Glossary of Relational Time Geometry (RTG) Terms

Revision date: 01 July 2025 | Authors: Mustafa Aksu, Grok 3, ChatGPT-4.5

Contents

1. Fundamental Building Blocks

Node: The sole primitive in RTG, carrying three intrinsic quantities: frequency \(\omega_i\), phase \(\phi_i\), and binary spin \(s_i = \pm i\). Everything emerges from node relationships (see Core Principles).
Frequency \(\omega_i\): Intrinsic “tick rate” of node \(i\), assigning energy \(E_i = \hbar \omega_i\).
Phase \(\phi_i\): Determines constructive vs. destructive interference with neighboring nodes.
Spin \(s_i\): Binary value \(s_i = \pm i\). Higher spins (e.g., 1, 3/2) emerge from coarse-graining clusters (see Deriving Physical Quantities in RTG).
Intrinsic time \(t_i\): \(t_i = \phi_i / \omega_i\), the local time experienced by node \(i\).

Wavefunction-style Resonance Strength \(\mathcal{R}_{ij}\): \[ \mathcal{R}_{ij} = \frac{3}{4} [1 + \cos(\phi_i – \phi_j)] (1 + s_i s_j) e^{-(\omega_i – \omega_j)^2 / (\Delta\omega^*)^2} \]
Values in [0, 3], maximal when phases and spins align, derived from wave-function overlap (see Two-Loop RG Derivation).
Beat-frequency distance \(r_{ij}\): \[ r_{ij} = \frac{2\pi c}{|\omega_i – \omega_j|} \]
Spatial separation emerges from frequency difference, modulated by resonance strength \(\mathcal{R}_{ij}\) and used in energy equations (e.g., Hamiltonian). Units: If \(\omega\) is in s⁻¹, then \(r_{ij}\) is in meters.
Hamiltonian (bond contribution): \[ E_{ij} = K |\omega_i – \omega_j| + J \mathcal{R}_{ij} + J_{\text{ex}} \sin(\phi_i – \phi_j) e^{-(\omega_i – \omega_j)^2 / \sigma^2} \]
Where \(K, J, J_{\text{ex}}, \sigma\) are RG-fixed constants, derived in Two-Loop RG Derivation.

Observer node: A designated node whose \((\omega_o, \phi_o, s_o)\) fixes all measurements.
Observer-dependence: Distances, durations, and interaction strengths use the observer’s beat-frequency frame: \(r_{io} = \frac{2\pi c}{|\omega_i – \omega_o|}\). For switching observers \(o \rightarrow o’\), use \(\phi_i^{(o’)} = \phi_i – \phi_{o’}\), \(\omega_i^{(o’)} = \omega_i – \omega_{o’}\).
Planck observer: A hypothetical observer with frequency \(\omega_P = \Delta\omega^* \approx (1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1}\), anchoring quantum-gravity limits (see Planck Observer Calibration).

Critical bandwidth (two-loop RG): \[ \Delta\omega^* = (1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1} \]

Derived via RG flow, marking thresholds for dimensional transitions (see Two-Loop RG Derivation).

Bandwidth window (\(\delta\omega / \Delta\omega^*\))	Effective spatial D	Typical structure
0.0 – 0.5	2	Planar resonance sheets
0.5 – 1.0	3	Stable shells (proton, nuclei)
1.0 – 1.7	4	Tightly-bound higher-D objects
1.7+	5+	Emergent 5D+ effects, under study

Stability / Equilibrium: Achieved when local energy variation satisfies \(\frac{\partial E}{\partial (\text{degrees})} = 0\).
Zero-arbitrariness: All constants determined via two-loop RG around \(\Delta\omega^*\); no empirical fitting.
Relational consistency: Every definition traces back to node-to-node relations; no external background space-time.

Photon node: \(\omega = 0\), lays down spatial extension but incurs no intrinsic time (e.g., light-like paths in emergent geometry).
Whirling frequency: Rotational component controlling the spatial extent of a node’s interference pattern, derived from phase gradients (e.g., orbital patterns in atoms).
Resonance lattice: A self-organized array of nodes with phase-locked \(\phi_i\) within a narrow bandwidth, forming an emergent space (see Core Principles).

Heat: Energy transfer due to frequency gradients between nodes (see Planck Observer Calibration).
Entropy: Measure of disorder in node resonance, defined as \(S = -k_B \sum p_{ij} \ln p_{ij}\), where \(k_B \approx 1.38 \times 10^{-23} \, \text{J/K}\) is the Boltzmann constant (see Planck Observer Calibration).
Mass: Emergent property from resonance interactions: \(m_i = \alpha \sum_j \frac{\mathcal{R}_{ij}}{r_{ij}^2}\), where \(\alpha\) is the mass calibration constant derived from proton mass fitting (see Deriving Physical Quantities in RTG).
Time dilation: Arises from phase offset \(\Delta\phi\) between nodes, scaling with relative velocity (see Deriving Physical Quantities in RTG).

Placeholder: Consider a future diagram showing node interaction → resonance → space-time emergence.

See also: Deriving Quantities in RTG, Planck Observer Calibration, RTG Cosmological Dynamics