α 2 π ⋅ 4.5 ( 1 ) 2 ⋅ ( 4.5 u n i t π u n i t ⋅ h c 2 ) ⋅ k L 0 1   kg ⋅ c 2 ⋅ ( m e m p ) = 2.4826212101 E   −   36 R p ( m p ) ≡ G n e w c 3 ( m p c 2 ) 2 ( ( e n e w ⋅ c ) 2 4 π ε 0 _ n e w ⋅ c ) = 4.5 ( 1 ) 2 ⋅ 2 π ( 1 ) ⋅ k T c α ⋅ 1 1   kg ⋅ c 2 = 8.11158917 E   −   37

where Rp (mp) denotes the dimensionless ratio of the gravitational force to the electrostatic force between two protons at the same separation. Although Equations (36) and (37) are not essential to the main purpose of the present paper, they are retained as possible clues for further development.

 2.3. Useful Equations

The following equations are useful for confirming the internal consistency of the present framework.

1 ε 0 _ n e w c ( Ω = kg C 2 ⋅ m 2 s ) = M min ( e n e w ) 2 ⋅ k L 0 ⋅ 2 α = 378.8493064 ( Ω ) = Z 0 _ n e w μ 0 _ n e w c ( Ω = Wb 2 kg ⋅ s m 2 ) = ( q m _ n e w ) 2 M min ⋅ k L 0 ⋅ 2 α = 378.8493064 ( Ω ) = Z 0 _ n e w m p c 2 ( J ) ⋅ 4.5 u n i t e n e w ⋅ c = 2 π ( 1 ) ⋅ k T c α ( J ) = 3.24014789 E   −   20 ( J ) m e c 2 ( J ) ⋅ q m _ n e w ⋅ c π u n i t = 2 π ( 1 ) ⋅ k T c α ⋅ c 2 ( J ) = 3.24014789 E   −   20 ( J ) m e c 2 ⋅ k L 0 = π u n i t ⋅ e n e w ⋅ c = 1.50474532 E   −   10 ( J m 2 s ) m p c 2 ⋅ k L 0 = q m _ n e w ⋅ c 4.5 u n i t = 2.76294215 E   −   07 ( J m 2 s ) h c 2 ⋅ k L 0 − 1 ( J ) = 2 π ( 1 ) ⋅ k T c α ( J ) = M min c 2 = 3.24014789 E   −   20 ( J ) e n e w ( C ) = h 4.5 u n i t π u n i t ⋅ m e m p = 1.597689670 E   −   19 ( C ) q m _ n e w ( Wb ) = h ⋅ 4.5 u n i t ⋅ π u n i t ⋅ m p m e = 4.147282338 E   −   15 ( Wb ) π u n i t 4.5 u n i t ⋅ h c 2 = m p c 2 ⋅ k L 0 ⋅ m e c 2 ⋅ k L 0 = 4.15752428 E   −   17 4.5 u n i t π u n i t ⋅ h c 2 = ( 2 π ( 1 ) ⋅ k T c / α ) m p c 2 ⋅ ( 2 π ( 1 ) ⋅ k T c / α ) m e c 2 = 8.530216942 E   −   17

To simplify the calculation, GN is defined as follows:

G N = G ⋅ 1   kg = 6.68917521 E   −   11 ( m 3 ⋅ s − 2 ) G N _ n e w ( m 3 s 2 ) = α c 4.5 ( 1 ) 4 π ( 1 ) ⋅ ( 4.5 u n i t ⋅ e n e w c ) ⋅ h c 2 m p c 2 = 6.6891752 E   −   11 ( m 3 s 2 ) G N _ n e w ( m 3 s 2 ) = α c 4.5 ( 1 ) 4 π ( 1 ) ⋅ ( 4.5 u n i t ⋅ e n e w ⋅ c ) 2 ⋅ π u n i t ⋅ e n e w ⋅ c m e c 2 = 6.6891752 E   −   11 ( m 3 s 2 ) G n e w c 3 = α 2 π ( 1 ) 9 4 ( 1 ) ⋅ ( 4.5 u n i t ⋅ e n e w ⋅ c ) 2 ⋅ k L 0 1   kg ⋅ c 2 = 2.48262121 E   −   36 ( s kg ) G n e w c 3 = α 2 π ( 1 ) 9 4 ( 1 ) ⋅ ( q m c π u n i t ) 2 ⋅ k L 0 1   kg ⋅ c 2 ⋅ ( m e m p ) 2 = 2.48262121 E   −   36 ( s kg ) G n e w c 3 = α 2 π ( 1 ) 9 4 ( 1 ) ⋅ ( 4.5 u n i t π u n i t h c 2 ) ⋅ k L 0 1   kg ⋅ c 2 ⋅ ( m e m p ) = 2.48262121 E   −   36 ( s kg ) R p ( m p ) = π ( 1 ) 2 ⋅ ( M min ) 3 m p ⋅ m e ⋅ h c 2 ⋅ ( m 4 s 3 ) = 8.11158917 E   −   37 G c 3 = α 4 ⋅ ( M min ) 3 m p ⋅ m e ⋅ ( m p c 2 ) 2 ⋅ ( m 4 s 3 ) = 2.48262121 E   −   36 ( s kg )

Although Equations (49)-(56) are not essential to the main purpose of the present paper, they are retained as possible clues for further development.

 3. Methods

In this section, we prepare the framework required for the proofs presented in Section 4. For this purpose, we first explain the redefinition procedure adopted in the previous work [16], and then introduce the structural quantities and relations that will be used in the subsequent proofs. In this paper, these quantities are not introduced as independent assumptions, but as preparatory elements for examining whether a consistent readout structure can be formulated in relation to E = hν and E = mc2.

3.1. c-Normalization

In this work, physical quantities are expressed in c-normalized form by introducing appropriate factors of c. This normalization places electromagnetic quantities, mass-energy quantities, and action-type quantities into a common structural form, so that the relation between E = hν and E = mc2 can be written consistently.

e n e w → e n e w ⋅ c q m _ n e w → q m _ n e w ⋅ c ε 0 _ n e w → ε 0 _ n e w ⋅ c μ 0 _ n e w → μ 0 _ n e w ⋅ c h → h ⋅ c 2 k g → k g ⋅ c 2 G → G / c 3 G N → G N / c

Although Equations (63) and (64) are not essential to the main purpose of the present paper, they are retained as possible clues for further development.

 3.2. Redefinition Procedure

In the redefinition procedure, macroscopic and microscopic quantities are related through a normalization of the unit Ω. Under this normalization, the unit of Jm2/s is fixed as follows:

4.4873976 3.1327945 ( 1 / A ⋅ m V ⋅ m ) = 4.5 π ( 1 / A ⋅ m V ⋅ m = 1 J ⋅ m 2 / s )

By redefining the macroscopic unit 1 Ω, the following relations are obtained.

( R k ) n e w = 25957.9968739 25812.807459 = 4.5 4.4873976 ⋅ π 3.1327945 = 1.005624705 ⋅ ( R k ) MKSA 1 ( Ω ) n e w = 1.005624705 ⋅ ( Ω ) MKSA = 1.005624705 ⋅ ( Wb C ) MKSA ( Z 0 ) n e w = 378.8493104 376.7303137 ⋅ ( Z 0 ) MKSA = 1.005624705 ⋅ ( Z 0 ) MKSA ( ε 0 ) n e w = 8.8541878 E   −   12 8.8046642 E   −   12 ⋅ ( ε 0 ) MKSA = 1 1.005624705 ⋅ ( ε 0 ) MKSA ( μ 0 ) n e w = 1.2637053 E   −   06 1.2566370 E   −   06 ⋅ ( μ 0 ) MKSA = 1.005624705 ⋅ ( μ 0 ) MKSA ( e ) n e w = 1.59768967 E   −   19 1.60217663 E   −   19 ⋅ ( e ) MKSA = 1 1.005624705 ⋅ ( e ) MKSA ( q m ) n e w = 4.14728234 E   −   15 4.13566770 E   −   15 ⋅ ( q m ) MKSA = 1.005624705 ⋅ ( q m ) MKSA ( ( e ⋅ c ) 2 ε 0 ⋅ c ) n e w = 1.005624705 ( 1.005624705 ) 2 = ( ( e ⋅ c ) 2 ε 0 ⋅ c ) MKSA ( ( q m ⋅ c ) 2 μ 0 ⋅ c ) n e w = ( 1.005624705 ) 2 1.005624705 = ( ( q m ⋅ c ) 2 μ 0 ⋅ c ) MKSA 3.3. Microscopic Quantities: m <sub>gen</sub> and s <sub>gen</sub>

The parameter kL0 is introduced as a coefficient associated with the dimensional factor (m2/s). On this basis, the microscopic quantities mgen and sgen are introduced as the quantities corresponding to the macroscopic units of length m and time s in the present formulation.

1 s g e n = k L 0 c 2 = 2.04499 E   −   14 ( s ) 1 m g e n = k L 0 c = 6.13073 E   −   06 ( m ) c = 1 m g e n 1 s g e n = 6.13073 E   −   06 ( m ) 2.04499 E   −   14 ( s ) = 299792458 ( m s ) k L 0 = ( 6.13073 E   −   06 ) 2 2.04499 E   −   14 ( s ) = s g e n ⋅ c 2 = 1837.94538044 ( m 2 s ) 3.4. Condition for the Minimum Mass

From E = mc2 and E = hν, we obtain

ν = m c 2 / h

Taking the selected mass to be the minimum mass (Mmin), the condition becomes

ν ( k L 0 / c 2 ) = ( M min c 2 ⋅ k L 0 ) / h c 2 = 1

This relation is adopted as the condition for the minimum mass used in the following section.

 3.5. Structural Quantities: SI, UI, UIP, EI, and EIP

We introduce the following quantities and their numerical values.

SI: Structural invariant.

SI = 4.5 u n i t π u n i t h c 2 = ( M min ) 2 m p ⋅ m e = 8.53021694 E   −   17 ( dimensionless )

UI: Generative action.

UI = h c 2 = π u n i t 4.5 u n i t ⋅ ( M min ) 2 m p ⋅ m e = M min c 2 ⋅ k L 0 = 5.95521486 E   −   17 ( J ⋅ m 2 ⋅ s − 1 )

UIP: Unit conversion factor.

UIP = 4.5 u n i t π u n i t = 1.432394488 ( J − 1 ⋅ m − 2 ⋅ s )

EI: Energy-pair quantity.

EI = ( m e c 2 ⋅ k L 0 ) ⋅ ( m p c 2 ⋅ k L 0 ) = 4.15752428 E   −   17 ( J 2 ⋅ m 4 ⋅ s − 2 )

EIP: Square-type unit structure.

EIP = ( 4.5 u n i t π u n i t ) 2 = 2.051753969 ( J − 2 ⋅ m − 4 ⋅ s 2 ) 3.6. Fundamental Relations

The following relations are introduced as part of the structural framework:

SI = UIP ⋅ UI = 8.53021694 E   −   17 ( dimensionless ) SI = EIP ⋅ EI = 8.53021694 E   −   17 ( dimensionless ) EIP = UIP 2

At this stage, these relations are presented as structural consistency relations. Their physical meaning will be discussed in the next section.

 4. Results

In this section, we prove the following three theorems under the condition that the macroscopic unit 1 kg and the speed of light c (m/s) are fixed. In this paper, “readout” denotes the macroscopic expression of an underlying structural quantity.

Theorem 1: Mmin, sgen, and mgen cannot be determined by the usual relations alone.

Theorem 2: The minimum-mass determination equation follows from the structural relations.

Theorem 3: When 4.5unit and πunit are introduced as connection coefficients, the resulting connection relations remain mutually consistent.

4.1. Proof for Theorem 1

Equations (89)-(91) do not determine Mmin, mgen, and sgen separately, because they involve three variables but provide only two independent equations. Therefore, an additional condition is required.

1 s g e n = k L 0 c 2 = h M min c 2 1 m g e n = k L 0 c = h M min c 2 c = 1 m g e n 1 s g e n = 299792458 ( m s )

Equation (80) alone does not determine the minimum mass Mmin. Therefore, an additional equation is required, as shown in the next section.

 4.2. Proof for Theorem 2

As shown in Section 4.1, Equations (89)-(91) do not determine Mmin,mgen, and sgen separately, because they involve three variables but provide only two independent equations. Therefore, an additional equation is required.

Conveniently, Equation (81), (84) and (85) are rewritten as follows.

SI = 4.5 u n i t π u n i t h c 2 = ( M min ) 2 m p ⋅ m e = 8.53021694 E   −   17 ( dimensionless ) EI = ( m e c 2 ⋅ k L 0 ) ⋅ ( m p c 2 ⋅ k L 0 ) = 4.15752428 E   −   17 ( J 2 ⋅ m 4 ⋅ s − 2 ) EIP = ( 4.5 u n i t π u n i t ) 2 = 2.051753969 ( J − 2 ⋅ m − 4 ⋅ s 2 )

Combining Equations (92)-(94), we obtain

M min c 2 k L 0 ( kg s ) = 4.5 u n i t π u n i t ⋅ m e c 2 ⋅ m p c 2 ( kg s ) = 1.762918 E   −   23 ( kg s )

Therefore, the minimum-mass determination equation is the additional equation required in the present framework. Once kL0 is given, Mmin is determined. This completes the proof.

 4.3. Proof for Theorem 3

We prove Theorem 3 in three steps.

Step 1: The ratio 4.5unit/πunit is set as a connection ratio.

Step 2: The product 4.5unit·πunit is set as a connection product.

Step 3: Under these settings, no contradiction is found among the resulting connection relations.

4.3.1. Consistency under the Connection Ratio 4.5unit/πunit

By Theorem 2, Equation (95) gives the relation between Mmin and kL0. In the present formulation, the ratio 4.5unit/πunit is introduced as the connection ratio that relates this mass-side structure to the product-side quantities. Under this setting, the relation among Mmin, kL0, me, and mp remains consistent.

4.3.2. Consistency under the Connection Product 4.5unit∙πunit

In the second step, Equation (96) is used to examine the connection product 4.5unit·πunit.

R k n e w = 4.5 u n i t ⋅ π u n i t ⋅ m p m e = 25957.997 ( Ω )

4.3.3. Consistency of the Two Connection Coefficients

From Sections 4.3.1 and 4.3.2, the connection ratio 4.5unit/πunit and the connection product 4.5unit·πunit are both consistent with the corresponding structural relations. Therefore, when 4.5unit and πunit are introduced as connection coefficients, no contradiction is found among the resulting connection relations.

 5. Discussions

Section 4 established the numerical and structural consistency of the present framework. In this section, we discuss the possible physical meaning of these results and their implications for the connection between E = mc2 and E = hν.

5.1. Emergence of the Invariant Mass-Side Combination kg·s

Although the readout of kg is not Lorentz-invariant, the product kg·s appears as a readout-invariant quantity.

γ = 1 / 1 − ( v c ) 2

where v is the velocity.

kg → kg ⋅ γ s → s / γ

Therefore,

kg ⋅ s → ( kg ⋅ γ ) ⋅ ( γ / s ) = kg ⋅ s

In the present framework, C, Wb, Js, and Ω are treated as readout-invariant quantities under Lorentz transformation, whereas kg alone is not. This difference motivates the introduction of the mass-side invariant combination kg·s.

5.2. Meaning of the Minimum-Mass Determination Equation

5.2.1. Physical Meaning of the Minimum Mass in the Context of the Solid-State Ionics

A possible physical meaning of the minimum mass may be suggested by interpreting kTc/α as an activation energy. In the context of solid-state ionics, the relation [17],

V t h − O C V = E a 2 e ⋅ ( 1 − t i o n )

where Vth, OCV, Eaand tion are the Nernst voltage, open-circuit voltage, ionic activation energy and ionic transference number, respectively. Equation (101) may be explained by Jarzynski’s equality [18].

For example, Vth is 1.15 V, the OCV is 0.80 V, Ea is 0.7 eV, and tion is 0.

1.15 V − 0.80 V = 0.7 eV 2 e ⋅ ( 1 − 0 )

Here, α is the interaction coefficient. Therefore,

α = 1 − t i o n

Substituting Equation (103) into Equation (101),

( V t h − O C V ) ⋅ 2 e = E a ⋅ α

The left side of Equation (104) represents the energy loss due to dissipation. Therefore,

k T c = E a ⋅ α

Accordingly, kTc is the dissipation energy. From Equation (105),

E a = k T c α

Thus, we suggest that Equation (106) expresses the activation energy of the space. Next, the electrochemical potential energy of one particle (η) is the sum of the potential energy (P.E.) and the Gibbs energy (μ). Under the activation energy (Ea), these values should change.

η = μ + P . E .

Under the activation energy Ea = kTc/α, the energy redistribution is expressed as

μ p a r t i c l e = μ w a v e − k T c α > 0 P . E . p a r t i c l e = P . E . w a v e + k T c α

where μparticle, μwave, P.E.particle and P.E.wave are the Gibbs energy in the particle situation, the Gibbs energy in the wave situation, P.E. in the particle situation and P.E. in the wave situation, respectively.

In Equation (108), the minimum Gibbs energy should be positive. Considering the phase-related normalization coefficient (2𝜋), the minimum mass can be defined as follows,

M min ( kg ) = 2 π ⋅ k T c α ⋅ c 2 ( kg ) = 3.6051507 E   −   37 ( kg )

5.2.2. Interpretation under Special Relativity

Once the concept of a minimum mass is introduced into special relativity, an additional conclusion follows on the mass side. As discussed in Section 5.1, the invariant quantity on the mass side is of the product type kg·s. For the minimum mass, the c-normalized product-type readout corresponding to Equation (89) is given by Equation (111), namely,

M min c 2 k L 0 ( kg ⋅ s ⋅ ( m s ) 4 ) = h c 2 ( kg ⋅ s ⋅ ( m s ) 4 )

However, this quantity alone does not determine a definite mass. The minimum-mass determination equation, Equation (95), supplies the complementary quotient-type readout,

M min c 2 k L 0 ( kg s ) = 4.5 u n i t π u n i t m e c 2 ⋅ m p c 2 ( kg s )

From Equations (111) and (112),

M min ( kg ) = h c 2 4.5 u n i t π u n i t m e c 2 ⋅ m p c 2 = 3.60515074 E   −   37 ( kg ) s g e n ( s ) = h / c 2 4.5 u n i t π u n i t ⋅ m e c 2 ⋅ m p c 2 = 2.0449900 E   −   14 ( s )

Consequently, Mmin and sgen can be determined from the mutual consistency of the product-type and quotient-type readouts. Therefore, the minimum mass is selected by the combined mass-side structure under special relativity.

 6. Conclusions

In this paper, we reexamined empirical equations for the cosmic microwave background temperature from the viewpoint of macro-micro correspondence. We showed that the usual relations alone do not determine Mmin, mgen, and sgen separately, and that an additional relation is required within the present formulation. From the structural relations considered here, the minimum-mass determination equation was obtained as a consistency condition. On this basis, we examined the role of 4.5unit and πunit as connection coefficients. With these coefficients introduced, the ratio 4.5unit/πunit and the product 4.5unit·πunit can be written consistently with the corresponding structural relations, and no internal contradiction is found within this connection scheme.

The discussion further indicates that E = mc2 and E = hν may be organized within a common readout-based structural framework. On the mass side, special relativity provides the invariant readout kg·s, while the minimum-mass determination equation provides the complementary readout kg/s. In the present formulation, the minimum mass is selected through the consistency of these two readouts. The quantity hc2, whose unit is J m2/s, is interpreted here as a c-normalized product-side quantity relating quantum action to the mass-side readout. In this limited sense, the present framework offers a way to compare E = mc2 and E = hν through a common mass-time readout structure, rather than treating them only as isolated expressions.

At the same time, the present results should be regarded as a structural consistency analysis rather than a final physical theory. The paper examines numerical and structural consistency, but does not provide a complete microphysical derivation of all quantities appearing in the analysis. In particular, the physical origin of 4.5unit and πunit remains an open question. Further investigation will be required to clarify the physical meaning of the minimum mass and of the connection coefficients introduced in the present framework.

 Appendix Addendum on Local Unit-Shell Readout

At first sight, UIP appears to depend on the MKSA system. This differs from the previous MKSA-dependence problem [16], where the equation for Tc was shown not to be merely an MKSA-specific numerical coincidence by defining the compatibility between macroscopic and microscopic units. The present LUS problem is the remaining reference-shell display problem: UIP appears as 4.5/π only in the MKSA reference readout, while in a general local unit shell, it must be written as 4.5unit(x)/πunit(x).

 Appendix A. Readout covariance of UIP

At first sight, UIP appears to depend on the MKSA system. The structural invariant SI is dimensionless and is independent of the choice of unit system. By contrast, UI is a readout quantity. In a coherent unit system x, the same symbolic expression is used for UI.

SI = UIP ( x ) ⋅ UI ( x ) = UIP ( x ) ⋅ h c 2 ( x ) = 4.5 u n i t ( x ) π u n i t ( x ) ⋅ h c 2 ( x )

The expression hc2 is not replaced when the unit system is changed. Only its readout value changes. For example, if the readout value of hc2 in the unit system x is 1000 times larger than in the MKSA reference system, then the readout value of UIP(x)must be 1/1000 times smaller. Their product remains the same dimensionless invariant SI.

This point should be distinguished from the displayed form of UIP. UIP has an explicit MKSA-readout dependence when it is written as a numerical coefficient. The simple numerical representative 4.5/π seems to appear only in the MKSA reference readout. In a different coherent unit system, UIP generally does not have the same numerical readout value. Nevertheless, the local unit-shell form of UIP is preserved:

UIP ( x ) = 4.5 u n i t ( x ) π u n i t ( x ) = 4.5 / ( A ( x ) ⋅ m ( x ) ) π ⋅ ( V ( x ) ⋅ m ( x ) )

Here, A(x), V(x), and m(x) denote the ampere, volt, and meter of the same unit system x, not fixed MKSA units. Hence 4.5unit/πunit is form-invariant, but its readout changes through the local unit shell. In MKSA, this shell is normalized and UIP appears as 4.5/π.

 Appendix B. MKSA Mass-Shell Representative in the Reference Readout

The apparent special role of the MKSA readout can be expressed by introducing the mass-shell representative number. In a local unit shell x, we define

N ( x ) = 1   kg ( x ) M min ( x )

Here, N(x) is the representative number that expresses the ratio between the macroscopic mass unit 1 kg(x) and the minimum-mass readout scale Mmin(x), when the units of m and s are fixed. In the MKSA reference readout, this number is

N MKSA = 1   kg M min = 2.77380912 E   +   36

Next, we consider the following relationship.

C ( x ) ∝ N ( x ) W b ( x ) ∝ N ( x ) V ( x ) = W b ( x ) s ∝ N A ( x ) = C ( x ) s ∝ N

where C(x), Wb(x) represent the charge, the magnetic charge using the unit system(x) when the units of m and s are fixed, respectively. Then, usingE = m(x)c2, the following relationship can be explained.

1   kg ( x ) ⋅ c 2 = C ( x ) ⋅ V ( x ) 1   kg ( x ) ⋅ c 2 = W b ( x ) ⋅ A ( x ) M min ( x ) ⋅ c 2 = e ( x ) ⋅ V ( x ) M min ( x ) ⋅ c 2 = q m ( x ) ⋅ A ( x )

where e(x) and qm(x) denote the electric charge of one electron and the magnetic charge of one magnetic monopole using the unit system(x) when the units of m and s are fixed. Then,

4.5 u n i t ( x ) = 4.5 / ( A ( x ) ⋅ m ) π u n i t ( x ) = π ⋅ ( V ( x ) ⋅ m )

So,

UIP ( x ) = 4.5 u n i t ( x ) π u n i t ( x ) = 4.5 / ( A ( x ) ⋅ m ) π ⋅ ( V ( x ) ⋅ m ) = 4.5 π ⋅ N ( x ) N ( x ) 1 / C Wb

In Equation (B13), the local-shell structure of UIP is explicitly shown.

Appendix C. Dependence of the kL0 branch on the mass-shell representative

In the present framework, kL0 is associated with the m2/s-type readout branch. However, this branch does not form an independent LUS problem, because kL0 is connected to the mass-shell representative through the product relation. After c-normalization, this relation is written as

UI = h c 2 = ( M min c 2 ) ⋅ k L 0

Therefore, the readout of UI is determined by the combined readout of Mmin and kL0. If the readout of kL0 changes through the local unit shell, this change can be absorbed into the effective readout of the mass-shell representative. In this sense, the kL0 branch is treated as dependent on the kg branch at the level of LUS readout.

 References Miyashita, T. (2020) Empirical Equation for the Gravitational Constant with a Reasonable Temperature. JournalofModernPhysics, 11, 1180-1192. https://doi.org/10.4236/jmp.2020.118074 10.4236/jmp.2020.118074https://doi.org/10.4236/jmp.2020.118074 Miyashita, T. 2020 Empirical Equation for the Gravitational Constant with a Reasonable Temperature Journal of Modern Physics 11 10.4236/jmp.2020.118074 Miyashita, T. (2021) Various Empirical Equations for the Electromagnetic Force in Terms of the Cosmic Microwave Background Temperature. JournalofModernPhysics, 12, 623-634. https://doi.org/10.4236/jmp.2021.125040 10.4236/jmp.2021.125040https://doi.org/10.4236/jmp.2021.125040 Miyashita, T. 2021 Various Empirical Equations for the Electromagnetic Force in Terms of the Cosmic Microwave Background Temperature Journal of Modern Physics 12 10.4236/jmp.2021.125040 Miyashita, T. (2021) Various Empirical Equations to Unify between the Gravitational Force and the Electromagnetic Force. JournalofModernPhysics, 12, 859-869. https://doi.org/10.4236/jmp.2021.127054 10.4236/jmp.2021.127054https://doi.org/10.4236/jmp.2021.127054 Miyashita, T. 2021 Various Empirical Equations to Unify between the Gravitational Force and the Electromagnetic Force Journal of Modern Physics 12 10.4236/jmp.2021.127054 Miyashita, T. (2020) Erratum to “Empirical Equation for the Gravitational Constant with a Reasonable Temperature” [Journal of Modern Physics 11 (2020) 1180-1192]. JournalofModernPhysics, 11, 1559-1560. https://doi.org/10.4236/jmp.2020.1110096 10.4236/jmp.2020.1110096https://doi.org/10.4236/jmp.2020.1110096 Miyashita, T. 2020 Erratum to “Empirical Equation for the Gravitational Constant with a Reasonable Temperature” [Journal of Modern Physics 11 (2020) 1180-1192] Journal of Modern Physics 11 2020 10.4236/jmp.2020.1110096 Miyashita, T. (2021) Erratum to “Various Empirical Equations to Unify between the Gravitational Force and the Electromagnetic Force” [Journal of Modern Physics 12 (2021) 859-869]. JournalofModernPhysics, 12, 1160-1161. https://doi.org/10.4236/jmp.2021.128069 10.4236/jmp.2021.128069https://doi.org/10.4236/jmp.2021.128069 Miyashita, T. 2021 Erratum to “Various Empirical Equations to Unify between the Gravitational Force and the Electromagnetic Force” [Journal of Modern Physics 12 (2021) 859-869] Journal of Modern Physics 12 2021 10.4236/jmp.2021.128069 Miyashita, T. (2022) Empirical Equation for a Fine-Structure Constant with Very High Accuracy. JournalofModernPhysics, 13, 336-346. https://doi.org/10.4236/jmp.2022.134024 10.4236/jmp.2022.134024https://doi.org/10.4236/jmp.2022.134024 Miyashita, T. 2022 Empirical Equation for a Fine-Structure Constant with Very High Accuracy Journal of Modern Physics 13 10.4236/jmp.2022.134024 Miyashita, T. (2018) Empirical Relation of the Fine-Structure Constant with the Transference Number Concept. JournalofModernPhysics, 9, 2346-2353. https://doi.org/10.4236/jmp.2018.913149 10.4236/jmp.2018.913149https://doi.org/10.4236/jmp.2018.913149 Miyashita, T. 2018 Empirical Relation of the Fine-Structure Constant with the Transference Number Concept Journal of Modern Physics 9 10.4236/jmp.2018.913149 Miyashita, T. (2023) A Fine-Structure Constant Can Be Explained Using the Electrochemical Method. JournalofModernPhysics, 14, 160-170. https://doi.org/10.4236/jmp.2023.142011 10.4236/jmp.2023.142011https://doi.org/10.4236/jmp.2023.142011 Miyashita, T. 2023 A Fine-Structure Constant Can Be Explained Using the Electrochemical Method Journal of Modern Physics 14 10.4236/jmp.2023.142011 Miyashita, T. (2023) Explanation of the Necessity of the Empirical Equations That Relate the Gravitational Constant and the Temperature of the CMB. JournalofModernPhysics, 14, 432-444. https://doi.org/10.4236/jmp.2023.144024 10.4236/jmp.2023.144024https://doi.org/10.4236/jmp.2023.144024 Miyashita, T. 2023 Explanation of the Necessity of the Empirical Equations That Relate the Gravitational Constant and the Temperature of the CMB Journal of Modern Physics 14 10.4236/jmp.2023.144024 Miyashita, T. (2023) Simplification of Various Empirical Equations for the Electromagnetic Force in Terms of the Cosmic Microwave Background Temperature. JournalofModernPhysics, 14, 1217-1227. https://doi.org/10.4236/jmp.2023.148068 10.4236/jmp.2023.148068https://doi.org/10.4236/jmp.2023.148068 Miyashita, T. 2023 Simplification of Various Empirical Equations for the Electromagnetic Force in Terms of the Cosmic Microwave Background Temperature Journal of Modern Physics 14 10.4236/jmp.2023.148068 Miyashita, T. (2024) Correspondence Principle for Empirical Equations in Terms of the Cosmic Microwave Background Temperature with Solid-State Ionics. JournalofModernPhysics, 15, 51-63. https://doi.org/10.4236/jmp.2024.151002 10.4236/jmp.2024.151002https://doi.org/10.4236/jmp.2024.151002 Miyashita, T. 2024 Correspondence Principle for Empirical Equations in Terms of the Cosmic Microwave Background Temperature with Solid-State Ionics Journal of Modern Physics 15 10.4236/jmp.2024.151002 Miyashita, T. (2024) Ratio of Gravitational Force to Electric Force from Empirical Equations in Terms of the Cosmic Microwave Background Temperature. JournalofModernPhysics, 15, 674-689. https://doi.org/10.4236/jmp.2024.155031 10.4236/jmp.2024.155031https://doi.org/10.4236/jmp.2024.155031 Miyashita, T. 2024 Ratio of Gravitational Force to Electric Force from Empirical Equations in Terms of the Cosmic Microwave Background Temperature Journal of Modern Physics 15 10.4236/jmp.2024.155031 Miyashita, T. (2024) Algorithms for Empirical Equations in Terms of the Cosmic Microwave Background Temperature. JournalofModernPhysics, 15, 1567-1585. https://doi.org/10.4236/jmp.2024.1510066 10.4236/jmp.2024.1510066https://doi.org/10.4236/jmp.2024.1510066 Miyashita, T. 2024 Algorithms for Empirical Equations in Terms of the Cosmic Microwave Background Temperature Journal of Modern Physics 15 10.4236/jmp.2024.1510066 Miyashita, T. (2025) Normalization Methods in Algorithms for Empirical Equations in Terms of the Cosmic Microwave Background Temperature. JournalofModernPhysics, 16, 390-409. https://doi.org/10.4236/jmp.2025.163022 10.4236/jmp.2025.163022https://doi.org/10.4236/jmp.2025.163022 Miyashita, T. 2025 Normalization Methods in Algorithms for Empirical Equations in Terms of the Cosmic Microwave Background Temperature Journal of Modern Physics 16 10.4236/jmp.2025.163022 Miyashita, T. (2025) Correction Procedure for Empirical Equations for Calculating the Cosmic Microwave Background Temperature. JournalofModernPhysics, 16, 968-984. https://doi.org/10.4236/jmp.2025.167051 10.4236/jmp.2025.167051https://doi.org/10.4236/jmp.2025.167051 Miyashita, T. 2025 Correction Procedure for Empirical Equations for Calculating the Cosmic Microwave Background Temperature Journal of Modern Physics 16 10.4236/jmp.2025.167051 Miyashita, T. (2025) Solution for the Interpretation Problems Regarding Empirical Equations for Calculating the Cosmic Microwave Background Temperature. JournalofModernPhysics, 16, 1861-1877. https://doi.org/10.4236/jmp.2025.1612085 10.4236/jmp.2025.1612085https://doi.org/10.4236/jmp.2025.1612085 Miyashita, T. 2025 Solution for the Interpretation Problems Regarding Empirical Equations for Calculating the Cosmic Microwave Background Temperature Journal of Modern Physics 16 10.4236/jmp.2025.1612085 Miyashita, T. (2017) Equilibration Process in Response to a Change in the Anode Gas Using Thick SM-Doped Ceria Electrolytes in Solid-Oxide Fuel Cells. JournalofTheElectrochemicalSociety, 164, E3190-E3199. https://doi.org/10.1149/2.0251711jes 10.1149/2.0251711jeshttps://doi.org/10.1149/2.0251711jes Miyashita, T. 2017 Equilibration Process in Response to a Change in the Anode Gas Using Thick SM-Doped Ceria Electrolytes in Solid-Oxide Fuel Cells Journal of The Electrochemical Society 164 10.1149/2.0251711jes Jarzynski, C. (1997) Nonequilibrium Equality for Free Energy Differences. PhysicalReviewLetters, 78, 2690-2693. https://doi.org/10.1103/physrevlett.78.2690 10.1103/physrevlett.78.2690https://doi.org/10.1103/physrevlett.78.2690 Jarzynski, C. 1997 Nonequilibrium Equality for Free Energy Differences Physical Review Letters 78 10.1103/physrevlett.78.2690