Discrete Approach to Examining the Equivalence Principle through Count Relations

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We examine the equivalence principle using a count formulation of the Pythagorean construction, escape velocity and the speed of light. The resulting dimensionless relation connects a relative motion count with mass and radial separation counts when the relative speed is identified with the escape speed. An alternative force calculation recovers the same relation under an equal-mass specialization, an acceleration-count assignment and a further identification of radial and motion counts. These conditions are explicit parts of the calculation. The count relation and its macroscopic escape-speed form establish an algebraic connection within the stated construction; they do not independently establish equality of inertial and gravitational masses.

Keywords: Equivalence Principle, Discrete, Planck, Motion, Gravitation, Mass

1. INTRODUCTION

Einstein’s discussion connects gravitational and inertial effects in rotating reference systems [1]. The weak equivalence principle concerns the universality of free fall. Modern tests compare the accelerations of test bodies of different compositions, as in the MICROSCOPE experiment [2]. Here we examine conditional count relations motivated by that principle, rather than a composition-dependent free-fall test.

The approach uses the count nomenclature of Measurement Quantization (MQ). Published MQ applications are discussed in [3–9]. The present calculation examines a proposed relation between motion, mass and separation counts; it does not use the existence of those applications as evidence that the equivalence principle has been derived.

Counts n_L, n_M and n_T refer to the fundamental measures l_f, m_f and t_f, with units m, kg and s. The counts are dimensionless. To distinguish motion from radial separation in this paper, n_Lm denotes the change count of an observed mass and n_L denotes radial separation. MQ defines its fundamental measures through a discrete-count normalization [9], distinct from the conventional Planck units [10]. Their numerical values in Appendix A are conditional calculations using the specified source inputs and MQ realization normalization. The fundamental measures are positive. Time, mass and radial counts used as divisors in this calculation are positive, and the motion count is nonnegative.

MQ proposes count descriptions associated with the speed of light, escape velocity and the uncertainty principle [9,11]. Their use here establishes algebraic relations within the stated construction; their physical realization requires additional justification. The earlier frame terminology is presented in [9], Appendix K. In current MQ terminology, the Internal Frame is the discrete count configuration domain. The System Frame is its continuous relational encoding, induced by the Frames mapping. Here the mapping is the kinematic transformation from discrete count configurations to their continuous relational encoding. MQ defines observable quantities only after this mapping.

Expressions containing the fundamental measures retain their dimensional scales. Relations written entirely in dimensionless counts expose the count structure used in the calculation. Under the Frames mapping, that structure may be realized as System Frame geometry. The absence of dimensional measures alone does not establish a physical interpretation.

With this foundation, we examine two constructions. The first assigns count variables to the two legs of a Pythagorean relation and adopts a normalized relation between local and observed separations. Its positive-root form has the dependence of longitudinal length contraction between inertial reference systems in Special Relativity (SR) [12]. That physical relation uses simultaneous endpoint measurements in the observer’s frame. The normalization and interpretation of the count variables here are additional assumptions; endpoint agreement alone does not derive them.

The second construction starts from escape velocity and substitutes the fundamental measures to express its squared speed as a ratio of mass and radial counts. We compare that ratio with the normalized motion-count ratio. The alternative force calculation then examines the additional count assignments required to recover the same expression.

This approach uses escape velocity to resolve a count relation between motion and radial separation. Here n_Lc is the dimensionless light-travel count of l_f over the same time interval as n_Lm. The symbols v and c denote the relative motion speed and speed of light, respectively, in m s⁻¹. The relation developed below is

\frac{n_{L m}^{2}}{n_{L c}^{2}} = 2 \frac{n_{M}}{n_{L}} .

(1)

Multiplying motion and light-travel counts by l_f gives lengths. Dividing both lengths by the same positive interval n_Tt_f gives their associated speeds. Their squared ratio is v²/c² because the common time interval cancels; Eq. (15) retains this interval explicitly. The mass-to-radius ratio on the right of Eq. (1) is developed in Sec. 3.

Equation (1) summarizes the count relation examined below. Its left side uses the motion and light-travel counts already introduced, while its right side uses the source mass and radial counts. Equation (20) recovers this relation by identifying the relative motion speed with escape speed. The force calculation recovers it only under its additional stated conditions.

The MQ relation c=l_f/t_f specifies the fundamental length-to-time scale. The equality 2n_Ln_Mn_Lm=n_T is an MQ bound construction ([9], Appx. B). It is not the general quantum uncertainty statement for arbitrary states. The standard uncertainty relation is an inequality; equality occurs only for states that saturate its bound [11,13]. No state-dependent uncertainty calculation is used in the equivalence argument below.

2. METHODS

The calculation uses the fundamental measures and count notation of Measurement Quantization (MQ). Classical expressions for the speed of light and escape velocity supply the dimensional relations. The following algebra is evaluated separately from the interpretation of count geometry and its proposed physical support.

Measurement Quantization [9] presents the framework’s proposed derivations and physical support. Its appendices provide the source context for the MQ relations used here. The present algebra is checked separately from those physical claims. The source publication also identifies an Online Resources collection.

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

The source MQ publication [9] identifies two supplementary calculation spreadsheets, Suppl. 1 and Suppl. 2. The numerical quantities stated here use the fixed 2022 inputs recommended by the Committee on Data for Science and Technology (CODATA) identified in Appendix A [14], together with the specified MQ relations and realization normalization. The spreadsheet descriptions do not establish independent observational validation of these calculations.

3. RESULTS

1. Calculations

We begin with the Pythagorean theorem. In Eqs. (2)–(3), a and b are dimensionless nonnegative leg variables and c is the dimensionless hypotenuse variable; this local use of c differs from the speed of light. The subscripts max, min and bound identify endpoint values. The positive dimensionless scale counts n_max and n_min are used in Eqs. (4)–(10), and n_bound is the hypotenuse count for each trial construction ([9], Appx. AU).

a^{2} + b^{2} = c^{2},

(2)

a_{\max}^{2} + b_{\min}^{2} = c_{bound}^{2} .

(3)

Equation (4) tests multiplication of the leg variables by their scale counts. Equation (5) instead tests division by those counts. They define separate trial constructions; dividing by unequal scale counts is not a uniform rescaling of Eq. (2).

{(a n_{\max})}^{2} + {(b n_{\min})}^{2} = n_{bound}^{2} .

(4)

{(\frac{a}{n_{\max}})}^{2} + {(\frac{b}{n_{\min}})}^{2} = n_{bound}^{2} .

(5)

Equations (6)–(7) evaluate the product construction at the two axis endpoints. At one endpoint the first leg is zero, and at the other the second leg is zero. Their positive-root values are the squares of the corresponding scale counts. Those values differ when the scale counts differ, so this trial does not give a common endpoint hypotenuse count.

n_{bound} = {({(0 n_{\max})}^{2} + {(n_{\min} n_{\min})}^{2})}^{1/2} = n_{\min}^{2} .

(6)

n_{bound} = {({(n_{\max} n_{\max})}^{2} + {(0 n_{\min})}^{2})}^{1/2} = n_{\max}^{2} .

(7)

For the divisor construction, Eqs. (8)–(9) give the exact endpoint value n_bound=1 because each nonzero leg is divided by its own positive scale count. Equation (10) adopts this unit value as the normalized relation for the interior count pairs. Its adoption is an additional constraint, rather than a consequence of checking the two endpoints.

n_{bound} = {({(\frac{0}{n_{\max}})}^{2} + {(\frac{n_{\min}}{n_{\min}})}^{2})}^{1/2} = 1 .

(8)

n_{bound} = {({(\frac{n_{\max}}{n_{\max}})}^{2} + {(\frac{0}{n_{\min}})}^{2})}^{1/2} = 1 .

(9)

Equation (11) replaces the trial scale counts by n_Lc and n_Ll. The latter is the dimensionless reference count for a separation resolved in the local frame. We assign a=n_Lm and b=n_Lo, where n_Lo is the dimensionless count associated with the observed realized separation. These assignments produce Eq. (12). Solving for the observed-to-local ratio gives Eq. (13), and taking its nonnegative root gives Eq. (14). For that root to be real, the squared motion-to-light count ratio must lie between zero and one.

{(\frac{a}{n_{\max}})}^{2} + {(\frac{b}{n_{\min}})}^{2} = 1 .

(10)

{(\frac{a}{n_{L c}})}^{2} + {(\frac{b}{n_{L l}})}^{2} = 1 .

(11)

{(\frac{n_{L m}}{n_{L c}})}^{2} + {(\frac{n_{L o}}{n_{L l}})}^{2} = 1 .

(12)

\frac{n_{L o}^{2}}{n_{L l}^{2}} = 1 - \frac{n_{L m}^{2}}{n_{L c}^{2}} .

(13)

n_{L o} = n_{L l} {(1 - \frac{n_{L m}^{2}}{n_{L c}^{2}})}^{1/2} .

(14)

Within the adopted construction, Eq. (14) has the positive-root dependence of longitudinal length contraction [12]. Equation (15) converts its dimensionless motion-to-light count ratio to a squared speed ratio using a common positive time interval. The comparison supplies no Lorentz transformation or independent derivation of the adopted normalization.

\frac{n_{L m}^{2}}{n_{L c}^{2}} = {(\frac{n_{L m} l_{f}}{n_{T} t_{f}})}^{2} {(\frac{n_{T} t_{f}}{n_{L c} l_{f}})}^{2} = \frac{v^{2}}{c^{2}} .

(15)

For the gravitational calculation, M=n_Mm_f is the source mass in kg and R=r=n_Ll_f is radial separation in m. We use the upper-count gravitational coupling, denoted by upright G, with units m³ kg^-1 s^-2, and its MQ relation G=c³t_f/m_f ([9], Appx. F, G). Equation (16) is the Newtonian escape-speed relation for a test body outside an isolated spherical source, with escape to infinity and a constant coupling. Its use as a physical escape calculation requires the weak-field, nonrelativistic regime. The nonnegative v is escape speed. Equation (17) substitutes the mass and separation counts and upper-count coupling. Using c=l_f/t_f gives Eq. (18); division by the light speed squared gives Eq. (19).

v = {(\frac{2 G M}{R})}^{1/2} .

(16)

v^{2} = \frac{2 G n_{M} m_{f}}{n_{L} l_{f}} = n_{M} m_{f} \frac{2}{n_{L} l_{f}} \frac{t_{f} c^{3}}{m_{f}} .

(17)

v^{2} = \frac{2 n_{M} t_{f} c^{3}}{n_{L} l_{f}} = \frac{2 n_{M} c^{3}}{n_{L} c} = \frac{2 n_{M} c^{2}}{n_{L}},

(18)

\frac{v^{2}}{c^{2}} = 2 \frac{n_{M}}{n_{L}} .

(19)

The ratio in Eq. (15) equals that in Eq. (19) when the relative motion speed is identified with the escape speed. Under that identification, the inertial and gravitational count expressions give

\frac{n_{L m}^{2}}{n_{L c}^{2}} = 2 \frac{n_{M}}{n_{L}} .

(20)

The gravitational branch uses the upper-count identity for G together with the Newtonian escape-speed model ([9], Appx. F, G). Historical unit constructions [10,15] supply context for the dimensional comparison. Cancelling the fundamental measures establishes only a count relation under the stated assumptions.

2. Alternate Approach

We next examine force. In Eq. (21), F is force in N, m is inertial mass in kg and a is acceleration in m s⁻²; this use of a differs from the preceding triangle variable. Equation (22) uses m=n_Mm_f and the acceleration-count assignment

a = n_{L m} l_{f} / (n_{T}^{2} t_{f}^{2}) .

(20a)

This assignment supplies the acceleration parameter for this construction; it is not a general trajectory law. Cancelling the dimensional factors gives Eq. (23).

F = m a,

(21)

F = (n_{M} m_{f}) (\frac{n_{L m} l_{f}}{n_{T}^{2} t_{f}^{2}}),

(22)

F = \frac{n_{M} n_{L m}}{n_{T}^{2}} \frac{m_{f} l_{f}}{t_{f}^{2}},

(23)

Equation (24) gives the gravitational force on test mass m due to source mass M, using the upper-count coupling. Equations (25)–(32) specialize to equal masses m=M=n_Mm_f and a shared fundamental mass measure. To examine the coupling dependence, Eqs. (25)–(26) evaluate the corresponding force with the finite-count coupling. Its dimensionless residual is

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

(23a)

for positive radial count. The momentum realization p_si=l_fm_f/(2t_f) has units kg m s⁻¹ and realizes the invariant dimensionless MQ coefficient θ_si. Substituting the finite-count coupling gives Eq. (25); using r=n_Ll_f gives Eq. (26).

F = \frac{G M m}{r^{2}},

(24)

F = \frac{Q_{L} r^{2} c^{3}}{r p_{s i}} \frac{n_{M}^{2} m_{f}^{2}}{r^{2}},

(25)

F = (Q_{L} n_{L}) l_{f} c^{3} \frac{1}{p_{s i}} \frac{n_{M}^{2} m_{f}^{2}}{r^{2}},

(26)

For Eq. (27), we return to the upper-count coupling. The residual-count normalization in Eq. (26) is assigned its upper-count limiting value, and the momentum realization is expanded in fundamental measures. At finite count the residual-count product is less than one half; the replacement by one half belongs to this upper-count evaluation.

F = \frac{1}{2} l_{f} \frac{l_{f}^{3}}{t_{f}^{3}} \frac{2 t_{f}}{l_{f} m_{f}} \frac{n_{M}^{2} m_{f}^{2}}{n_{L}^{2} l_{f}^{2}},

(27)

F = {(\frac{n_{M}}{n_{L}})}^{2} \frac{m_{f} l_{f}}{t_{f}^{2}} .

(28)

Equation (28) reduces the upper-count expression in Eq. (27) to the common fundamental force scale. Equating it to Eq. (23) gives Eq. (29) under the equal-mass and acceleration-count assignments already stated. Cancelling that positive common scale gives Eq. (30). Division by the positive mass count and multiplication by two gives Eq. (31), after which multiplication by the radial count gives Eq. (32). The factor two is retained on both sides.

\frac{n_{M} n_{L m}}{n_{T}^{2}} \frac{m_{f} l_{f}}{t_{f}^{2}} = {(\frac{n_{M}}{n_{L}})}^{2} \frac{m_{f} l_{f}}{t_{f}^{2}},

(29)

\frac{n_{M} n_{L m}}{n_{T}^{2}} = \frac{n_{M}^{2}}{n_{L}^{2}},

(30)

2 \frac{n_{L m}}{n_{T}^{2}} = 2 \frac{n_{M}}{n_{L}^{2}},

(31)

2 \frac{n_{L} n_{L m}}{n_{T}^{2}} = 2 \frac{n_{M}}{n_{L}},

(32)

The radial and motion counts are distinct. Equation (33) imposes a particular identification between them for the alternative force construction. Equation (34) is a separate time-count condition. Using c=n_Lcl_f/(n_Tt_f) and c=l_f/t_f for the same interval gives that condition. It does not supply a derivation of Eq. (33).

n_{L} = n_{L m} / 2 .

(33)

n_{T} = n_{L c} .

(34)

Substituting Eqs. (33)–(34) into the corrected Eq. (32) gives Eq. (35). The result agrees algebraically with Eq. (20), but the force branch additionally depends on the imposed radial-to-motion identification in Eq. (33).

\frac{n_{L m}^{2}}{n_{L c}^{2}} = 2 \frac{n_{M}}{n_{L}} .

(35)

Equation (34) constrains the time count used in the acceleration assignment when the common light-speed interval is imposed. The agreement of Eq. (35) with the escape-speed branch is conditional on both this constraint and Eq. (33), together with the equal-mass specialization and upper-count coupling.

3. Macroscopic Form

The escape-speed and force branches recover the same count expression under different conditions. The first identifies the relative motion speed with escape speed. The second also imposes an acceleration-count assignment, equal masses and the relation in Eq. (33). Making those conditions explicit separates the common algebraic form from the question of whether the constructions have the same physical interpretation.

To express the count relation in macroscopic units, we use the mass and separation assignments already given and the speeds v=n_Lml_f/(n_Tt_f) and c=n_Lcl_f/(n_Tt_f). Both speeds use the same positive interval n_Tt_f, as required by Eq. (15). Equation (36) pairs each count with its fundamental measure. The common time factor cancels, giving Eq. (37). No assignment of the time count to one is required. The conversion does not establish the physically admissible count domain.

\frac{n_{L m}^{2} l_{f}^{2}}{n_{T}^{2} t_{f}^{2}} \frac{n_{T}^{2} t_{f}^{2}}{n_{L c}^{2} l_{f}^{2}} = 2 \frac{n_{M} m_{f} l_{f}}{n_{L} m_{f} l_{f}} .

(36)

\frac{v^{2}}{c^{2}} = 2 \frac{M l_{f}}{R m_{f}} .

(37)

Multiplying Eq. (37) by the light speed squared and taking the nonnegative root gives the first expression in Eq. (38). The next two forms use l_f=ct_f. Substitution of the upper-count coupling G=c³t_f/m_f ([9], Appx. F) gives Eq. (39), where v_e identifies escape speed. This recovers the starting gravitational speed relation in macroscopic units.

v = {(2 \frac{M c^{2} l_{f}}{R m_{f}})}^{1/2} = {(2 \frac{M c^{3} l_{f}}{R c m_{f}})}^{1/2} = {(2 \frac{M c^{3} t_{f}}{R m_{f}})}^{1/2},

(38)

v_{e} = {(\frac{2 G M}{R})}^{1/2} .

(39)

The macroscopic reduction shows how the conditional count relation returns to the escape-speed expression. This is an internal consistency check of the substitutions and units. The reduction does not independently establish that inertial and gravitational masses are equal.

4. DISCUSSION

The equivalence principle motivates this investigation. In the normalized Pythagorean construction, the positive-root separation relation is written in terms of a squared motion-to-light count ratio. The escape-speed branch yields a mass-to-radial count ratio ([9], Appx. F, G, AV). Equation (20) identifies those ratios under the stated speed identification. This result depends on the adopted normalization and the interpretation of the count variables.

The alternative force calculation retains the upper-count coupling and specializes to equal masses. It also requires the acceleration-count assignment, the radial-to-motion identification in Eq. (33), and the light-time condition in Eq. (34). The latter follows from the speed-of-light count relation for a shared interval; it does not establish Eq. (33). These conditions are required for the recovery of Eq. (35).

The cancellation of fundamental measures leaves a dimensionless count relation ([9], Appx. AU, AV). Its physical interpretation requires the adopted normalization and realization conditions to hold for the compared quantities. The algebra connects motion and escape speed conditionally; it provides no prediction for composition-dependent differential acceleration of the kind tested by MICROSCOPE [2]. An independent derivation of equality between inertial and gravitational masses therefore requires further physical justification.

5. APPENDICES

APPENDIX A: TERMS

The entries below supplement the first-use definitions in the narrative. Previously introduced counts, fundamental measures and the residual retain their meanings from Secs. 1–3. Further MQ term and keyword context is provided in [9].

The coefficient θ_si and momentum realization p_si were introduced before Eq. (25). With the fixed 2022 CODATA inputs and stipulated realization normalization, the MQ calculation gives p_si=3.26239030504(25) kg m s^-1 [14]. Calculated uncertainties here are conditional on that normalization and the MQ relations; parenthetical digits state one standard uncertainty in the last displayed digits. The historical angular realization is denoted by Θ_si, in rad. Shwartz and Harris’s theoretical Table I gives a signal/idler angle of 3.26239 rad for a selected phase-matched configuration in diamond [16]. This configuration-specific calculated angle is not an experimental determination of the MQ coefficient or momentum realization. Related experiments address X-ray down-conversion [17] and X-ray/optical wave mixing [18]. The separate conserved-quantity discussion at future null infinity [19], Secs. 4.3–4.4, does not supply an angular-to-momentum realization for the present calculation.

In the International System of Units (SI), the exact value of the speed of light introduced in Sec. 1 is c=299792458 m s^-1. It is unchanged between the historical 2018 compilation [20] and the 2022 inputs used here [14]. Its fundamental-measure relation is given in Sec. 1.

The upper-count and finite-count coupling roles were introduced in Sec. 3. The conditional MQ calculation from the fixed 2022 CODATA inputs gives

G = 6.6740779383 (10) \times 10^{- 11} m^{3} k g^{- 1} s^{- 2} [14].

(39a)

At the electromagnetic demarcation, the same calculation gives

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

(39b)

The electromagnetic demarcation is the fixed interaction count used for this finite-count evaluation, distinct from the arbitrary radial count in the force calculation. These are calculated MQ quantities. Their conditional input-propagation uncertainties do not include an independently estimated model error. Comparison with laboratory measurements requires a specified coupling regime and an appropriate uncertainty model.

The symbols l_p, m_p and t_p denote the Planck units of length, mass and time, with units m, kg and s, respectively.

The mixed left-floor and right-ceiling brackets denote rounding to the nearest whole-unit value in this MQ notation. The dimensionless marker n_θ is obtained from half the interaction count at the electromagnetic demarcation. Using the recalculated count gives

n_{θ} = ⌊ \frac{84.6005456716}{2} ⌉ = 42 [14].

(39c)

The count uncertainty does not approach a half-unit boundary for the argument in the preceding expression. The nearest-whole result therefore remains the exact marker 42 on the adopted branch, without selecting a tie rule.

The dimensionless factor 2Q_Ln_L provides the finite-count normalization in the coupling discussed before Eqs. (25)–(28). It is associated with the Informativity differential, the non-Lorentz length contraction arising from count realization under the Frames mapping. Its upper-count limit is unity.

Let h denote the exact SI Planck constant and π the dimensionless circular constant. At the electromagnetic demarcation, italic ℏ denotes the reduced Planck quantity, calibrated to the exact SI action

h / (2 π) = 1.054571817 \dots \times 10^{- 34} J s [14].

(39d)

Upright ħ denotes the upper-count MQ quantity, whose conditional 2022 calculation is

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

(39e)

The two action realizations differ through the finite-count normalization; the gravitational expressions in this paper do not require them to be identical.

The symbol m_e denotes the electron mass, in kg.

The symbol a₀ denotes the Bohr radius associated with the atomic ground-state length scale, in m.

The dimensionless quantities

α_{f}^{- 1}, α_{p}^{- 1} and α_{c}^{- 1}

(39f)

are the fundamental, Planck-like and classical inverse fine-structure forms. The fundamental form

α_{f}^{- 1} = 42 θ_{s i}

(39g)

is expressed relative to the Internal Frame; the remaining forms belong to System Frame realizations. The classical and Planck-like forms differ by the finite-count normalization.

The symbol γ denotes the historical electromagnetic transformation parameter. In its dimensional SI convention it has units m⁶ s⁻⁴ C⁻². That historical approach uses an auxiliary electromagnetic input; this parameter does not enter Eqs. (1)–(39).

STATEMENTS AND DECLARATIONS

The author has no relevant financial or non-financial interests to disclose.

The author has no competing interests to declare that are relevant to the content of this article.

DATA AVAILABILITY

The calculation inputs and realization assumptions are identified in Appendix A. The cited MQ publication [9] provides the wider theoretical context and identifies supplementary calculation material.

REFERENCES

[1] A. Einstein, The Formal Foundation of the General Theory of Relativity (1914), Sec. 1. Reprinted in The Collected Papers of Albert Einstein, Vol. 6, English Translation of Selected Texts, pp. 30–32 (Princeton University Press, 1997).

[2] P. Touboul et al. (MICROSCOPE Collaboration), MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle, Phys. Rev. Lett. 129, 121102 (2022), doi:10.1103/PhysRevLett.129.121102.

[3]J.A. Geiger, Measurement Quantization Unites Classical and Quantum Physics, J. High Energy Phys. Gravit. Cosmol., Vol. 4, No. 2, 262–311 (Apr 2018). https://doi.org/10.4236/jhepgc.2018.42019.

[4]J.A. Geiger, Quantum Model of Gravity Unifies Relativistic Effects, Describes Inflation/Expansion Transition, Matches CMB Data, J. High Energy Phys. Gravit. Cosmol., Vol. 4, No. 4 (Oct 2018). https://doi.org/10.4236/jhepgc.2018.44038.

[5]J.A. Geiger, Measurement Quantization Describes Galactic Rotational Velocities, Obviates Dark Matter Conjecture, J. High Energy Phys. Gravit. Cosmol., Vol. 5, No. 2 (Apr 2019). https://doi.org/10.4236/jhepgc.2019.52028.

[6]J.A. Geiger, Physical Significance of Measure, J. High Energy Phys. Gravit. Cosmol., Vol. 6, No. 1 (Jan 2020). https://doi.org/10.4236/jhepgc.2020.61008.

[7]J.A. Geiger, Measurement Quantization Describes History of Universe – Quantum Inflation, Transition to Expansion, CMB Power Spectrum, J. High Energy Phys. Gravit. Cosmol., Vol. 6, No. 2 (Apr 2020). https://doi.org/10.4236/jhepgc.2020.62015.

[8] J.A. Geiger, Measurement Quantization Describes the Physical Constants, Int. J. Theor. Math. Phys., 11(1), pp. 29–59 (2021).

[9]J.A. Geiger, Measurement Quantization, Intl. J. Geom. Methods Mod. Phys. (Jan. 2023), doi:10.1142/S021988782350069X.

[10]M. Planck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu, Berlin, 5, p. 480 (1899).

[11]Heisenberg, W., Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik, 43 (3–4): 172–198 (1927), Bibcode:1927ZPhy.43.172H, doi:10.1007/BF01397280, S2CID 122763326.

[12] A. Einstein, Zur Elektrodynamik bewegter Körper, Annalen der Physik 322, 891–921 (1905), doi:10.1002/andp.19053221004.

[13] H.P. Robertson, The Uncertainty Principle, Phys. Rev. 34, 163–164 (1929), doi:10.1103/PhysRev.34.163.

[14] P.J. Mohr, D.B. Newell, B.N. Taylor and E. Tiesinga, CODATA Recommended Values of the Fundamental Physical Constants: 2022, Rev. Mod. Phys. 97, 025002 (2025), doi:10.1103/RevModPhys.97.025002. Preprint arXiv:2409.03787 (2024).

[15] G. Johnstone Stoney, On the Physical Units of Nature, Philosophical Magazine, Series 5, 11, 381–390 (1881), doi:10.1080/14786448108627031.

[16]S. Shwartz and S. E. Harris, Polarization Entangled Photons at X-Ray Energies, Phys. Rev. Lett. 106, 080501 (2011), arXiv: 1012.3499, doi: 10.1103/PhysRevLett.106.080501.

[17]S. Shwartz, R. N. Coffee, J. M. Feldkamp, et. al., X-ray Parametric Down-Conversion in the Langevin Regime, Physical Review Letters, 109, 013602 (July 6, 2012), doi:10.1103/PhysRevLett.109.013602.

[18]T. E. Glover et al., X-ray and optical wave mixing, Nature, 488, 7413, 603–608 (2012), doi:10.1038/nature11340.

[19]Po-Ning Chen, Mu-Tao Wang, Ye-Kai Wang, Shing-Tung Yau, Conserved quantities in general relativity -- the view from null infinity (Apr. 8, 2022), doi:10.48550/arXiv.2204.04010.

[20]NIST: CODATA Recommended Values of the Fundamental Physical Constants: 2018, (May 2019), https://physics.nist.gov/cuu/pdf/wall_2018.pdf, doi:10.1103/RevModPhys.93.025010.