Curvature Definition

Curvature for the Smallest Region

At the closed-Region scale, curvature accounting is no longer performed by raw CPP count. CPPs remain the primitive intrinsic curvature sources, but once CPPs close into stable Regions, the externally sampled curvature source is closed-Region inventory.

The primitive CPP halo/core geometry is:

\[ r_H = 51L_0, \qquad r_C = 8L_0 \]

The curvature of the smallest closed compound-Region geometry is defined by:

\[ m_0 = \left[ 4\left(2\sqrt{3}\,r_H\right)+r_C \right] = \left[ 4\left(2\sqrt{3}\cdot 51\right)+8 \right] \approx 714.6767295 \]

After projection by the energy bridge \(\Gamma\), the smallest closure-equivalent energy is:

\[ E_0 = \Gamma m_0^2 = \Gamma \left[ 4\left(2\sqrt{3}\cdot 51\right)+8 \right]^2 = 510{,}762.7855~\mathrm{eV} \]

The smallest Region defines the reference closed-Region inventory unit:

\[ I_{\mu,0}=1\mu, \qquad 1\mu \equiv 1~\text{smallest closed compound-Region inventory unit} \]

Using the EOTU-derived closed-Region energy values, the derived inventory ratios are:

\[ \mu_e = 1, \qquad \mu_p \approx 1836.9480, \qquad \mu_n \approx 1841.7886 \]

The CODATA/NIST electron-normalized values are used only for SI comparison:

\[ \mu_e = 1, \qquad \mu_p \approx 1836.15, \qquad \mu_n \approx 1838.68 \]

Curvature Propagation Coefficient

For compact baryonic matter, the curvature-propagation coefficient is built from the smallest closure reach, the zero-curvature resistance ratio, and the baryonic carrier execution count.

\[ C_{\mu,L_0} \approx c_{L_0}^{\,2} \left[ R_{\mathrm{eff},0} \left( \frac{\bar{A}_{L_0}}{E_0} \right) \left( 8\lambda_{pp}\lambda_p \right) \right] \]

The baryonic carrier execution count is:

\[ 8\lambda_{pp}\lambda_p = 8(3{,}370{,}896)(1836) = 4.9511720448 \times 10^{10} \]

Substitution gives:

\[ C_{\mu,L_0} \approx \left(1.667800087 \times 10^{26}\right)^2 \left[ 816 \left( \frac{4.747213536 \times 10^{-48}} {510{,}762.7855} \right) \left( 4.9511720448 \times 10^{10} \right) \right] \]

\[ C_{\mu,L_0} \approx 1.0444933545 \times 10^{13} ~L_0^3\,\mu^{-1}\,s^{-2} \]

Expressed in SI comparison units:

\[ C_{\mu,SI} \approx 6.070856966 \times 10^{-41} ~\mathrm{m^3}\,\mu^{-1}\,\mathrm{s^{-2}} \]

Curvature as a Function of Distance

The curvature response sampled at distance is the curvature-propagation coefficient multiplied by closed-Region inventory and divided by the square of the sampled separation.

\[ C_{SI}(r_m) = C_{\mu,SI} \frac{I_\mu}{r_m^2} \]

\[ C_{L_0}(r_{L_0}) = C_{\mu,L_0} \frac{I_\mu}{r_{L_0}^{2}} \]

Curvature Propagation Length

The curvature-length response per closed-Region inventory unit is defined by separating the propagation coefficient into a fabric propagation term and an inventory-coupled curvature length.

\[ \ell_\mu = \frac{C_{\mu,pp}}{c^2} \approx 3.75507 \times 10^{-40} ~L_0\,\mu^{-1} \]

\[ C_\mu = c^2\ell_\mu \]

For a closed inventory \(I_\mu\), the associated curvature length is:

\[ L_c = \ell_\mu I_\mu \]

The external curvature response is therefore:

\[ C(r) = \frac{c^2L_c}{r^2} = \frac{c^2\ell_\mu I_\mu}{r^2} \]

Curvature Propagation Resistance

Closed-Region inventory first defines curvature length. Distance sampling of that curvature length through the fabric propagation term gives the observed second-derivative response.

\[ E_0 = 510{,}762.7855~\mathrm{eV} \]

\[ \bar{A}_{L_0} = 4.747213536 \times 10^{-48} ~\mathrm{eV}\,L_0 \]

The resistance ratio is therefore a native inverse-\(L_0\) scale:

\[ \mathcal{R}_A = \frac{E_0}{\bar{A}_{L_0}} \approx 1.07592 \times 10^{53} ~L_0^{-1} \]

Curvature-Propagation Comparison to \(G\)

The measured gravitational constant \(G\) is obtained from the interaction of compact bodies. In EOTU, compact body inventory reduces to closed-Region inventory, while the interaction carrier for compact baryonic matter reduces to the proton-proton carrier.

\[ G_{\mathrm{obs}} = 6.67430 \times 10^{-11} ~\mathrm{m^3\,kg^{-1}\,s^{-2}} \]

\[ G_{\mathrm{calc}} = \frac{C_{\mu,SI}}{m_{0,SI}} = \frac{6.070856966 \times 10^{-41}} {9.10517 \times 10^{-31}} \approx 6.66748 \times 10^{-11} ~\mathrm{m^3\,kg^{-1}\,s^{-2}} \]

The percent error is: