Document version 1.3 — June 29, 2025 (Updated: Deepened Section 4, refined predictions)
Contents
1. The Linear Beginning (Photon Level – 1D)
Initially, the photon propagates linearly at \(c\): \[ x(t) = c \cdot t \]
- No intrinsic angular momentum.
- State in real numbers.
- Energy: \(E = h f\).
2. Approaching the Frequency Limit
Energy input increases frequency, constrained by \(c\). The critical threshold aligns with \(\Delta\omega^* = (1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1}\). The Planck frequency is \(f_P \approx \Delta\omega^* / (2\pi) \approx 2.31 \times 10^{22} \, \text{Hz}\), where instability emerges.
3. The Transition Trigger: Planck Frequency Instability
At \(\delta\omega \approx \Delta\omega^* = (1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1}\), linear propagation becomes unstable. The resonance kernel \[ \mathcal{R}_{ij} = \frac{3}{4} [1 + \cos(\phi_i – \phi_j)] (1 + s_i s_j) e^{-(\omega_i – \omega_j)^2 / (\Delta\omega^*)^2} \] predicts saturation as \(\delta\omega \to \Delta\omega^*\), enforcing phase-locking (\(\cos(\phi_i – \phi_j) \to 1\)) and forcing energy into orthogonal modes, triggering a 2D whirling transition via Euler’s formula: \[ e^{i\theta} = \cos\theta + i \sin\theta \]
4. Whirling Emergence: Photon Becomes a Node Pair
The photon stabilizes as a whirling node pair. Derivation:
- Angular Frequency (\(\omega_P\)): Energy \(E = h f\) redistributes into angular momentum \(L = I \omega_P\), with \(I = \frac{E}{c^2} r_{\text{node}}^2\). Setting \(L = \hbar\) and \(v = \omega_P r_{\text{node}} = c\), \(\omega_P \approx \Delta\omega^* \approx 1.45 \times 10^{23} \, \text{rad/s}\), reflecting the critical bandwidth’s role.
- Stable Radius (\(r_{\text{node}}\)): \(r_{\text{node}} = \frac{c}{\omega_P} \approx \frac{2.998 \times 10^8}{1.45 \times 10^{23}} \approx 2.07 \times 10^{-16} \, \text{m}\), ensuring \(v = c\).
The resonance kernel \(\mathcal{R}_{ij}\) drives this transition by enhancing node coupling as \(\delta\omega \to \Delta\omega^*\). The node pair is: \[ z(t) = r_{\text{node}} \cdot e^{i(\omega_P t + \phi)} \]
Whirling (i.e., rotational phase progression in the complex plane) represents the emergence of angular behavior from linear propagation.
5. Planck Observer’s Role
The “Planck observer” defines the transition at \(\delta\omega \approx \Delta\omega^*\), validating the 1D-to-2D shift and ensuring \(v \leq c\).
6. Implications for Fundamental Physics
- High-energy photons near \(\Delta\omega^*\) may show rotational polarization, manifesting as helicity-dependent delays or birefringence in gamma-ray spectra above \(10^{21} \, \text{Hz}\), testable by FERMI or CTA.
- Spin correlations at TeV energies could reflect 2D constraints.
- Mass generation from node desynchronization is possible.
- Further energy (\(\delta\omega > 1.5 \Delta\omega^*\)) may trigger 3D or higher axes, subject to high-frequency studies.
7. Conclusion: A Dimensional Emergence Model
In RTG, the photon’s 1D-to-2D journey at \(\Delta\omega^*\) offers a testable model. Dimensional emergence follows discrete thresholds: 0.0–0.5 \(\Delta\omega^*\) (1D), 0.5–1.0 (2D), 1.0–1.5 (3D), and beyond (4D corridors). Euler’s formula, \(\mathcal{R}_{ij}\), and the Planck observer integrate to provide predictive power for high-energy physics.
Insert diagram of 1D-to-2D transition (pending).