Deriving the Planck Unit Expressions

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

Measurement Quantization (MQ) supplies count-dependent gravitational and momentum relations from which expressions for the fundamental length, time and mass are derived. The derivation distinguishes finite-count values from upper-count limits and identifies the assumptions needed for their realization in the International System of Units (SI). The conventional Planck expressions are recovered by using the conventional gravitational and reduced Planck constants. An angular construction illustrates the geometry, while the momentum relations supply the action dimensions. The electromagnetic demarcation is the interaction count distance used for electromagnetic realization and is selected according to the phenomenon. Reproduction of an input constant at its associated demarcation is a calibration identity and does not independently verify the framework.

Keywords: Planck Unit Expressions, Planck Units, Deriving Planck Units, Fundamental Measures, Planck Momentum

1. INTRODUCTION

The modern Planck units combine the speed of light c, in m s⁻¹, gravitational coupling G, in m³ kg⁻¹ s⁻², and reduced Planck constant ℏ, in J s, into measures of length, time and mass [1]. Planck’s original natural-unit construction provides the historical antecedent [2]. The present paper examines how the modern algebraic forms follow within the count relations of MQ under the assumptions specified below.

The fundamental length l_f, fundamental mass m_f and fundamental time t_f have units m, kg and s, respectively [3, Appendix W]. They are the measures used to express the MQ count relations. Their derived values depend on the chosen SI realization and source inputs. Stoney units provide an earlier example of constructing characteristic measures from physical constants [4].

The Internal Frame describes the discrete count configuration relative to the fundamental measures. The System Frame gives its non-discrete relational encoding. Frames mapping relates these descriptions without identifying a continuous observable with an integer count. The Informativity differential denotes the non-Lorentz length contraction arising from count normalization under this mapping. Its count-dependent normalization is used below [3, Appendix D]. The related framework discussions [5–10] provide context for these assumptions.

The gravitational relation used here is obtained from the MQ geometric count construction and its momentum normalization [3, Apps. C–F]. Its upper-count limit, denoted by upright G, is G = c³ t_f/m_f. This relates the fundamental measures before their individual values are resolved, conditional on the count geometry and SI normalization adopted in MQ.

The dimensionless geometric coefficient θ_si is distinguished from the angular quantity Θ_si, whose value is θ_si rad, and the momentum realization p_si, in kg m s⁻¹. The adopted MQ numerical realization is (p_si)/(1 kg m s⁻¹) = θ_si [3, Appendices A, Q and S]. This is a convention connecting SI numerical coefficients; dimensional consistency alone does not establish it as a physical law.

The analysis therefore separates the finite-count normalization, its upper-count limit and the dimensional momentum relation. These distinctions determine the interpretation of the resulting length, time and mass expressions.

2. METHODS

The method combines the MQ count construction with the stated light-speed and escape bounds [3] and the uncertainty principle [11]. The standard quantum uncertainty relation constrains statistical dispersions [12]. Selecting its equality case and identifying the relevant quantities with MQ count measures are additional assumptions in the bound construction; the uncertainty relation alone does not establish discrete length, time or mass units.

The framework and its supporting derivations are described in the appendices of Measurement Quantization [3]. The current numerical realization uses the 2022 recommended values from the Committee on Data for Science and Technology (CODATA). These are the dimensionless fine structure constant α_c = 7.2973525643(11) × 10⁻³, electron mass m_e = 9.1093837139(28) × 10⁻³¹ kg and Bohr radius a₀ = 5.29177210544(82) × 10⁻¹¹ m [1]. Planck’s constant h = 6.62607015 × 10⁻³⁴ J s and the speed of light are exact SI inputs. The conventional reduced Planck constant is (h)/(2π), with π the circle constant.

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

Supplementary materials 1 and 2 (Suppl1.xlsx and Suppl2.xlsx in the linked Online Resources collection) collect the framework resources and related literature. The classical fine structure realization introduced above is distinguished from the Planck-like α_p and fundamental α_f realizations. All three fine structure realizations are dimensionless. The fundamental realization uses the Internal Frame description; the other two use System Frame descriptions. Parenthetical uncertainties propagate the specified source uncertainties through the adopted MQ relations. They exclude uncertainty in the physical validity of those relations and exclude source-table rounding. The latter is checked separately when it controls the quoted precision.

For reproducibility, let n_L be the dimensionless radial separation count and Q_L the dimensionless right-triangle remainder specified in (36). The demarcation inversions use the positive-real continuation of the count expressions. The fine-structure calculation uses α_p^-1 = 84 θ_si − ⌊42θ_si⌉, α_c^-1 = 2 Q_L n_L α_p^-1, and n_L = (276)/(θ_si) at that demarcation [3, Appendices AA, AD and AE]. The mixed brackets denote the nearest whole-unit operation. The selected branch has ⌊42θ_si⌉ = 137 and gives θ_si = 3.262390305045149. The numerical realization then uses m_f = (2p_si)/(c), l_f = (m_ea₀α_p)/(m_f) and t_f = l_f/c [3, Appendix BM, Eq. BM3; Appendices S and W]. These source relations fix the scale before the Planck-form identities are evaluated.

The tabulated central electron mass and Bohr radius are multiplied as printed. For uncertainty propagation their product follows (h)/(2πα_cc) [1], so they are not treated as independent measurements. The difference between this exact identity and the printed product is a deterministic rounding effect, not an extra statistically independent measurement error. The elementary-charge default is reported at its established uncertainty-supported precision.

3. RESULTS

The results first obtain the gravitational relation from the count geometry and then combine the momentum and action relations to resolve the fundamental measures. The geometric illustration is kept separate from the dimensional derivation.

1. Nomenclature

Upright symbols G and ħ denote the MQ upper-count-limit quantities. Italic forms denote finite-count realizations where indicated. Conventional constants used in (28)–(30) are identified separately. The default electromagnetic demarcation is 84.60161(22), derived from elementary charge in Appendix B. A phenomenon-specific demarcation takes precedence. In particular, the reduced Planck constant calculation retains the adopted blackbody demarcation count 84.6005393909226, at which the blackbody-derived reduced Planck relation is evaluated.

2. Frames of Reference

Alongside the radial count introduced under Methods, n_M is mass divided by m_f, and n_T is elapsed time divided by t_f; both are dimensionless. The dimensionless mass-motion count n_Lm is the change in realized position of the observed mass divided by l_f. The count-resolved radial distance is r = n_L l_f, in m. The nearest-whole-unit brackets select the integer values required in the calculation. Comparison with a laboratory constant requires the corresponding realization [3, Appendix N].

G = {2 Q_{L} n_{L} c^{3} \frac{t_{f}}{m_{f}} |}_{n_{L} = 84.6005393909226} = 6.6738448316 (10) \times 1 0^{- 11} m^{3} k g^{- 1} s^{- 2} .

(1)

G = \lim_{n_{L} \to \infty} 2 Q_{L} n_{L} c^{3} \frac{t_{f}}{m_{f}} = c^{3} \frac{t_{f}}{m_{f}} = 6.6740779382 (10) \times 1 0^{- 11} m^{3} k g^{- 1} s^{- 2} .

(2)

Equations (1) and (2) distinguish the finite-count and upper-count gravitational realizations. The 2010, 2014 and 2018 CODATA adjustments [13–15] provide the historical comparison sets. The values shown here use the 2022 input chain. Equation (2) lies 1.48 published standard uncertainties below the 2022 recommended gravitational value [1]. This comparison includes the propagated input uncertainty but no estimated uncertainty for the MQ model; proximity alone does not establish the count construction.

The geometry and count normalization below use Frames mapping in the sense already defined [3, Appendix K; 10]. The count-resolved radial parameter and the continuously encoded hypotenuse serve different roles in that construction.

3. Discreteness and Countability of Fundamental Measures

The fundamental-measure expressions follow after the bound conditions and the numerical realization convention have been selected [3]. The algebra below establishes the resulting solution under those conditions; it does not prove uniqueness among all possible physical frameworks.

At the limiting bound used in the MQ construction, the radial, motion and time counts are unity and the normalized mass ratio is one half [3, Appendices B, H, I and J]. This is the boundary of the strict escape inequality, rather than a value inside its strict domain. The half value is a normalized bound ratio, not a primitive fractional integer count.

The radial count and mass-motion count have different roles. The saturation condition used in the MQ uncertainty-bound construction is 2 n_L n_M n_Lm = n_T [3, Eq. 5 and Appendix B]. It is not the uncertainty relation for arbitrary quantum states. Its application here retains the bound assumptions stated under Methods.

4. Discrete Gravity

Figure 1 illustrates the right-triangle count geometry and the remainder introduced above. Its capital letters label vertices, and “Side a”, “Side b” and “Side c” label edges; the side label c does not denote the speed of light. The vertical leg is one reference unit. The Informativity differential normalization is 2 Q_L n_L. Equation (3) has acceleration dimensions. Multiplication by c/p_si in (4) gives units m kg⁻¹ s⁻². Identifying this normalized expression with a gravitational field per unit source mass in (5) is the MQ gravitational realization assumption; the units alone do not imply that identification.

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

(3)

\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}} .

(4)

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

(5)

Equation (5) uses the finite-count gravitational coupling G at the same count n_L as Q_L and r. The upper-count-limit coupling is obtained separately in (6). These realizations must remain distinct.

G = \lim_{n_{L} \to \infty} \frac{Q_{L} n_{L} l_{f} c^{3}}{p_{s i}} .

(6)

5. Discrete Solution to G

The momentum normalization is p_si = l_f m_f/(2 t_f) [3, Appendix S]. Its numerical realization is specified in the Introduction. Substitution into the gravitational relation gives (7).

At the upper count limit, Q_L n_L tends to 1/2 ([3], Appx. D). Equation (7) takes this limit before reducing to the fundamental measures.

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

(7)

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

(8)

6. Momentum at the Fundamental Measure

The momentum p, in kg m s⁻¹, is defined by (9). It is twice the bound momentum realization used in (15).

p = \frac{l_{f} m_{f}}{t_{f}} .

(9)

7. Reduced Planck Constant

Equation (10) is the blackbody calibration identity when the count is the solution of (12), using (h)/(2π) as its exact SI input [1; 3, Appendix AB]. For the finite-count numerical presentation, the adopted working count remains 84.6005393909226. Evaluation at that working count reproduces the displayed digits in (10), but its terminal count digits must not be read as an exact solution for every rounded input set. Equation (11) gives the distinct upper-count-limit action.

ℏ = \frac{p_{s i} l_{f}}{Q_{L} n_{L}} = \frac{h}{2 π} = 1.054571817 \dots \times 1 0^{- 34} Js .

(10)

ħ = \lim_{n_{L} \to \infty} \frac{p_{s i} l_{f}}{Q_{L} n_{L}} = 2 p_{s i} l_{f} = 1.05453498444 \times 1 0^{- 34} Js .

(11)

Solving the finite-count action relation gives (12). The positive root requires positive momentum and (h)/(2π) greater than 2 p_si l_f. The numerical value in (11) is rounded to the precision shared by the printed-input calculation and a calculation enforcing the exact CODATA electron-mass–Bohr-radius identity. Source-table rounding exceeds the propagated input uncertainty for this particular output.

n_{L} = \frac{p_{s i} l_{f}}{\sqrt{ℏ (ℏ - 2 p_{s i} l_{f})}} .

(12)

8. Angular Interpretation of the Reduced Planck Constant

Figure 2 illustrates an arc of length L, in m, at the fundamental radius. Equation (13) uses the dimensionless geometric coefficient to calculate this length. Equation (14) is a separate momentum relation whose action dimensions follow from division by length. The angular illustration supplies no dimensional equality between angle and momentum.

L = r θ_{s i} = l_{f} θ_{s i} .

(13)

p_{s i} = \frac{ħ}{2 l_{f}} .

(14)

Related X-ray studies concern a theoretical proposal for entangled-photon generation [16], measurements of X-ray down-conversion [17], and X-ray–optical sum-frequency generation [18]. These references concern their respective optical processes. They do not establish equality between the MQ angular quantity and its momentum realization.

9. Momentum and the Reduced Planck Constant

The normalization in Section 3.5 identifies the left-hand side of (15) with p_si, while (14) identifies the right-hand side. Multiplying this equality by twice the fundamental length gives (16), with units J s. The momentum definition (9) is consistent with the same normalization.

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

(15)

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

(16)

Thus (16) expresses the upper-count-limit action through the dimensional momentum relation, with the numerical convention already stated.

10. Resolving the Fundamental Measures

Equation (17) follows from the momentum normalization. The substitutions in (18)–(21) use (7), (16) and c = l_f/t_f [3, Appendix BN]. Positive measures select the roots in (22), (24) and (27). Once the action and gravitational relations are adopted, these roots are algebraic consequences; their recovery supplies no additional independent equation fixing the absolute SI scale.

Length

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

(17)

l_{f} m_{f} = (2 p_{s i} l_{f}) \frac{G}{l_{f} G} t_{f} = ħ \frac{G}{l_{f} G} t_{f} .

(18)

l_{f} m_{f} (c^{3} \frac{t_{f}}{m_{f}}) = ħ \frac{G}{c},

(19)

l_{f} c^{4} t_{f} = ħ G,

(20)

l_{f}^{2} = \frac{ħ G}{c^{3}},

(21)

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

(22)

Time

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

(23)

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

(24)

Mass

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

(25)

t_{f} = \frac{G}{c^{3}} \sqrt{\frac{ħ c}{G}} = (c^{3} \frac{t_{f}}{m_{f}}) \frac{1}{c^{3}} \sqrt{\frac{ħ c}{G}} .

(26)

m_{f} = {(\frac{ħ c}{G})}^{1 / 2} .

(27)

The source-conditioned MQ results are 1.61619991147(12) × 10⁻³⁵ m for the fundamental length, 5.39106261127(41) × 10⁻⁴⁴ s for the fundamental time and 2.17643254057(16) × 10⁻⁸ kg for the fundamental mass. These uncertainties propagate the current input chain under the framework assumptions. They do not represent independent measurements of these measures.

For comparison, the conventional Planck length l_p, time t_p and mass m_p are defined by (28)–(30), in m, s and kg. Here the italic constants in these three expressions are the conventional CODATA gravitational constant and the exact SI reduced Planck constant [1]. They are distinct from the upright MQ upper-count-limit values used above.

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

(28)

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

(29)

m_{p} = {(\frac{ℏ c}{G})}^{1 / 2} .

(30)

Using the 2022 conventional gravitational constant 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻² with the exact SI inputs gives the conventional Planck length 1.616255(18) × 10⁻³⁵ m, time 5.391246(61) × 10⁻⁴⁴ s and mass 2.176434(24) × 10⁻⁸ kg. These recalculated values and uncertainties use the conventional gravitational input; the MQ values above use the distinct framework dependence.

4. DISCUSSION

The resulting Planck-form expressions follow conditionally from the count geometry and dimensional realization relations [3, 5–10]. Their derivation is algebraically closed under those premises. Their numerical realization also uses the CODATA electron-mass–Bohr-radius identity, which already contains the exact SI reduced Planck constant. Therefore recovery of that input at the blackbody demarcation is not an independent prediction of its value.

The cited preprint on conserved quantities concerns mass, energy-momentum, angular momentum and center of mass at null infinity [19]. Its mathematical setting is distinct from the fundamental-bound construction used here, and it supplies no derivation of the adopted SI coefficient relation.

An empirical test would require observables and uncertainty estimates independent of the inputs that fix the fundamental measures and demarcations. The numerical gravitational comparison reported above is a limited consistency comparison. The present derivation establishes the stated conditional relationships, rather than experimental proof of a unique microscopic count structure.

5. APPENDICES

APPENDIX A: TERMS

This appendix indexes the notation established in the narrative and identifies the additional count convention used in Appendix B. Complete framework terminology is provided in Measurement Quantization [3].

c — The exact speed-of-light input introduced in the Introduction; its count relation is used when eliminating time.

G and G — The upper-count and finite-count gravitational realizations in (1)–(7). The conventional gravitational constant is specified separately for (28)–(30).

l_f, m_f and t_f — The fundamental measures introduced in the Introduction and resolved in (22), (24) and (27). Their source-conditioned uncertainties do not establish greater experimental precision than the conventional Planck measures.

l_p, m_p and t_p — The conventional Planck measures introduced immediately before (28)–(30).

n_L, n_M and n_T — The separation, mass-ratio and time counts introduced under Methods and Frames.

n_Lm — The mass-motion count used in the uncertainty-bound relation.

n_L — The current radial-count notation used throughout the count geometry.

n_Lc — The dimensionless count of fundamental lengths traversed by light in the motion construction.

n_θ — The dimensionless System Frame angular count obtained by taking the nearest integer to half the selected electromagnetic demarcation. For the demarcations used here this count is 42.

The mixed nearest-whole-unit brackets are defined under Methods. No argument used in the present calculations lies at a half-unit tie.

Q_L — The non-discrete remainder introduced under Methods and illustrated in Fig. 1.

2 Q_L n_L — The Informativity differential normalization used in the finite-count and upper-count-limit relations.

ℏ and ħ — The finite-count action and upper-count-limit action distinguished in (10) and (11). The conventional SI action is (h)/(2π).

m_e — The electron-mass source input specified under Methods.

a₀ — The Bohr-radius source input specified under Methods.

α_f, α_p and α_c — The fundamental, Planck-like and classical fine structure realizations distinguished under Methods. The fundamental realization uses the Internal Frame description.

γ_F — Fundamental gamma, the electromagnetic realization parameter used in Appendix B, with units C². It replaces the older gamma approach for the elementary-charge demarcation considered here.

APPENDIX B ELECTROMAGNETIC DEMARCATION FROM ELEMENTARY CHARGE

The elementary-charge inversion uses the revised charge relation and the exact SI elementary charge 1.602176634 × 10⁻¹⁹ C [1]. It determines a default electromagnetic demarcation from this input. The resulting reproduction of the input charge is dependent and is not an independent prediction. A demarcation relevant to a known phenomenon takes precedence over this default.

The charge construction uses fundamental, intermediate and conventional charge quantities e_f, e_p and e_c, in C. The charge-valued residual n_e, also in C, is specified by (32). The metric differential describes the discrete–continuous mismatch in realizing count structure. In this charge construction it is represented by the dimensionless residual Q_{n_θ}, obtained from 42 θ_si by removing its nearest-integer part. This residual is distinct from the integer count introduced in Appendix A. Equation (31) is a relation among numerical SI coefficients, with the units explicitly divided out.

\frac{e_{f}}{1 C} = \frac{\frac{l_{f}}{1 m} {(\frac{c}{1 m s ⁻ ¹})}^{2}}{4 (θ_{s i} - 1)}

(B.1)

Q_{n_{θ}} = 42 θ_{s i} - ⌊ 42 θ_{s i} ⌉, n_{e} = (84 \frac{e_{f}}{1 C} - ⌊ 42 \frac{e_{f}}{1 C} ⌉) (1 C)

(B.2)

e_{p} = \frac{n_{e}}{84 + \frac{42}{277 - 3 Q_{n_{θ}} - 7 {Q_{n_{θ}}}^{2} - 14 {Q_{n_{θ}}}^{3}}}

(B.3)

Equations (31)–(33) resolve the intermediate charge. Equation (34) gives the revised charge quantity used in the demarcation inversion. Its dimensions are C² because the fine structure inverse and count normalization are dimensionless.

γ_{F} = {α_{p}}^{- 1} {({(Q_{L} n_{L})}^{2} e_{p})}^{2}

(B.4)

Equation (35) combines the separate MQ charge conversion e_c = 2 Q_L n_L e_p [3, Appendix AM, Eq. AM15] with (34). The second equality follows by taking the positive square root. Equation (34) alone does not determine the conventional charge or the demarcation.

e_{c} = 2 Q_{L} n_{L} e_{p} = \frac{2}{Q_{L} n_{L}} \sqrt{\frac{γ_{F}}{{α_{p}}^{- 1}}}

(B.5)

Let K denote the dimensionless product of the remainder and radial count. The charge conversion fixes it through (36). For 0 < e_c < e_p, the product is strictly between zero and one half and is a strictly increasing function of the positive count. The inversion therefore has exactly one positive real root on this branch.

K = Q_{L} n_{L} = \frac{e_{c}}{2 e_{p}}, Q_{L} = \frac{1}{\sqrt{1 + {n_{L}}^{2}} + n_{L}}

(B.6)

Substitution of the remainder expression into the product relation and squaring on the positive branch gives (37). The stated domain excludes the upper-count-limit denominator and fixes the positive solution.

n_{L} = \frac{K}{\sqrt{1 - 2 K}} = \frac{e_{c}}{2 e_{p} \sqrt{1 - \frac{e_{c}}{e_{p}}}}

(B.7)

The 2022 inputs and the adopted SI coefficient convention give (38). The reported standard uncertainty is propagated conditionally through the full dependent input chain. Substituting the solved count into (35) recovers the exact charge input by construction; γ_F is consequently a derived quantity in this inversion, not an additional independent input.

n_{L} = 84.60161 (22)

(B.8)

The default in (38) follows the phenomenon-specific selection rule in Section 3.1. The numerical SI convention in (31) is part of the charge construction and must accompany its use.

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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