Deriving the Planck Unit Expressions

Planck units bring gravitation, quantum action and the speed of light into a common system of length, time and mass. Their familiar formulas can be obtained by dimensional analysis. The deeper question is what physical structure, if any, explains why those constants belong together. Measurement Quantization (MQ) approaches that question through discrete count geometry, a count-dependent gravitational realization and a dimensional momentum relation. The resulting derivation connects the three fundamental measures before their individual SI values are calculated.

The distinction matters. Deriving an expression from adopted physical relations is different from selecting a combination with the right dimensions, and both are different from independently predicting its numerical value. The development below follows the preprint’s complete equation chain while separating these three achievements. Its proposed geometric origin is developed in Discrete Gravity; its dimensional and numerical consequences are developed here.

What the conventional Planck construction establishes

Let c denote the exact speed of light, in m s⁻¹. Let italic G denote gravitational coupling, in m³ kg⁻¹ s⁻², and italic ℏ denote reduced action, in J s. In this opening conventional comparison, these are the conventional gravitational and reduced Planck constants. The conventional Planck length l_p, time t_p and mass m_p are then

l_{p} = {(\frac{ℏ G}{c^{3}})}^{1 / 2}

t_{p} = {(\frac{ℏ G}{c^{5}})}^{1 / 2}

m_{p} = {(\frac{ℏ c}{G})}^{1 / 2}

These powers are fixed once one asks for a monomial in the three constants having the dimensions of length, time or mass, with the conventional numerical normalization. This is already a derivation of the unit expressions in the dimensional sense. It does not establish a microscopic mechanism, a unique physical minimum or an observed particle having the Planck mass. Stoney’s earlier natural-unit construction provides a useful historical comparison.

The 2022 CODATA adjustment reports the conventional constants and their measurement uncertainties. Using its gravitational input gives a Planck length of 1.616255(18) × 10⁻³⁵ m, a Planck time of 5.391246(61) × 10⁻⁴⁴ s and a Planck mass of 2.176434(24) × 10⁻⁸ kg. Their uncertainties predominantly inherit the gravitational input. A physical derivation must explain more than these dimensional combinations while retaining an honest account of the information used to fix their scale.

Fundamental measures and the count geometry

MQ introduces the fundamental length l_f, mass m_f and time t_f, measured in m, kg and s. They are not assumed to equal the conventional Planck measures. The Internal Frame is the discrete count configuration domain relative to those measures. The System Frame is its non-discrete relational encoding, and Frames mapping relates the two descriptions. A continuously realized separation is therefore not identified indiscriminately with an integer count. The frames discussion develops this representational distinction.

Let n_L be the dimensionless radial separation count and Q_L the dimensionless right-triangle remainder. A unit orthogonal leg and a radial leg determine the hypotenuse through the exact count construction.

1 + {n_{L}}^{2} = {(n_{L} + Q_{L})}^{2}

The positive remainder can be written without subtracting nearly equal large quantities.

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

The geometry fixes the remainder once the radial count is chosen. It is not an independently adjustable force. Expanding the triangle identity gives Q_L² + 2Q_Ln_L = 1. Thus the remainder decreases with separation while its product with the count tends to a finite limit.

\lim_{n_{L} \to \infty} Q_{L} n_{L} = \frac{1}{2}, \lim_{n_{L} \to \infty} 2 Q_{L} n_{L} = 1

The Informativity differential is the non-Lorentz length contraction associated with count normalization under Frames mapping. The factor 2Q_Ln_L enters the finite-count realization used here. It approaches unity in the upper-count limit. This is distinct from the metric differential, which concerns the discrete–continuous mismatch in realizing count structure, and from motion-induced Lorentz contraction. The Informativity differential discussion provides the broader setting.

For completeness, n_M denotes mass divided by the fundamental mass, n_T elapsed time divided by the fundamental time, and n_Lm the observed mass’s change in realized position divided by the fundamental length. Each is dimensionless. The physical radial distance is r = n_Ll_f. The radial and motion counts have different roles and cannot be exchanged simply because both involve length.

The bound assumptions behind the construction

The preprint combines count geometry with light-speed and escape bounds and a selected uncertainty-bound equality. At the limiting construction, the radial, motion and time counts are unity, while the normalized mass ratio is one half. The latter is a bound ratio rather than a fractional primitive integer count. The escape equality denotes the boundary of the strict escape domain. The count relation used for the uncertainty saturation is

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

This relation belongs to the specified MQ bound construction. Standard quantum uncertainty constrains statistical dispersions; its equality case and its identification with these count measures are additional choices. The quantum uncertainty principle alone does not establish discrete fundamental units. The site’s Bounds to Measure discussion develops the construction, while Robertson’s published treatment supplies the conventional statistical context.

From a geometric remainder to gravitational realization

The crucial bridge to mass is the MQ gravitational construction. Its primitive curvature-resolved scalar expression has acceleration dimensions.

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

Let p_si denote the momentum realization, in kg m s⁻¹. Multiplication by the speed-to-momentum ratio gives the following dimensional chain.

\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}}

The expression now has units m kg⁻¹ s⁻². MQ identifies this realization with a gravitational field per unit source mass.

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

This identification is the physical gravitational premise. The right-triangle theorem fixes a geometric remainder; it does not, by dimensional consistency alone, prove the gravitational field equation. Under the adopted mapping and normalization, however, the construction supplies a relation containing the fundamental mass. This is how MQ moves beyond merely combining constants with prescribed dimensions.

Use upright G for the upper-count gravitational quantity and upright ħ for the upper-count action. Italic G and ℏ denote finite-count values at the indicated electromagnetic realization; when the conventional SI values are used, the prose identifies that different context explicitly. The distinction is part of the calculation, not a stylistic alternative. Taking the upper-count limit in the gravitational relation gives

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

The momentum normalization then resolves the same relation into 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}}

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

The second form makes the substitution c = l_f/t_f explicit. The limit has already been taken before this reduction. A finite-count gravitational coupling must not be substituted into the upper-count equation without restoring its count normalization. Related implications are developed in The Gravitational Constant.

Momentum supplies the action dimensions

Let p denote the full momentum associated with one fundamental length and time. It is twice the bound momentum realization.

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

The momentum normalization gives a relation between the fundamental measures, and the upper-count action supplies the second expression for the same momentum.

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

Multiplying by twice the fundamental length gives

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

The expression has units J s. It is this dimensional momentum relation that supplies action. An angular illustration cannot do so merely because an angle and a momentum have matching numerical coefficients in selected SI units.

Let θ_si be the invariant dimensionless geometric coefficient and Θ_si its angular realization, whose value is θ_si rad. Their common numerical convention with momentum is stated with the unit explicitly divided out.

\frac{p_{si}}{1 kg m s⁻¹} = θ_{si}

For an arc length L, in m, at the fundamental radius, the preprint’s angular illustration is L=rθ_si=l_fθ_si. The separate momentum relation is

p_{s i} = \frac{ħ}{2 l_{f}}

The angle and momentum constructions share a coefficient while retaining different dimensions. The related Angular Measure and Momentum discussion examines this connection. The X-ray studies discussed in the preprint concern entangled-photon generation, down-conversion and sum-frequency generation. They do not establish a dimensional equality between angle and momentum.

Resolving the fundamental length

The momentum normalization can first be written compactly as l_fm_f=2p_sit_f. This is the fundamental relation in the present derivation. Combining it with the upper-count action and gravitational expressions yields the substitution chain below.

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}

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

The gravitational replacement introduces the fundamental mass in the denominator. Its cancellation is decisive. Mass has entered through the gravitational realization, and its elimination leaves a relation in length and time. The resulting product is l_fc⁴t_f=ħG. Replacing the remaining fundamental time through the light-speed relation gives the squared length.

l_{f}^{2} = \frac{ħ G}{c^{3}}

The positive root resolves the length.

l_{f} = {(\frac{ħ G}{c^{3}})}^{1 / 2}

The familiar Planck-form powers have therefore emerged from the adopted gravitational and action relations. The expression is not a new independent equation fixing the scale after those relations have been used. It is their exact algebraic consequence.

Time and mass follow from the same solution

The time measure follows by dividing the length by the exact light-speed input. The preprint preserves the intermediate substitution to make that elimination visible.

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

t_{f} = {(\frac{ħ G}{c^{5}})}^{1 / 2}

Mass follows by returning to the gravitational relation. The first rearrangement exposes the gravitational factor required for that step.

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

Equivalently, the intermediate time expression may be expanded through the fundamental measures.

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}}

The positive solution is

m_{f} = {(\frac{ħ c}{G})}^{1 / 2}

All three measures are coupled. The gravitational relation supplies the mass-to-time connection, the momentum relation supplies the length–mass product, and the light-speed relation connects length and time. Their closure is the central algebraic result. The Newton–Planck relation and Fundamental Measures discussions develop the same shared structure.

Finite counts and the electromagnetic demarcation

The upper-count expressions must now be distinguished from measurement at finite separation. The electromagnetic demarcation is the interaction count selected for the phenomenon being realized. It is not a universal instruction to use the same finite count in every electromagnetic calculation. For the blackbody-derived reduced action, the preprint retains the working count 84.6005393909226. Its finite gravitational and upper-count expressions are

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}

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}

These numerical values are retained as the preprint’s specified 2022-input realization. Their parenthetical uncertainties belong to that source dependency chain and are not experimental determinations of the MQ quantities. The historical 2010, 2014 and 2018 CODATA adjustments provide comparison contexts; they must not silently replace the current input chain. The conventional 2022 gravitational recommendation is 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻². Numerical proximity does not establish the proposed geometry.

Let h be the exact SI Planck constant and upright π the circle constant. The exact conventional reduced action is obtained from their quotient. The finite-count MQ action at the blackbody calibration is

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

The corresponding upper-count action is

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

The limit of the count product supplies the factor of two in the latter expression. The displayed upper-count action is rounded to the precision shared by the printed-input calculation and the calculation enforcing the exact electron-mass–Bohr-radius identity. Source-table rounding exceeds propagated input uncertainty for this output. The absence of printed uncertainty digits does not mean that the derived action is exact. The finite count instead retains the denominator containing the remainder. Solving the finite relation gives

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

The solution uses the positive branch. Momentum must be positive and the finite action must exceed twice the momentum–length product. The adopted working count reproduces the displayed digits of the action calibration, but its terminal digits are not an exact root for every rounded source-input set. Recovering the action used to calibrate that count is a consistency identity rather than an independent prediction of the SI constant.

What removing the Informativity differential means

At a common finite count, the two constants carry reciprocal count dependence. The exact relationships implied by the gravitational and action expressions are

G = 2 Q_{L} n_{L} G, ℏ = \frac{ħ}{2 Q_{L} n_{L}}

Removing the finite-count factor means replacing it by its upper-count limit of unity. It gives upright G and ħ and the fundamental-measure expressions already resolved. It does not authorize replacing the finite gravitational value while leaving the finite action unchanged. Such a mixed substitution would combine different realization contexts.

The conventional Planck expressions are recovered by using the conventional constants, as shown at the start of the discussion. That substitution is logically separate from taking an MQ limit. The conventional measured gravitational constant is not defined as the finite MQ gravitational coupling at the blackbody demarcation. Thus removing the Informativity differential establishes the MQ upper-count form; it does not prove that conventional experimental inputs must numerically equal the MQ limits.

MQ’s proposed refinement is an explicit account of count-dependent realization that the conventional dimensional construction does not contain. Calling it a more precise description is defensible in that structural sense. Greater metrological accuracy would require independent validation of the model and its uncertainty budget. Smaller propagated input uncertainties alone cannot establish it.

Fixing the SI scale

The algebraic form is not the entire numerical derivation. The preprint fixes its SI realization through fine-structure relations and source inputs. Let α_c, α_p and α_f denote the dimensionless classical, Planck-like and fundamental fine-structure realizations. The fundamental realization uses the Internal Frame; the other two use System Frame descriptions. The coefficient is obtained on the selected fine-structure branch.

{α_{p}}^{- 1} = 84 θ_{si} - ⌊ 42 θ_{si} ⌉, {α_{c}}^{- 1} = 2 Q_{L} n_{L} {α_{p}}^{- 1}, n_{L} = \frac{276}{θ_{si}}

The mixed floor/ceiling brackets denote the nearest-whole-unit operation used by the source. On this branch the rounded coefficient product is 137 and the geometric coefficient is 3.262390305045149. This numerical branch condition is not a claim that every possible input realization selects the same integer without verification.

Let m_e denote electron mass and a₀ the Bohr radius. The source uses the 2022 inputs α_c = 7.2973525643(11) × 10⁻³, m_e = 9.1093837139(28) × 10⁻³¹ kg and a₀ = 5.29177210544(82) × 10⁻¹¹ m. Its dimensional scale relations are

m_{f} = \frac{2 p_{si}}{c}, l_{f} = \frac{m_{e} a_{0}}{m_{f}} α_{p}, t_{f} = \frac{l_{f}}{c}

The printed electron mass and Bohr radius are multiplied as printed. Their uncertainty propagation follows their shared CODATA dependency, rather than treating them as independent measurements.

m_{e} a_{0} = \frac{h}{2 π α_{c} c}

The exact SI input is h = 6.62607015 × 10⁻³⁴ J s. The scale chain therefore already contains the conventional reduced Planck constant through the electron-mass–Bohr-radius identity. That dependency explains why the blackbody recovery cannot be counted as an independent determination of the same action. The Online Resources collection contains the supplementary framework materials, including Suppl1.xlsx and Suppl2.xlsx.

The elementary-charge demarcation

The preprint’s charge construction supplies a default electromagnetic count when no phenomenon-specific selection takes precedence. Let e_f, e_p and e_c be fundamental, intermediate and conventional charge quantities in C. The charge-valued residual is n_e, also in C. The dimensionless residual Q_nθ is obtained by removing the nearest integer from the geometric coefficient product. It differs from the dimensionless angular count n_θ, which is the nearest integer to half the selected electromagnetic count and is 42 for the demarcations considered here.

The first charge expression is explicitly a relation between SI numerical coefficients. Dividing out the stated units is essential.

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

The intermediate charge follows from the adopted cubic construction.

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

Let γ_F be the fundamental electromagnetic realization parameter, in C². The revised charge relation is

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

The separate charge conversion is required to connect this parameter to the conventional input. On the positive branch the combined result is

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

Let K denote the dimensionless product of remainder and radial count. The charge conversion fixes it, while the triangle fixes the remainder.

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

For a conventional charge strictly between zero and the intermediate charge, this product lies strictly between zero and one half and increases with positive radial count. The branch therefore has one positive real root. Squaring after enforcing that domain gives

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

Using the exact SI elementary charge input 1.602176634 × 10⁻¹⁹ C with the specified 2022 chain gives the default n_L=84.60161(22). Its reported uncertainty is conditional on the full dependent input chain. The solved count reproduces the charge input by construction; the electromagnetic realization parameter is then derived, rather than an additional independently measured input. The blackbody count remains the relevant selection for the action calibration discussed above.

Precision, uncertainty and what can be tested

The preprint’s source-conditioned fundamental measures are 1.61619991147(12) × 10⁻³⁵ m, 5.39106261127(41) × 10⁻⁴⁴ s and 2.17643254057(16) × 10⁻⁸ kg. Those parenthetical forms are retained as reported outputs of that calculation. They describe propagated source uncertainty under its assumptions. They are not independent observations of three microscopic units and do not include an assessed uncertainty for the physical validity of MQ.

A separate supplied uncertainty realization assigns the same length central value an absolute standard uncertainty of 3.96317756063 × 10⁻⁴³ m and the mass central value a standard uncertainty of 2.3883 × 10⁻¹⁶ kg. Rounding these supplied quantities consistently gives 1.616199911(40) × 10⁻³⁵ m and 2.176432541(24) × 10⁻⁸ kg. These are input-profile uncertainties, not independent measurements newly inferred from the Planck-form identities. With exact light speed, the time propagation is especially transparent. Let u denote standard uncertainty for the indicated quantity.

u (t_{f}) = \frac{u (l_{f})}{c}

Recalculation from those original decimal inputs gives 5.39106261(13) × 10⁻⁴⁴ s after rounding the standard uncertainty to two significant digits and the central value to the same decimal place. The unrounded standard uncertainty is 1.3219737371211653 × 10⁻⁵¹ s. This is a different uncertainty path from the preprint’s much smaller parenthetical time uncertainty. The two must not be presented as interchangeable estimates from a single budget.

Dependencies survive algebraic substitution

A derived gravitational quantity and a derived action can share the same underlying length, mass and coefficient inputs. Propagating them again as independent observations would discard their covariance and can misstate the uncertainty. The uncertainty budget must follow the original independent inputs through the full calculation. The same principle applies to the electron mass and Bohr radius, and to exact cancellation in a product or ratio. The JCGM uncertainty guides describe this covariance-aware treatment.

Nearest-integer operations also require branch checks. A vanishing derivative inside one interval does not establish zero probability of crossing its boundary. Extra arithmetic digits provide guard precision, not extra experimental information.

The preprint’s angular picture, action calibration and charge inversion each answer a different question. The picture illustrates geometry. The action and charge inversions test internal consistency with their inputs. An empirical test instead needs observables that do not merely reproduce those calibration inputs, with a documented measurement and model comparison. The cited work on conserved quantities at null infinity concerns a different mathematical setting and supplies no derivation of the SI coefficient convention adopted here.

The significance of the derivation

The MQ proposal joins a discrete geometric remainder to gravitational realization, joins momentum to action, and uses their common fundamental measures to resolve the Planck-form powers. Mass participates from the gravitational step onward. It does not have to be appended as a disconnected dimensional quantity after length and time have been chosen.

The resulting account is more informative than dimensional analysis alone because it identifies where count geometry enters, where a physical realization is assumed, how finite-count normalization differs from its upper-count limit, and which numerical inputs establish the SI scale. Its merit must be assessed at those respective levels. Algebra establishes consistency under the premises; comparison with independent physical observations tests whether those premises describe nature.

The central result is a shared structure for fundamental length, time and mass. Conventional Planck expressions are recovered in form when conventional constants are used. MQ’s additional claim is that a discrete geometric construction explains the relationships and their finite-count realization. Preserving that distinction gives the derivation its clearest scientific meaning.

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