Conditional Calculation of the Elementary Charge Demarcation

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We calculate the elementary-charge demarcation as a dimensionless count n_L=84.6005456716(65) of fundamental lengths, with fundamental length l_f=1.61619991148(12)×10^-35 m, under the stated Measurement Quantization (MQ) calibration and 2022 recommended-constant input from the Committee on Data for Science and Technology (CODATA). The retained conventional gamma approach uses an auxiliary electromagnetic source quantity. With a common calibration, the charge, blackbody and charge-coupling expressions recover the same count by internal closure. The calculation distinguishes the discrete Internal Frame from its continuous System Frame realization and reduces the Bohr-radius relation to dimensionless counts. The numerical precision is conditional on the selected realization and calibration. Neither closure nor the cited X-ray angular calculation independently measures the demarcation or predicts elementary charge.

Keywords: Length Contraction, Elementary Charge, Planck, Count, Fundamental Measures, Demarcation

1. INTRODUCTION

Here “elementary-charge demarcation” names the electromagnetic demarcation evaluated through the elementary-charge construction, a fixed interaction count distance in the adopted MQ model [1]. We present the conventional gamma approach to the elementary-charge demarcation as a conditional MQ calculation. The symbol e_c denotes the classical MQ charge in C, and n_L is the count introduced in the abstract. Within MQ, observable separations are realized in the System Frame [1].

Special Relativity relates measurements made in inertial coordinate systems through Lorentz transformations [2]. The MQ frame definitions used below specify a separate model description.

The historical MQ construction uses gravitational curvature, the speed of light, escape velocity and Heisenberg’s uncertainty principle [3] ([4], Appx. I, J). We adopt its discrete count framework as a model assumption. The current Measurement Quantization Knowledge Base (MQKB) definitions [1] distinguish the Internal Frame, a discrete configuration domain defined relative to the fundamental measures, from the System Frame, the continuous relational encoding induced through the Frames mapping. Fundamental mass m_f in kg and fundamental time t_f in s complete the length measure introduced in the abstract. Reference [4], Appx. K, supplies historical framework context; the current definitions and realization distinctions follow [1]. The descriptions of physical constants in [5] are background applications of the MQ program, rather than independent validation of this calculation.

The blackbody ([4], Appx. AB), charge-coupling ([4], Appx. AE) and historical elementary-charge constructions are compared using the same source calibration. The dimensionless coefficient θ_si and dimensionless classical coupling α_c, defined with respect to the System Frame [1], enter this calibration. The speed of light is c=l_f/t_f, in m s⁻¹. The dimensionless residual is Q_L=√(1+n_L²)-n_L. Conventional γ is the auxiliary electromagnetic parameter in m⁶ C⁻² s⁻⁴. Equation (1) expresses the residual-count product in terms of charge and the calibrated source parameter; Eq. (2) previews the resulting count. The mixed left floor and right ceiling brackets denote rounding to the nearest whole-unit value. All arguments evaluated here lie away from half-unit ties.

The dimensionless mass and time counts n_M and n_T are relative to m_f and t_f, respectively, in the discrete Internal Frame configuration description [1]. The fundamental measures ([4], Appx. W) are the MQ measures used in the count relations, with comparison to Planck’s units [6]. Within the adopted MQ description, the Informativity differential describes the non-Lorentz contraction associated with the Frames mapping [1] ([4], Appx. D). Earlier MQ applications are reported in [4, 5, 7–11]. Those applications do not supply independent observations of the elementary-charge demarcation evaluated here.

Here the demarcation count describes realized separation. It is distinct from an orbital length count and from a primitive integer configuration of the Internal Frame.

The momentum realization p_si has units kg m s⁻¹ and is related to the fundamental measures in Eq. (5) [1]. The angular representation discussed in [4], Appx. A, concerns a separate realization. Sharing a numerical coefficient under stipulated realization maps does not establish a physical equivalence of their dimensions. Upright ħ denotes the upper-count-limit reduced Planck quantity in J s.

The retained conventional gamma approach is an auxiliary-input cross-consistency calculation. Current MQKB distinguishes it from the later fundamental parameter γ_F, classified there as a first-principles electromagnetic realization parameter [1]. That classification alone does not establish a derivation connecting it to Eq. (1). The candidate construction leaves required realization and dimensional information unspecified for the present use. Equation (17) therefore retains conventional gamma, resolved through an electromagnetic source quantity.

The purpose of this paper is to calculate the elementary-charge demarcation under a specified calibration and reduce its relation to the Bohr-radius count. The demarcation is interpreted as a realized separation within the adopted MQ model. The orbital reduction checks dimensional consistency and count cancellation under the same calibration; it supplies neither a new constraint on the exact International System of Units (SI) speed of light nor independent precision for electromagnetic measurements.

2. METHODS

The calculation uses the MQ count relations and conventional electromagnetic identities [4, 5, 7–11]. The dimensionless Planck-like coupling α_p, defined with respect to the System Frame [1], has inverse α_p^-1=84θ_si-⌊42θ_si⌉ ([4], Appx. T). Electron mass m_e in kg and Bohr radius a₀ in m enter the length calibration l_f=m_ea₀α_p/m_f. Italic ℏ is the electromagnetic-demarcation reduced Planck quantity in J s, calibrated to h/(2π), where h is the exact SI Planck constant in J s and π is the dimensionless circular constant [12]. The conventional identity m_ea₀=ℏ/(α_cc) eliminates the correlated electron-mass and Bohr-radius product. Together with Eq. (5), this gives l_f=ℏQ_Ln_L/p_si. These are shared calibration dependencies, rather than additional independent measurements.

Reference [4] presents the MQ construction and discusses comparisons with X-ray studies. Shwartz and Harris [13], Table I, calculate a signal and idler angle of 3.26239 rad for a specified diamond phase-matching configuration. This is a numerical optical solution, not a measurement of θ_si or p_si. Shwartz et al. [14] measure X-ray photon coincidence rates, while Glover et al. [15] measure X-ray and optical sum-frequency generation in diamond. These studies establish results for their respective optical processes; they do not determine the MQ demarcation or independently measure its inverse couplings. The numerical similarity with the calculated angle therefore provides no independent precision or validation for the present realization convention. Associated MQ resources are available through Online Resources.

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

Reference [4] describes two supplementary spreadsheets for the introductory paper and its applications. Their description establishes the availability of calculation aids, rather than additional independent experimental support. The current manuscript is accompanied by the fixed 2022 input ledger and reproducible calculations used in the equation audit.

3. RESULTS

1. Calculations

We obtain the coefficient by inverting the MQ fine-structure relation ([4], Appx. AD) subject to n_Lθ_si=276 ([4], Appx. AC). The source input is the 2022 CODATA inverse coupling α_c^-1=137.035999177(21) [12]. Parentheses indicate standard uncertainty in the last displayed digits. Equation (3) gives the inversion and Eq. (4) its positive admissible solution on the branch ⌊42θ_si⌉=137. The fixed SI realization convention assigns the coefficient’s numerical value to the momentum realization in kg m s⁻¹; this is a stipulated calibration, rather than a momentum measurement inferred from [13]. The convention and the length calibration in Sec. 2 are held fixed in propagating the source uncertainty. Equation (5) supplies the dimensional momentum relation used below.

The symbols e_p and e_f denote the Planck-like and fundamental charge representations in C. The dimensionless ratio r_q41 relates them in Eq. (6) and is evaluated in Eq. (16). Equation (6) is a formal conversion between charge representations. Equation (7) factorizes the historical auxiliary gamma charge expression ([4], Appx. AN, Eq. (AN.4)); cancellation gives Eq. (8) when θ_si≠1. This factorization does not derive a dimensional fundamental charge from the coefficient alone. Equation (9) rearranges Eq. (8) for the residual-count product, and Eq. (10) inserts its definition. The positive physical-charge branch and positive source parameter are used. Equations (11)–(13) report the calibrated count, its corresponding length and the residual-count product, with their common input dependence retained.

2. Internal Closure Comparison

Equation (14) obtains the count from the blackbody relation ([4], Appx. AB). Its denominator requires ℏ>2p_sil_f, so the finite-count electromagnetic quantity is used here; inserting the upper-count quantity from Eq. (5) makes the denominator vanish. Equation (15) expresses the imposed charge-coupling constraint ([4], Appx. AE). Under the calibration of Sec. 2, both expressions recover the count in Eq. (11). Their common dependencies account for this equality.

The marker n_θ is the dimensionless System Frame integer obtained by rounding half the demarcation count [1], as evaluated in Eq. (19). Equation (16) evaluates the historical charge-representation ratio, with nearest-whole-unit denominator 134 on this branch. Equation (17) resolves conventional gamma using Coulomb’s constant k_e, vacuum permittivity ε₀, vacuum permeability μ₀ and the upper-count MQ gravitational quantity G, in N m² C⁻², F m⁻¹, H m⁻¹ and m³ kg⁻¹ s⁻². These three electromagnetic routes are related identities. Equation (18) evaluates the same charge relation as Eq. (8) using that source parameter.

The conditional upper-count MQ calibration gives γ=1.19967242190(37) m⁶ C^-2 s^-4. All three source routes share the same 2022 coupling and exact SI action, charge and light-speed inputs [12]. Table 1 records the resulting closure. The original scalar gamma approximation is superseded here because it lacks a dimensional realization and fails to reproduce the retained source-constant route. Its diagnostic calculations remain in the accompanying record. The displayed uncertainties propagate the source coupling under the fixed realization convention and calibration; they do not quantify uncertainty in those assumptions.

Table 1. Conditional 2022 elementary-charge closure for the retained conventional gamma route [4], using the shared source calibration [12].

^a Every charge entry is the exact SI elementary-charge input recovered by the shared calibration. Parenthetical count uncertainties are conditional standard uncertainties. Rows and columns do not represent independent experiments.

Table 1 records algebraic closure of the source identities. Reusing the common coupling, action, charge and length calibration accounts for the identical entries.

The coincident demarcation results do not independently establish a physical explanation of length contraction or an observed separation at the calculated scale.

The metric differential ([4], Appx. T) is the representation mismatch between discrete count relations and their continuous System Frame encoding under the Frames mapping [1]. It motivates the coefficient marker used in this construction. Equation (19) applies nearest-whole-unit rounding to half the realized demarcation count. The resulting integer is used in the count expressions; it is not a dimensional momentum.

The noninteger demarcation is a realized System Frame separation. It does not redefine primitive configurations in the Internal Frame as noninteger counts.

The dimensionless fundamental coupling α_f, defined with respect to the Internal Frame [1], has inverse α_f^-1=42θ_si ([4], Appx. AA). The Planck-like coupling incorporates the metric differential ([4], Appx. T), while the classical inverse coupling includes the Informativity differential through α_c^-1=2Q_Ln_Lα_p^-1 ([4], Appx. D). Table 2 separates the current source input from the conditional MQ calculations. For historical context, the 2018 CODATA classical inverse coupling was α_c^-1=137.035999084(21) [16]. It is not used in the current calculation. The angular studies [13–15] do not report independent determinations of these inverse couplings, so their optical results are not used as comparison measurements in Table 2.

Table 2. Current inverse couplings under the fixed 2022 source input and MQ calibration.

All three quantities are dimensionless. The fundamental and Planck-like entries are calculations based on the same classical source input, rather than additional CODATA recommended constants or independent tests.

The orbital expression from [4], Appx. BL, is evaluated with the explicit classical coupling substitution. The dimensionless orbital length count is n_La=a₀/l_f, and the dimensionless electron mass count is n_me=m_e/m_f. Here the Bohr radius is the conventional length scale defined in [12], rather than a general measured radius of every atom. Equation (20) expresses that scale using the dimensional momentum realization. Equation (21) substitutes the orbital count definition. Equation (22) uses Eq. (5) and the coupling relation to cancel the fundamental measures. Multiplication by the electron mass count gives Eq. (23). Reusing the Bohr-radius identity already adopted in Sec. 2 makes this a consistency reduction, not an independent atomic-radius prediction.

The orbital count and the demarcation count describe different quantities. The cancellation in Eq. (23) relates the orbital and electron-mass counts to the Planck-like inverse coupling; it does not equate either count to the demarcation.

The resulting relation consists entirely of dimensionless quantities. The momentum realization entering Eq. (20) retains its dimensions until the fundamental measures cancel. The expansion interpretation discussed separately in [4], Appx. AX, is not tested by this cancellation.

4. DISCUSSION

The retained conventional gamma route recovers the elementary-charge demarcation under the stated shared calibration. The agreement among the charge, blackbody and coupling expressions establishes internal closure. A dimensional bridge and independently specified realization would be required to replace this route with γ_F. The accessible candidate does not supply those requirements for this calculation [1]. Reparameterizing or fitting the present closure would retain its dependence on the same inputs.

The Bohr-radius reduction expresses the adopted conventional length scale through dimensionless counts. It tests the selected coupling substitution and momentum realization, without independently establishing atomic stability or measuring an expansion parameter.

The physical interpretation remains a model claim. An independent test would require a specified observable and realization, an uncertainty budget for the model assumptions, and data not already used in the calibration. No such demarcation measurement is supplied here. The X-ray results discussed in Sec. 2 and the conditional digits in Tables 1 and 2 do not meet that requirement.

5. APPENDICES

APPENDIX A: TERMS

Current MQ definitions and realization distinctions follow MQKB [1]. Historical definitions and applications are provided in Measurement Quantization [4]. The entries below identify definitions and uses in this manuscript.

θ_si and p_si are distinguished in Sec. 1 and Eqs. (4)–(5). The optical comparison is discussed in Sec. 2 [13–15]. Reference [17], author preprint Secs. 2–4, concerns relativistic conserved quantities and gives separate definitions for linear and angular momentum. It is background context, not a realization map from angle to the dimensional momentum used here.

c is introduced in Sec. 1. The current calculation uses its exact SI value 299792458 m s⁻¹ [12].

The upper-count gravitational quantity is introduced before Eq. (17). Its conditional calibration gives G=6.6740779383(10)×10^-11 m³ kg^-1 s^-2. This is an MQ calculation under the fixed calibration, rather than the conventional CODATA recommended gravitational value.

The fundamental measures are introduced in Sec. 1 and evaluated under the calibration used in Table 1.

l_p, m_p and t_p are Planck’s units for length, mass and time, in m, kg and s, respectively [6, 1].

The fundamental counts are introduced in Sec. 1; the orbital and electron mass counts are introduced before Eq. (20).

The demarcation length count is used in Eqs. (1)–(19); the orbital count is used in Eqs. (20)–(23).

The spatial interpretation of the demarcation count is discussed in Sec. 1.

The dimensionless light-propagation count n_Lc describes a change in light position in units of l_f relative to an observer [1]. It is used in the background MQ descriptions and does not enter the calculation in Eqs. (1)–(23).

n_θ is the integer marker evaluated in Eq. (19).

The nearest-whole-unit bracket convention is stated before Eq. (1).

Q_L is introduced before Eq. (1).

The Informativity differential and its residual-count normalization enter Eqs. (7)–(23).

The finite-count reduced Planck quantity is introduced in Sec. 2; the upper-count quantity is introduced in Sec. 1. The upper-count calibration gives ħ=1.0545349844417293(56)×10^-34 J s. The finite-count quantity, not this upper-count value, enters Eq. (14).

m_e is introduced in Sec. 2 and its mass count is introduced before Eq. (20).

a₀ is introduced in Sec. 2 and its orbital count is introduced before Eq. (20).

The Planck-like coupling enters the Table 1 calibration; the fundamental inverse coupling is introduced with Table 2.

The retained conventional gamma approach and its distinction from γ_F are described in Sec. 1 and Eq. (17).

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 accompanying calculation and claims-validation records identify the fixed source inputs, calibration assumptions, conditional uncertainties, equation dependencies and source-specific evidence checks. Earlier comparison entries removed from the manuscript remain in the external revision record for provenance.

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