Discrete Expressions for the Gravitational Constant

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We derive a measurement quantization (MQ) expression for the gravitational constant

G

from the discrete Internal Frame and its continuous relational encoding in the System Frame. Using the 2022 Committee on Data for Science and Technology (CODATA) fine-structure input and exact International System of Units (SI) constants, the upper-count expression gives G=6.6740779383(10)×10^-11 m³ kg^-1 s^-2. Electron mass and the Bohr radius enter the initial expression but cancel through their exact adjusted-constant relationship. The quoted standard uncertainty is calculated by statistical propagation of the calibration input uncertainty under the stated MQ relations. It quantifies input uncertainty within the model. The resulting gravitational prediction can be compared with measurements that were not used in this calibration.

Keywords: Gravitational Constant, Gravity, Discrete Gravity, Measure

1. INTRODUCTION

Newtonian gravity relates the gravitational constant to force, F=Gm₁m₂/r², and escape speed, v_escape=√(2Gm₁/r). Here

F

is the force between masses

m_{1}

and

m_{2}

r

is their separation, and

v_{escape}

is the escape speed at distance

r

from the spherically symmetric source mass

m_{1}

. Thus

G

has units m³ kg⁻¹ s⁻².

In this paper we develop an MQ gravitational coupling using the discrete description of motion ([1], Appx. C) and the frame construction ([1], Appx. K). The introductory paper, Measurement Quantization [1], presents the framework and its numerical comparisons ([1], Table 2). The present derivation and its calibration are distinguished from an independent experimental determination of the coupling.

In MQ, the Internal Frame contains discrete count configurations, while the System Frame expresses observable quantities as continuous relational separations through the Frames mapping ([1], Appx. K). The Informativity differential is the count-dependent, non-Lorentz length contraction arising from this count structure; its associated normalization factor enters the gravitational expression ([1], Appx. D). It is distinct from a complete description of spacetime curvature. We use upright gravitational and MQ reduced-action symbols at the upper count limit and italic symbols for their electromagnetic-demarcation realizations, evaluated at the fixed interaction count for electromagnetic realization ([1], Appx. AB, AE, AN). These distinctions identify the realization used in a calculation.

For Eq. (1),

c

is the exact speed of light, 299792458 m s⁻¹ [2]. The fundamental measures

l_{f}

m_{f}

and

t_{f}

have units m, kg and s. The dimensionless radial count

n_{L}

gives

r = n_{L} l_{f}

, and

Q_{L}

is the dimensionless right-triangle residual. The coefficient

θ_{si}

is dimensionless;

p_{si}

is its momentum realization in kg m s⁻¹, with p_si=l_fm_f/(2t_f). The upper-count gravitational expression ([1], Appx. BM) uses these dimensional measures and is calculated here from the 2022 CODATA calibration [2].

G = \lim_{n_{L} \to \infty} \frac{Q_{L} n_{L} l_{f} c^{3}}{p_{si}} = \frac{l_{f} c^{3}}{2 p_{si}} = 6.6740779383 (10) \times 1 0^{- 11} m^{3} k g^{- 1} s^{- 2}

(1)

Equations (2)–(3) describe the right-triangle construction underlying the residual ([1], Appx. D). The normalization factor associated with the Informativity differential is

2 Q_{L} n_{L}

([1], Appx. D). Equation (1) uses half of this normalization,

Q_{L} n_{L}

, whose upper-count limit is one half.

The angular realization

Θ_{si}

carries radians and remains distinct from the coefficient and momentum realization ([1], Appx. P–S). Related experiments study X-ray down-conversion [3] and X-ray–optical wave mixing [4]. Bell states are maximally entangled two-photon polarization states in this setting; their basis [5] and interpretations of Bell correlations [6] provide background. For diamond (111), a 25 keV pump and equal 12.5 keV daughter energies, Shwartz and Harris [7] tabulate several theoretical Bell-state branches. One branch has equal signal and idler wavevector angles of 3.26239 rad relative to the crystal planes. Conserved quantities in general relativity [8] provide separate theoretical context.

This selected angular entry agrees with the MQ coefficient when both are displayed to five decimal places. The table provides neither an angular measurement nor an uncertainty supporting further MQ digits [7]. Equal daughter angles alone are insufficient to establish a Bell state; the model also imposes phase matching, which enforces energy and crystal-momentum conservation, and the requisite polarization amplitudes. MQ relates the angular and momentum realizations through a shared coefficient ([1], Appx. P–S), while Eq. (9) calibrates it from the fine-structure input ([1], Appx. AD).

Einstein’s mass–energy equivalence provides a useful analogy. Rest energy and mass, despite their different dimensions, are related by the squared speed of light, as derived within special relativity [9]. Within MQ, the angular and momentum realizations are interpreted as manifestations of a common relation ([1], Appx. P–S). Their proposed physical identity remains unexplained in the present framework. The analogy introduces this interpretation; it does not establish the physical angle–momentum connection.

The experimental studies address different observables. Shwartz et al. [3] measure photon-pair coincidence rates for diamond (220) with an 18 keV pump. Glover et al. [4] demonstrate X-ray–optical sum-frequency generation in diamond (111), using 8 keV X-rays and 1.55 eV optical photons. Their reported analyser rocking-curve width, 17 ± 2 µrad at full width at half maximum, characterizes instrumental acceptance. It is not an uncertainty assigned to the MQ coefficient or to the 3.26239 rad angular branch. Neither study measures the MQ bound-related momentum realization.

These experiments support photon-process descriptions with geometry-dependent energy–angle relations determined by phase matching [3, 4]. Within MQ, the shared coefficient preserves the distinct dimensions of its angular and momentum realizations ([1], Appx. P–S). Common normalization explains their numerical representation, while a physical connection between the modeled angle and the gravitational momentum realization remains a testable assumption. The present gravitational result therefore retains its propagated standard uncertainty.

The upper-count limit in Eq. (1) is a mathematical asymptote. At a finite positive radial count, the residual in Eq. (14) gives a smaller gravitational coupling. For distances at least as large as the Bohr radius, the calculated fractional deficit is below 2.4 × 10⁻⁵⁰ under the present calibration. The limit is therefore numerically sufficient at the precision quoted here, while the finite-count expression remains available.

At the electromagnetic demarcation ([1], Appx. AB, AE, AN), Eq. (7) gives

n_{L} = 84.6005456716 (65)

, and the finite-count coupling is G=6.6738448317(10)×10^-11 m³ kg^-1 s^-2. Assigning that count to an experimental observable requires the applicable realization and measurement geometry; using electromagnetic instrumentation alone does not establish that assignment.

2. METHODS

This paper applies the MQ count description to the gravitational coupling. Its boundary construction uses classical speed and escape-speed relations together with the uncertainty relation [10], as developed in Measurement Quantization [1]. Other publications apply the framework to classical, quantum, cosmological and physical-constant problems [11, 12, 13, 14, 15, 16]. Their publication establishes the provenance of these applications, rather than independent validation of the present gravitational prediction.

The MQ boundary construction combines the uncertainty relation at its bound, the length–time relation for the speed of light, and the mass–length relation at the escape-speed bound ([1], Appx. I–J). We adopt the resulting fundamental references and discrete count domain for this derivation. Identifying uncertainty spreads with these count measures and applying the boundary conditions are parts of that construction. They do not provide a separate experimental determination of the references. The published appendices and definitions are available in Measurement Quantization [1], with associated material in Online Resources.

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

Suppl. 1 provides spreadsheet solutions to the numbered equations of Measurement Quantization [1], including its appendices. Suppl. 2 provides solutions for the preprints, with each preprint on a separate worksheet tab. The calculations reported here use the explicit 2022 inputs and equations stated in this manuscript.

3. RESULTS

1. Calculations

The residual

Q_{L}

depends on the radial count. The product

Q_{L} n_{L}

in the gravitational expression is half of the normalization factor introduced above and tends to one half at the upper count limit. The finite-count and limiting expressions retain the typographical distinction introduced in the Introduction.

Figure 1 illustrates the residual-count construction. The dimensionless ratio Q_L/n_L supplies a scalar geometric factor for the mapped gravitational coupling. Equation (4) converts it using the fundamental length and elapsed-time measures, and Eq. (5) adds the mass normalization needed for G/r². This scalar relation does not supply a complete spacetime metric or curvature tensor.

The factor l_f/t_f² in Eq. (4) has acceleration units. The mass normalization in Eq. (5) is c/p_si=2/m_f, using c=l_f/t_f and the momentum realization ([1], Appx. T, V). Its units are kg⁻¹. The dimensionless mass count

n_{M}

refers to

m_{f}

; the boundary calculation uses the reference fraction n_M=1/2 ([1], Appx. I).

Equation (6) identifies the normalized MQ expression with the Newtonian coupling per squared distance and determines the finite-count

G

. In Figure 1, A, B and C label the triangle vertices, AC is the hypotenuse and BC is the radial leg. The side letters are local diagram labels. The overbarred quantities

\bar{AC}

and

\bar{BC}

in Eqs. (2) and (13) are dimensionless segment lengths measured in units of

l_{f}

. The radial-leg count

n_{Lb}

equals

n_{L}

; the unit leg fixes the reference in Eq. (2). Equation (3) subtracts the radial-leg count to obtain the residual. Numerical fundamental measures come from the calibration below.

\bar{AC} = {(1 + n_{Lb}^{2})}^{1 / 2}

(2)

Q_{L} = {(1 + n_{Lb}^{2})}^{1 / 2} - n_{Lb}

(3)

Equations (4)–(6) then supply the dimensional gravitational coupling from the residual construction.

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

(4)

\frac{Q_{L} l_{f}}{n_{L} t_{f}^{2}} \frac{c}{p_{si}} = \frac{Q_{L} c^{2}}{n_{L} t_{f} p_{si}} = \frac{Q_{L} l_{f} c^{2}}{n_{L} l_{f} t_{f} p_{si}} = \frac{Q_{L} c^{3}}{r p_{si}}

(5)

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

(6)

To calculate the coefficient and the electromagnetic-demarcation count, we use the constraint ([1], Appx. AC),

n_{L} θ_{si} = 276

(7)

In Eqs. (7)–(9),

n_{L}

is specifically the electromagnetic-demarcation count ([1], Appx. AE); the gravitational evaluation radius is chosen separately. Here

α_{c}

is the classically realized, dimensionless fine-structure constant. The mixed brackets

⌊ x ⌉

round the dimensionless argument

x

to the nearest whole-unit value. The calibrated branch gives

⌊ 42 θ_{si} ⌉ = 137

. The inverse fine-structure relation is ([1], Appx. AA),

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

(8)

Inverting Eq. (8) and substituting the positive-coefficient residual from Eq. (3) gives ([1], Appx. AD),

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} ⌋}

(9)

With

n_{L} θ_{si} = 276

and the 2022 input α_c^-1=137.035999177(21) [2], solving Eq. (9) on that rounding branch gives,

θ_{si} = 3.26239030504 (25)

(10)

Parenthetical digits denote one standard uncertainty in the final quoted digits. All current MQ numerical results use the shared 2022 calibration. Their reported standard uncertainties are calculated by propagating the fine-structure input uncertainty through the MQ dependency chain, retaining correlations between derived quantities. Exact SI definitions contribute no input uncertainty. This statistical calculation assumes the stated MQ relations and numerical realization; it does not itself establish the physical angle–momentum identification.

Next, we calculate the fundamental measures ([1], Appx. BM) and the gravitational coupling from this calibration.

Returning to Figure 1, we now set the radial evaluation distance to one metre. This selects a different

n_{L}

from the electromagnetic-demarcation count used to determine the coefficient.

The calculation begins with fundamental length and time. Equation (11) uses electron mass

m_{e}

, the Bohr radius

a_{0}

and the dimensionless Planck-like fine-structure form

α_{p}

, with α_p^-1=84θ_si-⌊42θ_si⌉. The historical source is [17]; the present calculation uses 2022 CODATA [2]. Here

ℏ

is the conventional reduced Planck constant in J s,

h

is the exact SI Planck constant in J s, and

π

is the circular constant. The exact adjusted-constant identity m_ea₀=ℏα_c^-1/c, with ℏ=h/(2π), removes electron-mass dependence while retaining covariance.

l_{f} = \frac{m_{e} a_{0} α_{p}}{m_{f}} = 1.61619991148 (12) \times 1 0^{- 35} m

(11)

t_{f} = \frac{l_{f}}{c} = 5.39106261131 (41) \times 1 0^{- 44} s

(12)

At the exact evaluation distance

r = 1 m

, the recalculated radial count is n_L=6.18735338926(47)×10³⁴. Equation (14), derived by rationalizing Eq. (13), avoids cancellation in the residual calculation. The rounded inputs displayed in Eq. (15) illustrate this substitution; Eqs. (16)–(18) use the unrounded shared-input calculation.

Q_{L} = \bar{AC} - \bar{BC} = \sqrt{1^{2} + {\bar{BC}}^{2}} - \bar{BC}

(13)

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

(14)

Q_{L} = \frac{1}{\sqrt{1 + {(6.18735338926 (47) \times 1 0^{34})}^{2}} + 6.18735338926 (47) \times 1 0^{34}}

(15)

Q_{L} = 8.08099955739 (62) \times 1 0^{- 36}

(16)

G = \frac{r Q_{L} c^{3}}{p_{si}} = \frac{1 m \times 8.08099955739 (62) \times 1 0^{- 36} \times {(299792458 m s^{- 1})}^{3}}{3.26239030504 (25) kgm s^{- 1}}

(17)

G = 6.6740779383 (10) \times 1 0^{- 11} m^{3} k g^{- 1} s^{- 2}

(18)

For the upper-count solution, we take the limit explicitly. The product

Q_{L} n_{L}

, half of the normalization factor associated with the Informativity differential, tends to one half; it is not exactly one half at finite count. Equations (19)–(21) therefore use upright

G

G = \lim_{n_{L} \to \infty} \frac{n_{L} l_{f} Q_{L} c^{3}}{p_{si}} = \lim_{n_{L} \to \infty} Q_{L} n_{L} \frac{l_{f} c^{3}}{p_{si}} = \frac{l_{f} c^{3}}{2 p_{si}}

(19)

G = \frac{1.61619991148 (12) \times 1 0^{- 35} m \times {(299792458 m s^{- 1})}^{3}}{2 \times 3.26239030504 (25) kgm s^{- 1}}

(20)

G = 6.6740779383 (10) \times 1 0^{- 11} m^{3} k g^{- 1} s^{- 2}

(21)

Equations (19)–(21) evaluate a mathematical asymptote. They do not assert an infinite physical count configuration; a finite admissibility bound would require separate domain information. At any specified finite radius, Eq. (14) gives the exact relative deficit 1-2Q_Ln_L=Q_L², derived from Eq. (3). At one metre it is 6.53026 × 10⁻⁷¹, so Eqs. (18) and (21) have the same displayed value despite their distinct finite and limiting meanings.

2. Comparison with Measurements

Table 1 compares the finite-count expression with its upper-count limit using the present calibration. The percentage deficit is one hundred times the fractional deficit above. Separately, CODATA 2022 recommends 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻² [2]. The MQ upper-count value is lower by 0.003327 percent, or 1.4804 times that recommended standard uncertainty. CODATA obtains its gravitational recommendation from a separate weighted mean of 16 measurements and expands their uncertainties by 3.9 to address inconsistencies [2]. Thus this comparison does not establish the MQ digits as experimental accuracy. The historical 2014 recommendation, 6.67408(31) × 10⁻¹¹ m³ kg⁻¹ s⁻² [18], remains identified in Table 2.

Table 1. Calculated percentage deficits of the finite-count gravitational coupling from its upper-count limit using the 2022 calibration. The distance choices follow ([1], Table M.1).

Distance	Electromagnetic demarcation	500 $l_{f}$	10000 $l_{f}$	$a_{0}$ ^a	1 m	13.794 Gly ^b
Deficit (%)	3.49272 × 10⁻³	9.99998 × 10⁻⁵	2.50000 × 10⁻⁷	2.33200 × 10⁻⁴⁸	6.53026 × 10⁻⁶⁹	3.83443 × 10⁻¹²¹

^a The Bohr radius is used as an illustrative evaluation distance [2].

^b The nonzero deficit at this illustrative distance is far below the quoted propagated standard uncertainty. The 13.794 Gly distance is not a new cosmological estimate.

The calculated deficit decreases with increasing radial count. Comparisons involving Planck-form expressions ([1], Appx. N) must consistently identify the electromagnetic-demarcation or upper-count realization of each MQ quantity. Mixing those realizations changes the calculated values.

At the Bohr radius and larger evaluation distances, the calculated finite-count deficits in Table 1 are below the uncertainty of the present gravitational result. Within MQ, the residual contributes to the continuous geometry of the System Frame even though it is not an independent whole-unit count in the Internal Frame ([1], Appx. D, K).

The first Table 1 distance uses the recalculated electromagnetic-demarcation count

n_{L} = 84.6005456716 (65)

([1], Appx. AB, AE, AN). Its deficit is 0.00349271655 percent.

At 500 and 10000 fundamental lengths, the deficits are respectively 9.99998 × 10⁻⁵ percent and 2.50000 × 10⁻⁷ percent to the displayed precision. Their proximity to simple decimal values follows the large-count behavior of the right-triangle residual. These calculations alone do not establish a relation to universal expansion.

Table 2 preserves the earlier CODATA and MQ Planck-form comparisons from ([1], Table 2). Its columns use the conventional Planck length

l_{p}

in m, time

t_{p}

in s, and mass

m_{p}

in kg. The row-label function

f

denotes the source calculation from the stated reduced-action and gravitational quantities. In its MQ rows, italic

ℏ

denotes finite-count reduced action and upright

ħ

its upper-count limit, both in J s. The CODATA rows use the conventional reduced Planck quantity introduced with Eq. (11). This table supplies historical context, rather than an independent identification of the MQ realizations or a replacement for the present calibration.

Table 2. Historical CODATA and MQ Planck-form comparison, reprinted from ([1], Table 2). The original calibration and numerical entries are retained; they are not the present 2022 recalculation.

Year	ħ (10^-34 J s)	l_p (10^-35 m)	t_p (10^-44 s)	m_p (10^-8 kg)	G (10^-11 m³ kg^-1 s^-2)
2010 ^a	1.054571726(47)	1.616199(97)	5.39106(32)	2.17651(13)	6.67384(80)
2014 ^a	1.054571800(13)	1.616229(38)	5.39116(13)	2.176470(51)	6.67408(31)
2018 ^a	1.054571817	1.616255(18)	5.391247(60)	2.176434(24)	6.67430(15)
f(ħ,G) ^b	1.0545349844(45)	1.6161716871(50)	5.3909684650(90)	2.1764705491(26)	6.6738448362(53)
f(ħ,G) ^b	1.0545349844(45)	1.6161999120(35)	5.3910626131(72)	2.1764325398(24)	6.6740779428(56)
f(ℏ,G) ^b	1.0545718176(46)	1.6162281374(12)	5.3911567628(97)	2.1764705491(26)	6.6740779428(56)
f(ℏ,G) ^b	1.0545718176(46)	1.6161999120(35)	5.3910626131(72)	2.1765085590(91)	6.6738448362(53)

^a CODATA publications for 2010, 2014 and 2018 [17, 18, 19].

^b The historical MQ rows use their original reduced-action and gravitational calibrations at an electromagnetic-demarcation count of

n_{L} = 84.6005496647 (07)

, the upper count limit, or a mixture of these realizations. These historical entries are not current 2022 results.

Individual measurements also test the prediction without a measured gravitational input. Li et al. [20] report 6.674184(78) × 10⁻¹¹ m³ kg⁻¹ s⁻² using time of swing, and 6.674484(78) × 10⁻¹¹ m³ kg⁻¹ s⁻² using angular-acceleration feedback on different apparatus. The MQ upper-count value lies below these results by 1.36 and 5.21 times their respective reported standard uncertainties. These residuals are scaled only by the respective experimental standard uncertainties; they are not combined-uncertainty significance tests. They expose agreement with one result alongside tension with another. Both measurements contribute to the CODATA recommendation [2], so the three comparisons are not statistically independent. An MQ count assignment to either apparatus requires a measurement prescription; the measurement method or CODATA edition alone does not identify a realization [1, 17, 18].

4. DISCUSSION

The fundamental relation ([1], Appx. V), which links the fundamental length, mass and elapsed-time measures to the momentum realization, its scalability ([1], Appx. AY), and the boundary construction ([1], Appx. I–J) supply the framework for the gravitational prediction. Shared-input agreement among its derived constants tests algebraic consistency. Independent physical tests must use observables not fitted in the calibration and combine the experimental and propagated prediction uncertainties, retaining relevant correlations. An independently measured angular realization could test the coefficient and reverse the fine-structure calculation ([1], Appx. AD), provided the same fine-structure input is not reused to obtain that angle. The proposed test requires a specified crystal geometry, photon energies, Bell-state branch and angular uncertainty.

Within MQ, the fundamental references are irreducible under the boundary construction ([1], Appx. I–J). The residual contributes to System Frame observables through the Frames mapping. The Internal Frame has no intrinsic continuous metric, but this does not imply the absence of observable gravitational curvature. The scalar coupling derived here is not by itself a complete general-relativistic metric, a curvature tensor or a law for field dynamics.

The upper-count MQ expression can be extended as follows, using the same limiting normalization.

G = \lim_{n_{L} \to \infty} \frac{Q_{L} r c^{3}}{p_{si}} = \lim_{n_{L} \to \infty} \frac{Q_{L} n_{L} l_{f} c^{3}}{p_{si}} = \frac{c^{3} l_{f}}{2 p_{si}} = \frac{c^{3} t_{f}}{m_{f}} = \frac{l_{f}}{t_{f}} \frac{l_{f}}{t_{f}} \frac{l_{f}}{t_{f}} \frac{t_{f}}{m_{f}}

(22)

Equation (22) can also be written as G=c²l_f/m_f. The expanded form displays three length-to-time factors and one time-to-mass factor; their product has the required gravitational units. These are algebraically equivalent forms under the stated MQ relations.

In count notation,

n_{T}

denotes the elapsed-time count relative to

t_{f}

. The ratios n_L/n_T and n_M/n_T are dimensionless count ratios. Their corresponding length and mass frequencies count fundamental units per second ([1], Appx. O) and are obtained by dividing the ratios by the fundamental elapsed time. Multiplication by the fundamental length or mass then gives the respective dimensional measure rate. The reciprocal of

t_{f}

is a rate in s⁻¹, not an elapsed-time count. Equation (22) is an algebraic coupling relation under these definitions; its numerical prediction depends on the calibrated fundamental measures.

For comparison, conventional Planck measures have their historical origin in [21] and modern definitions in [2]. The identity G=((l_p)/(t_p))³(t_p)/(m_p) follows from those definitions. It is not an independent determination of

G

, because the measures already contain that constant. The MQ expression supplies a prediction conditional on its stated geometry and calibration assumptions.

5. APPENDICES

APPENDIX A: TERMS

Complete set of term and keyword definitions in the paper entitled, Measurement Quantization ([1], Appx. BP–BQ).

The coefficient and momentum realization are introduced before Eq. (1), and the angular realization between Eqs. (1) and (2); their distinct dimensions and calibration are retained throughout ([1], Appx. P–S). The X-ray model comparison is [7], while [3, 4] and [8] provide the separate contexts described in the Introduction.

The exact speed of light is introduced before Eq. (1) [19, 2]. For light motion, its MQ count form is c=(n_Ll_f)/(n_Tt_f) ([1], Appx. I). Here the length and elapsed-time counts describe the light trajectory, while the gravitational evaluation count is identified by its radial context.

The upper-count gravitational value is given in Eq. (1). The finite-count expression at the electromagnetic demarcation gives G=6.6738448317(10)×10^-11 m³ kg^-1 s^-2 under the same calibration.

The fundamental measures and their units are introduced before Eq. (1). Their calibrated values are calculated in Eqs. (11)–(12) and the associated momentum relation; they are distinct from conventional Planck measures.

Conventional Planck measures are introduced with Table 2; its historical Planck-form entries retain their original calibrations.

The radial, mass and elapsed-time counts are introduced at their first use in the derivation. System Frame radial counts used for finite-distance evaluation need not be integer Internal Frame configurations.

The length count describes the separation used in the corresponding relation. Its radial role is specified before Eq. (1), and its electromagnetic calibration role before Eq. (8).

The radial evaluation count and electromagnetic calibration count are identified in their respective passages and are not interchangeable.

The dimensionless count $n_{Lc}$ denotes the count of $l_{f}$ corresponding to a change in the position of light relative to the observer, as retained in the source nomenclature.

The dimensionless count $n_{θ}$ is the System Frame count of $θ_{si}$ resolved from the Internal Frame demarcation and adjusted for the metric differential, the mismatch in realizing discrete count structure as a continuous observable. The recalculated half-count has nearest whole-unit value 42.

The nearest-whole-unit convention is defined before Eq. (8); it is distinct from ordinary floor and ceiling.

The right-triangle residual is defined by Eq. (3) and evaluated in the stable form of Eq. (14).

The normalization factor associated with the Informativity differential is introduced after Eq. (1). Its half-product enters the gravitational coupling, with finite-count values and the upper-count limit distinguished in the Results.

The conventional reduced Planck quantity is introduced with Eq. (11) and is exact through h/(2π). The MQ upper-count quantity is introduced with Table 2. Under the same 2022 calibration its value is ħ=2p_sil_f=1.0545349844417293(56)×10^-34 J s.

The electron mass is introduced with Eq. (11).

The Bohr radius is introduced with Eq. (11); it is the conventional atomic length scale used for the illustrative distance in Table 1.

The inverse fine-structure forms α_f^-1, α_p^-1 and α_c^-1 distinguish the fundamental, Planck-like and classical realizations ([1], Appx. AA). The fundamental form α_f^-1=42θ_si refers to the Internal Frame; the other forms refer to the System Frame. Their difference involves the non-Lorentz count-dependent Informativity differential, rather than an additional Lorentz contraction.

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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