Appendix A — Closed-Region Inventories

Closed-Region inventory ratios expressed relative to the smallest closure-equivalent unit.

Inventory Reference Scale

Closed-Region inventory ratios are expressed relative to the smallest closed compound-Region inventory unit. The electron Region defines the reference inventory scale.

\[ \mu_e \equiv 1 \]

Proton and neutron inventories are then expressed as ratios against this smallest closure-equivalent reference.

Electron Region Inventory

For the electron, the geometric mass factor is derived from the EZZZ closure geometry:

\[ m_f^{EZZZ} = 408\sqrt{3}+8 = 714.6767295 \]

With the SI energy bridge \(\Gamma = 1~\mathrm{eV}\), the corresponding electron energy is:

\[ E_e \equiv E_{EZZZ} = \Gamma \left( m_f^{EZZZ} \right)^2 = \left( 408\sqrt{3}+8 \right)^2 ~\mathrm{eV} \]
\[ E_e \approx 510{,}762.7855~\mathrm{eV} = 0.5107627855~\mathrm{MeV} \]

Proton Region Inventory

For the proton, the geometric mass factor is formed from the UUD mass term and the internal coherence completion term.

\[ m_p = m_p^{UUD} + m_{Z\Delta} + \Delta m_{\mathrm{int}} \]
\[ m_p^{UUD} = \sqrt{ \frac{\pi}{3} } \left( 51 \cdot 9 \right)L_0 = 30{,}061.2454 \]
\[ \Delta m_{\mathrm{int}} = 256g_{UUD} = 256(2.22474488) = 569.5347 \]
\[ m_p = 30{,}061.2454 + 0 + 569.5347 = 30{,}630.7801 \]

The projected proton energy is:

\[ E_p = \Gamma \left( 30{,}630.7801 \right)^2 \approx 938.2446895~\mathrm{MeV} \]

Neutron Region Inventory

For the neutron, the geometric mass factor is treated as a proton-electron close-coupled Region state with an additional internal closure burden.

\[ m_n \equiv m_p + m_e + m_{n,\mathrm{int}} \]
\[ m_n = 30{,}650.3113 + 0 + 20.8 = 30{,}671.1113 \]
\[ E_n = \Gamma \left( 30{,}671.1113 \right)^2 = 940.7170684~\mathrm{MeV} \]

The electron contribution is not counted as an added external mass-factor term in this expression because the electron Region is treated as an embedded phase participant inside the proton envelope. The additional contribution is the residual internal closure burden of the composite neutron state.

Closed-Region Inventory Units

Using the EOTU-derived energy values, each closed Region can be expressed as an inventory quantity using the native light-propagation coefficient.

\[ E_e = 510{,}762.7855~\mathrm{eV} \]
\[ E_p = 938.2446895~\mathrm{MeV} \]
\[ E_n = 940.7170684~\mathrm{MeV} \]
\[ u_x = \frac{E_x}{c_{L_0}^{\,2}}, \qquad c_{L_0} = 1.667800087 \times 10^{26}~L_0/s \]
\[ u_e = \frac{510{,}762.7855}{c_{L_0}^{\,2}} \approx 1.83625 \times 10^{-47} ~\mathrm{eV\,s^2} \]
\[ u_p = \frac{938{,}244{,}689.5}{c_{L_0}^{\,2}} \approx 3.37309 \times 10^{-44} ~\mathrm{eV\,s^2} \]
\[ u_n = \frac{940{,}717{,}068.4}{c_{L_0}^{\,2}} \approx 3.38198 \times 10^{-44} ~\mathrm{eV\,s^2} \]

EOTU-Derived Inventory Ratios

Using the electron closed-Region inventory as the dimensionless reference scale, the EOTU-derived inventory ratios are:

\[ \mu_e = 1 \]
\[ \mu_p = \frac{938.2446895}{0.5107627855} \approx 1836.9480 \]
\[ \mu_n = \frac{940.7170684}{0.5107627855} \approx 1841.7886 \]
\[ \mu_e = 1, \qquad \mu_p \approx 1836.9480, \qquad \mu_n \approx 1841.7886 \]

Measured Comparison Ratios

For measured comparison, the CODATA/NIST mass-energy values are expressed using the same electron-normalized inventory structure.

\[ m_ec^2 = 0.51099895069~\mathrm{MeV} \]
\[ m_pc^2 = 938.27208943~\mathrm{MeV} \]
\[ m_nc^2 = 939.56542194~\mathrm{MeV} \]

Using the electron mass-energy as the dimensionless comparison scale:

\[ \mu_e = 1, \qquad \mu_p = \frac{m_pc^2}{m_ec^2}, \qquad \mu_n = \frac{m_nc^2}{m_ec^2} \]
\[ \mu_p = \frac{938.27208943}{0.51099895069} \approx 1836.1527 \]
\[ \mu_n = \frac{939.56542194}{0.51099895069} \approx 1838.6837 \]
\[ \mu_e = 1, \qquad \mu_p \approx 1836.15, \qquad \mu_n \approx 1838.68 \]

Inventory Ratio Summary

Region EOTU-Derived Ratio Measured Comparison Ratio Role
Electron \(\mu_e = 1\) \(\mu_e = 1\) Smallest closure-equivalent reference unit.
Proton \(\mu_p \approx 1836.9480\) \(\mu_p \approx 1836.15\) Compact baryonic Region inventory.
Neutron \(\mu_n \approx 1841.7886\) \(\mu_n \approx 1838.68\) Proton-electron close-coupled composite inventory.