Last updated: 12 Aug 2025 | Authors: Mustafa Aksu, ChatGPT, Grok
Preamble: Analytic spins use \(s_i=\pm i\); code spins use \(\sigma_i=\pm1\) with \(s_i\equiv i\,\sigma_i\). Unless noted, formulas are given in the analytic convention. All \(\omega,\Delta\omega,\delta\omega\) are angular frequencies (rad·s\(^{-1}\)); convert to Hz by \(f=\omega/(2\pi)\).
Contents
1. Fundamental Building Blocks
Node — The unique primitive of RTG, storing frequency \( \omega_i \), phase \( \phi_i \), and binary spin \( s_i=\pm i \). Geometry and dynamics emerge from inter-node relations.
Frequency \( \omega_i \) — Intrinsic tick-rate; local energy \( E_i=\hbar\,\omega_i \).
Phase \( \phi_i \) — Controls constructive/destructive interference with neighbours.
Spin \( s_i=\pm i \) — Analytic: \(s_i s_j\in\{-1,+1\}\); gate value \(1+s_is_j\) is 2 (open) for anti-aligned \((+i,-i)\) and 0 (closed) for aligned. Code: \(\sigma_i=\pm1\) with \(s_i\equiv i\,\sigma_i\); implement gate as \(1-\sigma_i\sigma_j\), which opens for opposite \(\sigma\) (corresponds to anti-aligned analytic spins).
Intrinsic time \( t_i \) — Using unwrapped phase \( \tilde\phi_i \): \( t_i=\tilde\phi_i/\omega_i \); meaningful only under local phase-locking.
Spin Gate Truth Table
| Convention | Pair | Product | Gate Value |
|---|---|---|---|
| ±i (analytic) | (+i, −i) | +1 | \(1+s_is_j=2\) (open) |
| ±i (analytic) | (+i, +i) or (−i, −i) | −1 | \(1+s_is_j=0\) (closed) |
| ±1 (code) | (+1, −1) or (−1, +1) | −1 | \(1-\sigma_i\sigma_j=2\) (open) |
| ±1 (code) | (+1, +1) or (−1, −1) | +1 | \(1-\sigma_i\sigma_j=0\) (closed) |
2. Resonance & Interaction Metrics
Resonance kernel \( \mathcal{R}_{ij} \) — \( \mathcal{R}_{ij}=A_{ij}\,(1+s_i s_j) \) with \( A_{ij}=\tfrac{3}{4}[1+\cos(\phi_i-\phi_j)]\,\exp[-(\omega_i-\omega_j)^2/(\Delta\omega^\ast)^2] \). Range \(0\le\mathcal R_{ij}\le3\); maximum at \( \Delta\phi=0 \), open gate, \( \Delta\omega=0 \). The spin factor is a binary gate, not an SU(2) projector.
Beat-frequency distance \( r_{ij} \) — \( r_{ij}=\dfrac{2\pi c}{|\omega_i-\omega_j|} \); valid for \( |\Delta\omega|\neq 0 \) with weak phase decoherence; use a coarse-grained metric as \( \Delta\omega\to0 \).
Bond energy \( E_{ij} \) — \( E_{ij}=K’\,\dfrac{|\omega_i-\omega_j|}{\Delta\omega^\ast}+J\,\mathcal{R}_{ij}+J_{\mathrm{ex}}\sin(\phi_i-\phi_j)\,\exp[-(\omega_i-\omega_j)^2/\sigma_{\mathrm{exch}}^{2}] \). Here \(K’,J,J_{\rm ex}\) are energies; \( \mathcal R_{ij} \) is dimensionless. Default regulator \( \sigma_{\mathrm{exch}}\simeq\Delta\omega^\ast \) and is independent of the kernel’s \( \Delta\omega^\ast \) because it regularizes only the exchange channel.
Units & mapping: On a lattice with spacing \(a_{\rm lat}\) in \(d\) spatial dimensions, the frequency-gap TV coefficient maps as \(K’_{\rm TV}=K_{\rm lat}\,a_{\rm lat}^{\,1-d}\). The small-angle phase stiffness maps as \( \rho_s=\tfrac{3}{2}J\,a_{\rm lat}^{\,2-d} \). RTG typically operates in an emergent \(3+1\)D regime (so \(d=3\) for these maps).
Notation guard: All frequencies are angular; use \(f=\omega/(2\pi)\). \( \sigma_{\mathrm{exch}} \) (exchange regulator) is distinct from \( \sigma_{\mathrm{noise}} \) (CHSH noise) and from RG smooth cutoffs.
3. Observer Concepts
Observer node — Node whose \( (\omega,\phi,s) \) frame defines all measurements.
Observer dependence — Relational distances to observer \(o\): \( r_{io}=2\pi c/|\omega_i-\omega_o| \).
\(\Delta\omega^\ast\)-observer — Reference observer with \( \omega_{\rm ref}=\Delta\omega^\ast\approx 1.45\times10^{23}\,\mathrm{s}^{-1} \).
4. Emergent Dimensionality & Critical Bandwidth
Critical bandwidth — \( \Delta\omega^\ast=(1.45\pm0.08)\times10^{23}\,\mathrm{s}^{-1} \).
| \(\delta\omega/\Delta\omega^\ast\) | Effective D | Typical structure |
|---|---|---|
| 0–0.28 | 2 | Planar sheets |
| 0.28–0.70 | 3 | Stable shells (proton)* |
| 0.70–1.70 | 4 | 4‑D corridors (propagation channels) |
| >1.70 | 5+ | High‑D sector (speculative; active work) |
*Calibrated against the proton radius; see Mathematical Foundations §9. Thresholds carry \( \pm0.02 \) systematic in \( \delta\omega/\Delta\omega^\ast \) (scheme choice).
5. Stability & Guiding Principles
Stationarity — \( \partial E/\partial q=0 \) for continuous degrees; binary spins update discretely (e.g., cluster flips). To conserve total energy, route the flip energy \( \Delta E \) into the conjugate phase momentum \( \pi_\phi \) (kinetic compensation).
Scale-setting — Two‑loop RG fixes \( \Delta\omega^\ast \) and anchors ratios such as \( g=\tfrac{3J}{2K’} \); absolute \(K’,J,J_{\rm ex}\) are calibrated to observables (e.g., proton radius).
Relational closure — No background spacetime; all definitions use node‑to‑node data.
6. Special Node Types & Dynamic Factors
Photon object — Spin–anti‑spin pair \((+i,-i)\) with shared phase, carrier \( \omega_\gamma\neq0 \), and internal \( \Delta\omega=0 \); gate open in the analytic convention. Energy \(E=\hbar\omega_\gamma\); effectively massless as the internal gap vanishes.
Whirling frequency \( \Omega \) — Local precession scale \( \Omega^2=|\nabla\phi|^2 \) (code units); used in coarse‑grained orbital fits.
Resonance lattice — Self‑organised 3‑D array with \( \delta\omega\lesssim0.28\,\Delta\omega^\ast \) (2‑D→3‑D tipping). Stability is diagnosed by the Cayley–Menger tetrahedral volume \( V_4 \) rising beyond the 0.28 threshold (see Enriched Geometry §4).
7. Quick-Reference Constants
| Constant | Symbol | Value |
|---|---|---|
| Speed of light | \(c\) | \(2.998\times10^8\,\mathrm{m/s}\) |
| Planck constant | \(\hbar\) | \(1.055\times10^{-34}\,\mathrm{J\cdot s}\) |
| Critical bandwidth | \(\Delta\omega^\ast\) | \(1.45(8)\times10^{23}\,\mathrm{s}^{-1}\) |
| Boltzmann constant | \(k_B\) | \(1.381\times10^{-23}\,\mathrm{J/K}\) |
8. Additional Terms
Coarse‑grained operational definitions for simulations; see linked notes for derivations.
Heat — Energy flux between subsystems: \( \dot Q=\hbar\,[\langle\omega_1\rangle-\langle\omega_2\rangle]\,\mathcal R_{12} \), with \( \mathcal R_{12} \) gating transfer efficiency.
Entropy — \( S=-k_B\sum_{i
Mass — \( m_i=\big[\hbar\omega_i-\sum_j E^{\rm res}_{ij}\big]/c^2 \) with \( E^{\rm res}_{ij}=\hbar\,|\omega_i-\omega_j|\,\mathcal R_{ij} \); ties to inertia via the emergent metric (RTG Gravity I §3).
Time dilation (operational) — \( \Delta\tau\approx\Delta\phi/\Delta\omega \) under phase‑locking; GR‑like red‑shift and Shapiro analogues emerge via the metric construction.
Curvature penalty — Helfrich‑style energy for phase/spin curvature: \( U_{\rm curv}=\kappa_c\,a_{\rm lat}^2\sum_{\langle ijk\rangle}\left(1-\dfrac{\mathcal R_{ij}+\mathcal R_{jk}+\mathcal R_{ki}}{9}\right)^2 \) with \(\kappa_c\) in MeV·fm.
Emergent gauge symmetry — From coarse‑graining: U(1) from local phase shifts (window 0–0.28), soft‑spin SU(2) from binary spins (0.28–0.70), and U(1)\(^2\) (≈ SU(3) for small cosmic \(\epsilon\)) near 1.55–1.70; see Gauge Symmetries §§2–4.
See also: Core Principles | Two‑Loop RG Derivation | Planck Observer Calibration | Quantum Behaviours | Thermodynamics | Gauge Symmetries | RTG Gravity I | RTG Gravity II | Lattice → Continuum Workbook