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 follow the analytic convention. All \(\omega,\Delta\omega,\delta\omega\) are angular frequencies (rad·s\(^{-1}\)); convert to Hz by \(f=\omega/(2\pi)\). The exchange regulator \(\sigma_{\mathrm{exch}}\) is distinct from CHSH noise \(\sigma_{\mathrm{noise}}\) (see Glossary §2, Gauge Symmetries §1).
Contents
- 1. Introduction
- 2. Fundamental Entities — Nodes
- 3. Emergent Space-Time
- 4. Node Interactions & Resonance
- 5. Observer Dependence
- 6. Dimensionality & Frequency Regimes
- 7. Spin Flips, Energy Conservation & Curvature
- 8. Emergent Energy & Mass
- 9. Stability & Equilibrium
- 10. Mathematical Consistency & Scale-Setting
- 11. Simulation Insights
- 12. Applications & Future Research
- 13. Conclusion
1. Introduction
Relational Time Geometry (RTG) models the universe as a graph of oscillatory nodes. Space, time, and interactions emerge from node-to-node relations. Two-loop renormalisation-group (RG) analysis fixes the universal bandwidth \(\Delta\omega^\ast=(1.45\pm0.08)\times10^{23}\,\mathrm{s}^{-1}\) and anchors dimensionless ratios; absolute couplings are calibrated to observables such as the proton radius (\(0.84\pm0.01\,\mathrm{fm}\)). Molecular-dynamics runs reproduce this radius and CHSH \(\approx 2.827\) under ideal settings (Gauge Symmetries §8).
2. Fundamental Entities — Nodes
| Quantity | Symbol | Role |
|---|---|---|
| Frequency | \(\omega_i\) | Intrinsic tick-rate; local energy \(E_i=\hbar\omega_i\). |
| Phase | \(\phi_i\) | Controls interference via \(\cos\Delta\phi\). |
| Spin | \(s_i=\pm i\) (analytic); \(\sigma_i=\pm1\) (code) | Binary chirality. Analytic gate: \(1+s_is_j\) is 2 for anti-aligned \((+i,-i)\) and 0 for aligned. Code gate: \(1-\sigma_i\sigma_j\) opens for opposite \(\sigma\); mapping \(s_i\equiv i\,\sigma_i\) aligns conventions. |
Intrinsic time — Using unwrapped phase \(\tilde\phi_i\): \(t_i=\tilde\phi_i/\omega_i\); only meaningful under local phase-locking.
Spin Gate Truth Table
| Convention | Spin Pair | Spin 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) |
Mapping: \(s_i\equiv i\,\sigma_i\); open-gate conditions match across conventions.
3. Emergent Space-Time
Beat-frequency distance — \(r_{ij}=\dfrac{2\pi c}{|\omega_i-\omega_j|}\); valid for \(|\Delta\omega|\neq0\) with weak decoherence; as \(\Delta\omega\to0\) use a coarse-grained metric (RTG Gravity I §3).
Proper-time increments (operational) — Under phase-locking, \(\Delta\tau\approx\Delta\phi/\Delta\omega\). Global Lorentz factors are not assigned from \(\Delta\phi\) alone; red-shift and Shapiro-type delays arise via the emergent metric.
Photon object — Spin–anti-spin pair \((+i,-i)\) with shared phase, carrier \(\omega_\gamma\neq0\), and internal \(\Delta\omega=0\); gate open in analytic convention (\(1+s_is_j=2\)). In code, opposite \(\sigma\) (\(+1,-1\) or \(-1,+1\)) opens the \(1-\sigma_i\sigma_j\) gate. Energy \(E=\hbar\omega_\gamma\); effectively massless due to vanishing internal gap.
4. Node Interactions & Resonance
Resonance strength — \(\mathcal R_{ij}=A_{ij}(1+s_is_j)\) 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 selector, not an SU(2) projector.
Bond energy — \(E_{ij}=K’\,\dfrac{|\omega_i-\omega_j|}{\Delta\omega^\ast}+J\,\mathcal R_{ij}+J_{\mathrm{ex}}\sin(\phi_i-\phi_j)\,\exp[-(\omega_i-\omega_j)^2/\sigma_{\mathrm{exch}}^2]\). Here \(K’,J,J_{\rm ex}\) have energy units; \(\mathcal R_{ij}\) is dimensionless. Default \(\sigma_{\mathrm{exch}}\simeq\Delta\omega^\ast\) (see Mathematical Foundations §4). No gauge term is included here; for gauge coupling, see Mathematical Foundations §4.
Lattice→continuum mapping (overview): In \(d\) spatial dimensions with lattice spacing \(a_{\rm lat}\), the total-variation coefficient maps as \(K’_{\rm TV}=K_{\rm lat}\,a_{\rm lat}^{\,1-d}\) and the phase stiffness as \(\rho_s=\tfrac{3}{2}J\,a_{\rm lat}^{\,2-d}\). RTG typically operates in emergent \(3+1\)D (take \(d=3\) in these maps; see Lattice to Continuum Workbook §6).
5. Observer Dependence
Observer node — Defines relational distances \(r_{io}=\dfrac{2\pi c}{|\omega_i-\omega_o|}\).
\(\Delta\omega^\ast\)-observer — Reference observer with \(\omega_{\rm ref}=\Delta\omega^\ast\approx 1.45\times10^{23}\,\mathrm{s}^{-1}\).
6. Dimensionality & Frequency Regimes
Critical bandwidth — \(\Delta\omega^\ast=(1.45\pm0.08)\times10^{23}\,\mathrm{s}^{-1}\).
| \(\delta\omega/\Delta\omega^\ast\) | Emergent D | Structure |
|---|---|---|
| 0–0.28 | 2 | Planar sheets |
| 0.28–0.70 | 3 | Curved shells (proton) |
| 0.70–1.70 | 4 | 4-D corridors (propagation channels) |
| 1.55–1.70 | 3 or 4 | U(1)^2 phase shells (Gauge Symmetries §4) |
| >1.70 | 5+ | High-D anomalies |
Note: Thresholds carry \(\pm0.02\) systematic in \(\delta\omega/\Delta\omega^\ast\) (scheme choice); see Enriched Geometric Concepts §4 for Cayley–Menger diagnostics.
7. Spin Flips, Energy Conservation & Curvature
For a flip \(\sigma_i\to-\sigma_i\), the \(J\)-term changes by \(\Delta E_J=2J\,\sigma_i\sum_j A_{ij}\,\sigma_j\) (with \(A_{ij}\) as above). Simulations conserve total energy by routing \(|\Delta E_J|\) into the conjugate phase momentum \(\pi_{\phi_i}\) (kinetic compensation), keeping long-time drift small (\(\lesssim 4.3\times10^{-4}\)). A coarse-grained curvature penalty uses a Helfrich-style form \(U_{\rm curv}=\kappa_c\,a_{\rm lat}^2\sum_{\langle ijk\rangle}\left(1-\dfrac{\mathcal R_{ij}+\mathcal R_{jk}+\mathcal R_{ki}}{9}\right)^2\), where \(\kappa_c\) is in MeV·fm and \(a_{\rm lat}\) the lattice spacing. An alternative form \(U_{\rm curv}=\kappa_c\,(1-\mathcal R/3)^2/a_{\rm lat}^2\) (Mathematical Foundations §7) is simpler but less suited to emergent geometry; Helfrich-style is preferred (Enriched Geometric Concepts §6).
8. Emergent Energy & Mass
Total node energy includes kinetic, bond, and curvature contributions. A practical rest-mass estimator is \(m_i=\dfrac{\hbar\omega_i-\sum_j E^{\rm res}_{ij}}{c^2}\) with \(E^{\rm res}_{ij}=\hbar\,|\omega_i-\omega_j|\,\mathcal R_{ij}\); clamp \(m_i\ge0\) to avoid artifacts when coarse-graining. This connects to inertia via the emergent metric (RTG Gravity I §3).
9. Stability & Equilibrium
Equilibrium requires \(\partial H/\partial q=0\) for continuous variables \(q\in\{\omega,\phi\}\). Binary spins achieve stationarity via discrete updates (e.g., cluster flips); maintaining a modest acceptance (\(\approx 0.02\) per 100 ticks in typical runs) stabilises long-time drift.
10. Mathematical Consistency & Scale-Setting
Define \(\tilde J=J/(\hbar\Delta\omega^\ast)\), \(\tilde K=K’/(\hbar\Delta\omega^\ast)\), and the resonance coupling \(g=\bar J/K’\) with \(\bar J\equiv\tfrac{3}{2}J\). A representative two-loop flow uses \(\beta_g(g)\approx 0.72\,g-0.63\,g^2-0.011\,g^3\) with fixed point \(g^\ast\approx1.14\) (scheme-dependent). Example truncations for individual dimensionless couplings: \(\beta_{\tilde J}=-\tilde J+O(\tilde J^3)\), \(\beta_{\tilde K}=-\tfrac12\tilde K\,\tilde J+O(\tilde J^3)\). Absolute \(K’,J,J_{\rm ex}\) are calibrated after fixing \(\Delta\omega^\ast\).
11. Simulation Insights
- CHSH under aligned phase, open gate, optimal angles \((0,\pi/2,\pi/4,-\pi/4)\) with OU noise on \(\Delta\omega\): \(S(\sigma)=2\sqrt{2}\,e^{-\sigma^2}\) where \(\sigma\equiv \mathrm{sd}[\Delta\omega]/\Delta\omega^\ast\).
- Violation threshold \(\sigma_{\rm crit}\approx0.589\); at \(\sigma=0.5\), \(S\approx2.20\).
- Micro-canonical runs show energy drift \(\lesssim 4.3\times10^{-4}\) over 3000 ticks.
- Flip rate typically \(0.02\)–\(0.03\) per 100 ticks (sweet spot); broader \(0.02\)–\(0.30\) with curvature modulus \(\kappa_c\) in MeV·fm (Enriched Geometric Concepts §6).
- Volume scaling: \(L=32\) and \(L=40\) show consistent CHSH within errors.
Logging for reproducibility — Record \(\Delta\phi\), \(\Delta\omega/\Delta\omega^\ast\), \(V=\mathcal R_{12}/3\), the four \(E(\cdot,\cdot)\), CHSH \(S\), angle set, OU \(\tau\) (if used), and long-time energy drift.
12. Applications & Future Research
- Hadrons/nuclei: shell radii fits and flip–curvature stability (Mathematical Foundations §9).
- Quantum information: robust Bell violations for small \(\sigma\) with aligned phases and open gate (Gauge Symmetries §8).
- Cosmology: frequency-sweep red-shift scenarios without dark sectors (Cosmology v2.5 §3).
- Open questions: quantisation of ±i spins, curvature-modulus calibration, higher-D stability, effective gauge content (Gauge Symmetries §§2–4).
13. Conclusion
RTG provides a relational route from oscillatory nodes to emergent spacetime, matter, and interactions. The critical bandwidth \(\Delta\omega^\ast\) and two-loop flow anchor the framework; simulations and continuum mappings connect parameters to observables. Ongoing work targets curvature energetics, quantum spin representation, emergent gauge sectors, and cosmological tests.
See also: RTG Gravity I | RTG Gravity II | RTG Glossary | Two-Loop RG Derivation | Gauge Symmetries | Lattice → Continuum Workbook | Quantum Behaviours | Thermodynamics