How to Derive Physical Constants Using Quantization Ratios as a Function of the Discrete and Non-discrete Frames

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

Within the Measurement Quantization (MQ) framework, we examine physical-constant relations through the distinction between dimensionless configuration relations and their realized observables. The Internal Frame is the discrete count-based configuration domain. The System Frame is its non-discrete relational encoding, in which observables acquire their measurement interpretation through the Frames mapping. A fundamental expression describes configuration-domain relations; its Planck-like form is an intermediate realization, and its classical form additionally includes the contraction associated with discrete measure. We use gravitational curvature, elementary charge, coefficient counts and orbital velocity to illustrate these stages. The fine-structure example is evaluated with a fixed input from the 2022 adjustment by the Committee on Data for Science and Technology (CODATA). Inverting the classical relation determines the MQ coefficient, after which forward evaluation checks the consistency of the mapping and count normalization. This distinguishes configuration relations from laboratory quantities and separates calibration closure from an independent experimental comparison.

Keywords: Frame of Reference, Observer, Reference, Discrete, Measure

1. INTRODUCTION

Descriptions relative to an observer and an observed phenomenon remain essential for comparing physical measurements. In MQ, these operational reference choices apply to observables realized in the System Frame. The Internal Frame supplies the underlying configuration-domain count relations, while a conventional reference frame specifies how realized quantities are compared.

The two MQ frames introduced above have distinct roles. The Internal Frame carries dimensionless configuration relations; geometric and dimensionful observables belong to their System Frame realization. The symbols l_f, m_f and t_f identify fundamental length, mass and time in the stated representation. Their dimensional realizations carry m, kg and s. A configuration-domain use denotes the corresponding count description, and this reading is stated explicitly where needed below. We write c for the realized speed of light, with the exact value 299792458 m s^-1 [1].

The introductory presentation of Measurement Quantization [2] develops the count geometry used here ([2], Appx. D). Its right-triangle construction is a realized representation of count relations. One leg supplies a unit reference and the other a positive dimensionless length-count parameter n_L; the hypotenuse count is

\sqrt{1 + n_{L}^{2}} .

(1a)

The dimensionless residual is

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

(1b)

Discrete count structure and its non-discrete realization are connected by the metric differential ([2], Appx. T). This is the representation mismatch between discrete count relations and their continuous realization, rather than an additional physical deformation. The triangle does not assign an intrinsic geometry to the Internal Frame configuration domain.

A Frames mapping relation can connect a dimensionless configuration expression with a dimensionful realized quantity. The units are assigned in the realization, so the configuration expression is interpreted in its own domain before laboratory dimensions are compared. The mapping may also relate dimensionless quantities, as in the coefficient-count and fine-structure examples. For dimensional gravitational relations, p_si denotes the momentum realization in kg m s⁻¹. This distinction prevents a dimensional momentum from being confused with the invariant coefficient. The Informativity differential denotes the count-dependent non-Lorentz contraction in realized separations; its normalization factor is 2Q_Ln_L.

The invariant coefficient θ_si is dimensionless. In MQ, the radial expansion description uses 2θ_si ([2], Appx. AZ); this is a scalar coefficient rather than a rate with units supplied in advance. With the 2022 calibration developed in Sec. 3.2, the coefficient is θ_si=3.26239030504(25) and twice the coefficient is 6.52478061008(50). These are calculated values. Parenthetical digits give one standard uncertainty in the final reported digits. Table 1 separates the configuration relations, intermediate realizations and mapping ingredients. In its domain-ratio row, π is the dimensionless circular constant. The dimensionless coefficient-count marker n_θ refers to the count of θ_si selected by a specified configuration. The charge row uses r_q41 for the dimensionless mapping ratio evaluated in Eq. (8).

Table 1. Configuration relations and realization stages used in the examples of the metric differential. Updated from [2].

Example	Configuration relation	Realization stage	Count operation	Mapping ingredient
Gravitation	Residual count relation	Finite-count coupling	Residual normalization	$c / p_{si}$
Fine structure	$42 θ_{si}$	Planck-like inverse form	Nearest-whole residual	$42 θ_{si} - ⌊ 42 θ_{si} ⌉$
Elementary charge	Fundamental charge relation	Intermediate, then classical	Dimensionless charge ratio	$r_{q 41}$
Coefficient counts	Half the length-count parameter	Configuration marker	Nearest-whole-unit value	$n_{θ} = ⌊ n_{L} / 2 ⌉$
Orbital velocity	Dimensionless mass count	Velocity representation	Coefficient multiplication	$θ_{si}$
Observational domains	Dimensionless domain ratio	Domain comparison	Dimensionless coefficient	$(π / θ_{si})^{2 / 3}$

The invariant coefficient has an angular realization Θ_si in rad and the momentum realization already distinguished above. The X-ray papers [3–5] address specified crystal geometries and nonlinear optical processes. They provide physical context for the earlier comparison, rather than an independent measurement of the MQ coefficient or a test of the mapping. Common numerical coefficients do not equate radians with momentum units.

The charge row distinguishes a fundamental configuration relation, a Planck-like intermediate and a classical laboratory realization. The unit interpretation follows the stated stage of the Frames mapping.

Table 1 organizes the mapping ingredients without assigning kilograms or coulombs to the primitive configuration domain. The charge normalization and its classical realization are discussed with Eqs. (7)-(9).

The orbital-velocity and coefficient-count entries in Table 1 are interpreted in their respective configurations below. The velocity expressions are illustrative kinematic relations; no galaxy-specific data comparison is performed here.

The purpose of this paper is to explain how the configuration relations, intermediate mapped expressions and classical forms fit together. Its fine-structure example tests closure under a specified calibration, with the input and uncertainty treatment detailed in Sec. 3.2.

MQ applications are discussed in [6–11]. The generic dimensionless counts n_M and n_T refer to fundamental mass and time; n_L was introduced with the triangle construction. Count labels describe the configuration relations, while realized fundamental measures carry the units specified above. The source derivations use gravitational curvature, the speed of light, escape velocity and Heisenberg’s uncertainty principle [12].

This paper examines the relation between configuration-domain expressions and realized quantities ([2], Appx. K). Conventional reference-frame effects remain relevant to the comparison of observables. Einstein’s treatment [13] supplies the relativistic reference-frame context, while the MQ distinction concerns the count domain and its realization. These roles should be kept explicit when interpreting a frame relation.

2. METHODS

The method follows the count relations developed from gravitational curvature, light speed, escape velocity and uncertainty in the MQ source presentation. Each example identifies its configuration-domain expression, the mapping ingredient used to form an intermediate realization, and the additional normalization needed when a classical laboratory quantity is calculated.

The introductory paper Measurement Quantization [2] contains the source derivations, their proposed physical support, referenced appendices, and term definitions. A preprint is also listed in the Online Resources. Those sources provide the context for the equation relations retained here.

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

The source presentation identifies supplementary calculations for the introductory paper and related applications. The calculation used here is reproducible from the equations below and the specified 2022 CODATA inputs [1]; earlier supplementary values are not used as current calibration inputs.

3. RESULTS

We call the configuration-domain form fundamental. The Planck-like form is its intermediate System Frame realization through the Frames mapping. The classical form includes the Informativity differential introduced in Sec. 1. A comparison with a laboratory quantity is made at this final stage. These three forms are related, but their frame and normalization roles remain distinct.

The Planck-like stage alone does not include the Informativity differential. The fine-structure chain below makes the effect explicit and completes the classical evaluation. The gravitational relations and associated action normalization are developed in the source presentation [2]. The present calibration distinguishes evaluation at the electromagnetic demarcation, the fixed interaction count used in Sec. 3.2, from the upper-count limit and does not treat agreement with an input used in the inversion as a new measurement.

We first illustrate the metric differential in several configuration relations and their mapped forms.

1. Examples of the Metric Differential

Each example starts with the fundamental configuration relation or its corresponding count description. The mapping ingredient produces the Planck-like intermediate. Where a classical form is required, the Informativity differential must also be included. Configuration-domain symbols are read as counts; units apply when an observable realization is specified.

Gravitational curvature ([2], Appx. C). Equation (1) is the residual acceleration expression. In its dimensional realization, the multiplier

c / p_{si}

(1c)

in Eq. (2) has units kg⁻¹. The realized separation r=n_Ll_f has units m. Equation (3) expresses the gravitational coupling G, with units m³ kg⁻¹ s⁻², evaluated at the same finite count as the left side. The residual and momentum meanings were established above.

\frac{Q_{L} l_{f}}{n_{L} t_{f}^{2}},

(1)

\frac{Q_{L} l_{f}}{n_{L} t_{f}^{2}} (\frac{c}{p_{si}}),

(2)

\frac{Q_{L} c^{3}}{r p_{si}} = \frac{G}{r^{2}} .

(3)

Fine structure constant ([2], Appx. AA). The symbols

α_{f}^{- 1},

(3a)

α_{p}^{- 1}

(3b)

and

α_{c}^{- 1}

(3c)

denote the dimensionless fundamental, Planck-like and classical inverse forms. The fundamental form uses the configuration domain; the other two use its realization. Equations (4)–(6) use the first two. The mixed left floor and right ceiling brackets denote rounding to the nearest whole-unit value. The residual

Q_{n_{θ}}

(3d)

in Eq. (5) is the difference between the coefficient product and its nearest whole-unit value.

α_{f}^{- 1} = 42 θ_{si},

(4)

α_{p}^{- 1} = α_{f}^{- 1} + Q_{n_{θ}} = 42 θ_{si} + (42 θ_{si} - ⌊ 42 θ_{si} ⌉),

(5)

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

(6)

Elementary charge ([2], Appx. AM). Equation (7) gives the fundamental configuration-domain charge relation, denoted e_f. Its symbols are interpreted in the dimensionless Internal Frame description before laboratory units are assigned through realization. The Planck-like intermediate is denoted e_p. Equation (8) evaluates the mapping ratio introduced in Table 1 as the quotient between the specified count of θ_si and the nearest whole-unit value of that count. Every quantity in this ratio is dimensionless, and its nearest-whole-unit denominator must be nonzero. Its argument is determined by the charge configuration; the fine-structure marker resolved in Eq. (14) is not silently imposed on a different configuration.

e_{f} = \frac{1}{4 (θ_{si} - 1)} l_{f} c^{2},

(7)

r_{q 41} = \frac{(n_{θ} - 1) θ_{si}}{⌊ (n_{θ} - 1) θ_{si} ⌉},

(8)

e_{p} = r_{q 41} e_{f} .

(9)

Equation (9) evaluates e_p, the intermediate form associated with the configuration-domain count relation through the Frames mapping. It does not yet contain the Informativity differential. The laboratory realization e_c has units C and obeys e_c=2Q_Ln_Le_p ([2], Appx. AM). The normalization completes the transition from the intermediate to the classical form. This interpretation distinguishes a count description from a laboratory quantity; the mapping and charge normalization remain MQ realization assumptions, not a standalone proof of a charge identity in the International System of Units (SI).

Coefficient counts ([2], Appx. AC). Equation (10) specifies the nearest-whole-unit relation used for the fine-structure marker. The argument and result are dimensionless.

n_{θ} = ⌊ n_{L} / 2 ⌉ .

(10)

Orbital star velocity ([2], Appx. AF). In Eqs. (11) and (12), m_f denotes the nonnegative dimensionless fundamental-mass count used in the configuration relation, and θ_si has its invariant dimensionless meaning. The quantity under the square root is consequently dimensionless. The factor c supplies the realized velocity reference. The symbol v_bc names the baseline expression in Eq. (11); v_b names the coefficient-adjusted expression in Eq. (12). Both velocity representations have units m s⁻¹. This local count interpretation is retained when reading these equations. Their use as a model of an observed stellar orbit additionally requires a specified configuration and independent orbital data.

v_{bc} = c \sqrt{2 m_{f}} .

(11)

v_{b} = θ_{si} c \sqrt{2 m_{f}} .

(12)

2. Application of the Metric Differential to the Fine Structure Constant

For the fine-structure calculation, we use the charge-coupling route to the electromagnetic demarcation ([2], Appx. AE). The coefficient reported in Sec. 1 is calculated by inversion of Eq. (23) with the 2022 CODATA source

α_{c}^{- 1} = 137.035999177 (21)

(12a)

[1]. Equation (13) determines the corresponding demarcation length count. This realized interaction count is distinguished from a primitive integer configuration count.

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

(13)

Equation (14) evaluates the coefficient-count marker for this fine-structure configuration. It is obtained from half the demarcation length count with the nearest-whole-unit operation. The branch is stable under the propagated source uncertainty.

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

(14)

Equation (15) evaluates the fundamental inverse form. Equation (16) defines its metric-differential residual Δ_f, which is dimensionless and is the local name for the same correction represented by

Q_{n_{θ}}

(14a)

in Eq. (5). Adding that residual in Eq. (17) gives the Planck-like form, evaluated in Eq. (18) ([2], Appx. T).

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

(15)

Δ_{f} = 42 θ_{si} - ⌊ 42 θ_{si} ⌉ .

(16)

α_{p}^{- 1} = α_{f}^{- 1} + Δ_{f} = 42 θ_{si} + (42 θ_{si} - ⌊ 42 θ_{si} ⌉) .

(17)

α_{p}^{- 1} = 84 θ_{si} - ⌊ 42 θ_{si} ⌉ = 137.040785623 (21) .

(18)

To obtain the classical form, we use the Informativity differential at the demarcation count ([2], Appx. D). Equation (19) evaluates the residual-count product; twice this product is the normalization factor. Equation (20) defines Δ_(P-C), the dimensionless difference between the Planck-like and classical inverse forms.

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

(19)

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

(20)

Subtracting this difference gives Eq. (21). Equation (22) factors the same correction into the normalization multiplier. Substituting the Planck-like expression then gives Eq. (23), which returns the calibration input.

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

(21)

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

(22)

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

(23)

Table 2 shows the three inverse forms under the fixed 2022 calibration [1]. The earlier CODATA adjustment [14], the measurement-method review [15] and the rubidium-recoil determination [16] provide external context. The 2022 adjustment combines measurements with theoretical input and addresses their inconsistencies [1]; its recommended value is a calibration input here. In the table’s dimensional comparison expressions, m_e is the electron mass in kg, a₀ is the Bohr radius in m, ε₀ is the vacuum electric permittivity in F m⁻¹, and e is the conventional elementary-charge source in C. Italic ℏ denotes the electromagnetic-demarcation action in J s, calibrated here to the conventional exact source

h / (2 π),

(23a)

with h the Planck constant in J s. The coefficient-count calculations and these dimensional expressions are compared only after their realization and source dependencies are identified.

Table 2. Fundamental, Planck-like and classical inverse fine-structure calculations using the 2022 source input. X-ray studies provide physical context rather than independent validation of these values. Updated from [2].

Inverse forms	Values
Inverse forms	2022 source a	MQ (2022)	X-ray test	Difference
$α_{f}^{- 1} = 42 θ_{si}$	-	137.020392812(11)	Not tested	-
$α_{p}^{- 1} = \frac{m_{e} a_{0}}{m_{f} l_{f}}$	137.040785623(21)	137.040785623(21)	Not tested	0 (shared input)
$α_{c}^{- 1} = \frac{4 π ε_{0} ℏ c}{e^{2}}$	137.035999177(21)	137.035999177(21)	Not tested	0 (shared input)

a The classical source value is the 2022 CODATA recommendation [1]. The Planck-like comparison is derived from the same calibration. Each zero difference in these rows is an exact shared-input closure result; it is not evidence of two independent determinations.

Table I of Shwartz and Harris [3] lists calculated signal and idler angles of 3.26239 rad for a particular Bell-state configuration in diamond, using its (111) reciprocal-lattice vector, a 25 keV pump and degenerate 12.5 keV daughter photons. It is not an experimental determination of the MQ coefficient. The down-conversion experiment [4] and wave-mixing experiment [5] investigate their own optical processes. The conventional electromagnetic expression

α_{c}^{- 1} = 4 π ε_{0} ℏ c / e^{2}

(23b)

and the MQ Planck-like relation

α_{p}^{- 1} = m_{e} a_{0} / (m_{f} l_{f})

(23c)

([2], Appx. AA, BL) share the calibration used in Table 2. The exact conventional relation

m_{e} a_{0} = ℏ α_{c}^{- 1} / c

(23d)

eliminates the correlated electron-mass and Bohr-radius product before uncertainty is propagated. The dimensional calculation adopts the MQ momentum-realization convention in which the invariant coefficient supplies the SI numerical coefficient of p_si. Together with the source relation for fundamental length, these identities determine the realized fundamental measures, the finite-count and upper-count gravitational values, and the corresponding action values. Their numerical results below are conditional MQ calculations based on the common source inputs. The exact conventional identities and the assumed MQ realization relations have distinct evidentiary roles.

4. DISCUSSION

The metric differential characterizes the mismatch between discrete configuration relations and their continuous realization through the Frames mapping. The examples distinguish that mapping from the additional Informativity differential required to obtain the classical form. A configuration-domain count relation is interpreted before laboratory units are assigned to its realized observable. This resolves the charge and mass-count interpretation used in the examples without assigning SI dimensions to the Internal Frame itself.

The fine-structure derivation provides a complete example of the fundamental, Planck-like and classical stages. Under the common calibration, the upper-count gravitational value is

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

(23e)

while the value at the electromagnetic demarcation is

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

(23f)

The upper-count action is

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

(23g)

and the electromagnetic-demarcation action is calibrated to

ℏ = h / (2 π) .

(23h)

The source presentation discusses the fundamental measures and further applications ([2], Appx. BM). The quoted uncertainties propagate the common calibration input under the stated MQ realization assumptions; they do not include uncertainty in those assumptions or constitute new measurements of gravitational coupling or action. The finite-count gravitational value is distinguished from the macroscopic quantity recommended by CODATA [1]. Independent empirical tests must retain the measurement regime and its own uncertainty.

5. APPENDICES

APPENDIX A: TERMS

The full MQ term context is supplied in Measurement Quantization [2]. This appendix identifies the notation already introduced in the narrative and records its realization conventions without redefining the terms.

The invariant coefficient and its realization conventions are given in Sec. 1. The X-ray sources [3–5] concern their optical configurations; the conserved-quantity discussion [17], Sec. 4.4, concerns general relativity at null infinity. These contexts do not establish the MQ coefficient calibration or a universal mapping between angular and momentum units.
The exact speed-of-light source was introduced in Sec. 1. The generic count relation is
$c = n_{L} l_{f} / (n_{T} t_{f})$
(23i)
([2], Sec. 1). Its fundamental unit-count realization uses the exact speed-of-light source [14, 1] and equal length and time counts. This source relation does not identify a propagation count with the demarcation count of Eq. (13).
The finite-count and upper-count gravitational evaluations, their units and their numerical provenance are given in Sec. 4. Upright G denotes the upper-count limit and italic G the finite electromagnetic-demarcation evaluation.
The fundamental-measure representations are distinguished in Sec. 1. In the dimensional realization of the calibration,
$l_{f} = 1.61619991148 (12) \times 10^{- 35} m,$
(23j)

$m_{f} = 2.17643254057 (17) \times 10^{- 8} kg$
(23k)
and
$t_{f} = 5.39106261131 (41) \times 10^{- 44} s .$
(23l)
These dimensional realizations are kept separate from the dimensionless fundamental-mass count used in Eqs. (11) and (12).
The symbols l_p, m_p and t_p denote conventional Planck length, mass and time, in m, kg and s. They are not substituted for the fundamental measures in this calculation.
The generic length, mass and time count symbols are introduced in Sec. 1. Their local configuration and realized-parameter roles are specified where used.
The positive length-count parameter appears in the right-triangle construction and in the demarcation calculation. The realized separation is the count multiplied by fundamental length.
The source notation n_Lc denotes a dimensionless length count for light propagation. It is distinct from the demarcation count used in Sec. 3.2.
The coefficient-count marker is introduced in Sec. 1. Its configuration-dependent charge use appears in Eq. (8), while its fine-structure branch is evaluated in Eqs. (10) and (14).
The mixed nearest-whole-unit bracket convention is defined before Eq. (5). It retains the original argument and requires no half-unit offset.
The dimensionless residual count is defined in the triangle construction in Sec. 1.
The residual-count product of Eq. (19) enters twice in the normalization factor of the Informativity differential, as shown in Eqs. (20–23).
The electromagnetic-demarcation action and exact conventional source are introduced before Table 2. The upright upper-count action is reported in Sec. 4. The two evaluation regimes are not interchanged.
The electron mass is identified before Table 2. Its product with the Bohr radius is handled through the conventional exact relationship stated after that table.
The Bohr radius is identified before Table 2 and enters the same correlated product discussed there.
The three inverse fine-structure forms are introduced before Eq. (4). Their values and shared calibration appear in Eqs. (15–23) and Table 2.
The historical conventional electromagnetic transformation parameter γ has units m⁶ s⁻⁴ C⁻² in its dimensional realization and is distinct from a reference-cycle label. It is not required by the numerical calibration used here. Its source role in charge realization is discussed in [2], Appx. AN.

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 reported numerical results can be reproduced using the displayed relations and the 2022 CODATA input specified in Sec. 3.2 [1]. No new observational dataset is analyzed in this paper.

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