Contents
Notation guard
All \( \omega,\Delta\omega,\delta\omega \) are angular frequencies (rad·s\(^{-1}\)); convert to Hz by \( f=\omega/(2\pi) \). Spins: analytic \( s_i\in\{+i,-i\} \), code \( \sigma_i\in\{+1,-1\} \) with mapping \( s_i \equiv i\,\sigma_i \). Binary gate (open=1, closed=0): analytic \( G_{ij}=\frac{1+s_is_j}{2} \), code \( G_{ij}=\frac{1-\sigma_i\sigma_j}{2} \). Critical bandwidth: \( \Delta\omega^\ast=1.45(8)\times10^{23}\,\mathrm{s}^{-1} \) (uncertainty dominated by RG scheme spread).
I. Introduction
Classical geometry posits points and lines within a fixed background. RTG removes the background entirely: only oscillatory nodes with \( (\omega,\phi,s) \) exist, and geometry (distance, curvature, dimensionality) emerges from node‑to‑node relations. Distances follow beat frequencies; curvature follows phase–spin structure; dimensionality follows the local bandwidth ratio \( \delta\omega/\Delta\omega^\ast \).
II. Classical notion ↔ RTG analogue
| Classical | RTG counterpart | Key formula / definition |
|---|---|---|
| Point | RTG dot | \( z_i(t,s_i)=s_i\,e^{i(\omega_i t+\phi_i)} \) |
| Line | Resonance loop | \( r_{ij}=\frac{2\pi c}{|\omega_i-\omega_j|} \) |
| Plane / Space | Resonance lattice | \( Z_{\text{space}}(t)=\sum_i w_i\,s_i\,e^{i(\omega_i t+\phi_i)} \) |
| Curvature | Helfrich‑style penalty | \( U_{\text{curv}}=\kappa_c\,a_{\text{lat}}^2\sum_{\langle ijk\rangle}\left(1-\frac{\mathcal R_{ij}+\mathcal R_{jk}+\mathcal R_{ki}}{9}\right)^2 \) |
| Observer frame | Observer‑relative geometry | \( r_{io}=\frac{2\pi c}{|\omega_i-\omega_o|},\quad G_{\text{obs}}(t)=\sum_i s_i\,e^{i[(\omega_i-\omega_o)t+(\phi_i-\phi_o)]} \) |
| Higher‑D space | Bandwidth thresholds | \( \delta\omega/\Delta\omega^\ast\in[0,0.28)\Rightarrow D=2;\ [0.28,0.70)\Rightarrow D=3;\ [0.70,1.70)\Rightarrow D=4\ \text{(with 1.55–1.70 transitional \( \mathrm{U}(1)^2\) shells)};\ >1.70\Rightarrow D\ge5 \) |
| Inter‑plane link | Coupling proxy (dimensionful) | \( C_{12}(\alpha)=r_{12}^{-\alpha}\,\mathcal R_{12} \) |
\[ \delta\omega/\Delta\omega^\ast=\begin{cases} 0\text{–}0.28 & \Rightarrow D=2\ \text{(planar sheets)}\\ 0.28\text{–}0.70 & \Rightarrow D=3\ \text{(shells)}\\ 0.70\text{–}1.70 & \Rightarrow D=4\ \text{(propagation corridors; 1.55–1.70 shows transitional \( \mathrm{U}(1)^2\) shells)}\\ >1.70 & \Rightarrow D\ge 5\ \text{(high‑D anomalies)} \end{cases} \]
III. Detailed formulations
1. RTG dot
A spin‑tagged oscillator \( z_i(t,s_i)=s_i\,e^{i(\omega_i t+\phi_i)} \) with \( s_i\in\{+i,-i\} \). Binary gate: analytic \( G_{ij}=\frac{1+s_is_j}{2} \), code \( G_{ij}=\frac{1-\sigma_i\sigma_j}{2} \); gates open only for anti‑aligned analytic spins or opposite code spins.
2. RTG line (resonance loop)
Two‑dot superposition \( z_{\text{line}}(t)=e^{\,i\left(\frac{\omega_1+\omega_2}{2}t+\frac{\phi_1+\phi_2}{2}\right)}\cos\!\left(\frac{(\omega_1-\omega_2)t+(\phi_1-\phi_2)}{2}\right) \), with beat‑distance \( r_{12}=\frac{2\pi c}{|\omega_1-\omega_2|} \) as the length proxy.
3. RTG plane / space (resonance lattice)
Use the bounded resonance kernel \( \mathcal R_{ij}=A_{ij}(1+s_is_j) \) with \( A_{ij}=\frac{3}{4}\left[1+\cos(\phi_i-\phi_j)\right]\exp\!\left[-\left(\frac{\omega_i-\omega_j}{\Delta\omega^\ast}\right)^2\right] \) and \( 0\le \mathcal R_{ij}\le 3 \). Coherent sheets favor small phase differences; a practical heuristic is \( w_i\propto 1+\cos\Delta\phi_i \) .
4. Emergent curvature
The preferred coarse‑grained penalty is \( U_{\text{curv}}=\kappa_c\,a_{\text{lat}}^2\sum_{\langle ijk\rangle}\left(1-\frac{\mathcal R_{ij}+\mathcal R_{jk}+\mathcal R_{ki}}{9}\right)^2 \), which penalizes local departures from mutual resonance without breaking the binary gate convention, since \( \mathcal R_{ij} \) already enforces open‑gate conditions and yields dimensionless, bounded weights (normalization by 9 reflects three bonds each bounded by 3 in the triad).
5. Observer‑relative geometry
All measurements relative to observer node \( o \): distances \( r_{io}=\frac{2\pi c}{|\omega_i-\omega_o|} \) and field \( G_{\text{obs}}(t)=\sum_i s_i\,e^{i[(\omega_i-\omega_o)t+(\phi_i-\phi_o)]} \) .
6. Multi‑dimensional emergent geometry
A compressed representation of an \( n \)-dimensional configuration is \( G(t)=\sum_{j_1,\dots,j_n} s_{j_1}\cdots s_{j_n}\,e^{\,i\sum_{k=1}^n(\omega_{k,j_k}t+\phi_{k,j_k})} \), where \( n=D_{\text{eff}} \). The effective dimension increases when \( \delta\omega/\Delta\omega^\ast \) crosses \( 0.28, 0.70, 1.70,\dots \).
7. Inter‑plane node coupling
Compare resonance sheets via the dimensionful proxy \( C_{12}(\alpha)=r_{12}^{-\alpha}\,\mathcal R_{12} \) with \( \alpha\ge 0 \). Typical choices: \( \alpha=1 \) for long‑range propagation channels, \( \alpha=2 \) for emergent‑metric‑weighted couplings; calibrate \( \alpha \) to observables (e.g., proton radius fits).
IV. Numerical toy models of dimensional emergence
Geometric diagnostics track simplex volume growth as \( \delta\omega/\Delta\omega^\ast \) increases. For four nodes (tetrahedron), \( V_3=\frac{1}{6}\left|\,(P_1-P_0)\cdot\big((P_2-P_0)\times(P_3-P_0)\big)\,\right| \). In general, \( V_n^2=\frac{(-1)^{n+1}}{2^n(n!)^2}\det C \), with Cayley–Menger matrix \( C=\begin{pmatrix} 0 & 1 & 1 & \cdots & 1 \\ 1 & 0 & d_{01}^2 & \cdots & d_{0n}^2 \\ 1 & d_{10}^2 & 0 & \cdots & d_{1n}^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & d_{n0}^2 & d_{n1}^2 & \cdots & 0 \end{pmatrix} \). In small‑\( N \) scans, median‑based knees can appear \(\sim 0.2\) higher; full‑lattice MD confirms the RG thresholds to within \( \pm 0.02 \) (scheme‑dependent).
# Python snippet for Cayley–Menger volume (n-simplex)
import numpy as np, math
def cayley_menger_volume(n_points, dist_squared):
CM = np.zeros((n_points + 1, n_points + 1))
CM[0, :] = 1; CM[:, 0] = 1; CM[0, 0] = 0
k = 0
for i in range(1, n_points + 1):
for j in range(i + 1, n_points + 1):
CM[i, j] = CM[j, i] = dist_squared[k]; k += 1
det_CM = np.linalg.det(CM)
dim = n_points - 1
denom = (2**dim) * (math.factorial(dim))**2
V2 = ((-1)**(dim + 1)) * det_CM / denom
return math.sqrt(abs(V2))
# Example: Equilateral tetrahedron (4 points), side = 1
print(cayley_menger_volume(4, [1.0]*6)) # ~0.11785
V. Two‑loop RG connection
Define( g=\bar J/K^{\prime}=(3/2)J/K^{\prime} ). Representative two‑loop flow \( \beta_g(g)=0.72\,g-0.63\,g^2-0.011\,g^3 \) has non‑Gaussian fixed point \( g^\ast\approx 1.14 \). The same analysis yields \( \Delta\omega^\ast=1.45(8)\times10^{23}\,\mathrm{s}^{-1} \), which sets the observed dimensional thresholds and long‑wavelength beat scales.
VI. Practical take‑aways
- Curvature–field alignment. Large phase gradients inside open‑gate regions correlate with strong interaction zones; monitor \( \mathcal R_{ij} \) and \( U_{\text{curv}} \) jointly.
- Dimensional control. Tuning \( \delta\omega/\Delta\omega^\ast \) through \( 0.28 \) and \( 0.70 \) triggers robust 2D→3D and 3D→4D transitions; a \( \mathrm{U}(1)^2\) shell regime often appears for \( 1.55\text{–}1.70 \).
- Lab handle. Sweeping \( \delta\omega \) across \( 0.28\,\Delta\omega^\ast \) should onset phase‑dilation (a red‑shift analogue) predicted by the beat‑distance metric. Experimental analogs include dual‑frequency optical cavities and pump‑driven resonators exhibiting synthetic frequency shifts and coherence‑loss spectral deformation; measurable via delay shifts or spectral line deformation at \( 0.28\,\Delta\omega^\ast\approx 4.06\times10^{22}\,\mathrm{s}^{-1} \).
VII. References
- M. Aksu, ChatGPT, Grok. Core Principles & Foundations of RTG (2025).
- M. Aksu, ChatGPT, Grok. Mathematical Foundations of RTG (2025).
- M. Aksu, ChatGPT, Grok. Two‑Loop RG Derivation of the Critical Bandwidth (2025).
- M. Aksu, ChatGPT, Grok. From Lattice Hamiltonian to Continuum Action in RTG: Workbook (2025).
- M. Aksu, ChatGPT, Grok. RTG Glossary (2025).
VIII. Open questions
- How do inter‑plane couplings with \( \alpha>0 \) shape emergent gauge content near the 1.55–1.70 band (cf. \( \mathrm{U}(1)^2\))?
- What curvature moduli \( \kappa_c \) best stabilize mixed‑D domains without suppressing corridor formation?
- Can dual‑frequency cavity experiments map the predicted dilation onset curve \( \delta\omega/\Delta\omega^\ast \mapsto \) delay shift with sufficient dynamic range?