Mathematical Foundations of Relational Time Geometry (RTG)

Revision date: 12 Aug 2025 | Authors: Mustafa Aksu, Grok, ChatGPT

Notation guard: All \(\omega,\Delta\omega,\delta\omega\) are angular frequencies (rad·s\(^{-1}\)); convert to Hz via \(f=\omega/(2\pi)\). Exchange regulator \(\sigma_{\mathrm{exch}}\) is distinct from CHSH noise \(\sigma_{\mathrm{noise}}\) (Glossary §2).

1 Preamble

RTG models the universe as a network of oscillatory nodes. No background space or time is assumed; geometry and physics emerge from node relations. Two-loop renormalisation-group (RG) analysis fixes every dimensionless ratio, leaving only calibrations to observables. Monte-Carlo molecular-dynamics (MD) runs reproduce the proton radius \(0.84 \pm 0.01 \,\mathrm{fm}\) and Bell-pair CHSH \(\approx 2.827\), with total-energy drift below \(4 \times 10^{-4}\) over 3000 ticks.

2 Node Properties

Symbol	Definition	Notes (Units)
\(\omega_i\)	Intrinsic frequency	\(E_i = \hbar \omega_i\) [s\(^{-1}\)]
\(\phi_i\)	Phase angle	Carrier for interference via \(e^{i\phi}\) [rad]
\(s_i = \pm i\) (analytic)	Binary spin	Analytic: \(1+s_i s_j = 2\) for anti-aligned \((+i,-i)\), 0 for aligned. Code: \(\sigma_i=\pm 1\), gate \(1-\sigma_i\sigma_j\) opens for opposite \(\sigma\), matching analytic anti-aligned via \(s_i\equiv i\sigma_i\) (see §3 truth table).
\(t_i\)	Intrinsic time	\(t_i = \tilde{\phi}_i / \omega_i\); meaningful only under local phase-locking.

3 Wave-function-Style Resonance

\(\mathcal{R}_{ij} = A_{ij} (1+s_i s_j), \quad A_{ij} = \frac{3}{4}[1+\cos(\phi_i-\phi_j)]\,e^{-(\Delta\omega/\Delta\omega^*)^2}, \quad 0 \le \mathcal{R}_{ij} \le 3\).

Maximum at \(\Delta\phi=0\), analytic spins anti-aligned \((+i,-i)\) so gate=2, and \(\Delta\omega=0\). With code spins \(\sigma=\pm 1\), gate opens for \(\sigma_i \neq \sigma_j\), consistent via \(s_i \equiv i\sigma_i\).

4 Bond Hamiltonian

\[E_{ij} = K’ \frac{|\Delta\omega|}{\Delta\omega^*} + J \mathcal{R}_{ij} – J_{\mathrm{ex}}\cos(\Delta\phi – 2\pi a A_{ij})\,\exp[-(\Delta\omega/\sigma_{\mathrm{exch}})^{2}]\]

\(K’,J,J_{\mathrm{ex}}\) in MeV; default \(\sigma_{\mathrm{exch}}\approx\Delta\omega^*\) (Lattice to Continuum §2). Gauge term included only for explicit U(1) coupling (Gauge Symmetries §2).

5 Two-Loop RG & Critical Bandwidth

\(\Delta\omega^* = (1.45\pm 0.08) \times 10^{23}\,\mathrm{s}^{-1}\).

\(\tilde{J}=J/(\hbar\Delta\omega^*), \tilde{K}=K’/(\hbar\Delta\omega^*), g=\tilde{J}/\tilde{K}\). Two-loop: \(\beta(g) \approx 0.72g – 0.63g^2 – 0.011g^3\), fixed point \(g^* \approx 1.14 \pm 0.02\). With §10 couplings, \(J/K’\approx 0.27\) matches \(g^*\) after finite-size corrections.

6 Dimensionality & Scaling

\(\delta\omega\) = in-cluster sd of \(\omega\) after detrending local mean. ±0.02 sys. on \(\delta\omega/\Delta\omega^*\). See Enriched Geometric Concepts §4 for Cayley–Menger diagnostics.

\(\delta\omega/\Delta\omega^*\)	Emergent D	Structure
<0.28	2	Planar sheets
0.28–0.70	3	Curved shells (proton)
0.70–1.55	4	4-D corridors
1.55–1.70	3 or 4	U(1)2 phase shells (Gauge Symmetries §4)
>1.70	5+	High-D anomalies

7 Spin Flips, Energy Conservation & Curvature

For flip \(\sigma_i\to -\sigma_i\) with bond term \(+J\mathcal{R}_{ij}\):

• \(\Delta E_{\mathrm{bond}} = 2J\sigma_i\sum_j A_{ij}\sigma_j\) (\(J\) in MeV, \(A_{ij}\) dimensionless).
• Kinetic comp.: \(\pi_{\phi i} \leftarrow \pi_{\phi i} – \mathrm{sign}(\Delta E)\sqrt{2M_\phi|\Delta E|}\), \(M_\phi\) in MeV·s².
• Curvature (Helfrich-style): \(U_{\mathrm{curv}} = \kappa_c a^2 \sum_{\langle ijk\rangle} (1 – \frac{\mathcal{R}_{ij}+\mathcal{R}_{jk}+\mathcal{R}_{ki}}{9})^2\), \(\kappa_c\) in MeV·fm. Alt.: \(U_{\mathrm{curv}} = \kappa_c(1 – \mathcal{R}/3)^2/a^2\) (Core Principles §7) — simpler but less geometric.

8 Emergent Energy, Mass & Photons

• Phase mode — net \(\omega\approx 0\), \(\Delta\omega\approx 0\), spins unchanged, \(\omega(k)\approx ck\).
• Photon — spin–anti-spin pair (+i,−i), shared \(\phi\), \(\omega_\gamma\neq 0\), \(\Delta\omega=0\), gate open in analytic ±i, \(E=\hbar\omega_\gamma\).
• Mass — \(m_i=[\hbar\omega_i-\sum_jE^{\mathrm{res}}_{ij}]/c^2\), \(E^{\mathrm{res}}_{ij}=\hbar|\omega_i-\omega_j|\mathcal{R}_{ij}\); clamp \(m_i\ge 0\). Proton: \(\alpha\approx 938\,\mathrm{MeV\cdot fm}^2\) in \(\alpha\sum\mathcal{R}_{ij}/r_{ij}^2\) form (§9 derivation).
• Decoherence — aligned phase, open gate, optimal angles, OU noise on \(\Delta\omega\): \(S(\sigma_{\mathrm{noise}})=2\sqrt{2}e^{-\sigma_{\mathrm{noise}}^2}\), \(\sigma_{\mathrm{crit}}\approx 0.589\).

9 Stability & Equilibrium Examples

Proton (3-node triangle) equilibrates at \(r=0.84\pm 0.01\,\mathrm{fm}\); binding \(=E_{3\text{-node}}-3E_{\text{isolated}}\), \(\approx 48\pm 3\,\mathrm{MeV}\). Mass via \(\alpha \sum\mathcal{R}_{ij}/r_{ij}^2\), \(\alpha\approx 938\,\mathrm{MeV\cdot fm}^2\).

10 Mathematical Consistency & Scale-Setting

\(\tilde{J}=J/(\hbar\Delta\omega^*), \tilde{K}=K’/(\hbar\Delta\omega^*), \tilde{\sigma}_{\mathrm{exch}}=\sigma_{\mathrm{exch}}/\Delta\omega^*\). \(K’/J\approx 3.70\) (\(J/K’\approx 0.27\)) matches \(g^*\approx 1.14\). Flows: \(\beta_{\tilde{J}}=-\tilde{J}+O(\tilde{J}^3)\), \(\beta_{\tilde{K}}=-\frac12\tilde{K}\tilde{J}+O(\tilde{J}^3)\).

Coupling	Value (MeV)	Rel. err.
K′	12.0	±0.5
J	3.24	±0.12
Jex	2.20	±0.08

11 Simulation Benchmarks (Aug 2025)

Observable	Value	Run params
CHSH (\(\sigma_{\mathrm{noise}}=0\))	2.827±0.002	L=32, T=0
CHSH (\(\sigma_{\mathrm{noise}}=0.5\))	2.20±0.02	OU noise, aligned phase
Energy drift	<4.3×10-4	3000 ticks, Δt=5e-5
Flip rate	0.02–0.03 / ≤0.30 (thermostat)	\(\kappa_c=1\,\mathrm{MeV\cdot fm}\) (curvature control, Enriched Geometric Concepts §6)

12 Outlook

• Extend MD to 10⁶-node lattices; test high-D stability.
• Quantise ±i spins (path-integral + Grassmann; Gauge Symmetries §6).
• Cosmological frequency sweeps (Cosmology v2.5 §3).

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References: RTG Gravity I | RTG Gravity II | RTG Glossary | Two-Loop RG Derivation | Gauge Symmetries | Lattice to Continuum | Quantum Behaviours | Thermodynamics