An Operational Measurement Model for the Planck Length, Using a Torsion Balance and Silicon-28 Atom Counting

1. Introduction and Metrological Objective

The decisive issue should be stated at the outset. The familiar algebra

$$G={\displaystyle \frac{{\ell }_{\mathrm{P}}^2{c}^3}{\hslash }}$$

(1)

does not, by itself, establish that G is physically composed of a Planck length, a quantum of action, and the speed of light. If ${\ell }_{\mathrm{P}}$ has first been calculated from

$${\ell }_{\mathrm{P}}=\sqrt{{\displaystyle \frac{\hslash G}{c^3}}},$$

(2)

then Eq. (1) merely returns the information inserted into Eq. (2). The same problem arises for the Planck mass,

$$m_{\mathrm{P}}=\sqrt{{\displaystyle \frac{\hslash c}{G}}},$$

(3)

and the rearrangement

$$G={\displaystyle \frac{\hslash c}{m_{\mathrm{P}}^2}}.$$

(4)

Cahill was among the first to argue explicitly that the gravitational constant could be interpreted through a dynamically generated Planck-mass scale rather than treated as irreducibly fundamental. In his tetrad formulation he wrote:

“The actual distribution of energy throughout space-time causes the tetrads to assume vacuum expected values of the order of the Planck mass, $m_{\mathrm{P}}$. Thus the gravitational constant, $G=\hslash c/m_{\mathrm{P}}^2$, may be viewed not as a fundamental constant, but as a mass scale that is dynamically determined by the large-scale structure of the Universe.” [7,12]

The proposal is physically suggestive, but the algebra alone does not determine either G or $m_{\mathrm{P}}$ independently. Cohen made this limitation explicit in work presented in 1987 and published in the corresponding proceedings volume in 1988:

“Dimensional analysis let us write $G=\hslash c/m_{\mathrm{pl}}^2$, where $m_{\mathrm{pl}}$ is the Planck mass, $21.77\times {10}^{-9}\mathrm{kg}$, but this is of no help in determining G since there is no independent determination of $m_{\mathrm{pl}}$.” [8]

Hereafter $m_{\mathrm{P}}$ is used for the Planck mass, although $m_{\mathrm{pl}}$ is also common in the literature. Cohen correctly identified the essential logical defect: unless a Planck unit can be determined independently of G, reconstructing G from that unit is circular. McCulloch later restated the same objection in a modern context [9].

The objection is correct against purely algebraic relabelling. It is not correct against an operational determination of a Planck unit. The circle is broken when the following conditions are simultaneously met:

(i)

the gravitational experiment is analysed without inserting a tabulated value of G;

(ii)

the inertial content of the source body is established without using a mass standard whose relevant calibration presupposes the same gravitational information;

(iii)

the final observable relation contains no hidden occurrence of G; and

(iv)

every remaining quantity belongs to an independently realizable measurement chain.

A Cavendish-type experiment combined with Compton-scale mass representation and atom counting satisfies these conditions in principle [10,11,12]. The result is stronger than an alternative notation for G. It is a measurement architecture in which ${\ell }_{\mathrm{P}}$ is inferred first and G may subsequently be reconstructed through Eq. (1). Once this ordering is available, the traditional circularity objection has been answered.

2. Measurand and Operational Independence

The measurand is the Planck length ${\ell }_{\mathrm{P}}$ associated with the specified torsion-balance and source-body measurement model. The conventional relation ${\ell }_{\mathrm{P}}=\sqrt{\hslash G/c^3}$ is not used as the evaluation equation with a tabulated value of G. Instead, the gravitational response of the balance and the independently characterized inertial content of the source body are combined directly.

Operational independence from G requires that:

(i)

no recommended or previously measured value of G is entered as an input quantity;

(ii)

the source-body inertial content is obtained from non-gravitational measurements;

(iii)

the final measurement function contains no explicit or hidden dependence on G; and

(iv)

every input quantity has a stated realization and traceability route.

This criterion does not claim statistical independence from all measurements historically used in the SI or in determinations of other constants. It is narrower and testable: G is not an input to the measurement model, and the source body’s inertial content is not supplied merely as a conventional kilogram value.

3. Historical Motivation for an Independent Realization

Planck introduced natural units in 1899 by combining constants he regarded as universal [1]. His construction produced characteristic units of length, time, mass, and temperature. In modern reduced-constant notation, the best-known length is Eq. (2). Planck’s original convention employed h, rather than ℏ, so numerical factors involving $2\pi$ distinguish the historical and modern forms. That distinction does not alter the methodological point: dimensional analysis identifies a unique scale but does not by itself specify an experiment that realizes it.

Planck’s later treatment of thermal radiation placed these natural units within a broader physical framework [2]. Eddington went substantially further. After constructing the fundamental length from the velocity of light, the quantum of action, and the gravitational constant, he concluded:

“But it is evident that this length must be the key to some essential structure. It may not be an unattainable hope that some day a clearer knowledge of the processes of gravitation may be reached; and the extreme generality and detachment of the relativity theory may be illuminated by the particular study of a precise mechanism.” [3]

This is the complete passage rather than the frequently shortened form. Eddington’s expectation preceded quantum field theory and the modern quantum-gravity programme, yet it captured the central intuition that a length involving gravitation, relativity, and quantum action should not be dismissed as an accidental unit conversion. The historical wording and context are also reviewed in Ref. [12].

Bridgman supplied the necessary counterweight. His criticism was directed specifically at the physical interpretation of Planck’s natural units. After discussing the units constructed from the gravitational constant, the velocity of light, and the quantum of action, and after reproducing Eddington’s suggestion that the resulting extremely small length must reveal an essential structure, Bridgman concluded:

“Until some essential connection is discovered between the mechanisms which are accountable for the gravitational constant, the velocity of light, and the quantum, it would seem that no significance whatever should be attached to the particular size of the units defined in this way.”

This statement appears on pp. 101–102 of the 1922 edition of Dimensional Analysis [4]. It may be regarded as an early precursor to the later circularity objection. Bridgman’s concern was not merely that the Planck units arise from dimensional analysis, but that their physical significance remains unestablished so long as no independent connection exists between the mechanisms represented by G, c, and the quantum of action. In modern terms, if a Planck unit can only be calculated by inserting G, and is then used to reinterpret G, the construction has not supplied the “essential connection” Bridgman required. The operational measurement chain developed here is designed precisely to provide that missing connection by determining ${\ell }_{\mathrm{P}}$ without using G as an input.

Bridgman’s warning therefore remains valid. Dimensional analysis identifies a possible scale, but it cannot by itself establish that the scale corresponds to an independently measurable physical quantity. The central question is consequently not whether ${\ell }_{\mathrm{P}}$ can be written down, but whether it can be inferred from a measurement protocol that does not assume G in advance.

The operational construction reviewed below meets Bridgman’s requirement rather than bypassing it. It preserves the distinction between dimensional possibility and experimental realization, then supplies the missing realization.

4. Torsion-Balance Measurement Model

Let a torsion balance be characterized by an observed angular deflection $\theta$, free oscillation period T, source-body inertial mass M, and an apparatus coefficient $\mathcal{A}$. The coefficient $\mathcal{A}$ contains the complete geometrical and dynamical model of the balance, including lever arms, source–test separations, finite-body integration, density distributions, alignment, and multipole corrections. It has dimensions of volume. The gravitational measurement equation is

$$G={\displaystyle \frac{\mathcal{A}\theta }{M{T}^2}}.$$

(5)

For the simplified point-mass geometry used in earlier expositions,

$$\mathcal{A}=4{\pi }^{2}L{r}^2,$$

(6)

and Eq. (5) becomes

$$G={\displaystyle \frac{4{\pi }^{2}L{r}^2\theta }{M{T}^2}}.$$

(7)

Equation (6) is a special case, not a universal torsion-balance identity. A practical realization must determine $\mathcal{A}$ from the actual apparatus model and propagate all geometric uncertainties and covariances.

Figure 1 shows a modern Cavendish-type torsion-balance apparatus. The photograph is of a real instrument rather than a schematic rendering. In the illustrated setup the large source masses are lead spheres mounted on a rotating arm, while the suspended balance responds through the angular deflection $\theta$ and the free oscillation period T. For the specific measurement architecture proposed in this paper, isotopically enriched ²⁸Si spheres would be preferable in principle because the number of atoms in them can be established much more precisely through atom-counting methodology. However, high-purity ²⁸Si spheres are expensive, which is one practical reason why laboratory demonstrators and many existing Cavendish-style instruments instead employ more accessible materials such as lead. In all cases, the apparatus coefficient $\mathcal{A}$ must be evaluated from the complete finite-body geometry and the actual mass distribution.

Modern Cavendish apparatuses are commonly combined with modern electronics, including optical angle readout, digital data acquisition, and precise timing systems. These permit very accurate measurement of both the angular deflection $\theta$ and the oscillation time period T, thereby strengthening the practical feasibility of the proposed measurement model even though they do not remove the need for careful control of systematic effects.

Combining Eq. (5) with the conventional dimensional relation for the Planck length gives

$${\ell }_{\mathrm{P}}=\sqrt{{\displaystyle \frac{\hslash \mathcal{A}\theta }{M{T}^2{c}^3}}}.$$

(8)

At this stage G is no longer an input. The remaining task is to replace M by a non-gravitationally realized microscopic mass representation.

5. Eliminating Both the Source Mass and ℏ

For any total inertial mass M, define the reduced Compton wavelength

$${\overline{\lambda }}_M={\displaystyle \frac{\hslash }{Mc}},M={\displaystyle \frac{\hslash }{{\overline{\lambda }}_{M}c}}.$$

(9)

Insertion of Eq. (9) into Eq. (8) cancels ℏ:

$$\begin{array}{cc}\hfill {\ell }_{\mathrm{P}}& =\sqrt{{\displaystyle \frac{\hslash \mathcal{A}\theta }{\left(\hslash /{\overline{\lambda }}_{M}c\right)T^2{c}^3}}}\hfill \\ \hfill & ={\displaystyle \frac{1}{Tc}}\sqrt{\mathcal{A}\theta {\overline{\lambda }}_M}.\hfill \end{array}$$

(10)

The final expression contains neither G nor ℏ. It depends on the torsion-balance observables, the exact speed of light, and the aggregate reduced Compton wavelength of the source body.

The phrase “aggregate reduced Compton wavelength” must be interpreted correctly. A macroscopic source body is not asserted to possess a coherent free-particle Compton wave extending around the apparatus. In the present measurement model, ${\overline{\lambda }}_M$ is a composite bookkeeping quantity that represents the total inertial mass through an inverse-length scale:

$${\displaystyle \frac{1}{{\overline{\lambda }}_M}}={\displaystyle \frac{Mc}{\hslash }}.$$

(11)

If a body is idealized as a collection of constituents with additive rest masses $m_i$, then

$$M=\sum _i{N}_i{m}_i⟹{\displaystyle \frac{1}{{\overline{\lambda }}_M}}=\sum _i{\displaystyle \frac{N_i}{{\overline{\lambda }}_i}},$$

(12)

before binding-energy and material corrections are included. Equivalently,

$${\overline{\lambda }}_M={\left(\sum _i{\displaystyle \frac{N_i}{{\overline{\lambda }}_i}}\right)}^{-1}.$$

(13)

Thus the composite wavelength becomes smaller as the total mass becomes larger. It is not introduced as a literal matter wave of the macroscopic sphere, but as mass accounting expressed in inverse Compton-length units. This interpretation and aggregation rule follow the composite treatment developed in Ref. [12]. Its metrological role is to replace a kilogram entry by a microscopic inventory whose constituents can be established through non-gravitational measurements.

6. Non-Gravitational Construction of the Particle Compton Scales

The electron Compton scale can be connected to Compton scattering. For an incident photon of wavelength $\lambda$, a scattered photon of wavelength ${\lambda }^{\prime }$, and scattering angle $\varphi$,

$${\lambda }^{\prime }-\lambda ={\lambda }_e(1-cos\varphi ),$$

(14)

where ${\lambda }_e=h/\left(m_{e}c\right)$ is the ordinary electron Compton wavelength [16]. Solving Eq. (14) for the electron Compton wavelength gives

$${\lambda }_e={\displaystyle \frac{{\lambda }^{\prime }-\lambda }{1-cos\varphi }}.$$

(15)

The corresponding reduced electron Compton wavelength is therefore

$${\overline{\lambda }}_e={\displaystyle \frac{{\lambda }_e}{2\pi }}={\displaystyle \frac{{\lambda }^{\prime }-\lambda }{2\pi (1-cos\varphi )}}.$$

(16)

This is an electromagnetic scattering measurement and does not require gravitational calibration. Most importantly for the present construction, Eqs. (15) and (16) contain neither h nor ℏ: the electron Compton scale is obtained from the measured incident photon wavelength $\lambda$, the measured scattered photon wavelength ${\lambda }^{\prime }$, and the measured scattering angle $\varphi$.

Cyclotron-frequency ratios provide a direct non-gravitational route to charged-particle mass ratios. In a uniform magnetic field, the free cyclotron angular frequency is

$${\omega }_c={\displaystyle \frac{\left|q\right|B}{m}}.$$

(17)

For a proton and an electron referred to the same magnetic field, and using $|q_p|=|q_e|=e$, the field and charge cancel in the frequency ratio:

$${\displaystyle \frac{m_p}{m_e}}={\displaystyle \frac{{\omega }_{c,e}}{{\omega }_{c,p}}}.$$

(18)

Since a reduced Compton wavelength is inversely proportional to mass, the same measurement gives

$${\displaystyle \frac{{\overline{\lambda }}_p}{{\overline{\lambda }}_e}}={\displaystyle \frac{m_e}{m_p}}={\displaystyle \frac{{\omega }_{c,p}}{{\omega }_{c,e}}},$$

(19)

and therefore

$${\overline{\lambda }}_p={\overline{\lambda }}_e{\displaystyle \frac{{\omega }_{c,p}}{{\omega }_{c,e}}}.$$

(20)

This is not merely a formal possibility. Farnham, Van Dyck, and Schwinberg determined the electron’s relative atomic mass and the proton–electron mass ratio by Penning-trap mass spectroscopy, using cyclotron-frequency comparisons of particles alternately confined in the same highly uniform magnetic field [17]. Modern Penning-trap experiments continue to determine light-particle and light-ion masses from cyclotron-frequency ratios; for example, the LIONTRAP collaboration measured the proton’s atomic mass with a purpose-built multi-Penning-trap system [18,19]. In a practical realization, suitable reference ions and corrections for charge state and electronic binding energy may be required rather than a literal simultaneous comparison of a bare proton and electron. These corrections are electromagnetic and spectroscopic, not gravitational.

An interpretational qualification is essential. The electron Compton wavelength in Eq. (15) is connected directly to the measured Compton scattering shift. By contrast, the quantity denoted ${\overline{\lambda }}_p=\hslash /\left(m_{p}c\right)$ is used here as the inverse-mass scale associated with the proton mass determined from frequency ratios. The paper does not require, and does not claim, that the proton exhibits a directly observed free Compton wave in the same operational sense as the electron Compton wavelength obtained from scattering. The proton is itself a composite hadron, and ${\overline{\lambda }}_p$ functions in this construction as a compact mass-equivalent parameter. The essential point is that, once ${\overline{\lambda }}_e$ has been obtained from the measured Compton shift, the proton mass scale and hence ${\overline{\lambda }}_p$ can be fixed through measured cyclotron-frequency ratios without inserting either G or ℏ.

For a metrologically complete silicon-28 construction, the neutron contribution must also be included. It may be represented through an independently determined neutron-to-proton mass ratio, or through nuclear and atomic mass differences derived from non-gravitational spectroscopy and reaction-energy measurements. The specific laboratory chain may vary; circularity is avoided as long as it does not require G.

7. Silicon-28 as the Atom-Counting Realization

7.1. Why Silicon-28 Is the Natural Metrological Choice

Isotopically enriched ²⁸Si is the natural material for a practical realization of the non-circular measurement chain. The Avogadro project established how the number of atoms in an almost perfect single-crystal silicon sphere can be inferred from interferometrically measured volume, x-ray determination of the lattice parameter, isotope analysis, and detailed corrections for surfaces, defects, and impurities [13,14,15]. This is not a speculative counting method. It is an experimentally mature route developed in the redefinition of the kilogram.

Silicon crystallizes in the diamond-cubic structure. The conventional cubic unit cell contains eight atoms. This can be counted directly: the eight corner atoms contribute one atom in total, the six face-centred atoms contribute three, and four atoms lie fully inside the conventional cell. Hence

$$N_{\mathrm{cell}}=8.$$

(21)

If a is the lattice parameter, i.e. the edge length of the conventional cubic cell, and $V_{\mathrm{core}}$ is the corrected volume of the crystalline core, the ideal atom count is

$$N_{\mathrm{Si}}^{\left(0\right)}={\displaystyle \frac{8V_{\mathrm{core}}}{a^3}}.$$

(22)

A real realization requires corrections for point defects, vacancies, interstitials, substitutional impurities, residual isotope fractions, oxide and chemisorbed surface layers, adsorbed water, metallic contamination, voids, strain, and departures from an ideal sphere. It is convenient to write

$$N_{\mathrm{Si}}={\displaystyle \frac{8V_{\mathrm{core}}}{a^3}}+\Delta N_{\mathrm{defect}}+\Delta N_{\mathrm{surface}}+\Delta N_{\mathrm{impurity}}+\dots .$$

(23)

In a high-precision treatment, the number density, volume, isotope composition, and all correction terms must be propagated with their full covariance structure.

7.2. Microscopic Constitution of an Enriched ²⁸Si Source

A neutral atom contains 14 protons, 14 neutrons, and 14 electrons. For an ideal isotopically pure source,

$$N_p=N_n=N_e=14N_{\mathrm{Si}}.$$

(24)

A first-order pedagogical approximation neglects electron masses, the neutron–proton mass difference, and all binding-energy defects. The source is then represented as $28N_{\mathrm{Si}}$ proton-mass equivalents. This is not a claim that the silicon sphere possesses a physical proton-like Compton wave; it is the bookkeeping construction of Eqs. (12) and (13), applied to the total mass-equivalent inventory:

$${\overline{\lambda }}_M^{\left(0\right)}\approx {\displaystyle \frac{{\overline{\lambda }}_p}{28N_{\mathrm{Si}}}}.$$

(25)

This approximation is sufficient to display the logical cancellation, but not for precision metrology.

A more complete constituent representation is

$${\displaystyle \frac{1}{{\overline{\lambda }}_M}}={\displaystyle \frac{N_p}{{\overline{\lambda }}_p}}+{\displaystyle \frac{N_n}{{\overline{\lambda }}_n}}+{\displaystyle \frac{N_e}{{\overline{\lambda }}_e}}-{\displaystyle \frac{E_{\mathrm{bind},\mathrm{tot}}}{\hslash c}}+{\Delta }_{\mathrm{chem}}+{\Delta }_{\mathrm{thermal}}+{\Delta }_{\mathrm{defect}}.$$

(26)

Here $E_{\mathrm{bind},\mathrm{tot}}$ includes the nuclear mass defect and the electronic and condensed-matter binding contributions relative to the chosen free-particle decomposition. The material terms account for surface chemistry, impurities, vacancies, strain, and the thermodynamic state of the actual sphere.

An equivalent and often cleaner route is to use the experimentally determined neutral-atom mass of ²⁸Si:

$${\displaystyle \frac{1}{{\overline{\lambda }}_M}}=N_{\mathrm{Si}}{\displaystyle \frac{{m(}^{28}\mathrm{Si})c}{\hslash }}+{\Delta }_{\mathrm{mat}}.$$

(27)

This remains non-gravitational provided the atomic mass ratio and reference scale are established through mass spectrometry, spectroscopy, and frequency measurements without inserting G.

8. The silicon-28 Planck-Length Equation

Combining the diamond-cubic atom count with the leading 28-nucleon approximation gives

$${\overline{\lambda }}_M^{\left(0\right)}\approx {\displaystyle \frac{{\overline{\lambda }}_p{a}^3}{224V_{\mathrm{core}}}},$$

(28)

because

$$28N_{\mathrm{Si}}=28{\displaystyle \frac{8V_{\mathrm{core}}}{a^3}}={\displaystyle \frac{224V_{\mathrm{core}}}{a^3}}.$$

Substitution into Eq. (10) yields

$${\ell }_{\mathrm{P}}^{\left(0\right)}\approx {\displaystyle \frac{1}{Tc}}\sqrt{{\displaystyle \frac{\mathcal{A}\theta {\overline{\lambda }}_p{a}^3}{224V_{\mathrm{core}}}}}$$

(29)

for an ideal isotopically enriched ²⁸Si source.

Equation (29) is the central atom-counting result. It contains neither G, ℏ, nor a source mass entered in kilograms. Its inputs are the torsion-balance observables, the speed of light, a non-gravitationally established particle Compton scale, the silicon lattice parameter, and the corrected volume of the silicon crystal.

Replacing the first-order approximation by Eq. (26) gives the higher-order relation

$${\ell }_{\mathrm{P}}={\displaystyle \frac{1}{Tc}}\sqrt{{\displaystyle \frac{\mathcal{A}\theta }{{\displaystyle \frac{N_p}{{\overline{\lambda }}_p}+\frac{N_n}{{\overline{\lambda }}_n}+\frac{N_e}{{\overline{\lambda }}_e}-\frac{E_{\mathrm{bind},\mathrm{tot}}}{\hslash c}+{\Delta }_{\mathrm{mat}}}}}}$$

(30)

with $N_p,N_n,N_e$ supplied by Eqs. (22)–(24). As before, the appearance of ℏ in an intermediate binding-energy notation does not require ℏ to be inserted as an independent input if the correction is realized directly as a frequency, mass ratio, or inverse Compton length.

9. Input Quantities and Traceability Chain

Table 1 summarizes the principal input quantities. The gravitational interaction enters through the observed torsion-balance response, not through a supplied value of G.

A practical sequence is:

1.

determine ${\overline{\lambda }}_e$ from measured Compton-scattering wavelengths and angle;

2.

determine proton and neutron mass ratios through frequency-ratio, spectroscopic, and nuclear measurements;

3.

characterize an enriched single-crystal ²⁸Si source body;

4.

determine a, $V_{\mathrm{core}}$, isotope composition, defects, and surface corrections;

5.

construct the aggregate reduced Compton wavelength ${\overline{\lambda }}_M$;

6.

determine $\theta$, T, and the full apparatus coefficient $\mathcal{A}$;

7.

evaluate ${\ell }_{\mathrm{P}}$ from Eq. (10); and

8.

report the result with a GUM-consistent uncertainty statement.

The measurement chain contains no recommended value of G. Traceability for each input must be documented through its calibration hierarchy, measurement model, and uncertainty evaluation in accordance with the GUM and the International Vocabulary of Metrology [20,21].

10. Suitability and Limitations of Silicon-28

Silicon-28 does more than provide a convenient example. It supplies the strongest presently available material basis for the argument because the relevant atom-counting technology already exists. The lattice parameter of silicon can be measured by x-ray interferometry, the volume of a nearly perfect sphere can be determined optically, and the isotope composition can be established by mass spectrometry. Surface layers and crystal defects can be modelled and corrected at a level developed for primary metrology [13,14,15].

This does not mean that the proposed Planck-length determination would automatically exceed the precision of the conventional value inferred from G, c, and ℏ. The torsion balance remains a demanding gravitational experiment, and the source-body characterization introduces its own uncertainty budget. The importance of ²⁸Siis instead methodological: it removes the most obvious objection that the source mass must enter as a kilogram value. The source body’s inertial content is reconstructed from crystal geometry, atom counting, isotope composition, and non-gravitational microscopic mass relations.

11. Uncertainty Model

For the measurement function

$${\ell }_{\mathrm{P}}=f(\mathcal{A},\theta ,{\overline{\lambda }}_M,T)={\displaystyle \frac{1}{Tc}}\sqrt{\mathcal{A}\theta {\overline{\lambda }}_M},$$

(31)

the first-order sensitivity coefficients are

$$\begin{array}{cccc}\hfill {\displaystyle \frac{\partial {\ell }_{\mathrm{P}}}{\partial \mathcal{A}}}& ={\displaystyle \frac{{\ell }_{\mathrm{P}}}{2\mathcal{A}}},\hfill & \hfill {\displaystyle \frac{\partial {\ell }_{\mathrm{P}}}{\partial \theta }}& ={\displaystyle \frac{{\ell }_{\mathrm{P}}}{2\theta }},\hfill \end{array}$$

(32)

$$\begin{array}{cccc}\hfill {\displaystyle \frac{\partial {\ell }_{\mathrm{P}}}{\partial {\overline{\lambda }}_M}}& ={\displaystyle \frac{{\ell }_{\mathrm{P}}}{2{\overline{\lambda }}_M}},\hfill & \hfill {\displaystyle \frac{\partial {\ell }_{\mathrm{P}}}{\partial T}}& =-{\displaystyle \frac{{\ell }_{\mathrm{P}}}{T}}.\hfill \end{array}$$

(33)

The law of propagation of uncertainty is

$$u^2\left({\ell }_{\mathrm{P}}\right)=\sum _i\sum _j{\displaystyle \frac{\partial f}{\partial x_i}}{\displaystyle \frac{\partial f}{\partial x_j}}u(x_i,x_j),$$

(34)

where $x_i\in \{\mathcal{A},\theta ,{\overline{\lambda }}_M,T\}$ and $u(x_i,x_j)$ is the covariance [20]. If correlations are negligible,

$${\left[{\displaystyle \frac{u\left({\ell }_{\mathrm{P}}\right)}{{\ell }_{\mathrm{P}}}}\right]}^2={\displaystyle \frac{1}{4}}{\left[{\displaystyle \frac{u\left(\mathcal{A}\right)}{\mathcal{A}}}\right]}^2+{\displaystyle \frac{1}{4}}{\left[{\displaystyle \frac{u\left(\theta \right)}{\theta }}\right]}^2+{\displaystyle \frac{1}{4}}{\left[{\displaystyle \frac{u\left({\overline{\lambda }}_M\right)}{{\overline{\lambda }}_M}}\right]}^2+{\left[{\displaystyle \frac{u\left(T\right)}{T}}\right]}^2.$$

(35)

The exact SI value of c contributes no uncertainty.

The uncertainty of $\mathcal{A}$ must be evaluated from the actual geometry and finite-body gravitational integration. The uncertainty of ${\overline{\lambda }}_M$ must be propagated from the Compton-scattering data, frequency ratios, atom count, isotope composition, defect model, and binding-energy corrections. Important correlations include the common temperature dependence of the silicon lattice parameter and sphere volume, and common coordinate systems used in determining the torsion-balance geometry.

A realized experiment should report a complete input-quantity table, probability models, sensitivity coefficients, covariance matrix, effective degrees of freedom where applicable, combined standard uncertainty, and expanded uncertainty in accordance with the GUM. It must also document torsion-fibre anelasticity, damping, drift, electrostatic and magnetic backgrounds, seismic coupling, source positioning, and validation of the finite-body model used for $\mathcal{A}$.

The present work establishes the measurement model and traceability architecture; it does not claim a competitive numerical uncertainty. In an initial realization, the torsion-balance response and the determination of $\mathcal{A}$ are expected to dominate over the silicon atom-counting contribution, but that expectation requires experimental verification.

12. Validation and Consistency Tests

The generalized model must reproduce the idealized result when $\mathcal{A}=4{\pi }^{2}L{r}^2$. Independently characterized source bodies should also yield consistent values of ${\ell }_{\mathrm{P}}$ within their stated uncertainties. A material-dependent discrepancy would indicate an error in the source-body representation, binding or surface corrections, or the gravitational geometry model.

As an output-level validation, the independently evaluated ${\ell }_{\mathrm{P}}$ may be used to reconstruct

$$G_{\mathrm{reconstructed}}={\displaystyle \frac{{\ell }_{\mathrm{P}}^2{c}^3}{\hslash }}.$$

(36)

Agreement with conventional determinations of G would be a consistency test, not an input or calibration.

13. Metrological Interpretation

The circularity objection can now be separated into a valid restricted statement and an invalid unrestricted statement.

The valid statement is:

One cannot establish the physical primacy of a Planck unit by algebraically solving a Planck-unit definition for G when that Planck unit has itself been calculated from G.

The invalid unrestricted statement is:

A Planck unit can never be determined without first knowing G, and therefore every expression of G in Planck-scale variables is necessarily circular.

Equations (10), (29), and (30) provide explicit counterexamples to the unrestricted claim. A torsion balance supplies the gravitational interaction; atom counting and Compton-scale relations supply the source’s inertial content. Neither G nor ℏ remains as an independent input to the leading final equation.

This result does not prove that the Planck length is the smallest possible spatial interval. It does not prove spacetime discreteness, select a theory of quantum gravity, or guarantee competitive experimental precision. Those are separate questions. What it does prove at the level of operational construction is that the Planck length need not be confined to dimensional analysis. It can be attached to a concrete measurement procedure.

14. Relation to Conventional Determinations of G

Once ${\ell }_{\mathrm{P}}$ has an independent operational route, Eq. (1) changes status. It is no longer forced to be a circular rewriting. It can be interpreted as a reconstruction of the gravitational constant from an independently determined gravitational length, the speed of light, and the quantum action scale:

$$G={\displaystyle \frac{{\ell }_{\mathrm{P}}^2{c}^3}{\hslash }}.$$

(37)

Whether this should be regarded as an ontological decomposition of G is a further theoretical question. The operational result does not compel one metaphysical interpretation. It does, however, remove the standard logical veto against considering G a composite constant.

The stronger conclusion is therefore justified: the historical circle has been broken. Bridgman’s demand for an operation has been met, and Eddington’s expectation that the Planck length is connected to the structure of gravitation becomes experimentally meaningful. The silicon-28 construction reinforces that conclusion by grounding the method in an experimentally established atom-counting technology.

15. Conclusion

Planck’s natural units originated in dimensional analysis [1,2]. Eddington anticipated deeper gravitational content [3], while Bridgman correctly warned that dimensional combinations should not be mistaken for physical discoveries [4]. The later circularity criticism sharpened this warning: a Planck unit calculated from G cannot then be used to explain G without returning the same information [6,7,8,9].

That criticism has a definite boundary. It applies to algebraic inversion, not to an independent operational determination. A Cavendish-type torsion balance can be written directly as a Planck-length experiment. When the source mass is replaced by an aggregate reduced Compton wavelength constructed from atom counting and non-gravitational particle measurements, both G and ℏ disappear from the leading final relation [10,11,12].

For a diamond-cubic ²⁸Si source, the ideal atom count is $N_{\mathrm{Si}}=8V_{\mathrm{core}}/a^3$. Constituent-resolved or neutral-atom mass accounting then supplies the aggregate Compton scale. The resulting Planck-length equation is explicit, testable, and non-circular. Silicon-28 is particularly well suited to precision atom counting, and the established Avogadro-project methodology gives the measurement architecture a concrete metrological foundation.

The Planck length is therefore more than a number produced by combining dimensions. It is an indirectly measurable gravitational length. The traditional circularity objection does not survive the operational construction.

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