Enhancing Speculations on Dimensional Emergence in RTG

Relational Time Geometry (RTG) posits a universe built from oscillatory nodes—each with frequency ω, phase φ, and binary spin s—where spacetime isn’t a priori but emerges from resonance relations. Dimensions “appear” as effective structures when node clusters achieve coherent stability at certain frequency-spread thresholds (x = δω / Δω* ≈ 0.28, 0.70, 1.70). What are they exactly? In RTG, dimensions are relational attractors: the number of independent “axes” along which resonance can propagate without decohering, defined by the rank of the structure tensor Sμν from weighted bonds. They represent the fabric of reality as a dynamic web of coherence, not fixed geometry—speculatively, a resonance “scaffold” that self-organizes to minimize energy while maximizing stable configurations.

To enhance these ideas, we combine prior notes (coherence attractors, rerouting entropy, RG crossovers) with fresh speculations grounded in RTG mechanics and analogies from quantum gravity (e.g., spacetime from entanglement, why 3D is favored for stability). This paints reality as an emergent resonance network, where dimensions optimize information flow and stability.

1. Why Dimensions Appear: Resonance as Dimensional “Unlocks”

  • 2D sheets: x < 0.28
  • 3D shells: 0.28 ≤ x ≤ 0.70
  • 4D corridors: 0.70 ≤ x < 1.55
  • U(1)2 band: 1.55 ≤ x ≤ 1.70 (triplet coherence, SU(3)-like for small ε)
  • Higher-D / fractal: x > 1.70

U(1)2 as constraint θ1 + θ2 + θ3 = const reduces to two U(1) phases—per Gauge §4, enabling SU(3) approx.

Refined: Dimensions are the number of stable resonance axes in the relational web—fabric of reality as a coherence graph, where D optimizes entropy vs. decoherence. Mathematically, Deff ≈ rank of Sμν where eigenvalues λa > threshold from resonance variance; e.g., exp[−x²] suppression in kernel weakens at x≈0.28, allowing multi-axis λa.

2. How Dimensions Appear: Rerouting and RG Attractors

Curvature complements dimensions: post-threshold, variance stresses bonds, forcing rerouting (phase/spin adjustments) to relieve overload without breaking coherence. This “bends” the graph, emerging as curvature via Ucurv penalty:

U_curv = κ_c a_lat² ∑_{ijk} (1 - (ℜ_{ij} + ℜ_{jk} + ℜ_{ki})/9)²

Rerouting via spin flips (ΔE ~ 2J σ ∑ A σ) opens/closes gates to relieve stress. These mechanisms are resonance-level equivalents of elastic deformations in spacetime. Speculatively, they mirror induced gravity via fluctuation stabilization (cf. Sakharov models).

Thresholds as RG crossovers: The two-loop βg(g) fixed point (g* ≈ 1.14) sets Δω*, but crossings are attractors where entropy (log gated configs) > decoherence cost (∼ −log exp[−x²]), unlocking rank(Sμν). Crossovers near βg(g*) ≈ 0 where entropy > decoherence, stabilizing higher rank Sμν.

Prediction: Deff jumps when rank(Sμν) increases, with entropy S ~ log(gated configs) > decoherence ~ −log(exp[−x²]). Test: Eigen gaps Δλa < 0.1 max λ near x=0.28; require corr(Ucurv, flip rates) > 0.8 with p < 0.05 over ensembles.

3. Enhancing Speculations: Pathways Forward

To deepen understanding:

  • Simulate Tensor Rank: Code to track Deff vs x (enhanced with RTG kernel):
import numpy as np
import matplotlib.pyplot as plt

def d_eff_from_tensor(S, x, thresh_base=0.01):
    eigvals = np.linalg.eigvalsh(S)
    deco_var = np.exp(-x**2)
    thresh = thresh_base * deco_var * np.max(np.abs(eigvals))
    return np.sum(np.abs(eigvals) > thresh)

x_vals = np.linspace(0, 2, 100)
D_eff = []
for x in x_vals:
    w = np.exp(-x**2)
    S = np.diag([1, 1 + w*x, 1 + w*2*x, 1 + w*3*x])
    D_eff.append(d_eff_from_tensor(S, x))

plt.step(x_vals, D_eff)
plt.xlabel('x = δω / Δω*')
plt.ylabel('Effective Dimensional Rank')
plt.title('Speculative D_eff vs x')
plt.grid()
plt.savefig('d_eff_vs_x.png')
  • Track Entropy vs x: Visualize entropy peaks at thresholds, suggesting dimensions as entropy maxima for coherence:
import numpy as np
import matplotlib.pyplot as plt

def entropy_from_configs(N_nodes, x, gate_prob=0.5):
    open_gates = np.random.binomial(N_nodes*(N_nodes-1)//2, gate_prob)
    deco = np.exp(-x**2)
    S = np.log(open_gates + 1) * deco
    return S

x_vals = np.linspace(0, 2, 100)
S = [entropy_from_configs(100, xi) for xi in x_vals]
plt.plot(x_vals, S)
plt.xlabel('x = δω / Δω*')
plt.ylabel('Entropy (mock)')
plt.title('Entropy vs Disorder')
plt.axvline(0.28, ls='--', label='2D→3D')
plt.axvline(0.70, ls='--', label='3D→4D')
plt.legend(); plt.grid()
plt.savefig('entropy_vs_x.png')
  • Further Work:
    • Simulate eigenvalue gap Δλ at threshold to detect Sμν rank jumps.
    • Correlate Ucurv peaks with flip rates near thresholds to quantify rerouting.
    • KS test vs uniform Δφ: p > 0.05 accepts uniformity; p < 0.05 = non-uniform, fold into systematics.
    • Simulate entropy S vs x to confirm peaks at thresholds, using Nconfigs ~ binomial(open gates) * exp[−x²].

This model re-frames dimensions as dynamic coherence equilibria: resonance axes activated by variance and stabilized by entropy. With kernel suppression, gated spins, and curvature re-routing all playing roles, RTG gives a rich arena to probe the emergence of geometry from oscillatory foundations.


Scroll to Top