§2.6 — Curvature Coefficient

The compact p-p curvature coefficient defines the native propagation scale for closed-Region inventory curvature response.

Coefficient Role

The curvature coefficient \(C_{\mu}\) maps closed-Region inventory into external curvature response. It is used when compact matter is represented by its enclosed inventory rather than by unresolved CPP count.

\[ C(r) = C_{\mu}\frac{I_{\mu}}{r^2} \]

In this expression, \(I_{\mu}\) is the enclosed closed-Region inventory and \(r\) is the sampled separation. The coefficient \(C_{\mu}\) supplies the propagation scale connecting inventory to distance-sampled curvature response.

Native Compact p-p Coefficient

For compact baryonic matter, the p-p curvature propagation coefficient in native \(L_0\) form is:

\[ C_{\mu,L_0}^{pp} = 1.0444933545\times10^{13}\, L_0^3\,\mu^{-1}\,s^{-2} \]

This is the preferred native form for EOTU calculations because it keeps distance in lattice units and inventory in closed-Region units.

SI Bridge Form

The same coefficient may be expressed through the SI length bridge for measured comparison:

\[ C_{\mu}^{pp} \approx 6.070856966\times10^{-41}\, \mathrm{m^3}\,\mu^{-1}\,s^{-2} \]

The SI form is not the native construction. It is the measured-unit projection of the same curvature response scale.

Native-to-SI Conversion

The conversion follows from the \(L_0\)-to-meter bridge:

\[ 1\,L_0 = 1.797965855\times10^{-18}\,\mathrm{m} \]

Since the curvature coefficient carries cubic length units, the conversion applies as:

\[ C_{\mu}^{pp} = C_{\mu,L_0}^{pp} \left(1\,L_0\right)^3 \] \[ C_{\mu}^{pp} = \left(1.0444933545\times10^{13}\right) \left(1.797965855\times10^{-18}\right)^3 \] \[ C_{\mu}^{pp} \approx 6.070856966\times10^{-41}\, \mathrm{m^3}\,\mu^{-1}\,s^{-2} \]

This preserves the same coefficient while changing only the length unit used to display it.

Compact Carrier Basis

The compact baryonic carrier execution count is anchored on the baryonic portion \(8\), the proton-proton carrier \(\lambda_{pp}\), and one proton-scale carrier projection \(\lambda_p\).

\[ \lambda_p = 1836\,L_0 \] \[ \lambda_{pp} = \lambda_p^2 = 1836^2 = 3{,}370{,}896\,L_0 \] \[ 8\lambda_{pp}\lambda_p \]

This carrier count supplies the compact baryonic curvature channel used in the native coefficient construction.

Native Propagation Form

In native form, the compact curvature coefficient is expressed as a fabric propagation term multiplied by the curvature-length construction:

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

Here \(c_{L_0}\) is the native propagation coefficient, \(R_{\mathrm{eff},0}\) is the smallest closure reach, \(\bar{A}_{L_0}/E_0\) is the zero-curvature resistance ratio, and \(8\lambda_{pp}\lambda_p\) is the compact baryonic carrier execution count.

Use in Curvature Response

Once \(C_{\mu}\) is defined, compact curvature response can be evaluated directly from closed-Region inventory and sampled separation.

\[ I_{\mu} \rightarrow C_{\mu}I_{\mu} \rightarrow \frac{C_{\mu}I_{\mu}}{r^2} \]

This keeps large-scale curvature accounting tied to the same Region-inventory structure used by the post Freeze-Out framework. The coefficient is therefore a propagation bridge, not an independent replacement for Region geometry.

Summary

The compact p-p curvature coefficient is \(C_{\mu,L_0}^{pp}=1.0444933545\times10^{13}\,L_0^3\mu^{-1}s^{-2}\) in native \(L_0\) form.

Its SI bridge form is \(C_{\mu}^{pp}\approx6.070856966\times10^{-41}\,\mathrm{m^3}\mu^{-1}s^{-2}\). The native form is used for EOTU curvature calculations, while the SI form is used for measured comparison.