RTG → Geometry → GR

Version: v2.2 (Gold Standard, Pre-Axiom)
Date: 2025-11-25
Status: Conceptual and structural foundation, ready to be turned into Axiom Set v2.0.

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0. Core Idea in One Sentence

Relational Time Geometry (RTG) starts from purely relational oscillator nodes (frequencies, phases, spins) on a graph, with no spacetime assumed. Through a sequence of dynamical bifurcations and algebraic lifts, it produces:

• a complex structure via Neimark–Sacker (NS) bifurcation,
• a quaternionic/SU(2) frame field via frustration,
• a pre-geometric structure tensor,
• tetrads, Lorentzian signature, torsion and curvature,
• a continuum manifold as an inverse limit,
• and a low-energy Einstein–Hilbert effective theory.

Spacetime is not the starting point; it is the output.

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Phase I – Micro Ontology: Resonant Nodes (No Spacetime)

Status: Solid (simulated, conceptually clear)

Primitive entities:

• Nodes \( i \) with:
  • frequency \( \omega_i \in \mathbb{R}_+ \)
  • phase \( \phi_i \in S^1 \)
  • spin label \( s_i \in \{+i,-i\} \)
• Resonance kernel \( \mathcal{R}_{ij}[\omega,\phi,s] \)
• Couplings \( K_{ij} \), phase shifts \( B_{ij} \)

Schematic evolution:

\[\dot \phi_i = \omega_i + \sum_j K_{ij}\, \mathcal{R}_{ij}\, \sin(\phi_j – \phi_i – B_{ij})\]

No coordinates, no metric, no manifold. Just relational dynamics.

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Phase II – NS Bifurcation: Emergence of Complex Structure

Status: Proven for small graphs (K₃, K₄), scaling to large N = open RG problem

When coupling exceeds a critical value \( K_c \):

• A fixed point loses stability.
• A Neimark–Sacker (NS) bifurcation creates an invariant 2‑torus \( T^2 \).
• This gives an emergent complex structure \( \mathcal{J} \) with \( \mathcal{J}^2 = -\mathbf{1} \).

Conceptually:

\[\mathbb{R} \longrightarrow \mathbb{C}\]

Key scales from simulation:

• NS critical bandwidth: \( \Delta\omega^* \approx 1.45 \times 10^{23}\,\text{rad/s} \) (with uncertainty)
• Base lattice spacing: \( a_0 \approx 0.08\,\text{fm} \)

Emergent “speed of light”:

\[c \equiv v_{\text{lat}} = a_0 \Delta\omega^\ast\]

Here \( v_{\text{lat}} \) is the lattice maximal propagation speed. In RTG, \( c \) is a derived network constant, not an externally imposed background parameter.

Open task: characterize the RG flow of \( K_c(N) \), \( \omega(\mu) \), \( \Delta\omega(\mu) \) for large graphs.

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Phase III – Non-Abelian Lift: Frustration → SU(2) / Quaternions

Status: Conceptually clear, partly proven numerically

Triad frustration:

\[F_\triangle = (\phi_{ij} + \phi_{jk} + \phi_{ki}) \mod 2\pi \neq 0 \]

implies:

• No consistent global U(1) phase assignment.
• An obstruction that forces a non-Abelian lift to SU(2).

We associate to each coarse region a unit quaternion:

\[q_i(t) \in \mathbb{H} \simeq SU(2) \]

This gives an internal 4‑dimensional algebraic structure:

• 1 scalar direction (real part)
• 3 imaginary directions (like \( i, j, k \))

Division-algebra ladder (rungs):

• \( k = 0 \): \( \mathbb{R} \) – trivial
• \( k = 1 \): \( \mathbb{C} \) – NS bifurcation, complex structure
• \( k = 3 \): \( \mathbb{H} \) – triadic frustration, SU(2) regime
• \( k = 7 \): \( \mathbb{O} \) – conjectured octonionic high‑energy sector

This ladder underlies RTG’s story about dimensionality and algebra.

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Phase IV – Pre-Geometric Structure Tensor and Tetrads

Status: Conceptually solid; definitions now non‑circular

We must build geometry from pre-geometric quantities only, without assuming spacetime coordinates.

4.1 Phase-Geometry Displacement (Internal Indices Only)

For each edge \( \langle ij \rangle \) and internal direction \( a \in \{0,1,2,3\} \), define:

\[\xi_{ij}^{(a)} := \frac{v_{\text{lat}}}{\omega_i} (\phi_j – \phi_i)\,\hat{n}_{ij}^{(a)}\]

where:

• \( v_{\text{lat}} = a_0 \Delta\omega^* \) is the emergent lattice velocity.
• \( \hat{n}^{(a)}_{ij} \) are four orthogonal internal unit directions, derived from the quaternionic orientation \( q_i \)
  
  (real part and three imaginary axes).

Indices \( a,b \) are internal, not spacetime.

4.2 Pre-Geometric Structure Tensor \( S_{ab} \)

\[S_{ab} = \frac{1}{\mathcal{N}} \sum_{\langle ij \rangle} w_{ij}\, \xi_{ij}^{(a)} \xi_{ij}^{(b)}, \quad w_{ij} = \mathcal{R}_{ij}\]

This is a symmetric \( 4 \times 4 \) tensor in internal index space.

4.3 Eigendecomposition, Density, and Tetrads

\[S_{ab} = \sum_{A=0}^3 \lambda_A\, v^{(A)}_a v^{(A)}_b\]

Let \( \rho(x) \) be the coarse-grained node density (nodes per emergent volume). We tie the tetrad scale to density via a conformal factor:

\[e^A_{\ \mu}(x) = \Omega(x)\,\sqrt{|\lambda_A(x)|}\,v^{(A)}_\mu(x), \quad \Omega(x) \propto \rho(x)^{-1/3}\]

Thus:

• Higher node density \( \rho \) → smaller effective length scale.
• Lower density → larger length scale.

The conformal factor \( \Omega(x) \) is therefore physically the density-determined scale factor, linking thermodynamics (node density) to geometry (metric size).

At this stage, tetrad indices \( A \) are internal; \( \mu \) will become a spacetime index once the manifold structure is in place (Phase IX).

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Phase V – Signature and Dimensionality

Status: Conjectural but structured; clearly marked as such

5.1 Higgs vs Goldstone and “Radial vs Angular” Modes

The NS bifurcation yields:

• Radial mode:
  • controls amplitude of motion on the torus,
  • stiff / unstable direction,
  • Higgs-like amplitude mode (massive).
• Angular modes:
  • motion along the torus,
  • soft / marginal directions,
  • Goldstone-like phase modes (effectively massless).

In RTG’s interpretation:

• radial mode ↦ time-like direction,
• angular modes (after SU(2) lifting and ladder structure) ↦ space-like directions.

5.2 Signature from Stability – Conjecture 1

Linearize dynamics around the NS torus with Jacobian \( J \). Then:

Radial eigenvalue:
\[\lambda_0 \approx 1 + \gamma \Delta t, \quad \gamma < 0 \]

Angular eigenvalues:
\[\lambda_{1,2,3} \approx 1 + \alpha_j \Delta t, \quad \alpha_j \approx 0\]

Heuristically:

• Radial instability ↦ negative principal eigenvalue in \( S_{ab} \) ↦ time-like direction.
• Angular marginality ↦ positive eigenvalues in \( S_{ab} \) ↦ space-like directions.

Conjecture 1 (Signature):

For the k = 3 quaternionic ladder regime: \[\lambda_0 < 0, \quad \lambda_{1,2,3} > 0 ⇒ \eta_{AB} = \mathrm{diag}(-1, +1, +1, +1)\]

This conjecture is to be tested by:

• Floquet / normal-form analysis of the NS dynamics,
• Direct eigen-analysis of \( S_{ab} \) in RTG simulations.

5.3 From 2D Torus to 3D Space – Conjecture 2

A minimal NS bifurcation (e.g. on K₃) gives a 2D torus \( T^2 \) with two angular directions. RTG needs three spatial dimensions.

Proposed RTG mechanism:

• The k = 3 ladder rung (quaternionic regime) provides three independent angular directions via SU(2) generators.
• In large frustrated graphs, the pre-geometric structure tensor \( S_{ab} \) generically has rank 4, with one negative and three positive eigenvalues.

Conjecture 2 (3D space):

In the quaternionic/frustrated regime, rank(S_{ab}) = 4 generically, with one negative and three positive principal values, encoding 1 time and 3 space directions.

This is also to be validated numerically by studying the spectrum of \( S_{ab} \) for large graphs.

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Phase VI – Metric and Connection

Status: Standard differential geometry, contingent on Phases IV–V

Given:

• tetrads \( e^A_{\ \mu} \),
• signature \( \eta_{AB} = \mathrm{diag}(-1,+1,+1,+1) \),

we define:

Metric:

\[g_{\mu\nu} = \eta_{AB}\, e^A_{\ \mu} e^B_{\ \nu}\]

Spin connection:

\[A_\mu = q^{-1} \partial_\mu q\]

This is an SU(2) (or, in appropriate representation, SO(3,1)) gauge connection on the emergent manifold.

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Phase VII – Torsion: Micro Nonzero, Macro Vanishing via Energy Minimization

Status: Conceptually strong; energetic mechanism specified

At the microscopic lattice level, frustration \( F_{\triangle,i} \) at site \( i \) produces discrete torsion:

\[T^\lambda_{\ \mu\nu}(i) \sim \frac{1}{a_{\text{lat}}^2} \epsilon^{\lambda\mu\nu\rho} F_{\triangle,i,\rho}\]

This is analogous to Einstein–Cartan: torsion couples to spin density.

7.1 Frustration Energy and Macroscopic Torsion

Instead of relying on “random cancellation”, RTG uses an energetic argument:

Define a coarse-grained torsion field \( \bar T^\lambda_{\ \mu\nu}(x) \).

Define a frustration energy functional:
\[E_{\text{frust}} \propto \int (\bar T^\lambda_{\ \mu\nu} \bar T_\lambda^{\ \mu\nu})\, dV\]

The network relaxes dynamically to minimize \( E_{\text{frust}} \).

• The macroscopic state with \( \bar T = 0 \) is the ground state.
• Any nonzero large-scale torsion costs energy proportional to volume and is suppressed.

Thus:

• Microscale: torsion is real and important (spin, particle structure, mass gap).
• Macroscale: the energy-minimizing configuration is torsion-free, i.e. the Levi–Civita connection.

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Phase VIII – Holonomy and Curvature

Status: Standard once connection exists

Given a spin connection \( A_\mu \):

Holonomy around a loop:

H(\partial P) = \mathcal{P} \exp\left(\oint_{\partial P} A_\mu dx^\mu\right)

Curvature:

R_{\mu\nu} = \partial_\mu A_\nu – \partial_\nu A_\mu + [A_\mu, A_\nu]

In tetrad indices, this yields \( R^{AB}_{\ \ \mu\nu} \). We can define observables such as a “holonomy gap” (spread of \( \mathrm{Tr}(H) \) over loops) as a diagnostic of curvature and torsion fluctuations.

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Phase IX – Continuum Manifold as Inverse Limit

Status: Mathematically standard framework; convergence conditions to be tested numerically

RTG defines the continuum spacetime manifold \( \mathcal{M} \) as the inverse limit of refining graphs:

\mathcal{M} = \varprojlim (G_0 \leftarrow G_1 \leftarrow G_2 \leftarrow \cdots)

• Graphs \( G_n \) have lattice spacing \( a_n = a_0 / 2^n \).
• Each refinement step splits nodes into subnodes with consistency conditions.

Convergence conditions (schematic):

1. Frequency conservation under refinement:
   

   (1 / 2^d) \sum_{j \in \text{children}(i)} \omega_j^{(n+1)}
   =
   \omega_i^{(n)} + O(a_n^2)
       

   
   
2. Curvature bounded:
   

   \sup_n \|R_{\mu\nu\rho\sigma}(G_n)\|_{L^\infty} < \infty
       

   
   
3. Sobolev regularity:
   
   The fields (like \( \phi^{(n)} \)) satisfy regularity conditions so that derivatives exist in the limit.

If these hold, the inverse limit \( \mathcal{M} = \varprojlim G_n \) exists and carries a smooth differential structure. Numerically, one checks convergence by simulating at \( a_0, a_0/2, a_0/4 \) and looking for stable proton mass, binding energy, and curvature.

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Phase X – Low-Energy Gravity: Einstein–Hilbert Dominant

Status: Effective field theory logic, widely accepted

Once \( (\mathcal{M}, g_{\mu\nu}) \) and a Levi–Civita connection exist at large scales, the effective action can be written as:

S_{\text{eff}} = \int d^4x\,\sqrt{-g}\, \left[ \frac{1}{16\pi G_0}R – \Lambda_0 + \alpha_1 R^2 + \alpha_2 R_{\mu\nu}R^{\mu\nu} + \beta_1 T^2 + \cdots \right]

RTG’s claim:

• For energies \( E \ll \hbar \Delta\omega^* \sim 10\,\text{GeV} \), the Einstein–Hilbert term \( R \) is the dominant contribution.
• Higher-curvature and torsion terms are suppressed by powers of the UV scale (related to \( \Delta\omega^* \) or Planck-like scales).

Thus, GR is the IR limit of RTG.

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Phase XI – Quantum RTG (Sketch)

Status: Programmatic, not yet fully developed

To include quantum effects, RTG promotes node variables to operators on a Hilbert space.

• Define a Hamiltonian \( H \) reflecting the sinusoidal couplings (Kuramoto-like terms) at the quantum level.
• Define a graph-based path integral or partition function:
  

  Z = \int \mathcal{D}[\text{graphs}, \phi_i(t)]\, e^{i S_{\text{nodes}}/\hbar}
      

  
  

In this picture:

• NS bifurcations and frustration become nontrivial saddle points / instantons.
• They contribute to mass gaps and possible torsion anomalies at quantum level.
• “Quantum gravity” in RTG lives on top of the emergent classical GR background created in Phases I–X.

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Phase XII – Phenomenology and the RTG Radion

Status: Predictions and targets

The radial NS mode is a scalar controlling the amplitude (scale) of the emergent geometry.
A characteristic mass scale is:

m_{\text{radial}} \sim \hbar \Delta\omega^*/c^2 \sim \mathcal{O}(10\,\text{GeV})

We name this:

RTG Radion: the scalar mode that controls the overall scale of the emergent geometry (analogous to a radion/dilaton in extra-dimensional theories).

• It couples to the trace of the energy–momentum tensor.
• It is distinct from the Standard Model Higgs, though mixing is not forbidden in principle.
• It provides a concrete phenomenological target: a geometric scalar in the ~10–100 GeV range.

Other possible phenomenological signatures:

• Small corrections to proton charge radius (via micro torsion effects)
• Holonomy gap effects in strong-gravity or high-density regimes
• Subtle cosmological signatures in CMB phase statistics or high-energy spectra

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Phase XIII – What RTG Explains vs What GR Assumes

Feature	GR: usually…	RTG: here…
Spacetime manifold	Assumed as a background	Emerges as inverse limit of graphs
Lorentzian signature	Postulated	Conjecturally derived from NS stability and structure tensor spectrum
Metric \( g_{\mu\nu} \)	Fundamental field	Derived from tetrads built from \( S_{ab} \)
Tetrads / frames	Auxiliary or postulated	Primary geometric object in RTG
Connection (Levi–Civita)	Imposed by setting torsion = 0	Ground state of frustration energy (macro torsion vanishes dynamically)
Spin–torsion coupling	Optional via Einstein–Cartan	Built-in via microscopic frustration and torsion
Cosmological constant	Ad hoc constant (fine-tuning)	Scale-dependent via RG flow of the network
Quantum gravity	External quantization of GR	Emerges from quantized node dynamics and graph path integrals

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Phase XIV – Final Pipeline Summary (v2.2)

Phase I (Micro): (ω_i, φ_i, s_i), R_ij ⇒ (K > K_c) ⇒ NS Bifurcation (T²) Phase II (Algebraic): ℝ → ℂ F_Δ ≠ 0 ⇒ SU(2), q_i ∈ ℍ Phase III (Pre-Geometry): ξ_{ij}^{(a)} ⇒ S_{ab} ⇒ {λ_A, v^(A)} ⇒ e^A_μ(ρ) Phase IV (Signature/Torsion): NS stability ⇒ Conjecture η_{AB} = (-,+,+,+) F_Δ ⇒ T_micro, E_frust ∝ ∫ T² ⇒ T_macro = 0 Phase V (Continuum/Gravity): lim← G_n ⇒ (M, g_{μν}) A_μ ⇒ R^{AB}_{ μν} ⇒ S_EH + corrections Phase VI (Quantum/Phenomenology): Z[graphs] ⇒ quantum GR corrections RTG Radion (m ~ 10 GeV), plus other signals

This completes the RTG → Geometry → GR Wrap-Up v2.2, capturing the current conceptual structure, the solid parts, and the conjectural parts with clear test programs, ready to be distilled into Axiom Set v2.0.