Properties of Atoms

Atoms provide an especially useful setting in which to examine what changes when continuous physical expressions are reconstructed within Measurement Quantization. Conventional physics describes quantities such as elementary charge, the fine-structure constant, electron mass, and the Bohr radius through experimentally established relations and measured or defined constants. MQ does not replace those empirical descriptions. Instead, it asks whether the same quantities can be expressed through discrete counts, fundamental measures, and the mapping by which count relations are physically realized.

This distinction begins with the MQ treatment of frames of reference. The Internal Frame contains the discrete count relations underlying physical measure. These relations are realized through the Frames mapping in the System Frame, where dimensional quantities become physically observable. The distinction is therefore not simply between an observer and an observed object. It distinguishes the discrete relational description of measure from its continuous physical realization.

The relationship between these frames is central to the MQ description of atomic properties. A quantity can have an Internal Frame representation and a corresponding System Frame realization. When an underlying continuous relation does not coincide exactly with an admissible whole count, the resulting difference is retained by MQ as a Diophantine realization residual. The subsequent mapping of discrete count structure into a continuous observable also introduces the metric differential. At finite count separation, the associated geometric normalization is expressed through the Informativity differential.

The Fine-Structure Constant

The fine-structure constant $\alpha$ is a dimensionless physical constant characterizing the strength of electromagnetic interaction. In MQ, its inverse is especially useful because the fine-structure construction can be followed from a native Internal Frame expression through successive System Frame realizations. The native relation is ${\alpha }_f^{-1}=42{\theta }_{\mathrm{si}}$, where ${\alpha }_f^{-1}$ is the fundamental inverse fine-structure quantity and ${\theta }_{\mathrm{si}}$ is the invariant MQ scalar. The factor 42 is established in the MQ construction from one-half of the electromagnetic interaction count rather than inferred from the target fine-structure value at this stage.

Because $42{\theta }_{\mathrm{si}}$ does not coincide exactly with a whole count, MQ retains the residual $Q_{n\theta }=42{\theta }_{\mathrm{si}}-⌊42{\theta }_{\mathrm{si}}⌉$. The mixed floor-ceiling operator denotes realization to the nearest admissible whole count and replaces the explicit rounding notation used in earlier MQ treatments. Adding this residual gives the Planck-like System Frame quantity ${\alpha }_p^{-1}=84{\theta }_{\mathrm{si}}-⌊42{\theta }_{\mathrm{si}}⌉$.

The Planck-like form is not yet the classically realized fine-structure constant. At the electromagnetic demarcation, the interaction-length count is $n_L=\frac{276}{{\theta }_{\mathrm{si}}}$. The corresponding residual product is $Q_L{n}_L=n_L\left(\sqrt{1+n_L^2}-n_L\right)$, where $Q_L$ is the finite-count residual associated with that count. The Internal Frame to System Frame differential is ${\Delta }_{\mathrm{int}-\mathrm{sys}}={\alpha }_p^{-1}\left(1-2Q_L{n}_L\right)$.

Applying this finite-count correction produces the principal classically realized relation:

The two corrections have distinct roles. The whole-count residual transforms the native Internal Frame expression into its Planck-like realization. The finite-count normalization associated with the Informativity differential then maps that realization to the continuously expressed inverse fine-structure quantity observed in the System Frame.

Elementary Charge

Elementary charge provides a dimensional application of the same realization structure. In the International System of Units, the elementary charge $e$ is defined exactly as $1.602176634\times {10}^{-19}\mathrm{C}$. MQ distinguishes this SI-defined observable from three successive representations used in its derivation: the fundamental charge $e_f$, the Planck-like charge $e_p$, and the classically realized charge $e_c$.

The fundamental charge is $e_f=\frac{l_f{\theta }_{\mathrm{si}}^2}{m_f^2\left({\theta }_{\mathrm{si}}-1\right)}$, where $l_f$ and $m_f$ are the fundamental length and mass measures. Using the MQ fundamental relations, this is equivalently $e_f=\frac{l_f{c}^2}{4\left({\theta }_{\mathrm{si}}-1\right)}$, yielding $e_f=1.60512538253\left(81\right)\times {10}^{-19}\mathrm{C}$.

This is the fundamental representation rather than the observable elementary charge. MQ next applies the charge-count realization ratio $r_{\mathrm{q41}}$. It is defined by $r_{\mathrm{q41}}=\frac{\left(n_{\theta }-1\right){\theta }_{\mathrm{si}}}{⌊\left(n_{\theta }-1\right){\theta }_{\mathrm{si}}⌉}$, where $n_{\theta }$ is the applicable charge-count parameter. Its value is $r_{\mathrm{q41}}=0.998197579719\left(51\right)$, giving $e_p=r_{\mathrm{q41}}{e}_f=1.60223227199\left(59\right)\times {10}^{-19}\mathrm{C}$.

The remaining finite-count correction applies the same normalization factor encountered in the fine-structure construction. The classically realized charge is the principal result:

The resulting MQ classical realization is $e_c=1.6021763105\left(70\right)\times {10}^{-19}\mathrm{C}$. This value is not an independent measurement of elementary charge. It is the classically realized value produced by the MQ construction. Comparison with the exact SI value tests the numerical closure of that construction subject to the inputs and dependencies used to establish its parameters.

Atomic Scale

The connection to atomic structure follows through the Bohr radius $a_0$, the atomic unit of length associated with the characteristic ground-state scale of hydrogen. Within MQ, the fundamental length is related to electron mass $m_e$, the Bohr radius, the Planck-like fine-structure quantity ${\alpha }_p$, and fundamental mass $m_f$.

The MQ realization gives $l_f=1.616199912\left(18\right)\times {10}^{-35}\mathrm{m}$. The atomic scale is therefore connected to the same fundamental-measure system used in the electromagnetic construction. MQ records the dependence compactly as $a_0=f\left({\theta }_{\mathrm{si}}\right)$, where $f$ denotes the established functional dependence rather than a new independent physical parameter.

A Common Realization Structure

The significance of the atomic construction is that the fine-structure constant, elementary charge, and atomic length scale are not treated as isolated numerical coincidences. They are connected through the same ${\theta }_{\mathrm{si}}$, fundamental measures, count relations, and Frames mapping. The same framework is subsequently carried into gravitational quantities, including the gravitational constant.

This common dependence also determines how agreement should be interpreted. When a measured quantity is used upstream to determine an MQ parameter and is then reconstructed from that parameter, the result is an internal consistency or closure test. It is not an independent prediction. Stronger empirical tests arise when the established MQ structure is carried into observables that were not used to determine the parameters entering the calculation.

The properties of atoms therefore occupy an important position in Measurement Quantization. They connect experimentally established electromagnetic and atomic quantities to the discrete realization structure proposed by MQ. The central question is not whether conventional atomic physics should be replaced. It is whether the observed quantities can also be represented consistently as System Frame realizations of conserved count relations in the Internal Frame, and whether that same realization structure continues to hold when applied to independently measured phenomena.