Contents
Observer-Relative Frequency Dynamics
In RTG, all properties are relative to an observer. An observer can be defined as a node or set of nodes characterized by their own frequency ωobs(t)\omega_{\text{obs}}(t) and phase ϕobs(t)\phi_{\text{obs}}(t). Observers measure interactions within the resonance graph from their unique frames, making temporal derivatives of frequencies and beat frequencies observer-dependent.
Observer-Dependent Frequencies
For an observer in frame SS, the node frequency is ωi(t)\omega_i(t). In another observer frame S′S’, moving or rotating relative to SS, the frequency transforms as: ωi′(t′)=f(ωi(t),Δϕi,Δωobs)\omega_i'(t’) = f(\omega_i(t), \Delta \phi_i, \Delta \omega_{\text{obs}})
where ff is a transformation function dependent on observer-relative parameters (analogous to relativistic frequency shifts). The first-order temporal derivative becomes: dωi′dt′=∂f∂ωi⋅dωidt+∂f∂ϕi⋅dϕidt\frac{d\omega_i’}{dt’} = \frac{\partial f}{\partial \omega_i} \cdot \frac{d\omega_i}{dt} + \frac{\partial f}{\partial \phi_i} \cdot \frac{d\phi_i}{dt}
Scale-Dependent Frequency Zones
RTG’s fractal hierarchy implies that interactions at different scales are largely isolated, with dynamics confined to specific frequency ranges or zones. This scale-dependent isolation ensures local resonance patterns dominate within a scale, minimizing influence across scales, thus mimicking natural decoupling seen in physical systems.
Frequency Zones
A frequency zone ZkZ_k is defined as a frequency band: Zk={ωi(t)∣ωk,min≤∣ωi(t)∣<ωk,max}Z_k = \{ \omega_i(t) \mid \omega_{k,\text{min}} \leq |\omega_i(t)| < \omega_{k,\text{max}} \}
Nodes within the same zone resonate strongly, whereas cross-zone interactions remain weak. Resonance strength becomes: Rij(t)=c⋅kforce2πrij(t)⋅cos(Δϕij)⋅δsi,sj⋅e−rij/r0⋅(1+kg⋅d(Δωij)dt)⋅ζij(Zk)\mathcal{R}_{ij}(t) = \frac{c \cdot k_{\text{force}}}{2\pi r_{ij}(t)} \cdot \cos(\Delta \phi_{ij}) \cdot \delta_{s_i, s_j} \cdot e^{-r_{ij}/r_0} \cdot \left( 1 + k_g \cdot \frac{d(\Delta \omega_{ij})}{dt} \right) \cdot \zeta_{ij}(Z_k)
where the zone function is: ζij(Zk)={1if ωi(t),ωj(t)∈Zkϵif ωi(t)∈Zk,ωj(t)∈Zm,k≠m0if ∣ωi(t)−ωj(t)∣>Δωmax\zeta_{ij}(Z_k) = \begin{cases} 1 & \text{if } \omega_i(t), \omega_j(t) \in Z_k \\ \epsilon & \text{if } \omega_i(t) \in Z_k, \omega_j(t) \in Z_m, k \neq m \\ 0 & \text{if } |\omega_i(t) – \omega_j(t)| > \Delta \omega_{\text{max}} \end{cases}
with ϵ≪1\epsilon \ll 1, ensuring strong isolation between zones.
Zone-Specific Derivatives
Frequency derivatives are zone-specific: dωidt=kω∑j∈ZkRij(t)cos(Δϕij)−γωi(t)+kg⋅d(Δωij)dt\frac{d\omega_i}{dt} = k_\omega \sum_{j \in Z_k} \mathcal{R}_{ij}(t) \cos(\Delta \phi_{ij}) – \gamma \omega_i(t) + k_g \cdot \frac{d(\Delta \omega_{ij})}{dt}
Weak cross-zone coupling may be considered: dωidt→dωidt+ϵ∑m≠k∑j∈ZmRij(t)cos(Δϕij)\frac{d\omega_i}{dt} \to \frac{d\omega_i}{dt} + \epsilon \sum_{m \neq k} \sum_{j \in Z_m} \mathcal{R}_{ij}(t) \cos(\Delta \phi_{ij})
Higher-order derivatives within zones are: d2ωidt2=ddt(kω∑j∈ZkRij(t)cos(Δϕij)−γωi(t))\frac{d^2 \omega_i}{dt^2} = \frac{d}{dt} \left( k_\omega \sum_{j \in Z_k} \mathcal{R}_{ij}(t) \cos(\Delta \phi_{ij}) – \gamma \omega_i(t) \right)
Zone-Modulated Frequencies
Nodes within a zone exhibit modulated frequencies primarily within their own zone: ωi(t)=ω0,i+δisin(νit+ψi)+∑k∈Zmmiksin(μkt+θk)\omega_i(t) = \omega_{0,i} + \delta_i \sin(\nu_i t + \psi_i) + \sum_{k \in Z_m} m_{ik} \sin(\mu_k t + \theta_k)
with temporal derivatives: dωidt=δiνicos(νit+ψi)+∑k∈Zmmikμkcos(μkt+θk)\frac{d\omega_i}{dt} = \delta_i \nu_i \cos(\nu_i t + \psi_i) + \sum_{k \in Z_m} m_{ik} \mu_k \cos(\mu_k t + \theta_k)
Multilayer Observer Dynamics
In hierarchical structures, sub-node frequency evolution for observer S′S’ is: dωsub,i′dt′=kω,g∑kRsub,ik′(t′)cos(Δϕsub,ik′)−γCωsub,i′(t′)+kg⋅d(Δωsub,ik′)dt′\frac{d\omega_{\text{sub},i}’}{dt’} = k_{\omega,g} \sum_k \mathcal{R}_{\text{sub},ik}'(t’) \cos(\Delta \phi_{\text{sub},ik}’) – \gamma_C \omega_{\text{sub},i}'(t’) + k_g \cdot \frac{d(\Delta \omega_{\text{sub},ik}’)}{dt’}
Aggregated to parent nodes, frequencies become: ωi′(t′)=∑subωsub,i′(t′),dωi′dt′=∑subdωsub,i′dt′\omega_i'(t’) = \sum_{\text{sub}} \omega_{\text{sub},i}'(t’), \quad \frac{d\omega_i’}{dt’} = \sum_{\text{sub}} \frac{d\omega_{\text{sub},i}’}{dt’}
Implementation Notes
Numerically compute derivatives carefully to maintain stability across observer frames and zones. This generalized and scale-dependent framework for RTG supports advanced applications, including adaptive neural networks, where nodes themselves act as dynamic observers influencing network-wide resonance and feedback mechanisms.