Configuration Relations for Atomic Charge Orbital Geometry and Action

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We examine how elementary charge, the Bohr ground state orbital and a quantum of action are organized by discrete configuration relations in Measurement Quantization (MQ). The charge expressions are presented in configuration-domain form without an explicit conversion to coulombs. For the orbital calculation, the Planck-like coupling fixes the product of the orbital length ratio and electron mass ratio. The same normalized count relation supports geometric and action descriptions, whose physical realizations retain different dimensions. We also examine the dependence of these constructions on the invariant MQ coefficient associated with universal expansion. The calculations use the 2022 recommended constants from the Committee on Data for Science and Technology (CODATA) where physical numerical values are required. Discreteness of configuration is distinguished from the size of a realized particle mass in fundamental-mass units. The normalized identities are conditional on the MQ realization and do not independently establish charge quantization or orbital stability. Here action denotes a length–momentum scale. A dynamical action functional is not evaluated in these identities.

Keywords: Atomic Structure, Atoms, Discrete, Charge, Discrete Charge, Quantum of Electromagnetic Action

1. INTRODUCTION

The Internal Frame is the discrete count-based configuration domain. The System Frame is its continuous relational encoding, induced by the Frames mapping. We examine six propositions within this framework. The equations distinguish formal count identities from their physical interpretation.

O1: The charge construction is interpreted in whole configuration counts.

O2: The orbital-radius expression has no explicit time dependence when its defining coefficients remain fixed.

O3: The ground state orbital is investigated as a realization of discrete configuration structure.

O4: The metric differential is the mismatch in representing integer count structure continuously. Its fine-structure realization contributes to the coupling that constrains the product of orbital length and electron mass ratios.

O5: Geometric length and a quantum of action share a normalized configuration relation.

O6: A relation between finite quantization series of recurring whole-count ratios and stable orbital states is proposed.

These propositions build on earlier MQ studies of fundamental measures and their physical realizations [1–7]. Fundamental length, mass and time are used as reference scales. Their values are related to Planck units, with the MQ treatment distinguishing finite-count evaluations at a specified interaction separation from the upper count limit, the asymptotic large-count normalization. A reference scale does not by itself establish a minimum allowed value for every realized observable.

The analysis adopts the discrete configuration domain and its physical realization described in MQ [1, 7]. The electromagnetic demarcation, the fixed interaction-count scale, and the nearest-whole-unit branch are inherited from that framework rather than fitted to atomic data here. The numerical coefficient is calibrated using the measured fine-structure constant [8], so reproducing that input is not an independent prediction. The present calculation does not add extra dimensions or additional adjustable parameters. It does not compare MQ with loop quantum gravity [9, 10], string theory [11] or supersymmetry.

We use n_L, n_M and n_T as dimensionless ratios associated with fundamental length l_f in meters, mass m_f in kilograms and time t_f in seconds. The speed of light c has units m s⁻¹ and is

\frac{l_{f}}{t_{f}}

when the length and time reference counts agree. Integer configuration counts belong to the Internal Frame. The realized ratios used for an observable in the System Frame need not be integer configuration counts. The reference measures and their realizations are developed using the speed of light, escape velocity and Heisenberg’s uncertainty principle [7, 12]. Their numerical calibration and the derived constants must be assessed with the relevant input uncertainties [6, 8].

We use the Bohr orbital as a setting in which the geometry, electron mass ratio and quantum of action can be compared. The reductions identify the count product shared by these descriptions. A relation between their configuration structures is distinguished from equality of their dimensional physical quantities.

2. METHODS

The analysis follows the MQ organization of configuration relations and physical realizations. The classical relations for the speed of light, gravitational curvature, escape velocity and uncertainty provide its starting context. Configuration-domain expressions are not assigned International System of Units (SI) dimensions merely by using the numerical coefficient of a realized quantity.

The discrete configuration structure and the mapping to physical realizations are premises of this analysis. Earlier derivations and proposed empirical comparisons are presented in Measurement Quantization [7], together with the referenced appendices and definitions. Consistency with that framework is distinct from an independent experimental test of the present atomic relations. The cited Online Resources provide further background.

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

Supplementary calculations are referenced in Online Resources. Such calculations serve to reproduce an expression and its chosen inputs. Reproduction alone does not establish its physical interpretation; that requires an independently specified observable and an appropriate comparison with measurement.

3. RESULTS

1. Calculations

We begin with the historical charge construction and then the orbital and action relations. The invariant dimensionless coefficient θ_si appears in the count-dependent expressions. The charge construction uses an offset at 41 times this coefficient. This offset is distinct from the interaction count used for the electromagnetic demarcation. The Informativity differential describes the non-Lorentz, count-dependent contraction of realized length under the mapping discussed in [7], Appx. D.

In the intended MQ interpretation, the same invariant coefficient participates in the expansion and atomic relations. If it remains fixed, a construction depending only on fixed reference quantities has no explicit time dependence. Comparisons with early-epoch and present-day quantities must also account for the physical realization used ([7], Appx. V, BG and AY).

Universal expansion denotes the evolution of the overall spatial realization, while relative motions of galaxies involve additional dynamical contributions. A fixed configuration coefficient alone does not identify a constant or accelerating physical rate until its time and length realization has been specified.

The charge construction originates in [7], Appx. AM. The expressions for elementary charge e_c below are retained as configuration-domain expressions. They organize the relevant count factors without an explicit dimensional conversion to coulombs. Here Q_L is the dimensionless fractional residual associated with the length count n_L. The gravitational coefficient G appears in the historical starting expression and has units m³ kg⁻¹ s⁻² in an SI realization. Its physical finite-count or upper-count realization must be specified before interpreting the reduction as an SI calculation. The mixed brackets ⌊41θ_si⌉ denote the nearest whole-unit operation.

e_{c} = 2 Q_{L} n_{L} \frac{41 θ_{si}}{⌊ 41 θ_{s i} ⌉} \frac{1}{2 (θ_{si} - 1)} \frac{G θ_{si}}{c},

(1)

e_{c} = \frac{Q_{L} n_{L}}{2} (\frac{41 θ_{si}}{⌊ 41 θ_{s i} ⌉} \frac{c^{2} l_{f}}{θ_{si} - 1}) .

(2)

The count factors in (1) and (2) describe the configuration organization of charge. Holding the reference coefficients fixed removes explicit time dependence from this construction. The reduction concerns the selected configuration realization; it does not by itself establish a conversion to a measured SI charge or a general prohibition on fractional charge.

The proposed whole-count interpretation rests on the configuration domain ([7], Appx. J). The fundamental length and time reference counts can be taken as n_L=n_T=1 in the reference construction ([7], Appx. V, following Eq. (V.4)). This choice does not establish that every realized observable is an integer multiple of every reference measure.

For the orbital reduction, a₀ denotes the Bohr ground state radius in meters and m_e the electron mass in kilograms. The electron mass ratio satisfies m_e=n_Mm_f. The Planck-like coupling α_p and classical coupling α_c are dimensionless but distinct System Frame couplings. The former satisfies

{α_{p}}^{- 1} = 84 θ_{si} - ⌊ 42 θ_{si} ⌉

([7], Appx. AA). Here the mass ratio refers to the electron realization, rather than an integer configuration count. Using the orbital relation ([7], Appx. BL, Eq. (BL.1)) gives

a_{0} = \frac{m_{f} l_{f}}{α_{p} m_{e}} = \frac{m_{f} l_{f}}{α_{p} n_{M} m_{f}},

(3)

a_{0} = (84 θ_{si} - ⌊ 42 θ_{s i} ⌉) \frac{l_{f}}{n_{M}} .

(4)

Equations (3) and (4) have no explicit time dependence if their reference measures, coupling and electron mass ratio are fixed. The intended discrete interpretation concerns the admissible configuration structure. A physical particle can have a realized mass ratio below one without requiring that the reference mass be a universal minimum particle mass. The reduction alone does not determine the admissible set of such ratios. This algebraic property does not establish a stationary quantum state or dynamical stability.

The inverse Planck-like coupling in (4) includes the fine-structure realization of the metric differential. Here the offset is the difference between 42 times the invariant coefficient and its nearest whole-unit value. The classical inverse coupling additionally carries the normalization factor associated with the Informativity differential 2Q_Ln_L ([7], Appx. AA). At the electromagnetic demarcation the interaction count is

n_{L} = \frac{276}{θ_{si}} .

Let n_a be the dimensionless orbital length ratio, so that a₀=n_al_f. The next reduction isolates the product of the length and mass ratios. This interaction count and the orbital ratio refer to different realized lengths.

α_{p} = \frac{l_{f} m_{f}}{a_{0} m_{e}} = \frac{l_{f} m_{f}}{n_{a} l_{f} n_{M} m_{f}} = \frac{1}{n_{a} n_{M}},

(5)

{α_{p}}^{- 1} = n_{a} n_{M} .

(6)

**Fig. 1.** Shared configuration with distinct length and action realizations. Adapted from [1, Fig. 2].

Equation (6) fixes the product n_an_M rather than either ratio independently. For the geometric comparison, Θ_si is the angular realization in radians and L denotes arc length in meters. Figure 1 illustrates the common configuration relation underlying the length and action descriptions.

To compare the count identity with a circle, take the orbital radius in meters r=a₀ and the normalized subtended angle Θ_siα_pn_M. Their product defines the geometric arc used in (7)–(9). The fundamental-radius representation shown in Fig. 1 gives the same arc length.

a_{0} = {α_{p}}^{- 1} \frac{1}{n_{M}} l_{f} = \frac{1}{α_{p} n_{M}} \frac{L}{Θ_{si}} .

(7)

L = a_{0} Θ_{si} α_{p} n_{M} .

(8)

Equation (8) expresses the same arc using the orbital radius. The product α_pn_M is the normalization factor on the angular realization. With r=a₀,

L = r Θ_{si} (α_{p} n_{M}) .

(9)

Using (6), the normalized arc in (7)–(9) reduces to l_fΘ_si. The Bohr radius itself remains n_al_f. Thus the angular factor and orbital radius can change their representation while preserving the same configuration relation. The distinction between the Internal Frame and System Frame is needed in this interpretation. The historical coupling normalization is given in [7], Appx. T, Eqs. (T.4)–(T.6).

The action description uses the momentum realization p_si in kg m s⁻¹, with

p_{si} = \frac{l_{f} m_{f}}{2 t_{f}} .

Let C_pθ denote the existing momentum-to-angle realization factor, in kg m s⁻¹ rad⁻¹, so p_si=C_pθΘ_si. At the upper count limit, upright ħ denotes reduced Planck action in J s, and ħ=2p_sil_f ([7], Appx. L, Eq. (L.7)). The length and action descriptions of the common configuration then satisfy

C_{p θ} L = \frac{ħ}{2} .

The historical angular construction appears in [7], Appx. R, Eq. (R.1). This motivates their interpretation as related foundational realizations without assigning action the unit meters.

\frac{ħ}{2} = n_{a} l_{f} p_{si} (α_{p} n_{M}),

(10)

\frac{2 p_{si} l_{f}}{2} = n_{a} l_{f} p_{si} (α_{p} n_{M}),

(11)

{α_{p}}^{- 1} = n_{a} n_{M} .

(12)

Substituting the upper-count action relation into (10) yields (11). Cancelling the nonzero reference factors recovers the inverse Planck-like coupling in (12), consistent with (6). The following expressions reorganize the same action identity ([7], Appx. L).

ħ = 2 n_{a} l_{f} α_{p} n_{M} p_{si},

(13)

ħ = 2 n_{a} l_{f} α_{p} n_{M} \frac{l_{f} m_{f}}{2 t_{f}},

(14)

ħ = α_{p} a_{0} m_{e} c .

(15)

The count substitutions a₀=n_al_f, m_e=n_Mm_f and

c = \frac{l_{f}}{t_{f}}

produce (15). If π denotes the dimensionless circular constant, the corresponding upper-count Planck action h is 2πħ. Reorganizing the factors gives

h = a_{0} m_{e} (2 π α_{p} c) .

(16)

Equations (15) and (16) combine the orbital radius, electron mass, speed of light and Planck-like coupling into upper-count action. The same normalized configuration supports the geometric and action descriptions. Their physical dimensions differ because the realizations differ. This is the proposed foundation-level connection. The identities alone do not supply an equation of motion or select an atomic state.

The speed of light converts between length and time realizations, while the circular factor distinguishes Planck and reduced Planck action ([7], Appx. W, Eq. (W.4), and Appx. Z, Eqs. (Z.4)–(Z.6)). The Planck-like and classical couplings must retain their respective count normalizations ([7], Appx. AA, Eqs. (AA.3)–(AA.10)). In particular, the upper-count action used above is distinct from the electromagnetic-demarcation quantity calibrated with the exact SI input.

4. DISCUSSION

The charge expressions make the intended configuration-count organization explicit. This supports examining how a charge descriptor is related to discrete configurations. Its conversion to a physical charge measurement is not established solely by the displayed products. The interpretation of fractional contributions therefore requires a specified physical realization.

For the orbital reduction, the Planck-like inverse coupling

{α_{p}}^{- 1}

equals n_an_M. The reference length and mass cancel from this coupling identity. It constrains a product and does not independently select a unique electron mass or orbital radius. The arc relations in (7)–(9) and action relations in (10)–(16) use the same normalization in their respective realizations.

A realized mass below the reference mass is compatible with using that reference as a scale. With the 2022 CODATA electron mass and the MQ-calculated reference mass, the electron mass ratio is 4.1854656848×10^-23. This is a central-value ratio; its standard uncertainty depends on the covariance of the source inputs. In the intended MQ interpretation, mass remains associated with discrete configuration structure even when its realized ratio is below one. This motivates investigating how mass arises from the length and time organization, but a complete explanation is not supplied by the present reduction.

The diameter–age construction originates in [7], Appx. AZ, Eq. (AZ.7). With the current dimensional realization, the invariant coefficient enters the adopted MQ cosmological model relation. Here D_U is the physical universe diameter, reported in light-years, and A_U is physical age, reported in billions of years. For a consistent physical equation, convert the diameter to meters and the age to seconds, or express the speed of light in matching length and time units. The current dimensionless count relation then gives

D_{U} = 2 θ_{si} c A_{U} .

(17)

The earlier configuration-domain mass identification ([7], Appx. AZ, Eq. (AZ.8)) was written as

m_{f} = \frac{D_{U}}{A_{U} c} .

(18)

With physical diameter, age and speed expressed consistently, the right-hand side of (18) is dimensionless. The proposed conversion of this configuration identification to an SI mass therefore requires a separate mass realization. Changing light-years and billions of years to SI units alone does not supply it.

A change in the invariant coefficient changes the fundamental inverse coupling

{α_{f}}^{- 1} = 42 θ_{si}

defined for the Internal Frame. The metric differential separates the fundamental and Planck-like forms; the Informativity differential supplies the additional classical normalization ([7], Appx. AA). The resulting orbital relation depends on the same coefficient ([7], Appx. BL). Table 1 lists the retained quantization-series counts through 64. Its index n_θ is the dimensionless System Frame count of the invariant coefficient resolved from the Internal Frame demarcation after the metric differential. The instance n_θ=42 identifies the series discussed here. The integer length-count endpoints in Table 1 are series-enumeration labels, not substitutions for the physical interaction count or orbital ratio.

Table 1. Repeating quantization ratios for counts n_θ with respect to the System Frame. Adapted from [7, Table AC.2].

Beginning n_L	Ending n_L	Repeat Count
3	6	2
4	40	10
7	21	3
11	44	4
15	105	7
19	627	33
23	322	14
26	52	2
27	135	5
31	93	3
34	204	6
35	70	2
39	78	2
41	82	2
42	1050	25
49	147	3
50	200	4
53	265	5
58	116	2
61	6893	113
64	128	2

The connection between repeating series and additional orbital states remains a conjecture. A finite series of configuration solutions does not by itself specify which members represent stable orbitals. Establishing that selection requires a physical realization and a stability analysis.

Table 1 gives 25 repeats for the series indexed by 42. A possible connection to the Pythagorean relation 3²+4²=5² remains conjectural. The table alone does not establish a physical connection between that count and an orbital stability rule.

5. APPENDICES

APPENDIX A TERMS AND REFERENCE VALUES

The entries below recall first-use definitions and numerical reference values. Measurement Quantization [7] provides the earlier published framework.

Shwartz and Harris calculate signal and idler propagation angles of 3.26239 rad for one maximally entangled state in diamond, using a 25 keV pump and 12.5 keV signal and idler energies [13, Table I]. This is a configuration-dependent phase-matching result, rather than an experimental determination of a universal polarization angle. Related X-ray down-conversion and optical wave-mixing experiments are reported in [14] and [15]. The MQ momentum calculation based on the 2022 CODATA input is 3.26239030504(25) kg m s⁻¹ [8]. The angular and momentum realizations have different dimensions; similarity of their numerical values in chosen units is not an independent test of their physical identification. Chen et al. discuss conserved quantities at null infinity in general relativity [16, Secs. 4.1–4.4]; that context does not establish the MQ angle–momentum realization.

The speed of light introduced earlier has the exact SI value 299,792,458 m s⁻¹ [17, 8]. Equal reference length and time counts recover

c = \frac{l_{f}}{t_{f}} .

G at the upper count limit is 6.6740779383(10) × 10⁻¹¹ m³ kg⁻¹ s⁻². Italic G identifies a finite-count evaluation. At the electromagnetic demarcation

n_{L} = \frac{276}{θ_{si}},

its MQ calculation is 6.6738448317(10) × 10⁻¹¹ m³ kg⁻¹ s⁻² [8]. These are MQ calculations using the specified CODATA input, rather than CODATA recommendations.

The fundamental measures introduced earlier have the MQ-calculated values

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

using the fixed 2022 CODATA inputs [8]. The three uncertainties share the calibration dependency and must not be treated as independent.

l_p, m_p and t_p are Planck’s units for length in meters, mass in kilograms, and time in seconds. Their conventional definitions and recommended values are listed in [8]. A Planck mass is a reference scale and does not impose a lower bound on particle masses.

The count notation introduced earlier distinguishes discrete Internal Frame configuration counts from ratios used to represent physical observables in the System Frame. The electron mass ratio in the orbital reductions can be below one. A mapping rule explaining the associated discrete mass realization is not derived here.

The length ratio n_L applies to the realized separation under discussion. Its interpretation is fixed by that separation and its frame context.

A center-of-mass radial position is represented by the length count n_L introduced above.

A light-propagation length ratio may be distinguished as n_Lc when its context requires that notation.

At the electromagnetic demarcation, the series-index operation introduced above gives

n_{θ} = ⌊ \frac{n_{L}}{2} ⌉ = 42 .

The recalculated argument is 42.3002728358(32) and remains away from a half-unit boundary.

The mixed-bracket convention is the nearest-whole-unit operation already specified before (1).

The fractional residual Q_L has the realization meaning introduced with the charge expressions.

The factor 2Q_Ln_L supplies the Informativity differential normalization for the applicable interaction count. Its value and physical role depend on that count and the selected realization.

The electromagnetic-demarcation reduced Planck constant is ℏ = 1.054571817… × 10⁻³⁴ J s, equal to the exact SI input in this calibration. At the upper count limit, upright ħ has the MQ-calculated value 1.0545349844417293(56) × 10⁻³⁴ J s [8]. Parenthetical digits give the standard uncertainty in the final displayed digits. Reproduction of a calibration input is not independent validation.

The electron mass used in the orbital calculation is 9.1093837139(28) × 10⁻³¹ kg [8].

The Bohr radius used for comparison is 5.29177210544(82) × 10⁻¹¹ m [8]. It is a conventional input-dependent reference value rather than an independent test when the same source constants calibrate the MQ calculation.

The fundamental, Planck-like and classical inverse couplings are distinguished in the main text. The first uses the invariant coefficient in the Internal Frame; the latter two retain their different System Frame normalizations.

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 calculation inputs and their source uncertainties are identified in the cited MQ and CODATA sources [7, 8].

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