Version: 1.0 – April 2025
Authors: Mustafa Aksu, ChatGPT, Grok
Last updated: April 14, 2025
Contents
1. Domain of Objects: The Frequency-Phase-Spin Space \( \mathbb{F}_R \)
- An RTG object is a triple \( f = (\omega, \phi, s) \), where:
- \( \omega \in \mathbb{R}^+ \): Frequency (Hz) — represents temporal identity
- \( \phi \in [0, 2\pi) \): Phase (radians) — serves as a relational clock
- \( s \in \{+\frac{1}{2}, -\frac{1}{2}\} \): Spin — denotes intrinsic resonance parity (clockwise/counterclockwise)
These are the atomic entities of the RTG math system.
2. Fundamental Operations
- Frequency Addition (⊕): Beat Operation \( f_1 \oplus f_2 := |\omega_1 – \omega_2| \)
- Output: new beat frequency
- Commutative: \( f_1 \oplus f_2 = f_2 \oplus f_1 \)
- Not associative
- Frequency Scaling (⊗): Harmonic Multiplication \( n \otimes f := (n \cdot \omega, n \cdot \phi \mod 2\pi, s) \)
- Models harmonic stacking (overtones)
- \( n \in \mathbb{Z}^+ \)
- Spin is preserved in scaling
- Inversion (⊖): Anti-whirl \( \ominus f := (-\omega, \phi + \pi \mod 2\pi, -s) \)
- Represents reversed spin, frequency direction, and antiphase
3. Relational Distance Function
For two RTG objects: \( d(f_1, f_2) = \frac{2\pi}{|\omega_1 – \omega_2|} \)
- Interpreted as beat period (temporal separation)
- \( d(f_1, f_2) \to \infty \) as \( \omega_1 \to \omega_2 \)
4. Resonance Classes
Two nodes \( f_1, f_2 \) are in resonance if: \( \frac{\omega_1}{\omega_2} \in \mathbb{Q} \quad \text{and} \quad \Delta\phi = \phi_1 – \phi_2 = \frac{2\pi n}{m}, \; n, m \in \mathbb{Z} \quad \text{and} \quad s_1 = s_2 \)
- Resonance strength: \( \mathcal{R}(f_1, f_2) = \frac{1}{T_{\text{beat}}} \cdot \cos(\Delta \phi) \cdot \delta_{s_1, s_2} \)
5. Phase Geometry and Trigonometry
Phase Functions: \( \sin_t(f, t) = \sin(\omega t + \phi), \quad \cos_t(f, t) = \cos(\omega t + \phi) \)
Phase Distance: \( d_{\phi}(\phi_1, \phi_2) = \min(|\phi_1 – \phi_2|, 2\pi – |\phi_1 – \phi_2|) \)
Phase Addition: \( \phi_1 \oplus \phi_2 = (\phi_1 + \phi_2) \mod 2\pi \)
6. Composite Structures and Stability Rules
Stability Conditions for a Set of RTG Nodes \( \{f_i\} \):
- All \( \omega_i / \omega_j \in \mathbb{Q} \)
- All \( \Delta \phi_{ij} = 2\pi n / m \)
- All \( s_i = s_j \) or known pattern (e.g., balance of \( \pm \frac{1}{2} \))
- Beat loops form closed cycles
- Classes:
- Duon: 2-node stable system
- Trion: 3-node harmonic ring
- Shell: Nested beat structure
- Lattice: Grid of pairwise-resonant nodes
7. Temporal Derivatives (Preview)
\( \frac{d\phi}{dt} = \omega \quad ; \quad \frac{d\omega}{dt} \Rightarrow \text{temporal evolution or decay} \quad ; \quad \frac{ds}{dt} = 0 \text{ unless flipped by interaction} \)
These will later support RTG dynamics, fields, and interactions.
This formalism defines the core of RTG mathematics: frequency-first, relational, phase- and spin-driven. It replaces traditional position-based math with rhythm, resonance, and whirling node logic.