Assessing the Discreteness of Measure with a Proposed Form of Length Contraction

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

Measurement Quantization (MQ) interprets gravitational curvature through discrete reference counts and their continuous realization under the Frames mapping. Combining a saturated position–momentum uncertainty product, the speed of light and an upper-count escape boundary gives length and time counts of one and a mass count of one half. These counts are conditional on the selected boundary realization. A consistent finite-count extension retains unit length and time counts but changes the boundary mass count. We distinguish this boundary mass from the fundamental mass unit. We recalculate the fundamental measures and the Planck-unit comparisons using 2022 CODATA inputs, including their shared-input uncertainty. Published torsion-balance results motivate further tests of the proposed non-Lorentz length contraction, while their present uncertainty and apparatus dependence limit the strength of an MQ-specific interpretation. The fractional mass-boundary solutions do not by themselves establish admissible integer count configurations. The apparatus-specific mode assignment is retrospective.

Keywords: Discrete Measure, Frames of Reference, Measure, Length Contraction, Fundamental Measures, Planck Units

1. INTRODUCTION

The fundamental measures l_f, m_f and t_f are the MQ reference units of length, mass and time, in m, kg and s, respectively ([1], Appx. W). Their dimensionless counts are n_L, n_M and n_T. With positive counts, the saturated boundary case of Heisenberg’s uncertainty relation [2] reduces to Eq. (1) when these measures replace the corresponding Planck units ([1], Appx. B). The upper-count realization and the width identifications required for this equality are developed in Sec. 3.1.

2 n_{L} n_{M} n_{L} = n_{T} .

(1)

The upper-count escape condition similarly reduces to n_L>2n_M ([1], Appx. H); equality identifies its boundary. The speed of light c, in m s⁻¹, is

c = \frac{l_{f}}{t_{f}}

when n_L=n_T=1. The fundamental relation is l_fm_f=2p_sit_f ([1], Appx. V). Here θ_si is the invariant dimensionless MQ coefficient; p_si is its momentum realization, in kg m s⁻¹, and Θ_si is its angular realization, in rad. The adopted realizations assign the same numerical coefficient in their stated units while preserving their distinct physical dimensions.

For comparison, the conventional reduced-Planck expressions [3], descended from Planck’s unit construction [4], define the length l_p, mass m_p and time t_p, in m, kg and s. In Eqs. (2)–(4), G is the gravitational coupling, in m³ kg⁻¹ s⁻²; ħ is the reduced Planck constant, in J s, equal to Planck’s constant h divided by twice the dimensionless circle constant π. These are the conventional International System of Units (SI) quantities in the Planck-unit definitions. Later MQ realizations are distinguished by their stated count domain.

l_{p} = {(\frac{ħ G}{c^{3}})}^{1 / 2} .

(2)

m_{p} = {(\frac{ħ c}{G})}^{1 / 2} .

(3)

t_{p} = {(\frac{ħ G}{c^{5}})}^{1 / 2} .

(4)

The Planck expressions alone do not specify the reference counts or their realization in MQ. The additional hypothesis is that discrete reference structure produces a count-dependent non-Lorentz contraction, the Informativity differential ([1], Appx. D), while successive count updates supply the curvature construction. These consequences must be distinguished from the choice of dimensional units and tested physically. The frame distinction is developed in [1], Appx. K.

In this framework, the Internal Frame is the discrete integer count-configuration domain, and the System Frame is its non-discrete relational realization. The Frames mapping is the kinematic relation that assigns the former count structure to the latter observable separations ([1], Appx. K, W). MQ attributes physical constants and geometric effects to that realization [5]. These definitions distinguish a discrete reference structure from its continuous observable encoding; they do not supply independent measurement evidence by themselves.

A geometric relation can be expressed in counts when every participating length uses the same reference. In Eqs. (5)–(7), n_La, n_Lb and n_Lc denote the dimensionless length counts for the two perpendicular sides a and b and hypotenuse c of a right triangle. The side label c is distinct from the speed of light. The three equations retain the common length factor, expand it and then cancel it.

{(n_{L a} l_{f})}^{2} + {(n_{L b} l_{f})}^{2} = {(n_{L c} l_{f})}^{2},

(5)

n_{L a}^{2} l_{f}^{2} + n_{L b}^{2} l_{f}^{2} = n_{L c}^{2} l_{f}^{2},

(6)

n_{L a}^{2} + n_{L b}^{2} = n_{L c}^{2} .

(7)

Equation (7) is the Pythagorean relation among the counts. Cancellation of l_f removes its numerical scale from this relation, but the physical lengths still require that reference unit. The cancellation therefore distinguishes geometric count dependence from dimensional calibration; it does not establish discreteness on its own.

This paper examines the physical significance of discrete measure through the length contraction predicted by the MQ count geometry ([1], Appx. D). The momentum realization p_si, the fundamental measures and the classical expressions supply the dimensional normalization [4] ([1], Appx. I, J). We compare the resulting calculations with the 2010, 2014, 2018 and 2022 CODATA (the Committee on Data of the International Science Council) values, retaining the input editions and uncertainty dependencies. A calculated agreement is independent evidence only when the compared measurement has not supplied the quantity being reconstructed.

MQ uses count representations of established classical expressions together with the proposed Frames mapping. Its predictions require testing alongside instrumental and conventional physical explanations. The wider program is developed in [1, 5–10]; the present argument does not require adopting inflation theory [11], Loop Quantum Gravity [12–14], String Theory [15] or supersymmetry.

2. METHODS

We apply count notation to the classical expressions for the speed of light, escape velocity and gravitational coupling, and to the saturated uncertainty boundary. The assumed quantum-width identifications and count realization are stated beside their equations. The MQ applications in [1, 5–10] provide the broader context. The present derivation distinguishes a conditional count solution from experimental support for its physical interpretation.

The introductory publication Measurement Quantization [1] contains the externally cited appendices and extended definitions used here. Its geometric interpretation supplies the mathematical starting points; the measurements and their uncertainties must be assessed separately. A preprint and associated resources are available through Online Resources.

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

The associated supplementary spreadsheets provide the calculations for the introductory publication and applications across the MQ program. In the present paper, current numerical results use the specified 2022 CODATA inputs and the dependencies stated below. Historical comparison rows keep their original editions; the spreadsheets do not turn a source-driven calculation into an independent measurement.

3. RESULTS

We evaluate the coefficient through the conventional fine structure constant α, a dimensionless quantity. The 2022 CODATA inverse value is α^-1=137.035999177(21) [3]. Its MQ inversion gives θ_si=3.26239030504(25); parentheses denote one standard uncertainty in the final quoted digits. The adopted coefficient realizations follow the normalization already stated in the Introduction ([1], Appx. A, Q, S). The Bell-state context [16, 17] and the separate angular calculation of Shwartz and Harris [18] are discussed below. The complete inversion appears in Sec. 3.2, so the uncertainty quoted here is a propagated input uncertainty rather than a direct angular measurement.

At the upper count limit, the same momentum realization follows from Eq. (20) and the upper-count reduced Planck expression ([1], Appx. Q, P, L). These equivalent dimensional expressions do not identify an angle with a momentum. The conserved-quantity discussion of Po-Ning Chen et al. ([19], Sec. 4.4) also does not supply such a conversion. We therefore retain the stated realization convention throughout the calculation.

3.1 Resolving the properties of measure

Equation (8) concerns the state standard deviations σ_x and σ_p of the canonically conjugate position x, in m, and momentum p, in kg m s⁻¹. The widths have the same respective units. Measurements on separate, identically prepared ensembles can estimate those widths; a finite empirical variance product need not equal their population product. We therefore use Eq. (9) as the explicitly saturated quantum boundary. A pure Gaussian state with zero position–momentum covariance can realize this equality [20]. A classical conical description alone does not specify such a state.

The count reduction identifies the position width with a counted length and the momentum width with a counted mass times the corresponding speed, using positive counts. In Eqs. (10)–(11) the reference units are the Planck units from Eqs. (2)–(4), so the reduced Planck constant obeys

ħ = \frac{{l_{p}}^{2} m_{p}}{t_{p}}

. Substitution into the saturated Eq. (9) gives Eq. (12). Replacing the three Planck measures consistently by the fundamental measures gives Eq. (1) only with the corresponding upper-count reduced Planck realization ([1], Appx. B, V). The displacement and radial counts are identified in this boundary construction. These width and count identifications are physical assumptions. Changing the reduced Planck realization is an additional MQ hypothesis; it does not follow from a change of units in conventional quantum mechanics [20].

σ_{X} σ_{P} \geq \frac{ħ}{2},

(8)

σ_{x} σ_{p} = \frac{ħ}{2} .

(9)

\sqrt{{(n_{L} l_{p})}^{2}} \sqrt{{(n_{M} m_{p} \frac{n_{L} l_{p}}{n_{T} t_{p}})}^{2}} = \frac{ħ}{2} .

(10)

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

(11)

2 n_{L} n_{M} n_{L} = n_{T} .

(12)

Cancelling the positive dimensional factors gives Eq. (12), reproducing the count equality introduced in Eq. (1). This applies to the selected saturated widths, not to every quantum state. We next separate the angular analogy from the curvature construction before combining the latter with the escape boundary. Appendix B evaluates the alternative finite-count construction.

The equal-angle solution in Table I of Shwartz and Harris [18] uses a signal angle Θ_si of 3.26239 rad and an equal idler angle; its reported pump angle is 3.02079 rad. The calculation concerns diamond, its (111) reciprocal lattice vector, a 25 keV pump and equal 12.5 keV output photons. These are calculated phase-matching and current-density results for a maximally entangled Bell state ([1], Appx. A). In Eq. (13), the unscripted angle θ denotes this common signal and idler direction, in rad. The suffixes s, i and p label signal, idler and pump, while x and y identify Cartesian entries. The coefficients with these suffixes are dimensionless components of wave vectors normalized by a common fixed inverse-length unit. The reciprocal lattice vector G⃗ uses the same normalization and is distinct from the scalar gravitational coupling.

(n_{s x} \cos (θ), n_{s y} \sin (θ)) + (n_{i x} \cos (θ), n_{i y} \sin (θ)) = (n_{p x} \cos (2 π - θ), n_{p y} \sin (2 π - θ)) + \vec{G} .

(13)

Equation (13) applies the equal-angle, reflected-pump geometry used in [1], Appx. A. The pump reflection is an idealized MQ assignment; the published angles constrain it only at their reported precision. Equal output energies divide the pump energy, but this angular solution does not independently measure p_si or determine an absolute MQ mass unit. It supplies context for the adopted realization rather than the conditional minimum-count derivation.

The common length count in the saturated-width construction also serves as the radial count in the boundary calculation. Identifying these two roles is specific to this construction, rather than a general assertion that arbitrary displacements and radial separations coincide.

The angular analogy and dimensional cancellation alone do not establish a physically constrained reference. The next step uses the discrete-gravity construction to relate length and mass counts, then tests the consequences against the numerical and experimental record. Its physical interpretation remains subject to the assumptions already stated.

The curvature construction uses the right-triangle count geometry ([1], Appx. C). For a positive length count, define the geometric residual Q_L by

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

, using the positive square-root branch. It is dimensionless and is the difference between the realized hypotenuse count and the Internal Frame count. MQ treats a realized fractional residual as distinct from an additional whole reference increment. Figure 1 illustrates these roles.

Reference Framework—The observer’s reference AB fixes the unit side a in this triangle. This is a normalization for comparing counts, not a prior assignment of every physical length count to one.

Internal Frame—The side BC, labeled b, represents a count of the reference AB in the discrete configuration already defined in the Introduction.

System Frame—The side AC, labeled c, is the realized hypotenuse separation induced by the Frames mapping. Its length count includes the residual.

The side a in Fig. 1 is one reference increment; side b carries the variable count n_L. The Pythagorean hypotenuse gives the count n_L plus Q_L. This use of a unit transverse reference does not set every separation to a unit count. The unit-side illustration below is the special case in which b also has one count.

For the illustrative unit sides in Fig. 1, the hypotenuse is exactly the square root of two. The continuous geometric excess remains part of the System Frame description. Under the MQ Internal Frame admissibility hypothesis, it is not an additional realized fractional reference increment ([1], Appx. C). The proposed loss of the excess with each fundamental time increment is the interpretation that leads to the curvature expression. Figure 1 illustrates this Frames mapping.

Figure 1. Reference-count triangle — Fig. 1. Count of l_f between an observer and target. Adapted from [7].

Equation (14) expresses the residual length per squared fundamental time, divided by the radial count. Its dimensions are acceleration. Here the physical radius r, in m, is r=n_Ll_f. Multiplication by the speed-to-momentum ratio in Eq. (15) gives acceleration per unit mass; Eq. (16) identifies that expression with the finite-count gravitational coupling per squared radius. The normalization is the fundamental relation already defined ([1], Appx. V).

The geometry supplies a calculational model whose physical interpretation must be tested independently of the inputs used to calibrate it.

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

(14)

\frac{Q_{L} l_{f}}{n_{L} t_{f}^{2}} (\frac{c}{p_{s i}}) = \frac{Q_{L} c^{2}}{n_{L} t_{f} p_{s i}} = \frac{Q_{L} l_{f} c^{2}}{n_{L} l_{f} t_{f} p_{s i}} = \frac{Q_{L} c^{3}}{r p_{s i}} .

(15)

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

(16)

The exact residual geometry rearranges to Eq. (17). Its rationalized form in Eq. (18) provides a stable evaluation and the upper-count limit. The Informativity differential describes the non-Lorentz contraction associated with count realization ([1], Appx. D); it is distinct from the curvature interpretation of successive count loss. The product in Eq. (18) tends to one half. Consequently, the factor 2Q_Ln_L multiplying the upper-count coupling tends to one, as made explicit in Eq. (41).

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

(17)

Q_{L} n_{L} = \frac{n_{L}}{\sqrt{1 + {n_{L}}^{2}} + n_{L}}; \lim_{n_{L} \to \infty} Q_{L} n_{L} = \frac{1}{2} .

(18)

Solving Eq. (16) for the finite-count coupling and substituting the fundamental relation gives Eq. (19). At the upper count limit, Eq. (18) then gives Eq. (20). The finite expression and its limit describe different realizations of the same count-dependent relation.

G = \frac{Q_{L} r c^{3}}{p_{s i}} = \frac{Q_{L} n_{L} l_{f} c^{3}}{p_{s i}} = Q_{L} n_{L} l_{f} c^{3} \frac{2 t_{f}}{l_{f} m_{f}} .

(19)

G = {(\frac{l_{f}}{t_{f}})}^{3} (\frac{t_{f}}{m_{f}}) .

(20)

Upper-count-limit MQ realizations of G and ħ are upright; finite electromagnetic realizations are italicized. The electromagnetic demarcation is the fixed interaction count

n_{L} = \frac{276}{θ_{si}}

([1], Appx. AB, AE, AN). Its 2022-input value is 84.600545672(6). Ordinary laboratory separations rapidly approach the upper-count limit; this does not make the finite minimum-count coupling equal to that limit.

In Eq. (21), v is escape velocity, in m s⁻¹, and M is the attracting mass, in kg. We use M=n_Mm_f and the radius already defined above. Setting the boundary speed to c and using the upper-count coupling from Eq. (20) gives Eqs. (22)–(23) ([1], Appx. H, I). Appendix B keeps the finite-count dependence explicitly. Using the classical escape expression at a boundary speed equal to the speed of light is a formal boundary construction, not a derivation of relativistic strong-field dynamics or Planck-scale quantum gravity.

v = {(\frac{2 G M}{r})}^{1 / 2},

(21)

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

(22)

n_{L} = 2 n_{M} .

(23)

Equation (24) repeats the saturated count equality for this boundary. With n_L=n_T, it reduces to Eq. (25). Substituting the upper-count escape relation from Eq. (23) gives Eqs. (26)–(27); the positive mass-count branch gives Eq. (28). Thus

n_{M} = \frac{1}{2}

is a result of this stated combination of assumptions. It is not a boundary value obtained by evaluating the finite coupling at unit distance.

2 n_{L} n_{M} n_{L} = n_{T} .

(24)

2 n_{L} n_{M} = 1 .

(25)

2 (2 n_{M}) n_{M} = 1 .

(26)

n_{M}^{2} = \frac{1}{4},

(27)

n_{M} = \frac{1}{2} .

(28)

Substituting the mass count from Eq. (28) into the saturated count relation gives Eq. (29). Combining it with the light-count relation yields Eqs. (30)–(31), n_L=n_T=1. The positive-count domain excludes the negative square-root branch.

2 n_{L} \frac{1}{2} n_{L} = n_{T} .

(29)

n_{L} = \frac{n_{T}}{n_{L}} = 1 .

(30)

n_{L} = n_{T} = 1 .

(31)

The upper-count construction isolates a particular positive solution under its stated assumptions. A fundamental measure is a reference unit; the corresponding boundary count is a distinct quantity. The finite-count extension in Appendix B changes the boundary mass count while leaving the adopted fundamental mass expression intact. This distinction prevents a fractional boundary mass count from being treated as a universal minimum mass for elementary particles. The integer configuration requirement stated above is not established by either fractional mass-boundary solution. An additional admissibility or normalization rule would be needed to identify these algebraic values with allowed Internal Frame configurations.

We now consider the significance of fractional count differences. This is achieved through a series of comparisons. Consider first a fractional count difference.

5.25 l_{f} - 5.00 l_{f} = 0.25 l_{f} .

(32)

Under the MQ admissibility hypothesis, a fractional reference difference does not supply a realized additional Internal Frame reference increment ([1], Appx. C, J). Equation (32) illustrates the arithmetic after that hypothesis is imposed. It does not independently establish the hypothesis.

10.25 l_{f} - 5.00 l_{f} = 5.25 l_{f} .

(33)

Equation (33) contains a fractional numerical difference, and Eq. (34) reduces the comparison to the same arithmetic structure. Whether either difference is physically realized depends on the Frames mapping and its admissibility rule, rather than on subtraction alone.

5.25 l_{f} - 5.00 l_{f} = 0.25 l_{f} .

(34)

The examples are consistent with discrete, countable Internal Frame referencing under the MQ geometric interpretation. Universal applicability remains a physical hypothesis whose support must come from distinct measurements. The program context and references are given in the Introduction.

3.2 Numerical calibration of the measures

The previous section isolates conditional count boundaries. We now distinguish two numerical calibrations of the dimensional measures and the evidence for their realization. The conventional Planck expressions remain the reference transformations; the MQ count correction changes which coupling and reduced Planck realization enter them. Reference [18] supplies the calculated equal-angle branch. The down-conversion experiment [21] and X-ray/optical sum-frequency experiment [22] establish related nonlinear optical phenomena, not the adopted MQ angle–momentum calibration. The torsion-balance comparisons in Sec. 4 address the separate physical test.

Solving Eq. (19) for length gives the first form of Eq. (35). Its second form uses the corresponding upper-count limit, so the two G symbols must be read in their stated realizations. Equations (36)–(38) use the 2022 CODATA gravitational input G=6.67430(15)×10^-11 m³ kg⁻¹ s⁻² [3] as an upper-count calibration, together with exact c=299792458 m s⁻¹ and the coefficient from Sec. 3. This input-driven route is a benchmark estimate of the measures under that assignment, not a measurement of the MQ-derived upper-count coupling.

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

(35)

The exact speed relation converts Eq. (36) to the time in Eq. (37). Equation (20) then gives the mass in Eq. (38); the gravitational input cancels, leaving the same momentum-based mass expression. This cancellation determines the mass uncertainty and prevents the gravitational-input uncertainty from being counted twice ([1], Appx. F).

l_{f} = \frac{2 G p_{s i}}{c^{3}} = 1.616254 (36) \times 1 0^{- 35} m .

(36)

t_{f} = \frac{l_{f}}{c} = \frac{2 G p_{s i}}{c^{4}} = 5.39124 (12) \times 1 0^{- 44} s .

(37)

m_{f} = \frac{2 p_{s i}}{c} = \frac{t_{f} c^{3}}{G} = 2.17643254057 (17) \times 1 0^{- 8} kg .

(38)

The fine-structure route supplies a separate calibration ([1], Appx. AD, BM), using exact h=6.62607015×10^-34 J s [3]. The Planck-like inverse coupling

{α_{p}}^{- 1}

is dimensionless and is defined by

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

; its difference from the conventional inverse value is specified by

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

. The mixed brackets ⌊x⌉ denote rounding a dimensionless argument x to its nearest whole-unit value, distinct from floor ⌊x⌋ and ceiling ⌈x⌉. In this inversion the branch values are

⌊ \frac{84.600545672}{2} ⌉ = 42

and 137 for the corresponding coefficient argument. The demarcation count is the one already defined above. The rounded branch values remain unchanged over the propagated input uncertainty.

In Eq. (39), m_e is electron mass, in kg, and a₀ is the Bohr radius, in m. The CODATA identity

m_{e} a_{0} = \frac{ħ}{c α}

eliminates their shared adjusted-input dependencies exactly [3]. Combining it with the adopted mass and momentum realization gives the fundamental length below.

l_{f} = \frac{m_{e} a_{0} α_{p}}{m_{f}} = \frac{m_{e} a_{0} c α_{p}}{2 p_{s i}} .

(39)

l_{f} = \frac{m_{e} a_{0} α_{p}}{m_{f}} = \frac{ħ}{2 p_{s i}} .

(40)

\frac{m_{e} a_{0} α_{p}}{m_{f}} = \frac{ħ_{SI}}{2 p_{s i}} \frac{α_{p}}{α} .

(41)

ℏ = \frac{p_{s i} l_{f}}{Q_{L} n_{L}}, α_{c}^{- 1} = 2 Q_{L} n_{L} α_{p}^{- 1} .

(42)

l_{f} = \frac{ℏ Q_{L} n_{L}}{p_{s i}} .

(43)

For this fine-structure calibration, the fundamental length, time and mass are 1.61619991148(12) × 10⁻³⁵ m, 5.39106261131(41) × 10⁻⁴⁴ s, and 2.17643254057(17) × 10⁻⁸ kg. These differ from the gravitational-input benchmark in length and time, while the mass is identical. The quoted uncertainties propagate the 2022 inverse fine-structure input under the fixed MQ realization assumptions. Equation (40) gives the associated upper-count reduced Planck and gravitational realizations.

ħ = 2 p_{s i} l_{f}; G = \frac{l_{f} c^{3}}{2 p_{s i}} .

(44)

Evaluating the finite-count gravitational expression at the electromagnetic demarcation gives

G = 2 Q_{L} n_{L} G; n_{L} = \frac{276}{θ_{si}} .

(45)

Differentiating the implicit inversion with respect to θ_si gives 83.99413200 at the stated input, so the 2022 inverse fine-structure standard uncertainty of 0.000000021 propagates to a coefficient uncertainty of 0.0000000002500175 [3]. The same input dependence is retained through Eqs. (39–45) and all four MQ rows of Table 1. The length and time difference between this calibration and the gravitational-input benchmark is 33.27 parts per million (ppm), or 1.48 combined standard uncertainties when their shared coefficient dependence is included. This comparison assesses the two calibration inputs under the model; it is not a second independent measurement. Unquantified model or realization uncertainty is not included in either budget.

Table 1. CODATA values by input edition and MQ Planck-expression combinations recalculated from the 2022 fine-structure input. Parentheses give one standard uncertainty; exact SI values are identified explicitly.

Year or route	ħ (10⁻³⁴ J s)	lₚ (10⁻³⁵ m)	tₚ (10⁻⁴⁴ s)	mₚ (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… exact	1.616255(18)	5.391247(60)	2.176434(24)	6.67430(15)
2022 ^a	1.054571817… exact	1.616255(18)	5.391247(60)	2.176434(24)	6.67430(15)
2022 ^b1	1.0545349844417293(56)	1.61617168659(12)	5.39096846322(41)	2.17647054987(17)	6.6738448317(10)
2022 ^b2	1.0545349844417293(56)	1.61619991148(12)	5.39106261131(41)	2.17643254057(17)	6.6740779383(10)
2022 ^b3	1.054571817646… exact	1.61622813686(12)	5.39115676105(41)	2.17647054987(17)	6.6740779383(10)
2022 ^b4	1.054571817646… exact	1.61619991148(12)	5.39106261131(41)	2.17650855984(17)	6.6738448317(10)

^a Published CODATA recommended values for the labeled editions [23–25, 3]. These rows retain their own input years and published uncertainties; their central values are not converted into 2022 results.

b1–b4 The four combinations use Eqs. (2)–(4) with upper-count or electromagnetic-demarcation realizations. Row b1 uses upper-count ħ with electromagnetic G; b2 uses both upper-count values; b3 uses the SI ħ and upper-count G; b4 uses the SI ħ and electromagnetic G. These are correlated model transformations, not four independent measurements. The electromagnetic count is 84.600545672(6).

For b1–b4, uncertainty is propagated through the complete dependence on the same inverse fine-structure input. With exact SI h and c, the remaining uncertainty is conditional on the MQ equations and realization convention. It excludes unquantified model uncertainty. The length and time in b2 and b4 coincide, while their transformed masses differ. Equations (36–38) retain the separate calculation using CODATA G as an input; their larger length and time uncertainties must not be replaced with those of the fine-structure route.

The finite electromagnetic and upper-count MQ values of G differ by 34.9283854689(54) ppm, normalized to the finite value. This calculated separation is the prospective quantity to test. A CODATA recommendation is an adjusted estimate, not a single experiment or a hard measurement interval. Claims that a result falls outside a measurement range therefore require the stated comparison, covariance, uncertainty budget and coverage factor. The two values are count-dependent evaluations within MQ, rather than two gravitational constants recognized in conventional metrology.

4. DISCUSSION

The Measurement Standards Laboratory (MSL) compensated torsion-balance result is G = 6.67387(27) × 10⁻¹¹ m³ kg⁻¹ s⁻² [26]. Both 2022-input MQ G values lie within its one-standard-uncertainty interval, so this measurement cannot distinguish the two realizations. Its difference from the 2022 CODATA central value exceeds the MSL standard uncertainty, but the adjusted recommendation contains historical G data and is not an independent null measurement. The corrected International Bureau of Weights and Measures (BIPM) Mark II result is G = 6.67554(16) × 10⁻¹¹ m³ kg⁻¹ s⁻²; its free-minus-servo difference is 106 ppm with 104 ppm standard uncertainty [27]. Zero is just outside that one-standard-uncertainty interval, whereas it remains inside a two-standard-uncertainty interval. This is compatible with a positive mode difference, but does not identify its cause. The corrected BIPM mean is above both MQ values. Its difference from the upper-count value is 9.14 times the published BIPM standard uncertainty, without an allowance for unquantified model uncertainty. Agreement of mode ordering does not remove this absolute-value tension.

The 2026 National Institute of Standards and Technology (NIST) redetermination gives a combined Bayesian estimate G = 6.67387(38) × 10⁻¹¹ m³ kg⁻¹ s⁻² [28]. Using its published primary uncertainties and correlations, the copper and sapphire free-minus-servo contrasts are 56.79 ± 29.43 ppm and 149.72 ± 92.01 ppm. Both exclude zero at one standard uncertainty, but neither at two. These contrast budgets omit the additional configuration-specific effects modeled in the consensus analysis; they are not null tests under that complete model. Statistically evaluated Type A uncertainty alone yields narrower intervals but omits other contributions. The mode offset remains unexplained, and the consensus model allows unidentified apparatus biases. Both MQ values lie within the combined NIST uncertainty, so its absolute result does not distinguish them. The offsets and their sign do not establish MQ causation [28].

The count solution and the proposed non-Lorentz length contraction should therefore be assessed separately. This proposed contraction is distinct from the Lorentz contraction of relativistic motion [29]. The count geometry supplies conditional relations and two realization values. Assigning free deflection to the upper-count realization and electrostatic servo to the electromagnetic demarcation is a retrospective apparatus interpretation. Under that assignment, the prospective free-minus-servo target is +34.93 ppm, normalized to the servo value ([1, 5]). The wider MQ program provides additional comparisons [1, 5], but their evidentiary weight depends on independent source inputs, specified domains and competing explanations. A finite-count recalculation alone cannot establish a smaller fundamental mass unit, even though elementary-particle masses are many orders of magnitude smaller than that reference mass. This applies equally when comparing with Stoney units [30], Planck units and other unit systems. Appendix B records the boundary distinction explicitly. A discriminating test should fix the mode assignment, count scale and calibration before unblinding, compare the contrast with both zero and the MQ target, and control electrical and mechanical offsets. Both modes measure gravitational torque [27, 28]; the electrostatic readout alone does not establish the assumed electromagnetic count of that interaction.

5. APPENDICES

APPENDIX A NOTATION GUIDE

The symbols used in the calculation are introduced beside their first substantive use. This guide identifies those locations; extended MQ definitions remain in [1].

θ_si, p_si and Θ_si—Introduction; the current coefficient evaluation is in Sec. 3, and the distinct angular context is in Sec. 3.1.

c—Introduction; the local speed relation is used in Eqs. (37) and (45) ([1], Appx. V).

G—Introduction, Eqs. (19)–(20), and Sec. 3.2; Table 1 separates the conventional input from the two MQ realizations.

l_f, m_f and t_f—Introduction; the two calibrations are distinguished in Eqs. (36)–(41).

l_p, m_p and t_p—Introduction and Eqs. (2)–(4); Table 1 uses those Planck-expression transformations.

n_L, n_M and n_T—Introduction; the conditional upper-count boundary is in Eqs. (24)–(31), with the finite-count follow-up in Appendix B.

The radial and displacement roles of the length count are identified in Sec. 3.1; that identification is specific to the boundary construction.

The mixed nearest-whole, floor and ceiling brackets are explained beside the inversion in Sec. 3.2.

Q_L—Sec. 3.1 before Fig. 1; Eqs. (17)–(18) give the equivalent count geometry and limit.

The Informativity differential is introduced in the Introduction and distinguished from curvature beside Eqs. (17)–(18).

ħ and h—Introduction; the conventional SI value and MQ count realizations are distinguished in Table 1 and Eq. (42).

m_e and a₀—Sec. 3.2 beside Eq. (39); their product is eliminated using the CODATA identity.

α and α_p—Sec. 3 and Sec. 3.2; the inversion determines the current coefficient from the conventional inverse fine-structure input.

APPENDIX B FINITE COUNT BOUNDARY

The main calculation selects upper-count realizations to isolate the candidate boundary relation. This appendix tests its finite-count extension using the exact curvature expression in Eq. (19) and the reduced Planck realization ([1], Appx. L). Here G(n_L) and ℏ(n_L) denote those existing quantities evaluated at the specified length count; they are not new physical constants. The extension assumes that the finite reduced Planck realization also sets the saturated quantum width product at that count. Conventional quantum mechanics does not establish this scale-dependent identification.

G (n_{L}) = \frac{2 Q_{L} n_{L} c^{3} t_{f}}{m_{f}}; ℏ (n_{L}) = \frac{p_{s i} l_{f}}{Q_{L} n_{L}} .

(B.1)

Using the escape velocity and mass from Eq. (21), substitute the first expression in Eq. (42) and impose the boundary speed. The same fundamental relation then gives the finite-count escape condition below.

\frac{v^{2}}{c^{2}} = 4 Q_{L} n_{M} = 1 .

(B.2)

Next use the second expression in Eq. (42) in the saturated uncertainty product, with the width identifications from Sec. 3.1. This gives

\frac{{n_{L}}^{2} n_{M} {l_{f}}^{2} m_{f}}{n_{T} t_{f}} = \frac{{l_{f}}^{2} m_{f}}{4 t_{f} Q_{L} n_{L}} .

(B.3)

Cancel the positive dimensional factors and impose the light-count relation n_L=n_T. The resulting equation and Eq. (43) yield

4 Q_{L} {n_{L}}^{2} n_{M} = 1; {n_{L}}^{2} = 1; n_{L} = n_{T} = 1 .

(B.4)

At unit length count, Eq. (17) gives the positive excess as the square root of two minus one. Hence

n_{M} = \frac{1}{4 (\sqrt{2} - 1)} = \frac{\sqrt{2} + 1}{4} .

(B.5)

Using Eq. (38), this finite-count boundary corresponds to a mass of 1.31359323926(10) × 10⁻⁸ kg. Its mass count is 0.6035533906, larger than the upper-count boundary count of one half. The fundamental mass remains fixed by l_fm_f=2p_sit_f and the adopted momentum realization. A lower-count calculation therefore changes this boundary mass, rather than producing a smaller or more accurate fundamental mass unit. A rescaling of the fundamental mass would require an additional independently supported change to its normalization.

If only the gravitational coupling is changed while the upper-count uncertainty relation is retained, the same algebra instead gives

n_{L} = n_{T} = \frac{2}{\sqrt{5}}; n_{M} = \frac{\sqrt{5}}{4} .

(B.6)

That mixed realization has a length count below one and fails the assumed Internal Frame domain n_L≥1. Using both finite-count expressions avoids that particular mismatch, but remains conditional on the quantum-width and realization assumptions. Neither construction proves a universal particle-mass minimum. The small electron mass is therefore compatible with a large reference mass; it does not select one of these boundary prescriptions. Both fractional mass results require an additional admissibility rule, as discussed after Eq. (31); the finite-count extension does not independently establish an integer configuration.

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 calculations use the numerical source inputs specified in Sec. 3.2 and the published experimental results cited in Sec. 4. Associated MQ supplementary spreadsheets are described in Online Resources.

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