Classical Approach to Discrete Gravity

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We present a classical count construction of gravity within Measurement Quantization (MQ). A discrete Internal Frame count domain is represented as continuous relational structure in the System Frame through the Frames mapping. The construction yields finite-count gravitational and reduced Planck quantities and their upper-count limits. The loss of the fractional count and the fundamental momentum are calculated extensions of existing classical mechanics under the MQ interpretation. Neither has been directly confirmed experimentally. Numerical examples use the 2022 recommended constants from the Committee on Data of the International Science Council (CODATA), with the independently measured cosmic microwave background (CMB) temperature where required. The reported uncertainties propagate those inputs conditionally on the MQ assumptions. Numerical agreement and additional calculated digits do not establish independent experimental validation or electromagnetic unification.

Keywords: Discrete Gravity, Fundamental Measures, Measure, Planck, Gravity

1. INTRODUCTION

We examine how reference measures and their counts enter the MQ description of gravity ([1], Appx. I, J). Their physical significance is part of the framework’s interpretation. Background independence and loop quantum gravity provide broader context [2, 3], while the two MQ domains introduced in the abstract supply the interpretation used here. The equations establish consequences of this count construction; they do not by themselves establish a physical mechanism for the proposed count loss.

The fundamental measures are l_f, m_f and t_f, in m, kg and s of the International System of Units (SI), respectively ([1], Appx. W). Their dimensionless Internal Frame counts are n_L, n_M and n_T. The speed of light is c=l_f/t_f. The invariant dimensionless coefficient θ_si has fundamental momentum realization p_si in kg m s⁻¹. These are scalar magnitudes; the index si identifies the invariant coefficient and its realization. The dimensional fundamental relation is l_fm_f=2p_sit_f ([1], Appx. V). These realizations belong to the MQ interpretation of classical mechanics [1, 4-9]. The count construction also uses escape velocity and Heisenberg’s uncertainty principle [10].

Our discussion concerns radial separation. Figure 1 uses a Euclidean right triangle with vertices A, B and C and a right angle at B. Its legs a and b carry the unit reference count and radial count; its hypotenuse c carries the realized count. These figure labels are local geometric labels. The geometry is an input to the MQ construction. Its applicability to a realized gravitational field requires the physical interpretation specified below.

The dimensionless right-triangle residual is $${\mathit{Q}}_{\mathit{L}}=\sqrt{1+{\mathit{n}}_{\mathit{L}}^2}-{\mathit{n}}_{\mathit{L}}$$, with n_L>0. The geometric leg count is an integer; numerical evaluations also use the same positive-real continuation, including the noninteger electromagnetic demarcation, the fixed interaction count for electromagnetic realization ([1], Appx. AE). Let r=n_Ll_f be the radial reference separation in m and G(n_L) the finite-count gravitational coupling in m³ kg⁻¹ s⁻². Under the MQ residual-loss interpretation, Eq. (1) rearranges that coupling to give the radial field per unit source mass, with units m kg⁻¹ s⁻². Multiplication by the source mass gives an acceleration. This is a conditional coupling identity; the geometry alone does not supply a dynamical law for the residual loss.

The Informativity differential is the dimensionless departure 1-2Q_Ln_L from unity of the finite-count normalization discussed in [1], Appx. D. It is distinct from the relativistic effects discussed by Einstein [11]. The following equations and CODATA comparisons ([1], Appx. N) examine its calculated consequences under the MQ interpretation; they do not establish a directly measured contraction.

2. METHODS

The method uses the speed-of-light relation, escape-velocity relation and uncertainty principle [10] in the MQ count construction. These are inputs to the interpretation, rather than principles derived by the present equations. The calculation gives a radial gravitational coupling and its finite-count normalization; a relativistic field equation and a conservation law for the proposed residual loss are not derived here.

The introductory paper Measurement Quantization [1] presents the broader derivations, comparisons with measurements, referenced appendices and term definitions. Those comparisons motivate the framework; they do not constitute direct experimental confirmation of the residual loss or fundamental momentum used here. A preprint is available in Online Resources.

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

The introductory paper [1] describes two supplementary calculation spreadsheets and links to Online Resources. They provide context for the broader MQ program. The numerical results in this paper are the present 2022-input calculations; the spreadsheets are not used as independent experimental evidence.

3. RESULTS

1. Calculations

Figure 1 illustrates the right-triangle count construction ([1], Appx. C). The segment joining the figure’s endpoints A and C is its hypotenuse. The geometry calculates a noninteger separation from integer leg counts; a separate MQ interpretation is required to relate its residual to gravity.

In Fig. 1, a is the unit reference count and b carries the radial count n_L. The hypotenuse c carries n_L+Q_L, distinct from the speed-of-light symbol. The adapted image’s n_L label on the hypotenuse is historical shorthand for this realized count; its unit-leg label denotes only the reference count. For positive integer radial counts, the residual is smaller than one half. Applying the MQ residual-loss interpretation gives the primitive acceleration measure in Eq. (2); the geometry alone does not imply physical loss.

Equation (2) is a primitive count-resolved acceleration measure, with units m s⁻². Multiplication by c/p_si, with units kg⁻¹, gives the field per unit source mass in Eq. (3). Equation (4) follows from the speed-of-light and radial-separation relations already defined. The external X-ray papers address different optical questions. Shwartz and Harris calculate phase-matching angles for maximally entangled polarization states (Bell states), including 3.26239 rad for a specified diamond configuration [12], Table I. The later papers measure X-ray photon-pair coincidence rates [13] and X-ray/optical sum-frequency generation [14]. These results do not directly measure the MQ fundamental momentum or residual loss.

Using the finite-count coupling in the same realization gives the exact identity in Eq. (5).

Multiplying Eq. (5) by r²p_si gives Eq. (6). At finite count, Q_Ln_L is less than one half and tends to one half as the count increases without bound. The upper-count gravitational quantity, written as upright G, therefore requires the limit in Eq. (7), with the fundamental measures held fixed ([1], Appx. M). The Informativity differential introduced in Section 1 vanishes at that limit.

MQ quantities evaluated at the upper count limit are written upright.

2. Numerical Comparisons

The CODATA publications provide adjusted recommended constants for the numerical comparisons ([1], Appx. N) [15–18]. These recommendations combine experimental inputs and theoretical relations; derived constants are not independent measurements. Historical comparisons retain their dates, and the recalculated examples use the 2022 adjustment. The shared-input dependencies must be retained when interpreting agreement.

Table 1 evaluates the finite-count gravitational field relative to its upper-count value ([1], Appx. M). Its percentage difference is 100(1-2Q_Ln_L), which decreases with increasing separation. The atomic length a₀ denotes the conventional Bohr radius in m. These are calculated consequences of the normalization, conditional on its MQ interpretation, rather than measured contractions.

Table 1. Relative difference from the upper-count gravitational field, expressed as a percentage. The first column uses the recalculated electromagnetic-demarcation count; the two following columns use the stated exact counts. Gly denotes a billion light-years. Adapted from [1].

^a The Bohr radius introduced above is 5.29177210544(82) × 10⁻¹¹ m [18]. Including the 2022 covariance with inverse fine structure gives a difference of 2.33199653412(35)×10^-48 %. The table rounds this central value to six significant digits; parentheses in this note give its conditional standard uncertainty.

^b The final column in Tables 1 and 2 is the light-travel distance cA_U, where A_U is the present System Frame age in the MQ CMB reconstruction ([1], Appx. BI). The CMB temperature is T_CMB=2.72548(57) K, the combined estimate in Fixsen [19], Table 2. With the 2022 calibration, the stipulated numerical evaluation in SI units gives a conditional age of 13.7943(38) billion Julian years and the same numerical distance in Gly. A Julian year is exactly 31,557,600 s. This is a framework-dependent numerical reconstruction, not an independent observation of cosmic age or radius. Its conditional uncertainty is dominated by the temperature input.

The gravitational difference becomes very small at laboratory and astronomical distances. Table 1 retains its nonzero calculated central values rather than rounding them to zero. Table 2 instead compares the reduced Planck quantity with its value at the electromagnetic demarcation introduced in Section 1 ([1], Appx. L). The different reference choices explain the two tables without assigning an experimentally established measurement regime to either constant.

Table 2. Absolute relative difference of the reduced Planck quantity at the indicated count from its electromagnetic-demarcation value, expressed as a percentage. The first column uses the same demarcation and is zero by identity. Adapted from [1].

Planck’s proposal of natural units [20], Sec. 26, provides historical context. The conventional Planck length l_p, time t_p and mass m_p, with units m, s and kg, combine the reduced Planck constant ℏ, gravitational constant and speed of light in Eqs. (8)–(10). The conventional value ℏ=h/(2π) uses the exact SI Planck constant h in J s and dimensionless circular constant π. Substituting the specified finite-count and upper-count MQ quantities tests their numerical consequences; the combinations do not provide an independent measurement of the Informativity differential.

Equations (8)–(10) are the conventional Planck-unit definitions. For the MQ calculations, every component must carry its stated realization ([1], Appx. X). Equation (11) gives the finite-count reduced Planck quantity ℏ(n_L) in J s. In Eq. (12), the retained notation f(n_L) denotes only the gravitational coupling G(n_L), not the reduced Planck quantity.

Table 3 retains the historical 2010, 2014 and 2018 CODATA rows [15-17]. The 2022 row is recalculated from the displayed 2022 gravitational input and exact SI Planck constant [18]. Each MQ row is labeled by its ordered reduced-Planck and gravitational input pair, with upright symbols identifying upper-count quantities and italic symbols identifying conventional or electromagnetic-demarcation inputs. The four rows use the common 2022 inverse fine-structure input and the quantities in Eqs. (11)–(12) ([1], Appx. L, M, AE). Their uncertainties are conditional on the MQ realization and count assumptions. Shared-input dependencies are retained; these comparisons are not independent experimental confirmation.

Table 3. Historical CODATA reference values, recalculated 2022 Planck quantities, and MQ Planck-form combinations. Adapted from [1].

^a CODATA publications for 2010, 2014 and 2018 [15-17].

^b The four MQ rows use, in order, upper-count reduced Planck quantity with finite-count gravitational coupling; both upper-count quantities; the conventional exact reduced Planck input with upper-count coupling; and that same reduced Planck input with finite-count coupling. The finite count is n_L=84.6005456716(65). The reduced Planck entries are rounded to twelve decimal places in the displayed scale. Parentheses give one-standard-deviation conditional uncertainties on the last digits. c The 2022 row is calculated using G=6.67430(15)×10^-11 m³ kg⁻¹ s⁻² and exact h and c [18]; its reduced Planck entry is rounded, with no source uncertainty.

The historical rows document the earlier comparison. In the 2022 adjustment, G=6.67430(15)×10^-11 m³ kg⁻¹ s⁻² is obtained separately from the atomic-constant adjustment [18], Sec. XV.A.2. The upper-count MQ result lies 1.48 combined standard uncertainties below that recommendation; the electromagnetic-demarcation result lies 3.03 below it. The comparison includes the much smaller conditional MQ uncertainty. It neither establishes which MQ realization a gravitational experiment measures nor validates the residual-loss mechanism ([1]; [15–18]).

The finite-count and upper-count evaluations differ because of the normalization factor. Table 3 checks the resulting Planck-form combinations. These are calculated extensions of classical mechanics under the MQ interpretation. Direct experimental confirmation of the residual loss and fundamental momentum would require evidence beyond these input-dependent numerical comparisons.

4. DISCUSSION

The same MQ realization permits a consistency comparison with electromagnetic quantities ([1], Appx. AT). In Eq. (13), γ is the conventional electromagnetic transformation parameter in m⁶ C⁻² s⁻⁴ and ε₀ is vacuum electric permittivity in F m⁻¹. The equation uses the upper-count gravitational coupling. Its interpretation depends on how the transformation parameter is determined.

When the parameter in Eq. (13) is resolved from the same electromagnetic and gravitational inputs, the relation is an auxiliary-input consistency construction ([1], Appx. AS). It does not independently establish unification. The historical comparison of angular and momentum realizations ([1], Appx. S) does not equate their dimensions, and 2π is the stated electromagnetic normalization ([1], Appx. Z).

The Informativity differential depends on the radial count in the MQ length construction. This dependence is distinct from the relative-velocity dependence of special-relativistic length contraction [11], Secs. 2–4. The present right-triangle construction supplies no corresponding residual-loss law for mass or time. Establishing a physical length effect would require a test that distinguishes it from established relativistic effects and measurement systematics.

Within MQ, the fundamental relation connects the count construction with the realized quantities ([1], Appx. V). The broader constant derivations [9] extend that framework. Here the derived gravitational identities and electromagnetic consistency relation illustrate those dependencies, while the proposed physical interpretation remains subject to independent testing.

The purpose of this paper is to introduce the MQ calculation of discrete gravity and explain its assumptions and numerical dependencies. The 2022 inverse fine-structure input supports conditional relative uncertainties of order 10⁻¹⁰ to 10⁻¹¹ for the fundamental measures and gravitational quantities reported here. The upper-count reduced Planck calculation has a smaller conditional relative uncertainty because of its shared-input dependence. The CMB-based age and distance have a relative conditional uncertainty of about 2.79 × 10⁻⁴, dominated by Fixsen’s temperature input. Twelve or fourteen displayed digits therefore cannot be treated as a general claim of experimental precision. The supplementary documents give further applications of the framework.

5. APPENDICES

APPENDIX A: TERMS

The term definitions are provided in Measurement Quantization [1]. The entries below report numerical values and refer back to quantities introduced in the main narrative.

The coefficient introduced in Section 1 is θ_si=3.26239030504(25). The fundamental momentum has the same numerical coefficient in kg m s⁻¹ under the stipulated MQ realization. Both are calculations from the 2022 inverse fine-structure input [18]. The X-ray comparison is assessed with its experimental and theoretical scope in Section 3. Conserved quantities at null infinity in general relativity [21], Sec. 4.4, do not establish the angle-to-momentum realization used by MQ.

The speed of light introduced in Section 1 is exactly 299,792,458 m s⁻¹ [18]. The equality of its fundamental length and time counts follows from the stated MQ speed-of-light relation [1].

The upper-count coupling is G=6.6740779383(10)×10^-11 m³ kg⁻¹ s⁻². Its electromagnetic-demarcation realization is G=6.6738448317(10)×10^-11 m³ kg⁻¹ s⁻². These MQ calculations use the 2022 input ledger [18]; their parenthetical standard uncertainties are conditional on the MQ count and realization assumptions.

The fundamental measures introduced in Section 1 are l_f=1.61619991148(12)×10^-35 m, m_f=2.17643254057(17)×10^-8 kg and t_f=5.39106261131(41)×10^-44 s. Their uncertainties propagate the common 2022 inverse fine-structure input and remain conditional on the MQ normalization [18].

The Planck quantities in Table 3 use the definitions in Eqs. (8)–(10). The fundamental measures and Planck-form combinations are distinguished by the realization assigned to each component.

The length, mass and time counts introduced in Section 1 are dimensionless. Their values must be interpreted in the stated count domain.

The radial count used in the gravitational expressions satisfies the separation relation introduced before Eq. (1). It is not silently identified with a separate motion count.

The right-triangle count construction is shown in Fig. 1; the figure’s geometric labels are distinguished from the physical symbols in the equations.

The recalculated electromagnetic demarcation count is n_L=84.6005456716(65). Its nearest-whole half-count is ⌊n_L/2⌉=42. The mixed brackets denote nearest-whole-unit evaluation of a dimensionless count. The uncertainty does not approach a half-unit tie.

The exact integer 42 selects the branch used in the inverse fine-structure calibration. The reported numerical uncertainties retain this same branch.

The residual introduced before Eq. (1) remains nonzero at finite radial count. The physical loss interpretation is an MQ extension of classical mechanics rather than a consequence of the Pythagorean theorem alone.

The normalization factor 2Q_Ln_L and its departure from unity must be distinguished. The latter is the Informativity differential introduced in Section 1; the former supplies the factors in Eqs. (11)–(12).

The Planck constant used above is exactly h=6.62607015×10^-34 J s [18]. The conventional reduced Planck value is 1.054571817×10^-34 J s at the shown precision. The upper-count MQ quantity is upright ħ=1.0545349844417293(56)×10^-34 J s. Its small conditional uncertainty is a shared-input calculation, not a new experimental determination.

For the calibration, the electron mass m_e is 9.1093837139(28) × 10⁻³¹ kg [18]. The MQ fundamental-length relation contains the product of this mass and the Bohr radius. Replacing that product with its exact conventional relation to the reduced Planck constant, inverse fine-structure constant and speed of light removes the mass dependence from the present calibration. These adjusted constants are not treated as independent inputs.

The atomic distance in Table 1 depends on the Bohr radius introduced before that table. Its uncertainty calculation retains the covariance with inverse fine structure; it is not obtained by treating adjusted atomic constants as independent measurements.

In the broader MQ notation, α_f^-1, α_p^-1 and α_c^-1 denote the dimensionless fundamental, Planck-like and classical inverse fine-structure quantities. The first is defined with respect to the Internal Frame and the latter two with respect to the System Frame. Their current calculated values are 137.020392812(11), 137.040785623(21) and 137.035999177(21), respectively. The last value is the 2022 CODATA calibration input [18], not an independent output. The calibration includes the finite-count normalization ([1], Appx. D).

The conventional parameter in Eq. (13) is determined by the chosen electromagnetic and gravitational inputs. Its role differs from that of the fundamental parameter in later MQ derivations.

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 conventional numerical inputs are available in the published CODATA 2022 report [18] and the temperature study [19]. The present calculations are reported in Tables 1–3 and Appendix A. Broader MQ calculation materials are described in [1] and its Online Resources.

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