Appendix X — Solar-System and Halo Tests

Composition-based curvature inventory checks for Earth, Sun, orbital curvature, proton-electron curvature, and halo inventory.

Appendix X Framework

This appendix evaluates curvature inventory against observed curvature-response quantities. In this form, \(C_\mu I_\mu\) replaces the standard gravitational parameter \(GM\), while \(I_\mu\) replaces mass as the closed-Region source inventory.

\[ C_\mu I_\mu \longleftrightarrow GM \]

X.1 — Earth Curvature Inventory

Using common whole-Earth mass fractions, the bound closed-Region inventory is evaluated from representative isotopes for the dominant composition components.

Component Representative Isotope \(Z\) \(N\) \(\mu_{\mathrm{atom,raw}}\) \(\mu_{\mathrm{atom,bound}}\)
Fe 56 26 30 103,040.3060 102,079.5838
O 16 8 8 29,437.8928 29,187.2430
Si 28 14 14 51,516.3124 51,054.3838
Mg 24 12 12 44,156.8392 43,768.9658
S 32 16 16 58,875.7856 58,343.8703
Ni 58 28 30 106,716.2020 105,725.5087
Ca 40 20 20 73,594.7320 72,924.1621
Al 27 13 14 49,678.3644 49,238.1310

Earth bound closed-Region inventory, normalized to full Earth mass:

\[ I_{\oplus,\mathrm{bound}} \approx 6.5662 \times 10^{54} \]
\[ C_\mu I_{\oplus,\mathrm{bound}} = 6.070856966 \times 10^{-41} \left( 6.5662 \times 10^{54} \right) \approx 3.9862461010 \times 10^{14} ~\mathrm{m^3\,s^{-2}} \]

Observed Earth gravitational parameter comparison:

\[ GM_\oplus \approx 3.9860043551 \times 10^{14} ~\mathrm{m^3\,s^{-2}} \]
\[ \%\Delta = \frac{ 3.9862461010 \times 10^{14} - 3.9860043551 \times 10^{14} }{ 3.9860043551 \times 10^{14} } \times 100 \approx 0.00606\% \]

The surface acceleration using the same Earth radius is:

\[ a_{\oplus,\mathrm{calc}} = \frac{ 3.9862461010 \times 10^{14} }{ \left( 6.371 \times 10^{6} \right)^2 } \approx 9.82085~\mathrm{m\,s^{-2}} \]
\[ a_{\oplus,\mathrm{obs}} = \frac{ 3.9860043551 \times 10^{14} }{ \left( 6.371 \times 10^{6} \right)^2 } \approx 9.8203~\mathrm{m\,s^{-2}} \]

Earth acceleration error is approximately \(+0.00606\%\).

X.2 — Sun Curvature Inventory

For the Sun, the surface mass-fraction structure is evaluated using hydrogen, helium, and dominant heavy-element components.

\[ X_\odot = 0.7438, \qquad Y_\odot = 0.2423, \qquad Z_\odot = 0.0139 \]
Component Representative Isotope \(Z\) \(N\) \(\mu_{\mathrm{atom,raw}}\) \(\mu_{\mathrm{atom,bound}}\)
H 1 1 0 1,837.9480 1,837.9480
He 4 2 2 7,359.4732 7,304.1350
O 16 8 8 29,437.8928 29,187.2430
C 12 6 6 22,078.4196 21,897.7565
Ne 20 10 10 36,797.3660 36,484.0538
Fe 56 26 30 103,040.3060 102,079.5838
Si 28 14 14 51,516.3124 51,054.3838
N 14 7 7 25,758.1562 25,552.0699
Mg 24 12 12 44,156.8392 43,768.9658
S 32 16 16 58,875.7856 58,343.8703

Sun bound closed-Region inventory, normalized to full solar mass:

\[ I_{\odot,\mathrm{bound}} \approx 2.1851 \times 10^{60} \]
\[ C_\mu I_\odot = 6.070856966 \times 10^{-41} \left( 2.1851 \times 10^{60} \right) \approx 1.3265429556 \times 10^{20} ~\mathrm{m^3\,s^{-2}} \]

Observed solar gravitational parameter:

\[ GM_\odot \approx 1.3271244004 \times 10^{20} ~\mathrm{m^3\,s^{-2}} \]
\[ \%\Delta = \frac{ 1.3265429556 \times 10^{20} - 1.3271244004 \times 10^{20} }{ 1.3271244004 \times 10^{20} } \times 100 \approx -0.04381\% \]

For solar acceleration at Earth distance:

\[ a_{\odot \rightarrow \oplus,\mathrm{obs}} = \frac{ 1.3271244004 \times 10^{20} }{ \left( 1.495978707 \times 10^{11} \right)^2 } \approx 0.00593008~\mathrm{m\,s^{-2}} \]
\[ a_{\odot \rightarrow \oplus,\mathrm{calc}} \approx 0.00592749~\mathrm{m\,s^{-2}} \]

Sun-to-Earth acceleration error is approximately \(-0.04381\%\).

X.3 — Orbital Curvature

This section tests the two-body orbital curvature form using the Earth-Sun system.

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

For Earth and Sun:

\[ I_\odot = 2.184118884997422 \times 10^{60}, \qquad I_\oplus = 6.563147988462755 \times 10^{54} \]
\[ C_\mu = 6.070856966 \times 10^{-41}, \qquad r_{\oplus\odot} = 149{,}597{,}870{,}700~\mathrm{m} \]

The individual curvature sources are:

\[ \mu_\odot = C_\mu I_\odot = 1.3259473348 \times 10^{20} \]
\[ \mu_\oplus = C_\mu I_\oplus = 3.9843932685 \times 10^{14} \]

The Sun-sourced acceleration at Earth distance is:

\[ a_{\oplus \leftarrow \odot} = \frac{\mu_\odot}{r_{\oplus\odot}^{2}} = 0.00592482395~\mathrm{m\,s^{-2}} \]

The Earth-sourced acceleration at Sun distance is:

\[ a_{\odot \leftarrow \oplus} = \frac{\mu_\oplus}{r_{\oplus\odot}^{2}} = 1.78037453 \times 10^{-8}~\mathrm{m\,s^{-2}} \]

The two-body relative curvature acceleration is:

\[ a_{\oplus\odot} = \frac{ C_\mu \left( I_\odot + I_\oplus \right) }{ r_{\oplus\odot}^{2} } = 0.00592484176~\mathrm{m\,s^{-2}} \]

Observed comparison:

\[ a_{\oplus\odot,\mathrm{obs}} = 0.00593010133088442~\mathrm{m\,s^{-2}} \]

Percent error:

\[ \%\Delta \approx -0.08869\% \]

The two-body orbital curvature form reproduces the Earth-Sun relative acceleration using the same closed-Region inventories and the same curvature-propagation coefficient used in the single-body Earth and Sun tests.

X.4 — Electron-to-Proton Curvature

This section tests the two-body curvature form at the atomic scale using the proton-electron separation of the first shell.

\[ S_1 = 29{,}447{,}706~L_0, \qquad 1L_0 = 1.797965855 \times 10^{-18}~\mathrm{m} \]
\[ r_{pe} = S_1 = 5.294596989607863 \times 10^{-11}~\mathrm{m} \]

For proton and electron:

\[ I_p = 1836.9480, \qquad I_e = 1, \qquad C_\mu = 6.070856966 \times 10^{-41} \]

The individual curvature sources are:

\[ \mu_p = C_\mu I_p = 1.1151848562 \times 10^{-37} \]
\[ \mu_e = C_\mu I_e = 6.070856966 \times 10^{-41} \]

The proton-sourced acceleration at first-shell distance is:

\[ a_{e \leftarrow p} = \frac{\mu_p}{r_{pe}^{2}} = 3.9781490096 \times 10^{-17} ~\mathrm{m\,s^{-2}} \]

The electron-sourced acceleration at first-shell distance is:

\[ a_{p \leftarrow e} = \frac{\mu_e}{r_{pe}^{2}} = 2.1656296257 \times 10^{-20} ~\mathrm{m\,s^{-2}} \]

The two-body relative curvature acceleration is:

\[ a_{pe} = \frac{ C_\mu \left( I_p + I_e \right) }{ r_{pe}^{2} } = 3.9803146392 \times 10^{-17} ~\mathrm{m\,s^{-2}} \]

The coupled curvature measure is:

\[ \mathcal{C}_{pe} = \frac{ C_\mu I_p I_e }{ r_{pe}^{2} } = 3.9781490096 \times 10^{-17} ~\mathrm{m\,s^{-2}} \]

Since the electron is the smallest closed-Region inventory unit, \(I_e = 1\), the coupled curvature measure equals the proton-sourced acceleration sampled by the electron.

\[ \mathcal{C}_{pe} = a_{e \leftarrow p} \]

The proton contribution dominates the two-body curvature relation at first-shell distance. The electron contribution changes the relative two-body curvature acceleration by:

\[ 2.1656296257 \times 10^{-20} ~\mathrm{m\,s^{-2}} \]
\[ \frac{ \Delta a }{ a_{e \leftarrow p} } \times 100 \approx 0.05444\% \]

X.5 — Extended Halo Inventory Ratio

This section tests whether the extended halo inventory form reproduces the real-world excess-curvature phenomenon observed in pressure-supported dwarf galaxies. In these systems, the visible stellar inventory is too small to account for the measured stellar velocity dispersion.

In EOTU terms, the same excess is treated as halo curvature inventory. The calculation compares the enclosed inventory required by the observed velocity dispersion against the visible baryonic inventory. The difference between the required enclosed inventory and the visible baryonic inventory defines the halo-only contribution.

X.5.1 — Solar-System Halo Ratio

Solar-system orbital motion does not require a positive halo correction.

\[ \Phi = \frac{ a_{\mathrm{obs}} }{ a_{\mathrm{calc}} } \approx 1.000887715 \]
\[ \chi_h = \Phi - 1 \approx 0.000887715 = 0.0887715\% \]

This residual is below the composition and rounding uncertainty of the compact-inventory estimate and is not treated as an extended halo requirement for the solar-system regime.

X.5.2 — Galactic Halo Contribution

Galactic rotation curves require a positive halo correction when the observed velocity exceeds the compact baryonic-only velocity at the same radius. The fixed carrier relation provides the direct velocity test, while the halo-inventory form identifies the excess enclosed inventory implied by observed motion.

The fixed carrier value was applied across the SPARC comparison set without galaxy-by-galaxy adjustment.

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%

X.5.3 — Special Case: Segue 1

Segue 1 is used as a pressure-supported dwarf-galaxy test case because its visible baryonic inventory is small relative to the enclosed dynamical inventory inferred from stellar velocity dispersion.

For a pressure-supported dwarf system, the enclosed curvature inventory is:

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

The pressure-supported halo factor is:

\[ \kappa = \frac{\lambda_p}{\lambda_e} = \frac{1836}{816} = 2.25 \]
\[ C_\mu = 6.070856966 \times 10^{-41} \]

Therefore:

\[ I_{\mathrm{enc}}(r) = \frac{\lambda_p}{\lambda_e} \frac{ \sigma^2 r }{ C_\mu } \]
\[ \sigma = 3.7~\mathrm{km\,s^{-1}} = 3700~\mathrm{m\,s^{-1}} \]
\[ r = 28~\mathrm{pc}, \qquad 1~\mathrm{pc} = 3.0856775814913673 \times 10^{16}~\mathrm{m} \]
\[ r = 8.639897228175828 \times 10^{17}~\mathrm{m} \]

The uncorrected enclosed inventory is:

\[ I_0(r) = \frac{ \sigma^2 r }{ C_\mu } = \frac{ (3700)^2 \left( 8.639897228175828 \times 10^{17} \right) }{ 6.070856966 \times 10^{-41} } \approx 1.94832778496 \times 10^{65} \]

Applying the pressure-supported factor:

\[ I_{\mathrm{enc}}(r) = 2.25 I_0(r) \approx 4.38373751616 \times 10^{65} \]

Using the solar inventory reference:

\[ I_\odot = 2.184118884997422 \times 10^{60} \]
\[ \frac{ I_{\mathrm{enc}}(r) }{ I_\odot } \approx 200{,}709.6567 \]

Using visible baryonic inventory:

\[ I_b \approx 1000I_\odot \]
\[ \frac{ I_{\mathrm{enc}}(r) }{ I_b } \approx 200.7096567 \]

Compared against scale 200:

\[ \%\Delta = \frac{ 200.7096567 - 200 }{ 200 } \times 100 \approx 0.35483\% \]

The halo-only ratio is:

\[ \frac{ I_h }{ I_b } = 200.7096567 - 1 = 199.7096567 \]

Compared against reference ratio 200:

\[ \%\Delta = \frac{ 199.7096567 - 200 }{ 200 } \times 100 \approx -0.14517\% \]