Distance Corresponding to the Length Contraction Associated with a Measure of Charge Couplings

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

Measurement Quantization (MQ) gives the count n_L=84.6005456716(65) of fundamental lengths from the fine-structure input. The fundamental length is

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

using the stated MQ realization and the 2022 CODATA inputs. Parentheses denote standard uncertainties in the final digits, conditional on the stated MQ calibration. The count is obtained by inversion of the classical inverse fine-structure value, rather than by an independent measurement. The blackbody and elementary-charge expressions are evaluated with the same consistent source calibration. Under that calibration their count solutions coincide algebraically, and their agreement is an internal closure check. The calculation distinguishes the discrete Internal Frame from its continuous System Frame realization. A ground-state orbital starting expression is then evaluated with the explicit substitution of the classical MQ coupling. Cancellation of the dimensional measures gives a relationship between the orbital length count, electron mass count and the inverse coupling.

Keywords: Length Contraction, Charge Couplings, Fine Structure Constant, Planck, Count, Fundamental Measures, Demarcation

1. INTRODUCTION

We discuss a calculation of the distance associated with charge couplings. The electromagnetic demarcation is the fixed interaction count distance used in MQ. It is represented here by the length count introduced in the abstract, with realized separation n_Ll_f. The Internal Frame is the discrete configuration domain of MQ fundamental counts. The System Frame is the continuous relational encoding of those configurations, in which observables are realized. The Frames mapping is the kinematic transformation from the discrete configurations to continuous relational structure, preserving count and realization consistency ([1], Appx. K). Planck’s radiation work [2] is retained as historical context. The conventional ground-state orbital model is discussed by Bohr [3]. We consider the product expression recorded in the MQ source [1], Eq. (22), with the explicit coupling substitution developed in Eqs. (23–25). The resulting expression is dimensionless after the length and mass measures are written as counts.

The MQ approach is developed in [1, 4-9]. MQ uses counts of fundamental measures alongside dimensional quantities. The fundamental mass m_f has units kg and the fundamental time t_f has units s; the fundamental length was introduced in the abstract. The generic counts n_M and n_T are dimensionless primitive mass and time counts in the Internal Frame. The calculated demarcation count describes a realized separation in the System Frame. The length count n_L in this calculation refers specifically to the demarcation, rather than to a change in position.

The derivations using gravitational curvature, the speed of light, escape velocity and Heisenberg’s uncertainty principle [10] are used within MQ to construct the count framework employed here ([1], Appx. I, J). Heisenberg’s uncertainty principle alone does not establish that physical length consists of primitive integer counts. The role of physically significant references is developed in [1], Appx. K. In the current terminology, primitive count configurations belong to the Internal Frame, while their observable relational representation belongs to the System Frame. The applications to physical constants are discussed in [9].

The present calculation evaluates MQ count relationships using the specified conventional source inputs and the stated realization calibration. It determines the invariant coefficient by inversion of the measured fine-structure input. This parameter determination should be distinguished from an independent prediction of that input. Comparative assessments of inflation [11], background independence and Loop Quantum Gravity (LQG) [12, 13], and String Theory [14] are outside the scope of this paper; these approaches do not supply inputs to the calculation.

We begin with the charge-coupling relation established in the MQ source equations. The symbol θ_si denotes the invariant dimensionless coefficient. Equation (1) gives the demarcation count in terms of that coefficient.

n_{L} = \frac{276}{θ_{s i}} = 84.6005456716 (65) .

(1)

The MQ count realization defines a non-Lorentz contraction within its framework. Relativistic context is supplied in [15]; the moving-rod length comparison in special relativity is given in [16], Secs. 2–4. The Informativity differential is the MQ non-Lorentz length contraction arising from count normalization under the Frames mapping ([1], Appx. D). It quantifies the difference between count separation and its realized representation and is distinct from curvature. Here Q_L is the dimensionless right-triangle residual, and the dimensionless product Q_Ln_L is the count-normalization quantity used in the equations below. Equation (4) defines the residual for positive length count.

The purpose of this paper is to determine the charge-coupling count from the selected source input, relate that count to the Informativity differential, and examine its relationship to the orbital count expression. The comparison of demarcation formulas must preserve their shared source dependencies.

2. METHODS

This paper applies the MQ count relationships associated with gravitational curvature, the speed of light, escape velocity and Heisenberg’s uncertainty principle. Their interpretation as discrete measure is part of the MQ framework adopted for the calculation. Applications of that framework are presented in [4, 5, 6, 7, 8, 9, 1]. The calculations below test the conditional consistency of these relationships.

Measurement Quantization [1] supplies the framework derivations, comparisons, referenced appendices, and term definitions used here. Those sources motivate the discrete-count interpretation. The shared-input calculations in this paper do not constitute an independent empirical demonstration of fundamental discreteness or countability. A preprint is identified in Online Resources.

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

The supplementary spreadsheets described in Online Resources provide calculation context for the MQ expressions. The present results use the explicit 2022 CODATA inputs and calibration stated below. Reproducing a spreadsheet calculation checks its numerical implementation; it does not create an additional independent observation.

3. RESULTS

1. Calculations

The dimensionless quantization ratio r_q in Eq. (2) uses the coefficient count n_θ, a dimensionless System Frame coefficient-count marker evaluated in Eq. (22). It does not use the length count. The mixed brackets denote rounding to the nearest whole-unit value, with a left floor bracket and a right ceiling bracket; neither ordinary floor nor ordinary ceiling alone is intended ([1], Appx. AC). The ratio is not set equal to one. The separate length-count product in Eq. (3) has a whole-unit value.

r_{q} = \frac{n_{θ} θ_{s i}}{⌊ n_{θ} θ_{s i} ⌉} .

(2)

Equation (3) supplies the separate charge-coupling constraint. It should not be inferred by replacing the coefficient count in Eq. (2) with the length count.

n_{L} θ_{s i} = 276 .

(3)

Equation (3) relates the calculated demarcation length count to the dimensionless coefficient. In this construction the value 276 is a stipulated MQ count constraint, not a measured dimensional momentum or a product of two independent measurements. Together with Eq. (7), it determines the coefficient from the selected fine-structure source input.

The symbol α_c denotes the dimensionless classical MQ fine-structure coupling. We identify its source value with the 2022 CODATA conventional coupling for this inversion [17]. Shwartz and Harris [18] calculate phase-matching angles for a specified diamond configuration. The related down-conversion experiment [19] and X-ray/optical mixing experiment [20] concern different observables. These studies provide external context and are not substituted as momentum or fine-structure inputs. Equations (4)–(6) express the residual in the form needed for the inversion ([1], Appx. AD).

Q_{L} = \sqrt{1 + n_{L}^{2}} - n_{L},

(4)

Q_{L} θ_{s i} = \sqrt{θ_{s i}^{2} + (θ_{s i} n_{L})^{2}} - n_{L} θ_{s i},

(5)

Q_{L} = \frac{{(θ_{s i}^{2} + {(276)}^{2})}^{1 / 2} - 276}{θ_{s i}} .

(6)

Equation (7) gives the classical inverse fine-structure relation ([1], Appx. AA). Equation (8) rearranges it for the residual-count product. All factors in these two equations are dimensionless.

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

(7)

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

(8)

We substitute Eq. (6) into Eq. (7) and impose Eq. (3). The sole uncertain input to this inversion is the 2022 CODATA value

α_{c}^{- 1} = 137.035999177 (21)

[17], with α_c=7.2973525643(11)×10^-3 at reported precision. Equations (9) and (10) are algebraic rearrangements; Eq. (11) reports the recalculated dimensionless coefficient. The momentum realization operator ℛ_p maps the invariant dimensionless coefficient to a quantity with momentum units; its index identifies the momentum realization. Square brackets around a quantity report its SI units. Equation (12) introduces its momentum realization p_si, with units kg m s⁻¹, which remains dimensionally distinct from the coefficient. We adopt the MQ SI realization in which the numerical momentum value in kg m s⁻¹ inherits the invariant coefficient. This dimensional calibration is held fixed; it is not independently measured by the inversion.

2 n_{L} α_{c} = \frac{θ_{s i}}{(\sqrt{θ_{s i}^{2} + 27 6^{2}} - 276) (84 θ_{s i} - ⌊ 42 θ_{s i} ⌉)},

(9)

2 \cdot 276 α_{c} = \frac{θ_{s i}^{2}}{(\sqrt{θ_{s i}^{2} + 27 6^{2}} - 276) (84 θ_{s i} - ⌊ 42 θ_{s i} ⌉)} .

(10)

θ_{s i} = 3.26239030504 (25) .

(11)

p_{s i} : = ℛ_{p} (θ_{s i}) = \frac{l_{f} m_{f}}{2 t_{f}}, [p_{s i}] = kg m s^{- 1} .

(12)

Equations (13)–(15) give the recalculated count, realized distance and residual-count product. To evaluate the distance, we use the MQ fundamental-length calibration in [1], Eq. (22) and Appx. BM, Eq. (BM.3), together with the 2022 CODATA relationships [17]. The conventional Bohr-radius identity eliminates the correlated electron-mass and radius product from that calibration. The calculation uses the exact CODATA Planck constant

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

The reduced Planck quantity ℏ=h/(2π), in J s, supplies the electromagnetic-demarcation action calibration; π is the dimensionless circle constant. We propagate the single inverse-coupling uncertainty through the dependent quantities. No 2018 uncertainty is retained. The quoted uncertainties propagate the selected source input with the MQ count constraint and realization calibration fixed; they do not include an independently estimated uncertainty in those model assumptions.

n_{L} = \frac{276}{θ_{s i}} = 84.6005456716 (65) .

(13)

n_{L} l_{f} = 1.36731394425 (21) \times 1 0^{- 33} m .

(14)

Q_{L} n_{L} = 0.499982536417403 (27) .

(15)

2. Source Comparisons

For comparison we evaluate the blackbody and elementary-charge count expressions ([1], Appx. AB and AN) in Eqs. (16) and (17). The electromagnetic-demarcation action ℏ was introduced above. Upright ħ denotes its distinct upper-count-limit value. The symbol e_c denotes the classical elementary-charge quantity in C. The blackbody formula requires ℏ>2p_sil_f; substituting the upper-limit action would make its denominator zero. The speed of light is c=l_f/t_f=299792458 m s^-1, an exact source constant [17]. In Eq. (17), the conventional gamma parameter γ has units m⁶ C^-2 s^-4.

n_{L} = p_{s i} l_{f} \sqrt{\frac{1}{ℏ (ℏ - 2 p_{s i} l_{f})}} .

(16)

n_{L} (\sqrt{1 + n_{L}^{2}} - n_{L}) = \frac{l_{f} c^{2}}{2 e_{c}} \sqrt{\frac{2}{(84 θ_{s i} - ⌊ 42 θ_{s i} ⌉) γ}} .

(17)

Equation (17) gives an auxiliary-input count calculation using charge and the gamma parameter. It is evaluated with the exact SI charge input

e_{c} = 1.602176634 \times 10^{- 19} C

[17]. The underlying elementary-charge construction in [1], Appx. AM is not an additional independent input to the calculation used here. The three demarcation expressions must be evaluated with consistent measures and action normalization.

The conventional gamma parameter introduced before Eq. (17) is the auxiliary parameter used in [1], Appx. AR and AS. Equations (18)–(20) give three equivalent routes using Coulomb’s constant k_e in N m² C^-2, vacuum permittivity ε₀ in F m^-1, or vacuum permeability μ₀ in N A^-2. Upright G is the upper-count-limit MQ gravitational quantity, in m³ kg^-1 s^-2. For Table 1 these electromagnetic quantities are evaluated from the same 2022 CODATA coupling and exact action, charge and light-speed inputs [17]; they are not three independent measurements.

γ = 2 k_{e} G, [γ] = m^{6} C^{- 2} s^{- 4} .

(18)

γ = \frac{G}{2 π ε_{0}}, [γ] = m^{6} C^{- 2} s^{- 4} .

(19)

γ = \frac{μ_{0} G c^{2}}{2 π}, [γ] = m^{6} C^{- 2} s^{- 4} .

(20)

Equation (21) solves for elementary charge using the same gamma construction. Table 1 reports the consistent 2022-source recalculation. Because the source dependencies are shared, the charge result reproduces the exact input by algebraic closure; it does not supply an independent charge prediction. In Table 1, the unsubscripted source symbols identify the conventional fine-structure constant, reduced Planck action and elementary charge. Their adopted values are assigned to the classical coupling, electromagnetic-demarcation action and charge introduced above.

e_{c} = \frac{l_{f} c^{2}}{2 Q_{L} n_{L}} \sqrt{\frac{2}{(84 θ_{s i} - ⌊ 42 θ_{s i} ⌉) γ}} .

(21)

Table 1. Recalculated elementary-charge closure using the demarcation constructions from [1] and a common 2022 CODATA source calibration [17].

Source Constants	Gamma Source Measures
Source Constants	n_L of source	Charge via ɛ₀ (10⁻¹⁹ C)	Charge via μ₀ (10⁻¹⁹ C)	Charge via k_e (10⁻¹⁹ C)
α	84.6005456716(65)	1.602176634	1.602176634	1.602176634
ℏ	84.6005456716(65)	1.602176634	1.602176634	1.602176634
e^a	84.6005456716(65)	1.602176634	1.602176634	1.602176634

a The charge-associated count uses the defined SI charge in Eq. (17). All three rows share the calibrated fundamental measures; all gamma columns share the same coupling input. The charge entries are exact closure results within this construction, not independently measured values.

The three gamma routes in Table 1 are equivalent because the CODATA permeability, permittivity and Coulomb constant are dependent functions of the same coupling, action, charge and light speed [17]. The uncertainty calculation retains those dependencies rather than combining their uncertainties as independent contributions. Using the fundamental-length calibration of Eq. (14), the blackbody and auxiliary charge inversions recover the same count as Eq. (13). Their equality under this calibration replaces the older comparison of differently rounded source values.

The metric differential is the mismatch between discrete count structure and its continuous System Frame representation under the Frames mapping ([1], Appx. T). It is a representation mismatch rather than a physical deformation. The coefficient-count marker is associated with the nearest-whole-unit construction used here. Equation (22) rounds half the demarcation count to obtain 42. The input uncertainty does not approach a half-unit boundary. The coefficient count in Eq. (2) and length count in Eq. (3) remain distinct; the marker is not itself a physical length or momentum.

n_{θ} = ⌊ n_{L} / 2 ⌉ = ⌊ 84.6005456716 / 2 ⌉ = 42 .

(22)

A calculated noninteger demarcation count represents realized separation in the System Frame. It is not a primitive integer configuration of the Internal Frame. Equation (22) applies the nearest-whole-unit operation to obtain the integer marker used in the dimensionless coupling expressions.

The charge-coupling, blackbody and elementary-charge formulas have different source roles. Their count solutions coincide when the same 2022-source calibration and the supported normalization identities are used here. This conditional equality does not establish that independently measured phenomena would supply identical demarcations. The charge-coupling construction provides the reference count for the present internal comparison.

The source nomenclature distinguishes the dimensionless fundamental coupling α_f, the Planck-like coupling α_p, and the classical coupling already introduced ([1], Appx. AA). The fundamental form is defined with respect to the Internal Frame; the Planck-like and classical forms refer to the System Frame. Their inverse forms distinguish the marker contribution, the metric differential needed for System Frame representation, and the finite-count normalization associated with the Informativity differential. Table 2 updates the source comparison using the 2022 input [17]. The source-based and MQ classical values agree by the inversion used to determine the coefficient; the Planck-like comparison is likewise tied to the adopted fundamental-length calibration. These agreements are internal closure checks, not independent experimental tests.

Table 2. Recalculated inverse fine-structure forms using the structure of the comparison in [1] and the 2022 CODATA source input [17].

Inverse-coupling forms		Values
Inverse-coupling forms	2022 inputs a	MQ calculation	Historical MQ	Difference
α_f⁻¹=42θ_si	—	137.020392812(11)	137.02039	-
$α_{p}^{- 1} = 84 θ_{s i} - ⌊ 42 θ_{s i} ⌉$	137.040785623(21)	137.040785623(21)	137.04078	0 (closure)
α_c⁻¹=4πɛ₀ℏc/e²	137.035999177(21)	137.035999177(21)	137.03600	0 (closure)

a The 2022 column contains the conventional inverse coupling or a quantity derived from it using the stated calibration. The Historical MQ column preserves the comparison values transcribed in [1], Table AA.1. These are historical MQ-derived entries, not fine-structure measurements reported by [18-20]. Zero differences compare the 2022 and MQ calculation columns and denote algebraic closure of shared-input calculations, not zero experimental uncertainty.

The demarcation relation n_L=276/θ_si is dimensionless ([1], Appx. AE, Eq. (AE.1)). The coefficient is not a dimensional Planck momentum. Its momentum realization enters Eq. (12), while the count relation uses the invariant coefficient. This distinction is retained in the orbital calculation.

The demarcation count represents a separation in units of the fundamental length within the System Frame realization. The numerical comparison with the blackbody and elementary-charge constructions depends on the common input calibration, rather than on a dimensional identification of their source observables.

We now retain the orbital product expression recorded in the MQ source [1], Eq. (22), and deliberately use α_c in place of its source coupling. Here a₀ is the radius in m calculated under that substitution, m_e is electron mass in kg,

n_{L_{a}}

is the dimensionless orbital length count with

a_{0} = n_{L_{a}} l_{f},

and n_m=m_e/m_f is the dimensionless electron mass count. The orbital and electron mass counts here describe System Frame realizations, rather than primitive integer configurations. The calculated radius must be distinguished from the conventional CODATA Bohr radius used in the source calibration. Equation (23) is the stipulated starting expression; Eq. (24) writes its radius as a length count; Eq. (25) cancels the fundamental mass and substitutes Eq. (7). No additional finite-count factor is inserted into Eq. (23).

a_{0} = \frac{l_{f} m_{f}}{α_{c} m_{e}} .

(23)

n_{L_{a}} l_{f} = \frac{l_{f} m_{f}}{α_{c} m_{e}} .

(24)

n_{L_{a}} = \frac{m_{f}}{α_{c} n_{m_{e}} m_{f}} = \frac{1}{α_{c} n_{m_{e}}} = \frac{2 Q_{L} n_{L} (84 θ_{s i} - ⌊ 42 θ_{s i} ⌉)}{n_{m_{e}}} . .

(25)

Equation (25) gives

n_{L_{a}} n_{m} = α_{c}^{- 1} .

Thus the orbital length count differs from the demarcation count, and their relationship uses the residual-count factor through Eq. (7). The right side contains the coefficient obtained from the CODATA inverse-coupling input. It is not the reciprocal of the Planck-like bracketed expression, and it does not equate the residual-count product alone to the coupling.

Every factor in the final count relationship is dimensionless. The dimensional length and mass measures have canceled after the orbital expression was written as counts. This gives a count representation of the stipulated orbital relation; the cancellation alone does not establish that physical force has been eliminated or that the orbital radius is independently predicted.

4. DISCUSSION

Under the common 2022 CODATA-based calibration, the three demarcation calculations used here close on the same count. The older differences in their eighth digits cannot be retained as current results after this consistent recalculation. A comparison of independent source determinations would require their own input and uncertainty records. The present calculation instead establishes consistency of the selected count, action and gamma relationships.

An empirical test would require a separately specified observable, its measurement procedure, and input and uncertainty records that permit assessment of shared dependencies. The algebra in Eq. (25) establishes the count representation under the stipulated coupling substitution. It does not supply that independent test.

Within the MQ discrete gravitational description, curvature is related to realized count structure ([1], Appx. C). The orbital calculation developed here does not by itself establish an equivalent account of every force. It shows how the chosen dimensional orbital relation can be represented through counts; the physical interpretation of that representation remains distinct from the algebra checked in this paper.

5. APPENDICES

APPENDIX A: TERMS

The source symbol and term definitions appear in Measurement Quantization [1], Appx. BP and BQ. This appendix indexes the notation and source conventions already introduced in the narrative, without introducing alternative definitions.

The dimensionless coefficient is reported in Eq. (11), and its momentum realization is given in Eq. (12). Its angular realization carries radians and remains distinct from momentum. In [18], Table I, 3.26239 radians is a calculated signal/idler propagation angle relative to the atomic planes for a selected frequency-degenerate diamond configuration. It is not a measured universal coefficient or a momentum. The related experiments [19, 20] do not establish that identification. Reference [21] concerns conserved quantities at future null infinity in general relativity; it does not identify an angle with linear momentum or validate the MQ realization calibration.

The light-speed source and its fundamental length–time relationship are given before Eq. (17), using the exact 2022 CODATA input [17].

The upper-count-limit MQ gravitational quantity is calculated as

G = 6.6740779383 (10) \times 10^{- 11} m^{3} k g^{- 1} s^{- 2} .

The finite-count value at the common demarcation is

G = 6.6738448317 (10) \times 10^{- 11} m^{3} k g^{- 1} s^{- 2} .

These are MQ calculations based on the source calibration, not CODATA recommendations for gravitational coupling. Their displayed uncertainties are conditional input-propagation uncertainties under the fixed MQ calibration, rather than the experimental uncertainty of a gravitational measurement.

The fundamental length, mass and time used here are introduced in the abstract and Sec. 1. Their calculation uses the MQ calibration and the specified conventional source inputs; they remain distinct from conventional Planck units.

The conventional Planck length, mass and time are denoted l_p, m_p and t_p, respectively, with units m, kg and s. They are not substituted for the fundamental measures in the demarcation equations.

The generic fundamental mass and time counts are introduced in Sec. 1. The length count used in Eqs. (1)–(22) has the demarcation meaning stated there.

The demarcation length count is reported in Eqs. (1) and (13); the orbital length count is introduced separately before Eq. (23).

The earlier radial length subscript is represented by the current length subscript throughout this paper. A positional count and an orbital radius count are distinguished by their local context.

The source notation n_Lc denotes a dimensionless count for light propagation in the System Frame. It is not used as a replacement for the demarcation count in this calculation.

The coefficient-count marker is introduced before Eq. (2) and evaluated in Eq. (22). It is distinct from both the length count and the momentum realization.

The mixed nearest-whole-unit bracket operation is defined before Eq. (2). Its left floor and right ceiling brackets preserve the original argument without adding a half-unit offset.

The residual is introduced after Eq. (1) and defined mathematically in Eq. (4).

The residual-count product used for the Informativity differential is introduced after Eq. (1) and reported in Eq. (15).

The action normalization is introduced before Eq. (16). The electromagnetic-demarcation action is calibrated to the conventional exact CODATA value ℏ=h/(2π) [17]. Its upper-count-limit MQ value is

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

The two quantities are not interchanged in Eq. (16).

Electron mass and the electron mass count are introduced before Eq. (23). The conventional source mass is supplied by the 2022 CODATA adjustment [17].

The radius and its count are introduced before Eq. (23). That radius follows the explicit coupling substitution and is not silently identified with the conventional source Bohr radius.

The three inverse fine-structure forms are distinguished before Table 2. Their numerical relationships use the same source input and the two residual corrections described in Sec. 3.

The conventional gamma parameter is introduced before Eq. (17). Its three input routes have the same dimensions and shared source dependencies.

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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