§1.8 — Curvature

Curvature Departure

Within the EOTU framework, curvature is the fabric response to departure from the dormant-corridor mean amplitude. The dormant-corridor mean \(\bar{A}\) defines the zero-curvature reference level of the fabric. A lattice cell, transport packet, void, or closed Region contributes curvature only where it departs from this reference state.

\[ \Delta A = A-\bar{A} \]

This relation defines curvature as a departure from the dormant background rather than as an independent field. The fabric curvature response follows from the amplitude displacement and the closure geometry of the structure producing it.

Closed-Region Inventory

After stable Region formation, external curvature accounting is performed by closed-Region inventory. Raw CPP count is not the external curvature measure for compact matter. Once the electron, proton, and neutron have formed as closed Regions or close-coupled Region states, their external curvature contribution is expressed through inventory.

\[ I_{\mu} = \text{closed-Region inventory} \]

The smallest closure equivalent defines the reference inventory unit:

\[ \mu_0 = 1 \]

In the derived particle inventory, the electron corresponds to the smallest closure equivalent. The proton and neutron carry larger closed-Region inventory values. The neutron contributes inventory as a proton-electron close coupling and does not introduce a separate fundamental curvature carrier.

External Curvature Response

The external curvature response of compact matter is expressed by a curvature-propagation coefficient multiplied by the enclosed closed-Region inventory and sampled over distance.

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

Here \(I_{\mu}\) is the enclosed closed-Region inventory and \(r\) is the sampled separation. The inverse-square form appears in the far-field inventory limit because the same curvature inventory is distributed across expanding dormant-corridor shells.

Curvature Length

In native EOTU form, the propagation coefficient separates into a fabric propagation term and an inventory-coupled curvature length. Closed-Region inventory first defines a curvature length. Distance sampling of that curvature length through the fabric propagation term gives the observed second-derivative response.

\[ C_{\mu}=c^2\ell_{\mu} \] \[ L_c=\ell_{\mu}I_{\mu} \]

The same response may therefore be written as:

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

This form separates inventory, curvature length, propagation, and distance sampling. It keeps the response tied to closed-Region accounting rather than to a detached force constant.

Compact Baryonic Carrier

For compact baryonic matter, the carrier execution count is anchored on the baryonic portion, the proton-proton carrier, and one proton-scale carrier projection.

\[ 8\lambda_{pp}\lambda_p \]

The proton wavelength and proton-proton carrier are:

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

This gives the native compact curvature-propagation form:

\[ C_{\mu,L0} \approx c_{L0}^{\,2} \left[ R_{\mathrm{eff},0} \left( \frac{\bar{A}_{L0}}{E_0} \right) \left( 8\lambda_{pp}\lambda_p \right) \right] \]

In this expression, \(R_{\mathrm{eff},0}\) is the smallest closure reach, \(\bar{A}_{L0}/E_0\) is the zero-curvature resistance ratio, and \(8\lambda_{pp}\lambda_p\) is the compact baryonic carrier execution count.

Native Curvature Coefficient

The compact p-p curvature propagation coefficient in native length form is:

\[ C_{\mu,L0}^{pp} = 1.0444933545 \times 10^{13}\, L_0^3\,\mu^{-1}\,s^{-2} \]

The corresponding SI bridge form is:

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

The native coefficient expresses the curvature response in \(L_0\)-based units. The SI bridge form is used only for measured comparison.

Two-Body Curvature Relation

For two separated compact bodies, each body samples the far-field curvature response associated with the other. When separation is large compared with the resolved size of either body, the sampled response follows the same inventory-over-distance form.

\[ C_A(r_{AB}) = C_{\mu}\frac{I_{\mu,A}}{r_{AB}^{2}} \] \[ C_B(r_{AB}) = C_{\mu}\frac{I_{\mu,B}}{r_{AB}^{2}} \]

A coupled inventory-weighted curvature measure can also be written symmetrically:

\[ \mathcal{C}_{AB} = \frac{C_{\mu}I_{\mu,A}I_{\mu,B}}{r_{AB}^{2}} \] \[ \mathcal{C}_{AB} = \mathcal{C}_{BA} \]

This measure describes the shared curvature relation between two inventories. It should not be read as direct acceleration unless the sampled-body inventory is normalized out of the expression.

Scale Behavior

At small scales, curvature is resolved through Region geometry, wavelength coupling, corridor availability, and closure behavior. At large scales, the same fabric response is expressed through compact inventory curvature, orbital response, galactic carrier rotation, and extended halo inventory.

\[ \Delta A \rightarrow I_{\mu} \rightarrow L_c \rightarrow C(r) \]

The scale changes, but the accounting remains tied to departure from \(\bar{A}\), closed-Region inventory, effective reach, and carrier execution through the dormant-corridor fabric.

Summary

Curvature is the fabric response to departure from the dormant-corridor mean amplitude \(\bar{A}\). Closed Regions carry curvature through inventory \(I_{\mu}\), with the smallest closure equivalent defining the reference inventory unit \(\mu_0=1\).

The external compact-matter response is \(C(r)=C_{\mu}I_{\mu}/r^2\), with native compact p-p coefficient \(C_{\mu,L0}^{pp}=1.0444933545\times10^{13}\,L_0^3\mu^{-1}s^{-2}\). Across scales, curvature remains a Region-inventory response sampled through distance and transmitted through the dormant-corridor fabric.