EXTENDING THE PHYSICAL CONSTANTS

Measurement Quantization (MQ) describes physical observables through integer counts of fundamental measures and distinguishes the underlying count structure from its physical realization. In the Internal Frame, the primitive structure consists of discrete count relations. Through the Frames mapping, those relations are realized in the System Frame, where dimensional quantities such as distance, duration, mass, velocity, and acceleration become physically observable.

This distinction changes the role played by Planck-scale quantities. Conventionally, Planck length, mass, and time are constructed from combinations of $G$, $\hslash$, and $c$. MQ reverses that direction of explanation. Its fundamental measures are resolved through the count and mapping structure of the framework, after which familiar physical constants can be reconstructed from relationships among those measures.

A central quantity is the dimensionless invariant coefficient ${\theta }_{\mathrm{si}}$. Its angular and momentum realizations are separate physical quantities because equal numerical coefficients do not make quantities with different dimensions physically identical.

From Fine Structure to the Fundamental Measures

The fine-structure constant provides an empirical connection between atomic measurement and the MQ count structure. In the inverse construction, the measured inverse fine-structure constant fixes the physical solution for ${\theta }_{\mathrm{si}}$ once the Frames mapping, electromagnetic Fine Structure demarcation, and finite-count Informativity differential are included.

The momentum realization then provides the route to the fundamental mass through $m_f=\frac{2p_{\mathrm{si}}}{c}$, giving $m_f=\mathrm{2.17643253982(38)}\times {10}^{-8}\mathrm{kg}$.

The same count construction produces the Planck-equivalent inverse fine-structure realization:

This quantity is not the experimentally determined inverse fine-structure constant itself. It is an MQ realization produced by the discrete count relation. Maintaining that distinction is essential because the next step uses independently measured atomic quantities rather than treating the MQ result as an experimental measurement.

Using the electron mass $m_e$, Bohr radius $a_0$, Planck-equivalent fine-structure realization ${\alpha }_p$, and fundamental mass $m_f$, MQ resolves the fundamental length through $l_f=\frac{m_e{a}_0{\alpha }_p}{m_f}$, giving $l_f=\mathrm{1.6161999120(35)}\times {10}^{-35}\mathrm{m}$.

The result has a different logical status from the conventional CODATA Planck length. In MQ, $l_f$ occurs downstream of the count and fine-structure construction rather than being introduced through the conventional Planck-unit combination of $G$, $\hslash$, and $c$.

The exact defined value of the speed of light then relates fundamental length and time. The relation $t_f=\frac{l_f}{c}$ gives $t_f=\mathrm{5.3910626131(72)}\times {10}^{-44}\mathrm{s}$. The fundamental measures of mass, length, and time are consequently linked within a single derivational structure.

Recovering Gravitational and Quantum Constants

With $l_f$ and $m_f$ established, MQ reconstructs the gravitational constant. The upper-count realization is:

The upright $\mathrm{G}$ denotes the MQ upper-count realization. Finite-count realizations of $G$, including the electromagnetic Fine Structure demarcation realization, are distinct. The value above is a theoretical realization rather than an experimental measurement.

MQ also reconstructs the reduced Planck scale. In the dimensionally explicit upper-count formulation, ħ $=2l_f{p}_{\mathrm{si}}=\frac{l_f^2{m}_f}{t_f}$. The phase-action scale is also count-resolved, so its electromagnetic-demarcation and upper-count values are realizations of one count-dependent relation rather than two independently introduced constants.

Extending the Same Structure to Cosmology

The invariant structure is not restricted to microscopic quantities. MQ extends the same count relationships to normalized cosmological domains. These fixed domain constants are determined by ${\theta }_{\mathrm{si}}$ and are written upright rather than as italicized measured variables. The fundamental-domain partition is ${\mathrm{\Omega }}_{\mathrm{f}}=\frac{2}{{\theta }_{\mathrm{si}}^2}=\mathrm{18.7913577316(20)}\%$.

The visible-domain partition is ${\mathrm{\Omega }}_{\mathrm{vis}}=\frac{2}{{\theta }_{\mathrm{si}}({\theta }_{\mathrm{si}}^2+2)}=\mathrm{4.84883489533(53)}\%$, while the complementary unobserved-domain partition is ${\mathrm{\Omega }}_{\mathrm{uobs}}={\mathrm{\Omega }}_{\mathrm{obs}}-{\mathrm{\Omega }}_{\mathrm{vis}}=\mathrm{26.7887490004(13)}\%$.

A Connected System of Physical Constants

The broader result is not simply that individual numerical constants can be reproduced with additional digits. The important MQ claim is structural. The invariant coefficient ${\theta }_{\mathrm{si}}$, its dimensional realizations, the fundamental measures, the fine-structure relationship, gravitational coupling, reduced Planck scale, and cosmological-domain partitions participate in a connected derivational system.

The same structure extends to electromagnetic quantities including the coupling term $\gamma$, vacuum permittivity ${\epsilon }_0$, vacuum permeability ${\mu }_0$, and Coulomb's constant $k_e$. These relationships place electromagnetic and gravitational observables within the same fundamental-measure framework rather than treating their numerical constants as unrelated starting assumptions.

This construction does not remove the need for measurement. It increases the number of empirical comparisons available to the framework. A numerical agreement is informative only to the extent that the compared quantity was not independently inserted into the same derivation.

The distinction is especially important for $G$. The Newtonian constant of gravitation remains much less precisely determined than many other fundamental constants, making the current MQ treatment of the gravitational constant a sensitive point of comparison between count-resolved realizations and laboratory measurement.

The objective of extending the physical constants in MQ is consequently broader than numerical precision. It is to determine whether quantities conventionally introduced as separate constants can instead be connected through a smaller underlying system of fundamental measures, invariant count relations, and the Frames mapping. Every quantity derived without an additional adjustable parameter creates another point at which that proposed structure can be compared with observation.