§2 — Single-Body Curvature

Closed-Region inventory for atoms, larger bodies, compact systems, and galactic carrier rotation.

Single-Body Curvature Framework

Within the EOTU framework, curvature is the fabric response produced by closed Regions embedded in the dormant corridor network. Each closed Region contributes finite curvature inventory, and larger structures carry the summed inventory of their constituent atoms.

At distances external to the body, the internal geometry is no longer resolved directly. The body acts as a single composite Region whose far-field curvature response is governed by total closed-Region inventory and inverse-square spatial spreading.

For an Atom

For an atom, the raw closed-Region inventory is the sum of the proton, electron, and neutron Region inventories.

\[ \mu_{\mathrm{atom,raw}} = Z\mu_p + Z\mu_e + N\mu_n \]

Nuclear binding reduces the externally counted bound inventory by the binding energy expressed in smallest closure-equivalent units.

\[ \mu_{\mathrm{atom,bound}} = \mu_{\mathrm{atom,raw}} - \frac{E_{\mathrm{bind}}}{E_0} \]

Here \(Z\) is the proton and electron count, \(N\) is the neutron count, \(E_{\mathrm{bind}}\) is the positive nuclear binding energy, and \(E_0\) is the smallest closure-equivalent energy.

For a Larger Body — Simple Form

For a larger body built from atom counts, the total closed-Region inventory is the sum of the bound atomic inventories across all participating atom types.

\[ I_\mu = \sum_i n_i \left( Z_i\mu_p + Z_i\mu_e + N_i\mu_n - \frac{E_{\mathrm{bind},i}}{E_0} \right) \]

In this expression, \(n_i\) is the number of atoms of type \(i\), \(Z_i\) is the proton and electron count, \(N_i\) is the neutron count, and \(E_{\mathrm{bind},i}\) is the positive nuclear binding energy of atom type \(i\).

For a Larger Body with Composition Fractions

For a larger body with measured composition fractions, curvature is described by the closed-Region inventory of the participating atoms and inverse-square propagation across separation.

For component \(i\), the atom count is:

\[ n_i = \frac{M_{\mathrm{body}}f_i}{A_i u} \]

The closed-Region inventory is therefore:

\[ I_\mu = \sum_i \frac{M_{\mathrm{body}}f_i}{A_i u} \left( Z_i\mu_p + Z_i\mu_e + N_i\mu_n - \frac{E_{\mathrm{bind},i}}{E_0} \right) \]

The external curvature source is:

\[ C_\mu I_\mu \]

This is the EOTU curvature source compared against the observed gravitational parameter:

\[ GM \]

Here \(A_i\) is the representative isotope mass in atomic mass units, \(u\) is the atomic mass unit, \(f_i\) is the mass fraction of component \(i\), and \(E_{\mathrm{bind},i}\) is the EOTU nuclear binding energy.

Compact Solar-System Regime

Compact systems such as the solar system remain in the compact-inventory regime. Their orbital curvature is described by the closed-Region inventory of the participating bodies and inverse-square propagation across separation.

Extended halo inventory becomes active only where the enclosed curvature inventory grows with radius, as in galactic systems.

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

In the compact regime, the enclosed inventory remains effectively fixed once the sampled distance is outside the body.

Galactic Carrier Rotation

The galactic carrier curvature scale is the EOTU carrier acceleration associated with compact Region inventory at the galaxy level. The carrier wavelength is built from the nucleon interaction carrier.

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

The carrier scale is written as:

\[ \lambda_G(p) = 4\lambda_k\lambda_{pp}^{\,p} \]

For galaxy rotation, the galaxy-level structural carrier order is:

\[ p = 7 \]

The galaxy-level carrier scale is therefore:

\[ \lambda_G = 4\lambda_k\lambda_{pp}^{\,7} \]

The corresponding galactic carrier curvature scale is:

\[ g' = \frac{c^2}{\lambda_G} = \frac{c^2}{4\lambda_k\lambda_{pp}^{\,7}} = 1.617195308 \times 10^{-10} ~\mathrm{m\,s^{-2}} \]

The nucleon interaction carrier \(\lambda_{pp}\) is a dimensionless carrier interaction count, not a direct spatial length. Resolved Region wavelengths are expressed in \(L_0\)-based Region geometry, while interaction carrier products project back onto physical propagation length through the King coherence length \(\lambda_k\).

Carrier-Regime Velocity Relation

In the galactic carrier regime, the flat-velocity relation is:

\[ v_{\mathrm{calc}}^{\,4} = C_\mu I_{\mathrm{compact}} g' \]

Equivalently:

\[ v_{\mathrm{calc}} = \left( C_\mu I_{\mathrm{compact}} g' \right)^{1/4} \]

Here \(C_\mu\) is the curvature-inventory coupling constant, and \(I_{\mathrm{compact}}\) is the compact Region inventory associated with stars and gas.

\[ I_{\mathrm{compact}} = I_\odot \left( \frac{M_{\mathrm{compact}}}{M_\odot} \right) \]
\[ M_{\mathrm{compact}} = M_\star + M_{\mathrm{gas}} \]

SPARC galaxy masses are reported in solar units, so the inventory conversion is evaluated through \(I_\odot\). The fixed carrier value \(g' = 1.6172 \times 10^{-10}~\mathrm{m\,s^{-2}}\) is applied as a galaxy-level carrier acceleration and is not adjusted by galaxy.

SPARC Flat-Velocity Comparison

Using the fixed galactic carrier value, the EOTU carrier relation predicts SPARC flat velocities without galaxy-by-galaxy adjustment of the carrier scale.

SPARC Filter N Median \(v_{\mathrm{calc}}/v_{\mathrm{flat}}\) Within ±5% Within ±10% Within ±20%
Q = 1 87 0.9868 27 / 87 = 31.0% 51 / 87 = 58.6% 76 / 87 = 87.4%
Q ≤ 2 129 1.0176 40 / 129 = 31.0% 70 / 129 = 54.3% 110 / 129 = 85.3%
All \(v_{\mathrm{flat}} > 0\) 135 1.0180 40 / 135 = 29.6% 71 / 135 = 52.6% 111 / 135 = 82.2%

A gas-participation test improves the Q = 1 result to 78 of 87 within ±20%, indicating that the remaining scatter is dominated by structural participation rather than by a change in the carrier scale.