Fine Structure Constant Forms in Measurement Quantization

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

Within the adopted Measurement Quantization (MQ) framework, we describe three inverse forms of the fine-structure constant, the dimensionless electromagnetic coupling, using fundamental length, mass and time measures and count-dependent normalization. The Internal Frame is the discrete count domain, and the System Frame is its continuous relational realization. The fundamental count form becomes the Planck-like form after addition of the metric differential, the discrete-to-continuous representation mismatch. The classical realization follows using the normalization associated with the Informativity differential, MQ’s count-dependent non-Lorentz length contraction. The dimensionless invariant coefficient is calibrated by inversion from the 2022 recommended inverse fine-structure constant of the Committee on Data of the International Science Council (CODATA). The resulting classical value reproduces that input, establishing algebraic closure under the same calibration. The blackbody and charge-coupling count expressions are evaluated separately with shared dimensional inputs. These results are conditional on the half-count assignment, which uses the nearest whole-unit marker of half the length count, and on the selected charge-count branch. The numerical closure does not establish an independent prediction or a unique derivation of those assignments.

Keywords: Fine Structure Constant, Fundamental Measures, Planck Units, Measure, Discrete

1. INTRODUCTION

The fine-structure constant α is the dimensionless electromagnetic coupling in the low-energy limit. Sommerfeld’s 1916 analysis treated the fine splitting of hydrogenic spectral lines through relativistic corrections to the Bohr model [1]. Modern determinations use atom-recoil interferometry and electron magnetic-moment measurements interpreted with calculations from quantum electrodynamics (QED) [2–6]. These methods do not give mutually consistent central values at their reported uncertainties. The cesium and rubidium recoil discrepancy discussed by Fan et al. [5] remains part of the source context of the fixed 2022 adjustment [7].

MQ investigates whether the numerical value of α can be described through the count and realization structure associated with the fundamental measures. Fundamental reference notation is defined in the Internal Frame introduced above, where it represents count relations. The symbols l_f, m_f, and t_f denote their System Frame length, mass and time realizations, in m, kg and s, respectively. Relational structure is encoded as integer count configurations over this domain. The Frames mapping is the kinematic transformation that realizes this count structure as continuous relational quantities in the System Frame. The Internal Frame does not itself contain continuous geometric observables; these are defined after realization through the Frames mapping.

The derivation distinguishes three dimensionless inverse forms. The quantity

α_{f}^{- 1}

denotes the fundamental Internal Frame count form. The quantity

α_{p}^{- 1}

denotes the Planck-like form obtained after addition of the metric differential. The quantity

α_{c}^{- 1}

denotes the classical System Frame realization after the normalization associated with the Informativity differential. The conventional measured inverse α^-1 supplies the numerical calibration of the classical form in this paper.

The MQ construction is governed by the dimensionless invariant coefficient θ_si. The electromagnetic demarcation is the fixed electromagnetic interaction count distance used in MQ. At this demarcation, n_L denotes the positive, dimensionless continuous length-count parameter used in the realization normalization, measured in fundamental lengths l_f; integer Internal Frame configurations remain distinct. The dimensionless residual count Q_L is defined by the positive right-triangle root used below. The mixed brackets ⌊42θ_si⌉ denote the nearest whole-unit value of their argument. The difference from 42θ_si defines the metric differential, whereas 2Q_Ln_L provides the normalization associated with the Informativity differential. Their combined contribution yields the classical realization

{α_{c}}^{- 1} = 2 Q_{L} n_{L} (84 θ_{si} - ⌈ 42 θ_{si} ⌋) = 137.035999177 (21)

(1)

With the invariant coefficient determined by inversion from the 2022 CODATA inverse fine-structure constant [7], Eq. (1) reproduces that source value. This is a shared-input consistency calculation, rather than an independent experimental determination.

Within MQ, the metric differential described above characterizes the discrete–continuous representation mismatch under the Frames mapping. The Informativity differential is the count-dependent non-Lorentz contraction of realized separation. Its magnitude is the finite-count departure of the realization normalization from unity, rather than the normalization factor itself. The derivation uses the metric mismatch and contraction normalization in sequence to obtain the Planck-like and classical forms. Their interpretation uses the Internal Frame and System Frame meanings established above.

The objective of this paper is to derive the fundamental, Planck-like, and classical forms of the inverse fine-structure constant from the MQ fundamental measures and associated count relations, identify the distinct roles of the metric differential and Informativity differential, and evaluate the System Frame realization. The symbolic construction and the numerical calibration from the measured inverse fine-structure constant are distinguished throughout the calculation.

2. METHODS

This paper applies MQ fundamental-measure and count relations to the three inverse fine-structure forms. The published framework [8] provides the historical derivations summarized in Table 1. Current frame and symbol meanings are stated in this manuscript and its glossary; they are not attributed verbatim to the older publication. The half-count assignment and the selected charge-count branch are adopted from MQ, while the numerical coefficient is calibrated from the specified 2022 source [7].

Appendix BR of the published MQ paper [8] is a list of appendices. Its symbol and term lists are in Appendices BP and BQ. The present definitions use the current frame and quantity distinctions summarized in Appendix A of this manuscript. Table 1 points to the published body sections; the current structured registry is a separate source of MQ definitions.

The computational resources accompanying Measurement Quantization are available through Online Resources [8].

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

The published MQ paper describes two supplementary spreadsheets [8]. Supplementary Material 1 covers its calculations, and Supplementary Material 2 covers the paper and supporting MQ publications. The present numerical record is stated independently below.

The reported calculations use 80-digit working arithmetic, retain intermediate values and round only the quoted results. Their uncertainty propagation follows the common source dependency stated below. This precision concerns numerical evaluation; it does not add measurement information or include uncertainty in the MQ model assumptions.

The numerical evaluations of Eqs. (1), (2), (4)–(6), (8), (9), (11), and (15) use the 2022 CODATA inverse fine-structure constant 137.035999177(21) [7] as the measured input for inversion. The defined speed of light c, in m s⁻¹, and Planck constant h, in J s, supply the dimensional calibration in the International System of Units (SI). Here ℏ denotes the reduced Planck constant at the electromagnetic demarcation, in J s. For the electron mass m_e, in kg, and Bohr radius a₀, in m, the CODATA relation gives

m_{e} a_{0} = ℏ / (α c)

. These two quantities are therefore not treated as independent numerical inputs to their product. Parenthetical digits give the standard uncertainty in the final quoted digits. All calculated quantities sharing this input remain correlated.

The comparison with the CODATA source is a closure check under common calibration. Treating the calculated value and the same source input as independent would double-count that input. The Planck-like versus classical difference tests the adopted MQ normalization. An independent observational test requires a prediction specified before calibration, separately identified measurements and their covariance [7].

3. RESULTS

3.1 Supporting Expressions

The momentum calibration is the stipulated MQ SI realization of the invariant coefficient. Its numerical scale is adopted for this calculation, while its units and interpretation remain distinct from the coefficient. The blackbody demarcation is the electromagnetic count obtained from the blackbody construction ([8], Appx. AB). The charge-coupling demarcation is the electromagnetic count obtained from the charge-coupling relation ([8], Appx. AE). In Eq. (2), p_si is the momentum realization of the invariant coefficient, in kg m s⁻¹. The electromagnetic reduced Planck quantity introduced in Methods is distinct from the upright upper-count quantity. The blackbody count is evaluated from its own expression using the same dimensional calibration as the later charge-coupling calculation. Matching the conventional Bohr-radius relation to its MQ count form gives the adopted action normalization

ℏ = \frac{p_{si} l_{f}}{Q_{L} n_{L}}

([8], Eq. (BL.3); [9], Eqs. (20)–(21)). This matching fixes the action relation once the momentum realization and count calibration are specified; it does not independently measure their normalization.

n_{L} = p_{si} l_{f} \sqrt{\frac{1}{ℏ (ℏ - 2 p_{si} l_{f})}} = 84.6005456716 (65)

(2)

The fundamental relation ([8], Appx. V) correlates the fundamental length, mass and time with the momentum realization. Dividing this relation by the momentum-time product gives the dimensionless normalization in Eq. (3). The factor of two concerns the normalized correlated measures; it does not equate a length directly with a dimensionless coefficient.

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

(3)

For the electromagnetic count assignment, MQ adopts n_θ, the dimensionless nearest whole-unit marker of half the length count ([8], Appx. AB, AC). Equation (3) supplies a factor-of-two normalization for correlated fundamental measures, while Eq. (4) states the separate half-count assignment. The latter is used as a count relation within the model, rather than inferred from equality of quantities with different dimensions. It gives n_θ=42 on the adopted demarcation branch without asserting n_L=2n_θ for the continuous numerical count.

n_{θ} = ⌈ n_{L} / 2 ⌋ = ⌈ 84.6005456716 (65) / 2 ⌋ = 42

(4)

The inversion relation ([8], Appx. AD) is given in Eq. (5). Earlier determinations [10, 3] provide historical comparison values; the present calculation uses the 2022 source [7].

2 α_{c} = \frac{1}{n_{L} θ_{si}} \frac{θ_{si}}{\sqrt{{θ_{si}}^{2} + {(n_{L} θ_{si})}^{2}} - n_{L} θ_{si}} \frac{θ_{si}}{84 θ_{si} - ⌈ 42 θ_{si} ⌋}

(5)

On the adopted charge-count branch n_Lθ_si=276 ([8], Appx. AC, AE), inversion of Eq. (5) gives θ_si=3.26239030504(25) from the 2022 CODATA input [7]. Equation (4) gives n_θ=42 on this branch. These assignments delimit the calculation; consistency here is not a proof of uniqueness across other admissible branches. The angular realization Θ_si has units rad and is distinct from the dimensionless coefficient. The dimensional conversion factor

C_{p θ} : = \frac{p_{si}}{Θ_{si}}

has units kg m s⁻¹ rad⁻¹ and connects the distinct realizations under this calibration [11], Eq. (6). Sharing a numerical coefficient does not identify their physical dimensions or independently determine the conversion scale. The selected X-ray model comparison [12–13] is examined in Sec. 3.3. The charge-coupling length count is

n_{L} = \frac{276}{θ_{si}} = 84.6005456716 (65)

(6)

Equation (6) gives the charge-coupling demarcation ([8], Appx. AE). With the adopted shared calibration, the independently evaluated expression in Eq. (2) gives the same numerical count. This equality is conditional on the common source construction.

The difference between the continuous quantity 42θ_si and its nearest whole-unit realization is the metric differential ([8], Appx. T). In Eq. (7), Q_{n_θ} denotes that dimensionless difference.

Q_{n_{θ}} = 42 θ_{si} - ⌈ 42 θ_{si} ⌋

(7)

The positive right-triangle residual

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

gives the product in Eq. (8). For positive finite count, this product lies below one half; its limiting value is approached as the count increases. Accordingly, 2Q_Ln_L is a normalization below unity, rather than the size of the contraction itself.

Q_{L} n_{L} = n_{L} (\sqrt{1 + {n_{L}}^{2}} - n_{L}) = 0.4999825364172403 (27)

(8)

Beginning with the fundamental Internal Frame form, we add the metric differential to obtain the Planck-like form and then apply the Informativity differential normalization to obtain the classical System Frame realization.

3.2 Calculations

The fundamental form uses the assigned count n_θ=42 with the invariant coefficient ([8], Appx. AA). This gives

{α_{f}}^{- 1} = 42 θ_{si} = 137.020392812 (11)

(9)

Adding the metric differential ([8], Appx. T) gives the Planck-like form in Eqs. (10) and (11). Its relation to the Bohr-radius product expression ([8], Appx. BL) is a separate algebraic consistency comparison under the adopted calibration.

{α_{p}}^{- 1} = {α_{f}}^{- 1} + Q_{n_{θ}} = 42 θ_{si} + (42 θ_{si} - ⌈ 42 θ_{si} ⌋)

(10)

{α_{p}}^{- 1} = 84 θ_{si} - ⌈ 42 θ_{si} ⌋ = 137.040785623 (21)

(11)

The finite-count normalization 2Q_Ln_L acts on the Planck-like inverse form. The dimensionless correction Δ_(P-C) is the difference removed by this normalization, as given in Eq. (12). Subtracting it gives Eq. (13), and factorization yields Eq. (14). Substitution of Eq. (11) then gives Eq. (15).

Δ_{(P - C)} = {α_{p}}^{- 1} (1 - 2 Q_{L} n_{L})

(12)

α_{c}^{- 1} = α_{p}^{- 1} - Δ_{(P - C)}

(13)

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

(14)

{α_{c}}^{- 1} = 2 Q_{L} n_{L} (84 θ_{si} - ⌈ 42 θ_{si} ⌋) = 137.035999177 (21)

(15)

The calculated correction is Δ_(P-C)=0.0047864462024(15). It is 0.00349271655194(54)% of the Planck-like inverse form. Equation (15) reproduces the measured inverse fine-structure value used upstream. These dependent calculations test the internal realization sequence and do not furnish an independent experimental confirmation.

3.3 Source Calibration and Consistency Comparisons

Table 1 compares the fundamental, Planck-like and classical inverse forms. The Planck-like comparison uses

m_{e} a_{0} / (m_{f} l_{f})

([8], Appx. BL), whereas the classical comparison uses

4 π ε_{0} ℏ c / e^{2}

[7]. Here π is the dimensionless circular constant, ε₀ is vacuum permittivity in F m⁻¹, and e is the elementary charge in C. Both comparison expressions share the source calibration used to define the fundamental measures. The earlier 2018 input [10] is retained only as a historical source; the current numerical input is [7].

Table 1. Shared-input comparison of inverse fine-structure forms; historical MQ comparison context [14].

Expressions for α
Expressions for α	Source or implied value ^a	MQ calculation	Closure residual
$α_{f}^{- 1} = 42 θ_{si}$	-	137.020392812(11)	-
$α_{p}^{- 1} = m_{e} a_{0} / (m_{f} l_{f})$	137.040785623(21)	137.040785623(21)	0.000 000 0
$α_{c}^{- 1} = 4 π ε_{0} ℏ c / e^{2}$	137.035999177(21)	137.035999177(21)	0.000 000 0

^a The classical source is the 2022 CODATA recommendation [7]. The Planck-like source-column entry is implied by the adopted fundamental-length relation and is not a direct CODATA recommendation. Residuals are displayed at the table precision; no independent comparison uncertainty is assigned.

Shwartz and Harris calculate signal and idler propagation angles relative to diamond (111) planes for a 25 keV pump and frequency-degenerate 12.5 keV outputs [12]. One Table I configuration gives both angles as 3.26239 rad. The angular realization in this MQ calculation has coefficient 3.26239030504(25) in rad. The source angle depends on the selected crystal, photon energies, pump direction and polarization conditions; the table contains other solutions and supplies no experimental standard uncertainty for the MQ coefficient. The numerical proximity therefore does not identify a measured universal invariant. Related experiments study X-ray down-conversion with loss [13] and X-ray–optical sum-frequency generation [15]; neither supplies that missing calibration test.

The adopted length calibration uses the electron-mass–Bohr-radius product and the Planck-like form ([8], Appx. AD, BM, Eq. (BM.3)). The conventional product reduces to

ℏ / (α c)

[7], so this route retains the same inverse fine-structure input used to obtain the coefficient. Together with the Methods dependency analysis, this accounts for the closure in Table 1a.

Table 1a checks the algebra at its displayed precision. It does not discriminate between the discordant atom-recoil measurements [3, 6] or supply an independent test of MQ. The quoted propagated uncertainties account for the adopted source input, with the count assignments and realization map held fixed.

4. DISCUSSION

Within the adopted MQ construction, the Planck-like inverse form in Eq. (11) and the classical form in Eq. (15) differ by the finite-count correction in Eq. (12). The difference is therefore explained internally by the Informativity differential normalization. This statement concerns the two MQ realizations; the Planck-like value is not a separate conventional CODATA recommendation for the fine-structure constant. The usual electromagnetic expression remains the source identity used for comparison [7].

The fine-structure metric differential in Eq. (7) quantifies the nearest-unit mismatch on the selected count branch. Table 2 retains other geometric factors and unit descriptions from the historical MQ presentation [8]. Its rows provide source context; only the fine-structure and half-count rows are used in this derivation, and the unlike unit entries are not equations equating physical dimensions.

Table 2. Historical geometric factors and reported unit descriptions, adapted from [8].

Dimension	Initial units	Resulting units	Unit factor	Geometric factor
Gravitation	m s^-2	kg m s^-2	kg	$m_{f} / 2$
Fine Structure	non-dimensional	non-dimensional	-	42θ_si-⌊42θ_si⌉ ^a
Elementary Charge	m³ s^-2	C (kg m³ s^-2) ^b	kg	$\frac{41 θ_{si}}{⌊ 41 θ_{si} ⌉}$ a
Counts n_θ to n_L	non-dimensional	non-dimensional	-	$n_{θ} = ⌊ \frac{n_{L}}{2} ⌉$
Orbital Velocity	m s^-1	m s^-1	-	θ_si a
Observational Domains	non-dimensional	non-dimensional	-	${(\frac{π}{θ_{si}})}^{2 / 3}$ ^a

^a The coefficient is dimensionless throughout these factors. Its angular realization is a separate radian-valued quantity; the X-ray comparison [12–13, 15] is discussed in Sec. 3.3. The unit columns preserve the historical descriptions, rather than asserting a dimensional equality or redefining the current frame meanings. The original charge-row description uses the square of charge in electrostatic units. This is a source convention for the historical table, not the SI definition of the elementary charge.

For the present fine-structure calculation, the half-count assignment of Eq. (4) gives n_θ=42, and the charge-coupling relation gives n_Lθ_si=276. These are distinct count roles. The metric differential uses the nearest-unit remainder of 42θ_si, while the Informativity differential uses the residual normalization at the continuous count n_L.

The fundamental normalization motivates the half-count assignment as a relation between correlated MQ counts, but the adopted rule is a separate input to this construction ([8], Appx. AC). The published quantization table also contains neighboring counts with the same marker 42 ([8], Table AC.1). The marker alone therefore does not select the 276 branch. The calculation establishes consistency of the selected branch. It does not establish a unique division-by-two rule for every admissible realization. A derivation of that rule from the count structure, followed by a test using data not used in calibration, would be needed for the stronger physical conclusion. The related charge realization construction also leaves the uniqueness of its selected class basis open [9].

5. APPENDICES

APPENDIX A: DEFINITIONS OF SYMBOLS

The following glossary cross-references definitions introduced in the main text and supplies additional MQ concepts needed for the appendix. Statements about realization, geometry, curvature and dynamics describe the framework’s interpretation; the fine-structure calculation establishes only the algebraic and numerical consequences developed above. The historical definitions in [8], Appx. BP, BQ, provide source context.

The fine-structure constant α retains its electromagnetic-coupling meaning introduced in Sec. 1. The recommended numerical value is published by CODATA based on the least-squares adjustment of the best available experimental and theoretical data.

m_e retains the electron-mass meaning introduced in Methods.

The Bohr radius a₀ introduced in Methods is a length in m used in the conventional source identity [7].

The coefficient θ_si, angular realization Θ_si and momentum realization p_si have the distinct meanings introduced in the main text. The recalculated coefficient is 3.26239030504(25), and the corresponding momentum realization is 3.26239030504(25) kg m s^-1 under the stipulated MQ calibration. The angular realization has the same coefficient in rad; the value 3.26239 rad from [12] is a tabulated model comparison rather than a direct measurement of this coefficient. References [13, 15] retain the related X-ray context. The conversion factor C_pθ retains the definition and units given in Sec. 3.1 [11].

The measures l_f, m_f and t_f introduced in Sec. 1 are the dimensioned System Frame realizations of the MQ fundamental references. Their values in this calculation depend on the adopted normalization and dimensional calibration [8, 7]. Their conditional propagated uncertainties do not establish experimentally measured units more precise than conventional Planck units for length, mass and time.

The normalization 2Q_Ln_L introduced in Sec. 1 is associated with the Informativity differential. Its departure from unity gives the contraction correction used in Eqs. (12)–(15), while the normalization itself remains dimensionless.

The gravitational coupling is denoted by G, in m³ kg⁻¹ s⁻². The upper-count quantity is upright, whereas the finite electromagnetic-demarcation quantity is italic. Under the adopted MQ calibration, the upper-count expression gives
$G = 6.6740779383 (10) \times 10^{- 11} m^{3} k g^{- 1} s^{- 2}$
, and its finite-count normalization gives
$G = 6.6738448317 (10) \times 10^{- 11} m^{3} k g^{- 1} s^{- 2}$
([8], Appx. E, F; [7]). These are conditional MQ calculations, not CODATA recommendations. The older inputs [10, 3] belong to the historical comparison.

The speed of light introduced in Methods is exactly c=299792458 m s^-1 [7]. The fundamental measures satisfy
$c = l_{f} / t_{f}$
([8], Appx. W; [16]).

l_p, m_p and t_p — are Planck’s units for length, mass, and time, in m, kg and s, respectively.

The symbols n_L, n_M and n_T denote MQ counts associated with the fundamental length, mass and time measures. Integer configurations belong to the Internal Frame. The noninteger n_L evaluated in Eqs. (2) and (6) is the continuous count parameter used in this paper’s realization normalization; the nearest whole-unit marker is specified separately in Eq. (4).

n_L describes the count of l_f representative of a change in position of an observable measured with respect to the observer’s frame of reference. In this paper its numerical realization parameter is the continuous quantity distinguished above.

The count marker n_θ uses the half-count assignment explained beside Eqs. (3) and (4). The recalculated half count is 42.3002728358(32), whose nearest whole-unit value is exactly 42. This interpretation remains a model count assignment, rather than a physical equality of length and angle.

h retains the Planck-constant meaning introduced in Methods.

The reduced Planck constant introduced in Methods has the exact SI value fixed by h and the circular constant [7]. It is evaluated at the electromagnetic demarcation in Eq. (2). The distinct upper-count MQ quantity is upright and is
$ħ = 1.0545349844417293 (56) \times 10^{- 34} J s$
under the same calibration. Its small propagated uncertainty reflects correlated model inputs, not a newly measured fundamental constant.

The three inverse forms
$α_{f}^{- 1}$
,
$α_{p}^{- 1}$
and
$α_{c}^{- 1}$
retain the meanings given in Sec. 1. Their current values are those in Eqs. (9), (11) and (15). The Planck-like and classical forms are related by the finite-count normalization, and their shared-input comparison is summarized in Table 1a.

The dimensionless fundamental gamma γ_F belongs to the current MQ realization definitions. It is distinct from the historical electromagnetic parameter γ, which is resolved from physical-constant inputs in the older construction [8]. Neither parameter is used as an independent numerical input in this paper.

The vacuum permittivity introduced in Sec. 3.3 has the 2022 CODATA recommended value
$ε_{0} = 8.8541878188 (14) \times 10^{- 12} F m^{- 1}$
[7]. Its conventional relation is
$ε_{0} = 1 / (μ_{0} c^{2})$
, where μ₀ is vacuum permeability in N A⁻². These recommended electromagnetic quantities share the adjusted fine-structure input and are not independent checks of Eq. (15).

The permeability introduced above has the recommended value
$μ_{0} = 1.25663706127 (20) \times 10^{- 6} N A^{- 2}$
[7]. This published uncertainty is retained as the recommendation; it is not replaced by a different value propagated from rounded source digits.

Coulomb’s constant k_e, in N m² C⁻², is calculated conventionally as
$k_{e} = 1 / (4 π ε_{0})$
. From the fixed 2022 input construction it is
$k_{e} = 8.9875517862 (14) \times 10^{9} N m^{2} C^{- 2}$
[7]. It retains the shared fine-structure dependence.

The elementary charge introduced in Sec. 3.3 has the exact SI value
$e = 1.602176634 \times 10^{- 19} C$
[7]. It is not an additional measured input in this calibration.

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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