§4 — Halo Inventory Profile

Excess enclosed curvature inventory required by observed motion beyond compact baryonic inventory.

Halo Contribution

The halo contribution is defined as the excess enclosed inventory implied by observed motion after the compact baryonic inventory has been counted. The most direct measured quantity is orbital velocity, so the required enclosed inventory is evaluated from the motion relation.

\[ v^2(r) = \frac{C_\mu I(r)}{r} \]

Solving for the observed-motion inventory requirement gives:

\[ I_{\mathrm{req}}(r) = \frac{v_{\mathrm{obs}}^{2}(r)\,r}{C_\mu} \]

The baryonic inventory enclosed within the same radius is:

\[ I_b(r) \]

The halo inventory requirement is therefore:

\[ I_h(r) = I_{\mathrm{req}}(r) - I_b(r) = \frac{v_{\mathrm{obs}}^{2}(r)\,r}{C_\mu} - I_b(r) \]

Baryonic-Only Velocity

The baryonic-only velocity at the same sampled radius is:

\[ v_b^2(r) = \frac{C_\mu I_b(r)}{r} \]

This gives the direct comparison between observed motion and compact baryonic inventory.

\[ \chi_h(r) = \frac{I_h(r)}{I_b(r)} \]

Substituting the velocity forms gives:

\[ \chi_h(r) = \frac{v_{\mathrm{obs}}^{2}(r)} {v_b^{2}(r)} - 1 \]

Observed motion does not require a positive halo contribution when:

\[ \chi_h(r) \le 0 \]

Observed motion requires a positive halo contribution when:

\[ \chi_h(r) > 0 \]

Extended Halo Inventory

Closed-Region inventory defines the compact baryonic curvature source:

\[ \mu_A = C_\mu I_A \]

At distances external to a compact body, the enclosed inventory remains effectively constant:

\[ I_{\mathrm{enc}}(r) \approx I_A \]

The orbital velocity relation is therefore:

\[ v^2(r) = \frac{C_\mu I_A}{r} \]

For a compact source:

\[ v(r) \propto r^{-1/2} \]

Extended systems are governed by the enclosed inventory profile:

\[ I_{\mathrm{enc}}(r) = I_{b,\mathrm{enc}}(r) + I_{h,\mathrm{enc}}(r) \]

with orbital velocity:

\[ v^2(r) = \frac{C_\mu I_{\mathrm{enc}}(r)}{r} \]

Radial Inventory Profile

If the enclosed inventory follows a radial power relation,

\[ I_{\mathrm{enc}}(r) \propto r^n \]

then:

\[ v(r) \propto r^{(n-1)/2} \]

A compact source has \(n = 0\), producing a declining orbital curve. A flat galactic rotation curve has \(n = 1\), requiring enclosed inventory to grow approximately linearly with radius. Rising curves require \(n > 1\), while slowly declining outer curves require \(0 < n < 1\).

Inventory Profile Condition Velocity Behavior
Compact source \(n = 0\) \(v(r) \propto r^{-1/2}\)
Flat outer rotation curve \(n = 1\) \(v(r) \propto r^{0}\)
Rising curve \(n > 1\) \(v(r)\) increases with radius
Slowly declining extended curve \(0 < n < 1\) \(v(r)\) decreases more slowly than compact-source falloff

Halo Inventory as an Enclosed Profile

The halo inventory is not a fixed multiplier applied to a compact body. It is an extended radial curvature inventory whose effect depends on the enclosed inventory profile of the system being sampled.

\[ I_{h,\mathrm{enc}}(r) = I_{\mathrm{enc}}(r) - I_{b,\mathrm{enc}}(r) \]

This form applies to galactic rotation curves, dwarf-galaxy dynamics, cluster-scale motion, and lensing systems where the observed curvature exceeds the compact baryonic inventory.