Unifying Gravity with Electromagnetism

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We relate the upper-count gravitational coupling to vacuum electric permittivity through the dimensionless fundamental gamma parameter γ_F and its mapped charge-squared realization. The construction combines the charge-count relation with the fine structure calibration and the residual-count normalization. Fundamental-unit expressions in the Internal Frame represent configuration relations and acquire physical dimensions only through the Frames mapping. Calculations use the 2022 CODATA inverse fine structure input and the stipulated MQ realization calibration. The resulting electromagnetic comparisons retain shared-input covariance and are conditional consistency tests. The charge and electric-constant results differ from their SI/CODATA references by 4.62 conditional standard uncertainties. The coupling identity is an algebraic result within the construction; it does not establish common field dynamics.

Keywords: Unification, Gravity, Electromagnetism, Discrete, Classical, Classical Unification

1. INTRODUCTION

The construction uses the canonical dimensionless fundamental gamma parameter γ_F [1, 2]. Upright G denotes the upper-count gravitational coupling in m³ kg⁻¹ s⁻², and ε₀ vacuum electric permittivity in F m⁻¹. The fundamental mass realization m_f is in kg, and π is the dimensionless circular constant. We write γ_F,sys for the mapped charge-squared realization of γ_F, in C². The relation to be derived is

G = \frac{4 γ_{F, s y s}}{π ε_{0} m_{f}^{2}} .

(1)

Equation (1) follows by combining the upper-count gravitational expression with the mapped electric-constant solution. We first separate configuration counts from their physical realizations, then derive the finite-count gravitational normalization and the charge construction. Their common realization parameter connects the two calculated couplings [1, 2].

Measurement Quantization (MQ) describes configuration relations through discrete counts and their continuous System Frame realizations [3–9]. The distinction between the Internal Frame and the System Frame establishes the domain of each expression. It does not assign physical coordinates or dimensions to the configuration domain.

This analysis uses classical expressions with the MQ count relations and a specified realization calibration. The charge polynomial is part of that construction and is stated explicitly below. The comparison presented here does not assess alternative approaches such as inflation [10], Loop Quantum Gravity (LQG) [11, 12] or string theory [13].

The underlying account of the speed of light, escape velocity and Heisenberg’s uncertainty principle [14] is developed in [3], Eqs. (3)–(5) and Appendices B, G–I. Those arguments motivate the count interpretation. The present CODATA comparisons evaluate the resulting calculated quantities under a common calibration; they do not independently establish the discreteness premise ([3], Appx. J).

Dimensionless length, mass and time counts n_L, n_M and n_T encode configuration relations. Fundamental length, mass and time notation l_f, m_f and t_f in the Internal Frame is purely representational; it introduces no length, mass, time or other physical quality into the configuration domain. Only after the Frames mapping do their System Frame realizations carry m, kg and s. Subsequent equations containing physical units refer to those realizations. The realized speed

c = l_{f} / t_{f}

is exactly 299,792,458 m s⁻¹ [15]. We distinguish the dimensionless coefficient θ_si from momentum realization p_si in kg m s⁻¹ and angular realization Θ_si in rad. The realized fundamental relation is l_fm_f=2p_sit_f.

2. METHODS

The method has two stages. We use the MQ count and residual relations to obtain the gravitational and charge expressions, then evaluate their realizations using the 2022 inverse fine structure input [15]. The coefficient and electromagnetic demarcation count are solved together through Eqs. (6), (14) and (15). The fundamental measures use the same calibration, and all downstream uncertainties retain the dependence on that input.

The supporting count interpretation and its proposed physical applications are developed in Measurement Quantization [3]. Its appendices provide the background derivations and definitions cited here. This paper examines the coupling relations and their conditional numerical comparisons; it does not treat a shared-input match as an independent test of that interpretation. A preprint and supplementary background material are available through Online Resources.

https://www.researchgate.net/publication/361148724_OnlineResources

The earlier supporting material includes Suppl. 1, which contains calculations for Measurement Quantization [3] and its appendices, and Suppl. 2, which contains calculations for the main text of [3] and related MQ applications. The current results below use the fixed 2022 calibration and the updated charge solution [15, 1, 2].

3. RESULTS

The gravitational and electric-constant expressions use the same fundamental realizations but different count factors. Their comparison therefore requires both the residual normalization and the charge-count construction. We retain the finite-count factors until the relevant limit or mapping is taken.

Within the MQ interpretation, the right-triangle residual Q_L is a dimensionless fraction of a fundamental length count. The radial count is positive and satisfies r=n_Ll_f, where r is a separation in m. The stipulated residual-loss interpretation connects this count geometry with gravitational curvature ([3], Appx. C). The finite-count normalization and the non-Lorentz contraction associated with the Informativity differential are distinct from the cosmological expansion relation discussed separately below. The equations below examine those conditional relations. Fundamental references have no intrinsic curvature; curvature concerns realized measures in the System Frame.

Reducing an expression to dimensionless counts separates its count geometry from the units assigned to realized quantities. The universe diameter D_U is measured in m and its System Frame age A_U in s. Their dimensionless counts are

n_{L u} = D_{U} / l_{f}

and

n_{T u} = A_{U} / t_{f} .

The current diameter-age relation gives the following sequence ([16], Eq. (87)). This is an MQ cosmological relation, not an independent measurement of the universe’s diameter or a determination of an observational horizon.

D_{U} = 2 θ_{s i} A_{U} c .

(2)

\frac{n_{L u} l_{f}}{c} = 2 θ_{s i} n_{T u} t_{f} .

(3)

n_{L u} = 2 θ_{s i} n_{T u} .

(4)

Equations (2)–(4) separate the speed-of-light scale from the dimensionless diameter-age ratio. In the length-contraction example ([3], Appx. AU), n_Ll is the local separation count in the framework of the observable and n_Lo the corresponding count observed in the observer’s framework. The motion count n_Lm and light-travel count n_Lc use the same interval. Their ratio has magnitude no greater than one and is distinct from the radial count in the gravitational expressions.

n_{L o} = n_{L l} \sqrt{1 - \frac{n_{L m}^{2}}{n_{L c}^{2}}} .

(5)

The Informativity differential ([3], Appx. D) supplies a separate non-Lorentz normalization. Equation (6) gives the residual-count product. Twice that product is the finite-count realization factor; 1-2Q_Ln_L is its fractional departure from unity. The upper-count value is obtained only by taking the limit, not by substituting it at finite count.

Q_{L} n_{L} = n_{L} (\sqrt{1 + n_{L}^{2}} - n_{L}) .

(6)

Under the same MQ residual-loss interpretation, the count geometry gives Eq. (7). Here G(n_L) is the finite-count gravitational coupling in m³ kg⁻¹ s⁻². Both sides describe a radial field per unit source mass, with units m kg⁻¹ s⁻² ([3], Appx. C).

\frac{Q_{L} c^{3}}{r p_{s i}} = \frac{G (n_{L})}{r^{2}} .

(7)

To connect this gravitational expression with the electric constant, we next construct the charge realization and identify the factor shared by the mapped coupling expressions. The conclusion will depend on those count relations and the realization calibration.

1. Calculations

We begin with the finite-count gravitational relation ([3], Appx. C). Equation (8) substitutes the radial separation and speed-of-light definitions. Equation (9) uses the momentum realization in the fundamental relation. With the fundamental realizations held fixed, Eq. (10) takes the formal large-count limit to define the upper-count normalization. Realized counts range from one to finite upper bounds; infinity here denotes only mathematical normalization.

G (n_{L}) = \frac{Q_{L} c^{3} r}{p_{s i}} = \frac{Q_{L} l_{f}^{3}}{p_{s i} t_{f}^{3}} n_{L} l_{f} = \frac{Q_{L} n_{L} l_{f}^{4}}{p_{s i} t_{f}^{3}} .

(8)

G (n_{L}) = 2 Q_{L} n_{L} \frac{c^{3} t_{f}}{m_{f}} = 2 Q_{L} n_{L} \frac{l_{f} c^{2}}{m_{f}} .

(9)

G = li m_{n_{L} \to \infty} G (n_{L}) = \frac{c^{3} t_{f}}{m_{f}} = \frac{l_{f} c^{2}}{m_{f}} .

(10)

The elementary-charge construction [1, 2] supplies the electric-constant solution. Let e_f be the fundamental charge representation, n_e its count remainder, e_p its Planck-like representation and e_c the mapped classical charge. The residual n_θ is defined by Eq. (11). These fundamental-unit expressions are representational in the Internal Frame; charge units belong to the mapped realization. Mixed brackets denote nearest-whole-unit evaluation of the dimensionless representational argument.

e_{f} = \frac{l_{f} c^{2}}{4 (θ_{s i} - 1)}, n_{θ} = 42 θ_{s i} - ⌊ 42 θ_{s i} ⌉ .

(11)

The coefficients in the cubic charge denominator specify the realization construction developed in [1], Section 3.5. The integer 42 and the signed remainder n_θ have distinct roles. The Diophantine realization residual records the difference between a real-valued representational count product and its nearest whole-count value. These numerical representations do not assign physical qualities to the Configuration Domain.

The construction considers the nontrivial proper divisors of 42. Its nearest-whole realization classes select 3 as a primitive generator and 7 as a second primitive generator. The count 6 repeats the class generated by 3; 21 is the terminal member before the reference count 42 in the class generated by 7. Neither supplies an additional independent state. The count 2 does not provide the required repeating class. The first stable composite member 14 of the class generated by 7 is retained as an independent realization state ([1], Section 3.5, Table 2).

This selected basis assigns 3 to the linear term, 7 to the quadratic term and 14 to the cubic closure in Eq. (12). The explanation motivates the adopted construction; a general theorem establishing the admissibility and uniqueness of this selection remains open [1]. Agreement with an electromagnetic benchmark does not itself prove the selection principle.

n_{e} = 84 e_{f} - ⌊ 42 e_{f} ⌉, e_{p} = \frac{n_{e}}{84 + \frac{42}{277 - 3 n_{θ} - 7 n_{θ}^{2} - 14 n_{θ}^{3}}} .

(12)

The charge and inverse fine structure factors enter the fundamental realization parameter as follows. The energy normalization is stated separately.

The charge count realization. Equation (12) applies the nearest-count remainder and the polynomial divisor to obtain the Planck-like charge representation. These are the count corrections used in the subsequent parameter construction [1, 2].

The Informativity differential. The factor 2Q_Ln_L normalizes the Planck-like charge representation to the classical realization, e_c=2Q_Ln_Le_p. The fractional departure of this factor from unity expresses the non-Lorentz normalization effect, distinct from curvature. Physical evaluations of both charge realizations have units C after Frames mapping; their internal expressions remain representational.

The inverse fine structure construction. The three inverse fine structure quantities introduced here are dimensionless. The dimensionless fundamental inverse quantity

α_{f}^{- 1} = 42 θ_{s i}

leads to the Planck-like inverse quantity

α_{p}^{- 1}

in Eq. (13). The classical inverse quantity

α_{c}^{- 1}

supplies the 2022 calibration input. Its residual-normalized relation to Eq. (13), together with the interaction-count relation, determines the coefficient [1, 2].

α_{p}^{- 1} = 84 θ_{s i} - ⌊ 42 θ_{s i} ⌉ .

(13)

At the electromagnetic demarcation, the charge-coupling count relation and the classical inverse fine structure relation are ([3], Appx. AD–AE; [1, 2])

n_{L} = \frac{276}{θ_{s i}} .

(14)

α_{c}^{- 1} = 2 Q_{L} n_{L} (84 θ_{s i} - ⌊ 42 θ_{s i} ⌉) .

(15)

Using

α_{c}^{- 1} = 137.035999177 (21)

[15], Eqs. (6), (14) and (15) give θ_si=3.26239030504(25) and n_L=84.6005456716(65). The nearest-count branches are 137 for 42θ_si and zero for 42e_f in the stipulated numerical realization. They remain unchanged over the propagated input uncertainty. This calibration precedes the evaluation of the charge and fundamental parameter.

The fundamental length, mass and time representations used in Eq. (16) retain the count relation and the selected momentum normalization. The conversion to SI action is specified by the Bohr-radius matching in Eqs. (20)–(21), together with the charge realization and fundamental gamma construction of [1, 2]. Physical dimensions apply after Frames mapping. These relations specify the adopted normalization but do not independently validate the realization-class selection.

e_{f} = \frac{θ_{s i}^{2} l_{f}}{m_{f}^{2} (θ_{s i} - 1)} = \frac{l_{f}^{3}}{4 t_{f}^{2} (θ_{s i} - 1)} .

(16)

Particle and wave energy normalization. Let E_l and E_m denote the realized electromagnetic and baryonic energy expressions in J. Equation (17) retains the stated upper-count normalization ([3], Appx. Z). The finite-count energy ratio contains 4πQ_Ln_L and tends to 2π at the upper count limit. These are the selected MQ energy normalizations. Their ratio does not establish two different measured energies for the same physical process.

E_{l} = 2 π E_{m} .

(17)

Substituting the charge construction and the residual-count factor into the Planck-like inverse fine structure expression gives the fundamental realization parameter [2, Eq. (30)]. Equation (18) collects the factors; Eq. (19) expands them using Eqs. (12) and (13). The parameter is dimensionless in the configuration domain. The physical coupling equations use its mapped charge-squared realization.

γ_{F} = α_{p}^{- 1} ((Q_{L} n_{L})^{2} e_{p})^{2} .

(18)

γ_{F} = (84 θ_{s i} - ⌊ 42 θ_{s i} ⌉) {((Q_{L} n_{L})^{2} \frac{84 e_{f} - ⌊ 42 e_{f} ⌉}{84 + \frac{42}{277 - 3 n_{θ} - 7 n_{θ}^{2} - 14 n_{θ}^{3}}})}^{2} .

(19)

To make this normalization explicit, a₀ denotes the Bohr radius in m, m_e the electron mass in kg, and

ℏ = h / (2 π)

the conventional SI reduced Planck constant, with h the exact SI Planck constant in J s. The conventional Bohr-radius identity [15] and its MQ count representation ([3], Eq. (BL.3)), written with the momentum p_si rather than its numerical coefficient, give the following matching relation.

a_{0} = \frac{ℏ}{α_{c} m_{e} c} = \frac{p_{s i} t_{f}}{α_{c} m_{e} Q_{L} n_{L}} .

(20)

Using ct_f=l_f, equality of these representations fixes the SI action normalization. This conventional ℏ is distinct from the upper-count MQ action in Eq. (28).

ℏ = \frac{p_{s i} l_{f}}{Q_{L} n_{L}} .

(21)

Combining

α_{c}^{- 1} = 2 Q_{L} n_{L} α_{p}^{- 1}

and e_c=2Q_Ln_Le_p with the mapped realization of Eq. (19) gives

γ_{F, s y s} = \frac{α_{c}^{- 1} (Q_{L} n_{L}) e_{c}^{2}}{8} .

(22)

The 2022 CODATA electromagnetic identities are evaluated here with the predicted mapped charge, rather than substituting the exact SI comparison charge. With m_fc=2p_si and Eq. (21), the permittivity expression becomes

ε_{0} = \frac{α_{c}^{- 1} e_{c}^{2}}{4 π ℏ c} = \frac{4 γ_{F, s y s}}{π l_{f} m_{f} c^{2}} .

(23)

Equation (19) supplies the adopted Frames mapping for fundamental gamma; Eqs. (20)–(23) connect that construction to the conventional electromagnetic identities [15]. The resulting permeability follows from Eq. (34). These identities relate the mapped constants but are not an independent determination of the charge coefficients.

The mapped electric-constant solution [2, Eq. (31)] is Eq. (24). Multiplying it by the upper-count gravitational expression in Eq. (10) cancels the fundamental length and the speed-of-light scale, giving Eq. (25) and the introductory relation (1). This cancellation establishes the coupling identity under the stated mapping. Field equations, common dynamics and an independent physical identification of the two interactions are not derived by the cancellation.

ε_{0} = \frac{4 γ_{F, s y s}}{π l_{f} m_{f} c^{2}} .

(24)

ε_{0} G = \frac{4 γ_{F, s y s}}{π m_{f}^{2}}, G = \frac{4 γ_{F, s y s}}{π ε_{0} m_{f}^{2}} .

(25)

For comparison, Eq. (26) repeats the upper-count energy normalization in Eq. (17) ([3], Appx. Z). It is not an additional step in deriving Eq. (25).

E_{l} = 2 π E_{m} .

(26)

Equation (26) concerns energy normalization. The updated coupling relation in Eq. (25) follows from the charge solution; the energy ratio alone does not derive that coupling relation. At finite count the residual normalization remains explicit.

The factor 2π is also the dimensionless circumference-to-radius ratio of a circle. This geometric interpretation does not identify an angle with momentum or action. In Eq. (27), L is an arc length in m and the angular realization is in rad; the last equality selects r=l_f ([3], Appx. R).

L = r Θ_{s i} = l_{f} Θ_{s i} .

(27)

The momentum identity in Eq. (28) follows separately from the fundamental relation and the upper-count reduced Planck quantity, upright ħ in J s ([3], Appx. S). The angular and momentum realizations can share a numerical coefficient under the MQ calibration while retaining distinct physical dimensions.

p_{s i} = \frac{l_{f} m_{f}}{2 t_{f}} = \frac{ħ}{2 l_{f}} .

(28)

The revised equations connect the count construction to realized charge, electric permittivity and gravitational coupling through the stated mapping. A numerical match tests this construction conditional on its realization assumptions. It does not by itself establish a shared physical field or an independently verified mechanism.

2. Comparison with 2022 CODATA

Table 1 gives the calculated percentage departure of the finite-count gravitational field from its upper-count limit. It tests the normalization in Eqs. (7)–(10), rather than directly measuring the proposed contraction.

Table 1. Percentage departure from the upper-count gravitational field at the stated separation. The first column uses the 2022-calibrated electromagnetic demarcation count; other columns use the stated evaluation distances. Adapted from [3], Table M.1.

	$n_{L} l_{f}$	$500 l_{f}$	$10000 l_{f}$	a₀^a	1 m	13.794 Gly
Difference	$3.49272 \times 10^{- 3}$ %	$9.99998 \times 10^{- 5}$ %	$2.50000 \times 10^{- 7}$ %	$2.33200 \times 10^{- 48}$ %	$6.53026 \times 10^{- 69}$ %	$3.83443 \times 10^{- 121}$ %

a The atomic length a₀ is the Bohr radius introduced above, 5.29177210544(82) × 10⁻¹¹ m [15]. Its covariance with inverse fine structure is retained in the uncertainty evaluation. Let R_∞ denote the Rydberg constant in m⁻¹ and u standard uncertainty; ρ denotes a correlation coefficient and cov covariance. Equations (30)–(31) below state the exact relationship and covariance definition.

b Small nonzero calculated central values are retained. The 13.794 Gly column is a fixed comparison distance, where Gly denotes a billion light-years and a Julian year is exactly 31,557,600 s. It is not recalculated here as an independent estimate of cosmic age.

At a fixed realized separation and source mass, the finite-count field and coupling share the factor in Eq. (9). Table 1 displays its percentage departure from the upper-count result ([3], Table M.1). Equation (29) repeats the residual-count product so the field comparison can be read directly from the same normalization.

Q_{L} n_{L} = n_{L} (\sqrt{1 + n_{L}^{2}} - n_{L}) .

(29)

The finite-count and upper-count values are different quantities. For the atomic column, the Bohr radius and inverse fine structure input are correlated. Equations (30)–(31) state the exact defining relation and the covariance used in propagating that column’s uncertainty.

a_{0} = \frac{α_{c}}{4 π R_{\infty}} = \frac{1}{4 π R_{\infty} α_{c}^{- 1}} .

(30)

cov (a_{0}, α_{c}^{- 1}) = ρ u (a_{0}) u (α_{c}^{- 1}) .

(31)

NIST reports the 2022 CODATA correlation −0.99998 [17]. Combining this reported coefficient with the rounded uncertainties 8.2 × 10⁻²³ m and 2.1 × 10⁻⁸ gives a covariance of about −1.72 × 10⁻³⁰ m. This is a reconstruction from rounded tabulated data, not an unrounded covariance-matrix entry. The Rydberg uncertainty is small but nonzero; the calculation uses the reported correlation rather than declaring that constant exact.

Table 2 compares the updated predictions with 2022 CODATA [15]. Here μ₀ is vacuum magnetic permeability in H m⁻¹, k_e the Coulomb constant in N m² C⁻² and e the exact defining SI elementary charge in C. The MQ calculation uses the new charge construction, rather than deriving conventional gamma from an electromagnetic input. The inverse fine structure input still calibrates the fundamental measures, so the outputs remain correlated with CODATA electromagnetic reference quantities.

Table 2. Updated charge and electromagnetic predictions from the fundamental realization parameter, compared with 2022 CODATA [15].

Physical Constants	Updated fundamental parameter predictions
	Charge (10⁻¹⁹ C)	ε₀ (10⁻¹² F m⁻¹)	μ₀ (10⁻⁶ H m⁻¹)	kₑ (10⁹ N m² C⁻²)
MQ	1.60217663261(30) a	8.8541878035(20)	1.25663706345(28)	8.9875518017(20)
2022 CODATA [15]	1.602176634 exact	8.8541878188(14)	1.25663706127(20)	8.9875517862(14)

a The exact SI charge is used only as a comparison value. Parentheses give conditional one-standard-deviation uncertainties propagated from the common inverse fine structure input. Comparing the predicted electric constant with its CODATA reference requires propagating their difference with that shared input; treating their errors as independent is inappropriate.

The mapped electric constant determines the associated magnetic and Coulomb constants by their exact defining relations. The three calculation forms are

e_{c} = 2 Q_{L} n_{L} e_{p} = 1.60217663261 (30) \times 1 0^{- 19} C .

(32)

ε_{0} = \frac{4 γ_{F, s y s}}{π l_{f} m_{f} c^{2}} = 8.8541878035 (20) \times 1 0^{- 12} F m^{- 1} .

(33)

μ_{0} = \frac{1}{ε_{0} c^{2}}, k_{e} = \frac{1}{4 π ε_{0}} .

(34)

The results in Eqs. (32)–(34) use the fixed 2022 input and the stipulated realization calibration. With the conditional uncertainty in Eq. (32), the calculated charge differs from the exact SI defining charge by 4.62 standard uncertainties. Propagating the electric-constant prediction minus its CODATA reference with their shared inverse fine structure input likewise gives 4.62 conditional standard uncertainties. These comparisons assess the supplied construction under that calibration; they are not independent experimental significance tests [15, 1, 2]. The propagated uncertainties exclude uncertainty in the polynomial choice and the realization normalization. The 4.62 comparison therefore records conditional tension with the reference values; it does not establish improved physical accuracy.

The magnetic and Coulomb results follow algebraically from the predicted electric constant. Their numerical proximity consequently does not provide three independent tests. The charge polynomial and the numerical realization are assumptions of the supplied construction; a general proof of the selected polynomial basis remains an open question in [1]. For a separate gravitational benchmark, CODATA recommends 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻² [15]. The calculated upper-count coupling is lower by 1.48 combined standard uncertainties, using the conditional MQ uncertainty and CODATA’s separately adjusted gravitational uncertainty. This numerical comparison does not establish the mapping assumptions. The electromagnetic-demarcation value is a different finite-count quantity and is not a like-for-like laboratory benchmark.

4. DISCUSSION

The count construction organizes the gravitational and electromagnetic expressions through the same fundamental realizations [3]. Equations (10) and (24) establish their stated coupling relation, while Eqs. (11)–(16) and (18)–(23) supply its charge-dependent factor. The numerical comparison evaluates that combined construction; it does not derive an independently verified field-level unification.

The revised connection between the electric and gravitational constants depends on the charge realization, the fine structure count relation and the fundamental measures. It replaces the auxiliary electromagnetic input of the conventional gamma route with the supplied fundamental charge construction. Its remaining empirical question is whether that charge construction and realization normalization withstand tests independent of the calibration input.

The Internal Frame configuration domain has no physical, dimensional, geometric or temporal qualities. Expressions written there using fundamental-unit notation are representations of relational constraints. The Frames mapping supplies their System Frame realizations and physical dimensions. The right-triangle construction is a geometric representation of realized counts; it does not introduce a physical triangle into the configuration domain. The energy normalization remains distinct from the revised coupling relation.

The separation of configuration relations from their mapped observables is essential to this comparison. It permits a common account of the calculated couplings while preserving their distinct physical dimensions and finite-count behavior. Further tests must address the polynomial choice, the realization normalization and the proposed finite-count behavior using evidence independent of the calibration input.

5. APPENDICES

APPENDIX A: TERMS

The following symbol reference identifies the quantities already introduced in this paper and their locations. Current MQ definitions and the realization distinctions govern their use; the published background definitions are in Measurement Quantization [3].

θ_si,p_si,Θ_si. The coefficient used above is θ_si=3.26239030504(25), solved from the 2022 inverse fine structure input [15]. Its momentum realization has the same numerical coefficient in kg m s⁻¹ under the stipulated MQ calibration. The angular realization remains distinct. The 3.26239 rad value in [18], Table I, is a calculated signal/idler propagation angle for diamond (111), a 25 keV pump and degenerate 12.5 keV photons. The table also contains other solutions; this numerical correspondence does not independently determine the universal MQ coefficient or its propagated precision. The coincidence-count measurements in [19] and sum-frequency-generation measurements in [20] address their respective optical processes, not an independent calibration of the MQ momentum. The conserved quantities discussed at null infinity in [21], Section 4.4, concern asymptotic general relativity; that construction supplies no calibration used here.

c,π. The speed-of-light value introduced in Section 1 is an exact SI definition, also reported by the historical 2018 CODATA source [22]. The present calculations use the 2022 input ledger [15]. The circular constant is introduced in Section 1.

G,G(n_L). The coupling defined in Section 1 has the upper-count calculated value

G = 6.6740779383 (10) \times 10^{- 11} m³ kg⁻¹ s⁻².

At the electromagnetic demarcation, the finite-count value is

G = 6.6738448317 (10) \times 10^{- 11} m³ kg⁻¹ s⁻².

Both use the common 2022 inverse fine structure calibration [15].

l_f,m_f,t_f. The calculated fundamental realizations use the common calibration [15]. Their values are

l_{f} = 1.61619991148 (12) \times 10^{- 35} m,

m_{f} = 2.17643254057 (17) \times 10^{- 8} kg

and

t_{f} = 5.39106261131 (41) \times 10^{- 44} s .

They remain distinct from conventional Planck quantities.

n_L,n_M,n_T. The length, mass and time counts introduced in Section 1 encode configuration relations. Physical roles and dimensions belong to their System Frame realizations.

n_Ll,n_Lo,n_Lm. The motion count in Eq. (5) is distinguished from the radial count used in Eqs. (6)–(10).

n_L,r. The radial count already used above replaces the historical subscript Lr. Its meaning remains separation relative to a center of mass.

n_Lc. The light-travel count in Eq. (5) uses the same interval as its motion count.

n_θ. n_θ uses the residual convention of the current elementary-charge and electric-constant derivations [1, 2]. The integer electromagnetic half-count remains 42 and is not denoted by this residual symbol.

⌊x⌉. Here x denotes the dimensionless argument placeholder, not an additional physical variable. The mixed brackets have the nearest-whole-unit meaning introduced before Eq. (11). The calibrated branches and their uncertainty stability are stated after Eq. (15).

Q_L. The residual introduced in Section 3 is the quantity used in Eqs. (6) and (29).

Q_Ln_L. The residual-count product appears in Eqs. (6) and (29). The normalization is twice this product; its departure from unity is the Informativity differential introduced in Section 3.

h,ℏ,ħ. The conventional reduced Planck quantity is

ℏ = h / (2 π),

with the SI convention introduced before Eq. (20) and exact

h = 6.62607015 \times 10^{- 34} J s

[15]. Its value is

1.054571817 \times 10^{- 34} J s

at the shown precision. The MQ upper-count quantity used in Eq. (28) is upright

ħ = 1.0545349844417293 (56) \times 10^{- 34} J s.

Its uncertainty is a conditional shared-input calculation.

a₀. The atomic comparison in Table 1 uses the Bohr radius introduced before Eq. (20).

m_e. The electron mass occurs in the conventional and MQ Bohr-radius matching in Eq. (20); its first-use meaning and units are given immediately before that equation.

α_{f}^{- 1}, α_{p}^{- 1}, α_{c}^{- 1} .

The fundamental, Planck-like and classical inverse fine structure quantities are dimensionless. Their 2022-calibrated values are 137.020392812(11), 137.040785623(21) and 137.035999177(21), respectively [15]. The last is the calibration input. In fundamental-unit expressions all Internal Frame quantities remain representational; physical dimensions belong only to mapped realizations.

γ_F,γ_F,sys. The fundamental parameter introduced in the abstract is the configuration-domain quantity collected in Eqs. (18)–(19). The mapped charge-squared realization enters the physical coupling equations (1), (22), (23), (24), (25) and (33).

D_U,A_U,n_Lu,n_Tu. The diameter-age quantities and their count forms occur in Eqs. (2)–(4). Their scope is the MQ cosmological relation introduced in Section 3.

e_f,n_e,e_p,e_c,e. The charge representations, count remainder and mapped classical charge are introduced before Eq. (11). The exact SI comparison charge is introduced before Table 2. These quantities remain distinct in Eqs. (11)–(25) and (32).

ε₀,μ₀,k_e. The mapped electric permittivity is introduced in Section 1; the associated magnetic permeability and Coulomb constant are introduced before Table 2 and used in Eqs. (33)–(34).

E_l,E_m,L. The two selected energy normalizations are introduced before Eq. (17). The realized arc length is introduced before Eq. (27). Their subscripts and meanings do not denote the local/observed count pair in Eq. (5).

R_∞,u,ρ,cov. The Rydberg constant and statistical notation are introduced in the note to Table 1 and used in Eqs. (30)–(31). Their meaning and the rounded covariance reconstruction are unchanged.

STATEMENTS AND DECLARATIONS

The author has no relevant financial or non-financial interests to disclose.

The author has no competing interests to declare that are relevant to the content of this article.

DATA AVAILABILITY

The manuscript has data included as electronic supplementary material.

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