G Scaling Between Similarity Levels

1. Why G Must Scale Between Similarity Levels

The Dimensional Argument

The gravitational constant $G$ has dimensions:

$$[G] = \frac{L^3}{M \cdot T^2}$$

Between similarity levels, all three constituent dimensions change:

• Length ($L$): Atomic distances are vastly smaller than solar system distances
• Mass ($M$): Nucleon mass is vastly smaller than stellar mass
• Time ($T$): Atomic orbital periods are vastly shorter than planetary orbital periods

Since $G$ is built from length, mass, and time, its numerical value must change when expressed in the natural units of a different similarity level. This is not a physical assumption $-$ it is arithmetic. A quantity with dimensions cannot remain numerically unchanged when all of its constituent dimensions scale.

Analogy

The speed of light has a different numerical value in m/s versus km/hr. The physics doesn't change $-$ the number changes because the units change. The same principle applies to $G$ between similarity levels: the fundamental gravitational coupling may be universal, but its numerical expression in $SL_{-1}$ units differs from its expression in $SL_{0}$ units.

2. The Kepler Constraint

Deriving the Master Relationship

Kepler's Third Law governs orbital dynamics at every similarity level:

$$T^2 = \frac{4\pi^2 r^3}{GM}$$

This can be rearranged as:

$$T^2 \cdot G \cdot M = 4\pi^2 r^3$$

Between $SL_{0}$ (solar system) and $SL_{-1}$ (atomic), each quantity scales as a power of the distance scaling factor $k$:

Substituting into Kepler's law at $SL_{-1}$:

$$(T_0 / k^a)^2 \cdot (G_0 \cdot k^c) \cdot (M_0 / k^b) = 4\pi^2 (r_0 / k)^3$$

$$T_0^2 \cdot k^{-2a} \cdot G_0 \cdot k^c \cdot M_0 \cdot k^{-b} = 4\pi^2 r_0^3 \cdot k^{-3}$$

Since $T_0^2 \cdot G_0 \cdot M_0 = 4\pi^2 r_0^3$ at $SL_{0}$, dividing gives:

$$k^{-2a + c - b} = k^{-3}$$

Therefore:

$$\boxed{c = 2a + b - 3}$$

This is the Kepler constraint $-$ a rigid mathematical relationship between the four scaling exponents. Given any two, the third is determined. There is exactly one degree of freedom once distance scaling ($k$) is fixed.

Implications

• You cannot independently choose all four exponents
• If distance scales as $k$, only two of ($a$, $b$, $c$) are free
• The constraint holds regardless of the physical mechanism (gravitational shadowing, magnetic contributions, or any other force law that produces Keplerian orbits)

3. Scaling Exponents from Observation

Reference Objects

To determine the actual scaling exponents, we compare analogous objects between $SL_{0}$ and $SL_{-1}$:

Calculating the Exponents

Distance scaling factor:

$$k = \frac{r_{Oort}}{r_{Bohr}} = \frac{1.165 \times 10^{16}}{5.29 \times 10^{-11}} = 2.20 \times 10^{26}$$

Time scaling exponent ($a$):

$$\frac{T_0}{T_{-1}} = \frac{3.15 \times 10^7}{8.55 \times 10^{-16}} = 3.68 \times 10^{22}$$

$$k^a = 3.68 \times 10^{22} \quad \Rightarrow \quad a = \frac{\log(3.68 \times 10^{22})}{\log(2.20 \times 10^{26})} = \frac{22.57}{26.34} \approx 0.857$$

Mass scaling exponent ($b$):

$$\frac{M_0}{M_{-1}} = \frac{1.99 \times 10^{30}}{1.673 \times 10^{-27}} = 1.19 \times 10^{57}$$

$$k^b = 1.19 \times 10^{57} \quad \Rightarrow \quad b = \frac{\log(1.19 \times 10^{57})}{\log(2.20 \times 10^{26})} = \frac{57.08}{26.34} \approx 2.167$$

G scaling exponent ($c$) from the Kepler constraint:

$$c = 2a + b - 3 = 2(0.857) + 2.167 - 3 = 1.714 + 2.167 - 3 = 0.881$$

Comparison to empirical G ratio:

$$\frac{G_{-1}}{G_0} = \frac{3.81 \times 10^{13}}{6.674 \times 10^{-11}} = 5.71 \times 10^{23}$$

$$k^c = 5.71 \times 10^{23} \quad \Rightarrow \quad c_{empirical} = \frac{\log(5.71 \times 10^{23})}{\log(2.20 \times 10^{26})} = \frac{23.76}{26.34} \approx 0.902$$

The Kepler constraint predicts $c \approx 0.88$; the empirical value gives $c \approx 0.90$. The close agreement (within the uncertainty of reference object selection) confirms the framework is internally consistent.

4. The Complete Scaling Framework

Scaling Laws Between Similarity Levels

For the distance scaling factor $k$ between $SL_{0}$ and $SL_{-1}$:

Constraint: $c = 2a + b - 3$ (always satisfied)

Why the Exponents Are Not Simple Fractions

Unlike the idealized case ($k$, $k^{3/2}$, $k^3$), the actual exponents are not simple integer fractions. This reflects the physical reality that:

1. Nucleons are settled objects $-$ iron star analogs, not main-sequence star analogs. Their mass-to-size ratio differs from active stellar systems.
2. Magnetic forces contribute at $SL_{-1}$ $-$ planetron orbits are governed by gravity plus magnetic interactions, not gravity alone. This modifies the effective mass-distance-time relationships.
3. Reference objects are imperfect analogs $-$ the Sun and proton are not exact self-similar counterparts (the Sun is mid-evolution; nucleons are fully settled).

The Kepler constraint ensures internal consistency regardless of these complications.

5. Physical Interpretation

What G Scaling Means Mechanically

There are two complementary ways to understand G scaling in the AAM framework:

Interpretation A: Dimensional Bookkeeping

$G$ is a conversion factor between units of length, mass, and time in gravitational dynamics. Between similarity levels, all three base dimensions change. The numerical value of $G$ changes accordingly $-$ not because the underlying physics changes, but because the "ruler," "clock," and "scale" are all different.

This is analogous to how the speed of light has different numerical values in different unit systems, while the physics remains identical.

Interpretation B: Effective Gravitational Coupling (Preferred)

The fundamental gravitational shadowing mechanism (Axiom 1) is universal and scale-invariant. However, at $SL_{-1}$, the effective orbital dynamics include contributions beyond pure gravitational shadowing:

1. Nucleon magnetic dipoles $-$ nucleons are settled iron stars with strong magnetic fields from their internal rotation. This creates dipole interactions with orbiting planetrons.
2. Magnetic force geometry $-$ for co-rotating bodies in the equatorial plane (the typical planetron configuration), the magnetic dipole interaction is repulsive (parallel dipoles side-by-side repel). This partially opposes gravitational attraction.
3. Net effect $-$ the combination of gravitational attraction and magnetic repulsion produces orbital dynamics that differ quantitatively from pure Keplerian gravity. When fit to a Keplerian model, this appears as a modified $G$.

This interpretation preserves the principle that fundamental laws are scale-invariant while acknowledging that the effective dynamics at each similarity level include scale-dependent contributions (magnetic properties, density, composition).

Comparison: Nucleon Pairs vs Planetrons

In both cases, the magnetic repulsion between co-rotating bodies in the equatorial plane contributes to the force balance. For nucleon pairs, this creates the stable "hand-mixer" configuration. For planetrons, it means gravity must be stronger than a naive Keplerian calculation would suggest, which manifests as an apparently enhanced $G_{-1}$.

6. Magnetic Contributions at $SL_{-1}$

Why Magnetic Forces Matter More at Atomic Scale

At $SL_{0}$ (solar system), planetary magnetic moments are negligible compared to gravitational forces $-$ the Sun's gravitational pull dominates orbital dynamics entirely.

At $SL_{-1}$, the situation differs fundamentally:

1. Nucleons are iron stars $-$ fully settled, iron-rich composition with aligned internal magnetic domains
2. Rotation is rapid $-$ nucleon rotation at THz frequencies produces strong magnetic dipoles
3. Distances are tiny $-$ magnetic dipole force scales as $r^{-4}$, falling off much faster than gravity ($r^{-2}$), but at fm-scale separations the magnetic force is significant

Dipole Field at the Equatorial Plane

For a magnetic dipole with moment $\vec{m}$ oriented along the rotation axis, the field at the equatorial plane (where planetrons orbit) is:

$$\vec{B}_{equator} = -\frac{\mu_0}{4\pi} \frac{\vec{m}}{r^3}$$

The field points opposite to the dipole moment at the equator. A co-rotating planetron with its own dipole aligned in the same direction experiences an anti-aligned local field $-$ resulting in repulsive interaction.

Force Decomposition

The total radial force on a planetron:

$$F_{total} = F_{gravity} - F_{magnetic,repulsive}$$

$$F_{total} = \frac{G_0 M_{nucleus} m_{planetron}}{r^2} - f_{mag}(r)$$

where $f_{mag}(r) \propto r^{-4}$ for dipole interactions.

For the orbit to match observed frequencies, $F_{total}$ must equal the centripetal requirement. When we fit this to a pure Keplerian model (attributing everything to gravity), the effective gravitational constant becomes:

$$G_{eff} = G_0 + \Delta G_{magnetic}(r)$$

The distance-dependent magnetic correction $\Delta G_{magnetic}$ means that $G_{eff}$ is not truly constant even within a single similarity level $-$ it varies with orbital radius. This may explain why different planetrons (at different radii) show slightly different effective $G$ values, and why the overall "average" $G_{-1}$ differs from $G_0$.

Open Question

The precise decomposition of $G_{-1}$ into gravitational and magnetic contributions requires:

1. Knowledge of the nucleon's magnetic dipole moment at $SL_{-1}$
2. The planetron magnetic moments (expected to be negligible due to tidal locking)
3. A force-balance calculation at each planetron's orbital radius

This decomposition is a priority for future work and would provide independent validation of $G_{-1}$ from first principles rather than empirical fitting.

7. Consistency with Validated Work

Validation Audit

All validated AAM results use empirically determined $G_{-1}$, anchored to observed spectral data. None depend on ideal geometric scaling alone.

8. Summary

Core Results

1. G must scale between similarity levels $-$ dimensional necessity from $[G] = L^3 M^{-1} T^{-2}$
2. The Kepler constraint $c = 2a + b - 3$ rigidly connects the scaling exponents of distance, time, mass, and $G$
3. Observed exponents from empirical data: $a \approx 0.86$ (time), $b \approx 2.17$ (mass), $c \approx 0.88$ ($G$) $-$ all satisfy the Kepler constraint
4. Physical interpretation: The fundamental gravitational coupling is scale-invariant. The apparent scaling of $G$ reflects (a) dimensional unit changes between SLs, and (b) magnetic force contributions at $SL_{-1}$ from settled iron-star nucleons.
5. Validated work is consistent $-$ all spectral, ionization, and stability validations use empirically anchored $G_{-1}$ values that satisfy the Kepler constraint.

What This Establishes

• The scaling laws between similarity levels are constrained but not rigid $-$ one degree of freedom exists within the Kepler constraint
• The exponents are determined by observation, not by geometric idealization
• The framework is internally consistent: Kepler's law, dimensional analysis, and empirical measurements all agree
• Future first-principles work on the gravitational-magnetic force decomposition can independently verify $G_{-1}$

9. References

AAM Axioms

• Axiom 1: Space, matter, motion $-$ gravitational shadowing mechanism
• Axiom 8: Magnetic properties of matter $-$ nucleon dipole interactions
• Axiom 10: Self-similarity across scales $-$ scaling laws and organizational progression

Related Validations

• Hydrogen Spectral Analysis: Validated $G_{-1} = 3.81 \times 10^{13}$
• Planetary Resonance Migration: N-body orbital stability confirmation