Minimal RTG Model: Three Nodes and the Emergence of Interaction

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1. Purpose

To explore the origin of interaction in RTG by modeling the minimal configuration: three whirling nodes with no pre-existing space or time. We aim to demonstrate how interaction, temporal structure, and emergent spatial geometry arise from intrinsic frequency and phase dynamics alone. A single change in one node triggers the cascade that seeds all relational dynamics.

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2. Initial Conditions

Let three nodes exist from the beginning:

• Node A: \(f_A = (\omega_A(t), \phi_A(t), s_A)\)
• Node B: \(f_B = (\omega_B, \phi_B(t), s_B)\)
• Node C: \(f_C = (\omega_C, \phi_C(t), s_C)\)

Assume:

• All three nodes initially exist with static frequencies and phases: \[\omega_i = \omega_{i0},\phi_i = \phi_{i0}, s_i = +\frac{1}{2}.\]
• At some point, Node A undergoes a slight intrinsic change in its frequency: \[\omega_A(t) = \omega_{A0} + \epsilon(t)\].
• All spins are aligned: \(\delta_{s_i, s_j} = 1\).

This small asymmetry in Node A initiates the entire chain of relational emergence.

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3. Relational Phase and Frequency

Define relative quantities:

• \(\Delta \omega_{ij}(t) = \omega_j – \omega_i(t)\)
• \(\Delta \phi_{ij}(t) = \phi_j(t) – \phi_i(t)\)

Assume phase evolution: \(\phi_i(t) = \int \omega_i(t’) dt’ + \phi_{i0}\)

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4. Emergence of Beat Patterns

For each node pair \((i,j)\), define the beat pattern: \(\Psi_{ij}(t) = \cos(\phi_i(t)) + \cos(\phi_j(t))\)

The beat amplitude \(\cos(\Delta \omega_{ij}(t) t + \Delta \phi_{ij}(0))\) defines:

• Distance: \[r_{ij}(t) = c \cdot \frac{2\pi}{|\Delta \omega_{ij}(t)|}\]
• Resonance Strength: \[\mathcal{R}_{ij}(t) = \frac{c}{2\pi r_{ij}(t)} \cdot \cos(\Delta \phi_{ij}(t)) \cdot \delta_{s_i, s_j}\]

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5. Temporal Gradient and Relational Acceleration

The small change in Node A introduces a temporal gradient: \[\frac{d\omega_A}{dt} \neq 0 \Rightarrow \frac{dr_{AB}}{dt} \neq 0, \quad \frac{dr_{AC}}{dt} \neq 0\]

Compute acceleration: \[a_{ij}(t) \approx -\frac{r_{ij}(t)^2}{c} \cdot \frac{d^2 \omega_i}{dt^2}\]

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6. Emergent Quantities

For each node pair \((i,j)\):

• Distance: \[r_{ij}(t) = c \cdot \frac{2\pi}{|\omega_i(t) – \omega_j|}\]
• Force: \[F_{ij}(t) = \hbar \cdot \frac{k_{ij}(t)}{2\pi r_{ij}(t)^2}, \quad k_{ij} \propto \cos(\Delta \phi_{ij}(t)) \cdot \delta_{s_i, s_j}\]
• Proper Time: \[t_i = \frac{\phi_i(t)}{\omega_i(t)}\]

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7. Emergent Geometry

With three active beat distances \(r_{AB}(t), r_{AC}(t), r_{BC}(t)\), we form a relational triangle:

• Angle Definition: Resonance ratios define angle at each node.
• Triangle Conditions: If beat distances form a closed triangle, geometry emerges.
• Curvature: Phase or frequency imbalances distort triangle closure, indicating curvature.

This three-node structure serves as the seed of spatial dimensionality in RTG.

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8. Interpretation

This three-node system forms a complete minimal relational unit. If all node parameters remain static—no phase drift, frequency variation, or spin mismatch—no interaction occurs. The system is perfectly frozen.

But if even one node changes—just slightly—this asymmetry seeds interaction. The change ripples relationally, affecting all distances and forces. Geometry, motion, and time emerge.

The universe doesn’t begin with a bang, but with a ripple.

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8.5 Sustained Interaction through Frequency Oscillations

For sustained, stable interaction—such as those seen in bound systems, fields, and resonant structures—nodes must exhibit oscillating frequency profiles: \[\omega_A(t) = \omega_0 + \delta \cdot \sin(\nu t)\]

In this case:

• Beat frequencies between nodes oscillate.
• Distances and resonance strengths fluctuate within bounds.
• Periodic attraction and repulsion cycles emerge, mimicking oscillatory field behavior.
• The system can maintain stable interaction energy, supporting the emergence of structures such as atoms or molecules.

This harmonic dynamic reflects the physical intuition of stable forces as rhythmic resonance phenomena, grounding field persistence in the oscillatory nature of the nodes.

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8.6 The Necessity of Frequency Oscillation for a Stable Universe

A key insight from this model is that a static ensemble of nodes with constant frequencies and phases leads to no interaction, no force, no time, no space—no evolution.

A universe composed only of nodes with fixed frequencies and phases is relationally inert.

In contrast, a stable, structured universe—with persistent forces, evolving geometry, and dynamic systems—requires nodes with oscillating frequencies. These oscillations:

• Induce sustained beat frequencies, giving rise to stable distances.
• Support periodic force patterns, allowing for bound systems.
• Allow relational clocks to emerge, defining proper time and motion.

Thus, oscillating whirling nodes are not just common—they are necessary. The entire unfolding of spacetime and structure depends on this rhythmic fluctuation.

This rhythmic dynamic provides the heartbeat of RTG’s universe. It bridges the formation of particles, the persistence of fields, and the propagation of causal structure. Oscillation is not just a feature—it is the foundation.

In RTG, a stable universe emerges not from chaos, but from the gentle pulse of oscillating frequencies.

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9. Path Forward

Future extensions include:

• Modeling stable three-node structures as proto-particles.
• Defining angle, area, and torsion from beat ratios.
• Exploring entanglement-like triadic synchronizations.
• Analyzing energy transfer and entropy as relational heat.

This minimal three-node model lays the groundwork for RTG’s higher layers: from particles to fields, thermodynamics to cosmology. It encodes the essential blueprint of emergence: from asymmetry, all structure grows.