§3 — Two-Body Curvature

Orbital sampling and coupled inventory-weighted curvature between two closed-Region bodies.

Two-Body Curvature Framework

Two-body curvature separates into two related but distinct forms. In an orbital relation, each body samples the far-field curvature of the other. In a shared curvature interaction, both closed-Region inventories participate in a coupled curvature measure.

The orbital form produces a relative acceleration relation. The coupled form produces a symmetric inventory-weighted curvature measure and should not be read as direct acceleration unless one inventory factor is normalized or divided out.

Two Bodies Orbiting

For two separated bodies in orbital relation, each body samples the far-field curvature of the other. When the separation is large compared with the resolved size of either body, the relative orbital acceleration of the pair is represented by the summed closed-Region inventory.

The acceleration of each Region is sourced by the other Region's closed-Region inventory:

\[ a_{A\leftarrow B} = \frac{C_\mu I_B}{r_{AB}^{2}}, \qquad a_{B\leftarrow A} = \frac{C_\mu I_A}{r_{AB}^{2}} \]

Adding those individual responses gives the relative acceleration magnitude:

\[ a_{AB} = \frac{ C_\mu \left( I_A + I_B \right) }{ r_{AB}^{2} } \]

This is the EOTU equivalent of the standard two-body relative acceleration form:

\[ a_{AB} = \frac{ G \left( M_A + M_B \right) }{ r_{AB}^{2} } \]

In this relation, \(I_A\) and \(I_B\) are the closed-Region inventories of the two bodies, \(C_\mu\) is the curvature-propagation coefficient, and \(r_{AB}\) is the separation between the bodies.

The summed-inventory form applies when the physical question is orbital sampling: the relative acceleration of the pair across their shared separation.

Two Bodies Interacting

For two bodies in a shared curvature interaction, the relation is not only orbital sampling. Both closed-Region inventories participate in the coupled curvature measure.

\[ \mathcal{C}_{AB} = \frac{ C_\mu I_A I_B }{ r_{AB}^{2} } \]

This measure treats both inventories as active participants in the shared curvature relation. It is symmetric under exchange of the two bodies.

\[ \mathcal{C}_{AB} = \mathcal{C}_{BA} \]

Unlike the orbital acceleration form, \(\mathcal{C}_{AB}\) is a coupled inventory-weighted curvature measure. It should not be read as a direct second-derivative acceleration unless one inventory factor is normalized or divided out by the sampled body inventory.

\[ \frac{\mathcal{C}_{AB}}{I_A} = a_{A\leftarrow B}, \qquad \frac{\mathcal{C}_{AB}}{I_B} = a_{B\leftarrow A} \]

These normalizations recover the individual accelerations. Once an acceleration has been evaluated, an additional arbitrary Region-inertia multiplier or divisor is not part of the relation.

SI Force Comparison

Using the SI-equivalent mass bridge (M=m_{0,\mathrm{SI}}I), the closed-Region response has the same far-field inverse-square form as the standard gravitational force relation:

\[ M_A=m_{0,\mathrm{SI}}I_A, \qquad M_B=m_{0,\mathrm{SI}}I_B \] \[ F_G(r) = G_{\mathrm{EOTU}} \frac{M_A M_B}{r_{AB}^{2}} \]

This is a comparison bridge for the far-field curvature response. It is not a separate fitted interaction multiplier and does not define the complete local interaction of overlapping Regions.

Orbital Form Compared with Coupled Form

The orbital relation and the coupled relation answer different questions. The orbital form describes relative acceleration produced by summed source inventory. The coupled form describes the strength of a shared curvature interaction between two participating inventories.

Relation Expression Use
Orbital sampling \[ a_{AB} = \frac{ C_\mu \left( I_A + I_B \right) }{ r_{AB}^{2} } \] Relative orbital acceleration of a separated two-body system.
Coupled interaction \[ \mathcal{C}_{AB} = \frac{ C_\mu I_A I_B }{ r_{AB}^{2} } \] Symmetric inventory-weighted curvature measure for a shared interaction.

The difference is the inventory role. Orbital sampling uses the summed source inventory. Coupled interaction uses the product of the participating inventories.

Curvature Strength and Geometric Reach

Effective Region Radius describes a geometric influence envelope associated with Region closure. Gravitational strength instead follows from (C_\mu) and closed-Region inventory. These quantities are not interchangeable.

  • For the proton, (N_p^2=36) is a geometric reach index, not a gravitational attraction multiplier.
  • The product (N_p^2N_p^2=1296) does not determine proton-proton gravitational strength.
  • The proton inventory (I_p\approx1836.9480) is the source-strength quantity used in the far-field curvature response.
  • A geometric influence envelope is not automatically a hard cutoff of the inverse-square response.

Boundary of Applicability

These relations determine the inverse-square gravitational component between separated closed Regions. They do not establish the complete behavior of overlapping or closely approaching Regions. Local curvature response may also depend on closure geometry, phase compatibility, corridor response, and overlap constraints.

The far-field gravitational component remains attractive for positive closed-Region inventories. Additional short-range attraction, exclusion response, or repulsion requires a separately derived local interaction law. The complete near-field Region interaction remains an open mathematical derivation.