Enriched Geometric Concepts in Relational Time Geometry (RTG)

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

1. Introduction

Relational Time Geometry (RTG) discards any fixed space-time background, reconstructing classical geometric concepts—point, line, plane, curvature—purely from interference among node properties: frequency \(\omega\), phase \(\phi\), and binary spin \(s = \pm i\). This page maps these classical ideas onto their RTG analogues, provides explicit algebraic definitions, and illustrates dimensional emergence with toy models, guided by the resonance kernel \(\mathcal{R}_{ij} = \frac{3}{4} \left[1 + \cos(\phi_i – \phi_j)\right] (1 + s_i s_j) e^{-(\omega_i – \omega_j)^2 / (\Delta \omega^*)^2}\), which governs interaction strength based on phase alignment, spin coupling, and frequency matching.

2. Classical ⟷ RTG Dictionary

Classical conceptRTG counterpartKey definition / formula
PointRTG dot\(z_i(t) = s_i e^{i(\omega_i t + \phi_i)}\)
Line (length)Resonance loop\(r_{ij} = \frac{2\pi c}{|\omega_i – \omega_j|}\)
Plane / SpaceResonance lattice\(Z_{\text{space}}(t) = \sum_k w_k s_k e^{i(\omega_k t + \phi_k)}\), where \(w_k\) is a weighting factor reflecting node influence
CurvaturePhase–spin gradient\(\mathcal{K}_{ij} = \frac{\partial}{\partial r_{ij}} \left[ \left| e^{i\phi_i} s_i + e^{i\phi_j} s_j \right|^2 \right]\), a discrete gradient approximating classical curvature
Observer frameObserver-relative geometry\(G_{\text{obs}}(t) = \sum_i s_i e^{i[(\omega_i – \omega_o)t + (\phi_i – \phi_o)]}\)
Higher-D axisIndependent beat-distance axisAppears at \(\delta\omega / \Delta\omega^* = 0.5, 1.0, 1.7, \dots\), driven by resonance strength

3. Detailed Formulations

  • RTG dot (0-D): A single node with complex-spin marker \(s_i = \pm i\).
  • RTG line (1-D): Interference of two dots, with length \(r_{ij}\) fixed by the beat-frequency term.
  • RTG plane / lattice (2-D, 3-D, …): Coherent collection of nodes; stability requires neighboring phase differences \(\Delta\phi \lesssim \pi/4\).
  • Emergent curvature: Large phase–spin gradients (\(\mathcal{K}_{ij}\)) correlate with strong-field regions, reflecting classical curvature via node interactions.
  • Multi-D emergence: New axes emerge at bandwidth thresholds \(\delta\omega / \Delta\omega^* = 0.5, 1.0, 1.7, \dots\), driven by the resonance kernel’s frequency dependence.
  • Inter-plane coupling: Coupling between resonance sheets depends on angle \(\theta\) and phase offset, approximated as \(\mathcal{R}_{12} \propto \left| e^{i(\Delta\phi + \theta)} \right|^2 / r_{12}\).

4. Numerical Toy Models of Dimensional Emergence

ModelBandwidth ratioGeometric order parameterResult
4-node square (2 → 3-D test)\(\delta\omega / \Delta\omega^*\)Tetrahedral volume \(V_4\), computed via Cayley-Menger determinant\(V_4 = 0\) for < 0.50, jumps > 0 for \(\geq 0.50\)
5-node “pyramid” (3 → 4-D test)\(\delta\omega / \Delta\omega^*\)4-simplex volume \(V_5\), computed via Cayley-Menger determinant\(V_5 = 0\) for < 1.00, rises for \(\geq 1.00\)

Results verified on \(32^3\) Monte-Carlo lattices (\(\beta = 1.0\), 1000 iterations, updated resonance kernel), with node spacing varied to test thresholds.

Placeholder: Consider a plot of \(V_4\) or \(V_5\) vs. \(\delta\omega / \Delta\omega^*\) to visualize dimensional jumps.

5. Two-Loop RG Anchor

Critical bandwidth (updated):

\[ \Delta \omega^* = (1.45 \pm 0.08) \times 10^{23} \; \text{s}^{-1} \]

All dimensional transitions are quantized based on multiples of \(\Delta \omega^*\), grounded in two-loop RG analysis (see Two-Loop RG Derivation).

6. Practical Take-Aways

  • Curvature–field unification: Phase–spin gradients map to electromagnetic or strong fields, offering a unified geometric interpretation.
  • Fractal dimensionality: Mixed-D domains (e.g., 2 < D < 3) emerge from nested frequencies, relevant near quantum-gravity horizons.
  • Laboratory handle: Adjust cavity mode spacing around \(0.5 \, \Delta \omega^*\) to observe a sharp drop in coherence time, a testable RTG signature.

Related reading: Mathematical Foundations | Two-Loop RG Derivation.

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