CONNECTING THE NEWTON AND PLANCK CONSTANTS

Measurement Quantization (MQ) identifies a direct relationship between Newton's gravitational constant G and the reduced Planck constant ℏ. Although the two constants conventionally enter physics in very different regimes, G in gravitation and ℏ in quantum mechanics, MQ derives both from the same underlying system of fundamental measures.

The central MQ relation is

4 G θ_{s i}^{2} = ℏ c^{3}

MQ NOTATION AND PHYSICAL REALIZATION

Equations on this page follow conventional mathematical typesetting, so symbols such as G and ℏ appear in italics. MQ also uses typography to distinguish a specified physical realization when that distinction is being discussed explicitly. In that notation, an italic symbol identifies a measure at a finite physical demarcation, while an upright symbol identifies the corresponding upper-count-bound realization. Thus G and ℏ are used normally in the equations here, while upright G and ℏ are reserved for narrative passages that explicitly demonstrate the upper-count notation. The physical distinction remains the measurement demarcation, not the default appearance of a variable in an equation.

where c is the speed of light and θ_si is the fundamental constant. Here G and ℏ are upright because the expression resolves their upper-count-bound values. The relation follows directly from the MQ expressions $G = \frac{c^{3} l_{f}}{2 θ_{s i}}$ and $ℏ = 2 θ_{s i} l_{f}$ , where l_f is the fundamental measure of length.

The relationship is therefore more than a dimensional comparison within MQ. The gravitational constant and reduced Planck constant are related through l_f, c and θ_si. Together with the fundamental measures of mass m_f and time t_f, these quantities form the common measurement structure from which MQ develops its descriptions of the physical constants. The role of θ_si is developed further on the Fundamental Constant page.

THE INFORMATIVITY DIFFERENTIAL

The Newton-Planck relationship depends on an important distinction between count structure and physical realization. In the Internal Frame, physical relations are represented through discrete counts of the fundamental measures. Through the Frames mapping, conserved count structure is realized in the System Frame, where dimensional quantities become physically observable. This framework is developed on the Frames of Reference page.

At finite count, the mapping introduces a geometric correction associated with discrete measure. MQ calls this correction the Informativity differential. It is distinct from Lorentz contraction associated with relative motion and from gravitational effects described by General Relativity.

For the length relation, Q_L denotes the fractional count residual and n_L denotes the corresponding length count. Their exact relation is

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

As n_L increases toward its upper count limit, Q_L decreases. The product 2 Q_Ln_L consequently approaches unity. The finite-count contribution therefore approaches its limiting form as the upper count bound is approached.

PHYSICAL MEANING

The finite-count correction is not introduced as an adjustable parameter. It follows from the discrete count relation itself. Its influence recedes toward the limiting form as the realized count increases.

PLANCK FORM AND PHYSICAL REALIZATION

This distinction is important when G and ℏ are combined in Planck-unit expressions. Conventional Planck units are dimensional combinations of c, G and ℏ. MQ introduces an additional distinction between a Planck-form quantity and the corresponding quantity after the count-dependent Frames mapping has been taken into account.

Accordingly, MQ does not propose that the SI definition of the conventional reduced Planck constant changes. Since the 2019 revision of the SI, the Planck constant h has the exact value 6.62607015 × 10⁻³⁴ J s, and the conventional reduced Planck constant ℏ = h/(2 π) follows from that definition. MQ instead distinguishes that conventional quantity from the corresponding quantity resolved at another measurement demarcation.

The same distinction is applied to G. MQ resolves a value at the electromagnetic demarcation and a different value at the upper count bound. The electromagnetic-demarcation value is

G_{E M} = 6.6738448362 (53) \times 10^{- 11} m^{3} {kg}^{- 1} s^{- 2}

whereas the MQ upper-count value is

G = 6.6740779428 (56) \times 10^{- 11} m^{3} {kg}^{- 1} s^{- 2}

The separation between these MQ realizations is 34.9284 ppm. This is an MQ result, not a CODATA or experimentally established division of G into two physical constants. The distinction and its experimental implications are examined in detail on the Gravitational Constant page.

WHY THE GRAVITATIONAL CONSTANT MATTERS

The experimental status of G makes this distinction testable in a way that was not available when the original Newton-Planck discussion was written.

The 2022 CODATA adjustment is based on 16 determinations of G. Because those measurements are mutually inconsistent at the level implied by their originally reported uncertainties, CODATA multiplied their uncertainties by a common expansion factor of 3.9 before determining the recommended value. The resulting CODATA value remains G = 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻². The dispersion is an established experimental problem. It does not by itself demonstrate the MQ interpretation.

A particularly important development occurred in 2026 when Schlamminger and colleagues reported the first replication of a high-precision determination of G using the BIPM torsion balance at NIST. The apparatus measures the gravitational torque by two methods, free deflection and electrostatic servo. The experimenters obtained a combined value

G_{N I S T} = (6.67387 \pm 0.00038) \times 10^{- 11} m^{3} {kg}^{- 1} s^{- 2}

with a relative standard uncertainty of 57 ppm. The result is approximately 235 ppm below the earlier BIPM determination made with the same underlying torsion-balance apparatus after its transfer and modification at NIST.

The peer-reviewed experiment does not attribute this discrepancy to MQ. Schlamminger and colleagues emphasize the continuing possibility of unrecognized instrumental effects, and their work documents extensive changes, corrections and systematic investigations undertaken during the replication. The result is nevertheless important to MQ because the framework had already distinguished finite electromagnetic and upper-count gravitational realizations of G before the NIST result was published.

The apparatus-specific identification of free-deflection and electrostatic-servo measurements with particular MQ realizations is a subsequent analysis of the experimental configuration. It should therefore be distinguished from the earlier MQ prediction of distinct gravitational realizations. The quantitative mode analysis, covariance treatment and remaining systematic uncertainties are presented on the Gravitational Constant page rather than repeated here.

INTERPRETIVE CAUTION

The 2026 experiment does not attribute its result to MQ. The apparatus-specific association of the two measurement modes with particular MQ realizations is a subsequent analysis and should remain distinct from the earlier MQ prediction of distinct gravitational realizations.

RESOLVING THE NEWTON AND PLANCK CONSTANTS

The same count-dependent structure applies to the reduced Planck constant. MQ writes the finite-count relation as

ℏ = 2 Q_{L} n_{L} ℏ

where ℏ denotes the reduced Planck quantity realized at a finite measurement demarcation and ℏ denotes the corresponding upper-count-bound realization. Because 2 Q_Ln_L approaches unity at the upper count bound, the finite-count value approaches the upper-count value as n_L increases.

MQ presently resolves the upper-count reduced Planck constant as

ℏ = 1.054534984 (26) \times 10^{- 34} m^{2} kg s^{- 1}

For comparison, the conventional reduced Planck constant derived from the exact SI Planck constant is

ℏ = 1.054571817 \dots \times 10^{- 34} J s

These are not presented by MQ as two experimental determinations of the same conventional SI constant. The first is the MQ upper-count realization. The second is the conventional reduced Planck constant used in quantum mechanics. Their distinction follows from the MQ treatment of measurement demarcation through the Informativity differential.

At the upper count bound, MQ therefore writes the Newton and Planck quantities using the same underlying length measure

G = \frac{c^{3} l_{f}}{2 θ_{s i}}

ℏ = 2 θ_{s i} l_{f}

Eliminating l_f between these two expressions gives the principal relation presented at the beginning of this discussion

4 G θ_{s i}^{2} = ℏ c^{3}

Within MQ, the relationship means that the gravitational constant and reduced Planck constant are not independent at the upper count bound. Both are realizations of the same underlying measurement structure.

A COMMON STRUCTURE BENEATH THE PHYSICAL CONSTANTS

The Newton-Planck relationship is one instance of a broader MQ program in which physical constants conventionally introduced as independent quantities are expressed through the fundamental measures, measurement counts and θ_si. The same framework enters MQ treatments of the Fine Structure Constant, Fundamental Constant, Fundamental Measures, Frames of Reference and Gravitational Constant.

The significance of the Newton-Planck relation is therefore structural. MQ connects a quantity governing gravitational coupling with a quantity defining the quantum scale through a shared discrete measurement geometry. The experimentally established dispersion among measurements of G, together with the 2026 BIPM/NIST replication, provides a means by which aspects of that interpretation can be compared with measurement rather than remaining solely an algebraic consequence of the framework.

CURRENT NOMENCLATURE

Earlier MQ publications use notation and terminology that have since been superseded. Current work uses Q_L for the fractional length-count residual and n_L for the corresponding length count. The former n_Lr notation is no longer used.

Likewise, current MQ descriptions use the Internal Frame, System Frame and Frames mapping. Older references to a Target Frame or Measurement Frame should be interpreted through the current definitions rather than carried forward into the present nomenclature.

AIP Advances
8: Resolving the Hubble tension 68.261 and 73.510 km s Mpc

International Journal of Geometric Methods in Modern Physics
7: Measurement Quantization

Journal of High Energy Physics, Gravitation and Cosmology
6: Physically Significant Units of Measure
5: Measurement Quantization Unites Classical and Quantum Physics
4: Quantum Model of Gravity Unifies Relativistic Effects, Describes Inflation/Expansion Transition, Matches CMB Data
3: Measurement Quantization Accounts for Galactic Rotational Velocities and Obviates Dark Matter
2: Measurement Quantization Describes History of Universe -
Quantum Inflation, Transition to Expansion, CMB Power Spectrum

International Journal of Theoretical and Mathematical Physics
1: Measurement Quantization Describes the Physical Constants

The Newton and Planck constants