Simplest Relation Between Fundamental Units of Length, Mass, and Time

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

Planck unit expressions combine the speed of light c in m s⁻¹, gravitational coupling G in m³ kg⁻¹ s⁻² and reduced Planck quantity ℏ in J s to obtain measures of length, mass and time. We examine the Measurement Quantization (MQ) fundamental relation, l_fm_f=2p_sit_f, where l_f, m_f and t_f are fundamental length, mass and time in m, kg and s, and p_si is the momentum realization in kg m s⁻¹ of the dimensionless coefficient θ_si. The argument compares count-resolved coupling expressions under specified bound and normalization assumptions. Its angular realization is kept distinct from momentum, allowing a numerical comparison with a published X-ray configuration. The numerical results are MQ calculations calibrated with the 2022 inverse fine-structure input and exact source constants. They establish consistency within those assumptions, rather than an independent measurement of the fundamental measures.

Keywords: Fundamental Units, Planck Units, fundamental relation, Measures, Discrete

1. INTRODUCTION

We examine the relation between the three fundamental measures introduced in the abstract. Planck’s unit expressions [1] provide the conventional reference. MQ instead organizes the count and realization assumptions explicitly ([2], Appx. V). Equation (1) states the momentum form used throughout this paper; the following calculation examines its consistency with the accepted coupling normalization. Here, “simplest” refers to the displayed product relation under the stated MQ assumptions; no uniqueness or minimality theorem over other physical descriptions is asserted.

l_{f} m_{f} = 2 p_{s i} t_{f} .

(1)

In the current MQ notation, the dimensionless coefficient introduced in the abstract has the angular realization Θ_si in rad and the momentum realization used in Eq. (1) ([2], Appx. A, P, Q; historical discussion in [3]). The Internal Frame is the discrete count configuration domain; the System Frame is the non-discrete relational encoding of that count structure in which observables are realized. The Frames mapping carries the discrete count configurations into their continuous relational encoding. The angular comparison concerns the signal and idler propagation angles relative to the atomic planes in the frequency-degenerate X-ray configuration calculated by Shwartz and Harris ([2], Appx. A; [4], Table I). Their printed value 3.26239 rad is a model solution, not a measurement of the momentum in Eq. (1). The selected branch uses diamond (111) planes, a 25 keV pump and 12.5 keV signal and idler photons [4].

The realization maps retain the same numerical coefficient for the angular and momentum representations ([2], Appx. A, Q). The resulting quantities retain their different units. Equal numerical coefficients under those maps do not make angle and momentum the same physical quantity. In the specified realization in the International System of Units (SI), Θ_si=θ_si rad and p_si=θ_si kg m s^-1.

MQ describes quantities using dimensionless counts n_L, n_M and n_T of fundamental length, mass and time, respectively. The length count is used in propagation and in the residual normalization associated with the comparison geometry. A separate radius-count label is introduced before Eq. (2) to keep its role explicit. The MQ approach and its applications are developed in [5, 6, 7, 8, 9, 3].

The calculation uses count-resolved measures in the Internal Frame and compares their System Frame realizations. Fundamental measures serve as reference units for this construction. Their proposed physical interpretation depends on the assumptions and evidence developed in the cited MQ work; the algebra here does not by itself establish that every measured separation is an integer count.

Planck’s expressions organize conventional constants to isolate each dimension. The MQ comparison below additionally specifies a finite-count normalization, a bound-count selection and a relation between angular and momentum realizations. Those assumptions must remain visible when interpreting the resulting fundamental relation. The numerical calibration supplies an action scale as well as the speed of light and the coefficient.

2. METHODS

We use the classical escape-speed expression as a bound comparison, the defined speed of light and the count-resolved gravitational-curvature relation. The resulting equalities are checked within the MQ normalization, rather than used to assert a new measurement. The momentum scale is an MQ bound choice. The position–momentum uncertainty principle [10] does not alone determine that scale without specifying a quantum state and its uncertainties. Setting the classical escape speed to the speed of light is used here as an algebraic comparison; no relativistic horizon calculation or physical validity at the fundamental length follows from that substitution.

The count assumptions, their proposed physical support and the referenced appendices are developed in Measurement Quantization [2]. That work also supplies the wider term and symbol context. A preprint and associated material can be found in Online Resources.

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

The published MQ work describes two supplementary spreadsheets [2]. Suppl. 1 covers the introductory paper and its appendices; Suppl. 2 covers this paper and other applications. For the numerical results here, the uncertain calibration input is the 2022 inverse fine-structure constant recommended by the Committee on Data (CODATA) of the International Science Council, with value 137.035999177(21) [11]. The speed of light is exactly 299,792,458 m s⁻¹. The exact Planck constant is

h = 6.62607015 \times 10^{- 34} J s,

and the corresponding italic exact reduced source input is ℏ=h/(2π), where π is the circular constant. MQ-derived quantities are calculated from these inputs and the stipulated count and realization normalization; they are not separate CODATA recommendations.

3. RESULTS

1. Calculations

The speed of light in vacuum satisfies c=n_Ll_f/(n_Tt_f) for the counts introduced above. The dimensionless radius count is n_Lr, with separation r=n_Lrl_f in m, and the source mass is M=n_Mm_f in kg. We retain the radius subscript to distinguish its role from the count in the Informativity differential. The residual is

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

and the realization normalization factor is 2Q_Ln_L. The Informativity differential is the non-Lorentz contraction in realized separation associated with the finite-count departure 1-2Q_Ln_L of that factor from unity; the departure approaches zero at the upper-count limit. Italic G and upright G denote finite-count and upper-count gravitational couplings, respectively, in m³ kg⁻¹ s⁻². For a matched finite-count realization, MQ uses G=(2Q_Ln_L)G ([2], Appx. M, Eqs. (M.3)–(M.4)). Equation (2) rewrites the classical escape expression using this coupling normalization. The factor in its second denominator converts the coupling; the radius remains the same separation. With v_e denoting escape speed in m s⁻¹, we set v_e=c and use the bound assumptions n_Lr=n_L=n_T=1 and n_M=1/2 ([2], Appx. I, Eqs. (I.5), (I.8); Appx. J). The separate count roles are retained before making that substitution.

v_{e} = {(\frac{2 G M}{r})}^{1 / 2} = {(\frac{2 G n_{M} m_{f}}{2 Q_{L} n_{L} n_{L r} l_{f}})}^{1 / 2},

(2)

\frac{n_{L}^{2} l_{f}^{2}}{n_{T}^{2} t_{f}^{2}} = \frac{2 G n_{M} m_{f}}{2 Q_{L} n_{L} n_{L r} l_{f}},

(3)

G = \frac{Q_{L} n_{L} n_{L r} n_{L}^{2} l_{f}^{3}}{n_{M} m_{f} n_{T}^{2} t_{f}^{2}} = 2 Q_{L} n_{L} \frac{l_{f}^{3}}{t_{f}^{2}} \frac{1}{m_{f}} = 2 Q_{L} n_{L} c^{3} \frac{t_{f}}{m_{f}} .

(4)

Squaring Eq. (2) gives Eq. (3). Solving for the finite-count coupling gives the first member of Eq. (4); the factors of two in Eq. (3) cancel before using the bound counts. Its remaining members follow from the stated bound and c=l_f/t_f. For the second form of the coupling, the count-resolved gravitational-curvature expression in Eq. (5) has acceleration units ([2], Appx. C). The radius count and the residual count refer to the same separation in this comparison, although their subscripts retain their different roles.

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

(5)

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

(6)

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

(7)

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

(8)

Multiplying Eq. (7) by the squared radius yields Eq. (8). At the stated matched bound, equating this form with Eq. (4) gives Eq. (9); cancellation of the positive residual, counts and speed-of-light factors gives Eq. (10) ([2], Appx. V). This is a consistency check within the accepted MQ coupling normalization. The normalization is an input to this comparison, so the escape route is not an independent proof of the fundamental relation.

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

(9)

l_{f} m_{f} = 2 p_{s i} t_{f} .

(10)

2. Angular Comparison

Arc length construction at the fundamental radius — **Fig. 1.** Arc length L_si at radius l_f subtending angular realization Θ_si. Adapted from [5].

The momentum identity supplies a bound-related scale. To compare it with an angular quantity, we use the dimensional realization conversion in Eq. (16). Figure 1 provides the arc-length geometry used in Eqs. (11) and (12). The published X-ray model supplies a separate angular comparison; the mapping does not establish that its photons have the MQ bound-related momentum.

Figure 1 labels the arc length L_si in m at radius l_f. Equation (11) uses coherent SI radian measure for that arc relation. The conversion factor C_pθ=p_si/Θ_si has units kg m s⁻¹ rad⁻¹. Upright ħ denotes the upper-count MQ reduced Planck quantity in J s, distinct from the exact reduced source input in Methods. At this radius, the upper-count momentum identity gives Eqs. (11) and (12) after division by the conversion factor; no action quantity is identified directly with arc length. In Eqs. (13)–(15), p denotes the same momentum realization introduced in Eq. (1), in kg m s⁻¹. It is retained as historical notation in these three equations, rather than as an additional parameter. Equation (13) rearranges the finite-count gravitational relation. Equation (14) takes its upper-count limit, and Eq. (15) uses both quantities at their upper-count limits ([2], Appx. S, Eqs. (S.2)–(S.5), and Appx. L, Eq. (L.7)).

L_{s i} = r Θ_{s i} = l_{f} Θ_{s i} = \frac{ħ}{2 C_{p θ}},

(11)

Θ_{s i} = \frac{ħ}{2 l_{f} C_{p θ}} .

(12)

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

(13)

\frac{Q_{L} c^{3}}{r p} = \frac{G}{r^{2}},

(14)

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

(15)

At the upper-count limit, the residual-count product tends to one half and Eq. (15) gives the same momentum realization as Eq. (1). Equation (12) gives the angular quantity after the dimensional conversion. Equation (16) states the relation between these representations and preserves their units. It is a realization mapping, not an independent prediction of an X-ray beam momentum.

p_{s i} = C_{p θ} Θ_{s i} .

(16)

Table 1 reports the coefficient and its angular and momentum realizations calculated with the 2022 calibration. The entries share their numerical value through the stipulated realization maps. This shared input must be distinguished from the published X-ray model value discussed below.

Table 1. MQ coefficient and realized quantities calculated from the 2022 calibration. Updated from [2].

Dimension	Expression	Results	Units
Angular realization	$Θ_{s i} = p_{s i} / C_{p θ}$	3.26239030504(25) a	radians
Momentum realization	$p_{s i} = ħ / (2 l_{f})$	3.26239030504(25) b	kg m/s
Coefficient	$θ_{s i}$	3.26239030504(25)	-

a The angular entry is the MQ realization. Shwartz and Harris report the separate calculated propagation angle 3.26239 rad in their Table I [4]. The earlier MQ angular context is discussed in [2]. Reference [12] measures X-ray down-conversion coincidence rates in a lossy medium; [13] reports X-ray and optical sum-frequency mixing in diamond. These distinct experiments do not measure the selected angle in [4] or the momentum realization in this paper. These references do not turn the MQ table entry into an independent measurement.

b The numerical coefficient is obtained by inversion of the MQ fine-structure relation with the 2022 source input [11]. All three rows share that coefficient and its conditional source uncertainty. Parenthetical digits denote one standard uncertainty in the last reported digits. The count branch and the SI realization normalization are fixed model assumptions. The 2018 source [14] remains historical context.

The current coefficient is θ_si=3.26239030504(25). Its angular realization rounds to the printed Shwartz and Harris value at that source’s precision [4]. Their Table I lists numerically solved propagation angles, without an experimental standard uncertainty for 3.26239 rad. The comparison therefore concerns the printed model value; it cannot establish equality to the additional digits in Table 1 or independently validate the momentum relation. MQ also uses the angular comparison in selecting its electromagnetic demarcation, the fixed interaction count used for electromagnetic realization ([2], Appx. AC). The selection is therefore part of the model calibration context, which further limits an interpretation of this rounding agreement as an independent test.

4. DISCUSSION

The fundamental relation connects length, mass and time through the momentum realization. With c=l_f/t_f, it fixes m_f=2p_si/c. This does not by itself fix the length or time scale. An action input and its finite-count normalization are also required. Using the exact reduced source input at the electromagnetic demarcation gives l_f=Q_Ln_Lℏ/p_si and then t_f=l_f/c. The upright upper-count reduced Planck quantity in Eq. (15) is a derived MQ quantity, rather than the exact source input substituted without normalization.

The 2022 inversion gives the electromagnetic demarcation count n_L=276/θ_si=84.6005456716(65). With this count and the calibrated momentum realization, the fundamental mass, length and time are

m_{f} = 2.17643254057 (17) \times 10^{- 8} kg,

l_{f} = 1.61619991148 (12) \times 10^{- 35} m

and

t_{f} = 5.39106261131 (41) \times 10^{- 44} s .

These are conditional MQ calculations using the 2022 inverse fine-structure input and exact h and c. The X-ray model angle provides the separate comparison already described; it is not an independent source of those calculated uncertainties.

Equations (2)–(10) organize the classical bound comparison and the discrete coupling in the same finite-count regime before comparing them. The explicit normalization avoids equating a finite-count coupling to its upper-count limit. Because that normalization is accepted from the MQ construction, the resulting relation is an internal consistency result. The angular construction then uses the upper-count momentum identity and the specified dimensional conversion.

The roles of the inputs determine what can be concluded. The source inverse fine-structure value calibrates the coefficient; exact action and speed-of-light inputs supply the required dimensional scale. Reusing that calibration in the momentum and angular realizations does not provide independent agreement between measurements. Testing the proposed physical significance requires observables that do not merely reuse the same normalization and source input. For comparison, CODATA recommends a Newtonian gravitational constant of 6.67430(15)×10^-11 m³ kg⁻¹ s⁻² [11]. The calculated upper-count coupling is lower by 2.2206172×10^-15 m³ kg⁻¹ s⁻², or 1.48 times the CODATA standard uncertainty. This comparison is conditional on identifying the MQ upper-count coupling with that observable. It supports neither equality of the central values nor exclusion of MQ at that uncertainty level. The much smaller propagated MQ uncertainty describes sensitivity to the chosen calibration input; it does not include uncertainty in the physical model or its realization assumptions.

5. APPENDICES

APPENDIX A: TERMS

Broader historical term and symbol context is given in Measurement Quantization [2].

The coefficient and realizations are introduced in the abstract and introduction; Table 1 reports their shared calibration. The X-ray model comparison [4] and the distinct experiments [12, 13] are separate from the discussion of conserved quantities at null infinity in asymptotically flat general relativity ([15], Secs. 4.3–4.4). That work concerns angular momentum and center of mass, rather than an identification of a propagation angle with linear momentum. It is not evidence for the MQ realization conversion.

Methods gives the exact speed-of-light input [11], also present in the historical source [14]. The count form and selected bound are stated before Eq. (2).

Upright G is the upper-count gravitational coupling. The MQ calculation based on the 2022 inputs gives

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

Italic G denotes a finite-count realization. At the electromagnetic demarcation the recalculated value is

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

The fundamental measures introduced in the abstract are distinguished from conventional Planck units. Discussion gives their recalculated values and action normalization.

l_p, m_p and t_p are Planck’s units for length, mass, and time, in m, kg and s, respectively.

The dimensionless counts of the fundamental measures are introduced in the opening narrative and used explicitly in the bound assumptions before Eq. (2).

The roles of the length count n_L in propagation and residual normalization are specified before Eq. (2), separately from the radius count.

The radius count n_Lr retains its separate label as introduced before Eq. (2). Numerical equality with the length count is imposed only in the selected comparison geometry.

n_Lc describes the dimensionless count of l_f representative of a change in position of light measured with respect to the observer’s frame of reference.

The coefficient-count marker n_θ is obtained from half the electromagnetic demarcation count. Using the 2022 calculation, n_θ=⌊84.6005456716/2⌉=42. The mixed left floor and right ceiling brackets denote the nearest whole-unit value.

The residual introduced before Eq. (2) enters the curvature chain in Eqs. (5)–(8).

The Informativity differential is distinguished from the realization normalization before Eq. (2). The differential vanishes at the upper-count limit; the normalization approaches unity.

The MQ upper-count reduced Planck quantity is

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

It enters Eqs. (11), (12) and (15). Its uncertainty follows the shared calibration; the exact reduced source input is given in Methods.

m_e is the mass of an electron in kg.

a₀ denotes the Bohr radius in m, a length scale for the hydrogen ground state in the Bohr model.

The symbols

α_{f}^{- 1},

α_{p}^{- 1}

and

α_{c}^{- 1}

denote the fundamental, Planck-like and classical inverse fine-structure forms. The fundamental form is an Internal Frame quantity; the Planck-like and classical forms are System Frame quantities. The fundamental coefficient relation is

α_{f}^{- 1} = 42 θ_{s i} .

The inversion uses the classical form after its count and residual corrections; it is not an identification of all three forms.

The inherited symbol γ denotes the conventional electromagnetic transformation parameter used in the historical gamma approach. It is not used in this paper’s calculations.

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 reported here use the inputs and assumptions stated in Methods. The cited MQ work describes electronic supplementary spreadsheets; their availability and correspondence to this revision require verification.

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