In the RTG framework, the origin of structure is traced to the instability of a perfectly symmetric initial state. This primordial state — the “Void” — is not emptiness but a condition of absolute neutrality: net energy, phase, and spin vanish, and every excitation is canceled by its opposite. Analogies exist in modern physics, such as vacuum fluctuations in quantum field theory (Furry & Oppenheimer 1934; modern quantum vacuum studies).
Exact neutrality is inherently unstable. A perfectly balanced state contains within it the seeds of spontaneous symmetry breaking. In RTG this instability manifests as the division into node pairs — fundamental oscillators carrying intrinsic frequency, phase, and binary spin. Each pair is created as a balanced unit (+i and −i), conserving a “spin charge” (total s = 0), so the global sum remains zero. This resembles a conserved quantum number, analogous to baryon number or lepton number in particle physics.
Contents
Random but Symmetric Distribution
The first nodes are distributed not in space–time (which does not yet exist), but in frequency–phase–spin space. Their distribution is symmetric in total — every +i spin is matched by a −i partner — but random in placement. This randomness produces locally uneven concentrations: in some frequency–phase regions +i spin nodes dominate, while in complementary regions their −i counterparts dominate. The total balance is preserved, but the pattern is heterogeneous, like statistical fluctuations around a mean-zero distribution.
If one seeks an analogy to “matter–antimatter,” in RTG the reverse-spin structures play that role: mirror-symmetric but not annihilating, instead coexisting as complementary populations in the resonance network. Crucially, at every region of frequency–phase space there exists some mixture of all node types; no region is purely “matter” or purely “antimatter.” What emerges is a globally symmetric but locally fluctuating relational field.
Resonant Interaction
Once formed, nodes interact through resonance, a drive to reduce mismatch in frequency and phase. The governing relation is the RTG kernel, motivated by symmetry, bandwidth constraints, and positive-definiteness:
[
\boxed{\mathcal{R}{ij} = \tfrac{3}{4}[1 + \cos(\Delta\phi{ij})](1+s_i s_j)\exp!\Big[-\big(\tfrac{\Delta\omega_{ij}}{\Delta\omega^*}\big)^2\Big]}
]
– The cosine term reflects U(1) phase symmetry (favoring alignment).
– The exponential expresses suppression of mismatched frequencies, derived from the finite correlation bandwidth Δω* (RG v1.3.1 fixed point).
– The spin gate ensures pair complementarity: only opposite-spin pairs (+i, −i) open the resonance channel (Forces v1.1 §2).
The corresponding relational energy is minimized as
[
E_{ij} \;\sim\; -J \, \mathcal{R}_{ij},
]
with J the resonance strength. Thus coherent alignments are energetically favored, and networks evolve naturally toward states of maximal resonance and minimal stress.
Physical Implications
This mechanism illustrates a universal principle: systems evolve toward lower effective energy by maximizing coherence. Just as coupled oscillators synchronize to dissipate less energy (Kuramoto-type dynamics), nodes in RTG synchronize to reduce relational stress. Globally, the sum remains zero — the universe’s energy budget is conserved — but locally, resonance clusters form, stabilizing into trions, shells, or higher-dimensional “corridors.”
Such resonance-driven structures underpin multiple RTG predictions. For example, bandwidth-driven clustering explains structure formation without invoking dark matter, and high-δω black holes evaporate faster via 4-D corridor effects (Cosmology v2.8 Appendix B), producing signatures distinct from Hawking evaporation in GR.
Physics Justification
The Void’s instability can be linked to quantum vacuum fluctuations (virtual pairs arising from “nothing,” cf. arXiv:hep-th/0503203) and RG instabilities (critical bandwidth Δω* as fixed point, RG v1.3.1). Node pairs conserve total spin charge (like matter–antimatter), while local randomness introduces fluctuations (cosmological analog of primordial density perturbations).
The kernel itself encodes three essential physical principles:
1. Cos Δφ — U(1) phase invariance, ensuring constructive interference dominates.
2. Exp[-(Δω/Δω*)²] — finite correlation length, bounding influence in frequency space.
3. (1 + s_i s_j) — spin gating, enforcing complementary symmetry (Gauge Symm v1.4).
import numpy as np, matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
# Toy node pair from Void
nodes = np.array([[0,0,0], [1,0,0]]) # +i at origin, -i at (1,0,0)
fig = plt.figure(); ax = fig.add_subplot(111, projection='3d')
ax.scatter(nodes[:,0], nodes[:,1], nodes[:,2], c=['blue','red'], s=100)
ax.plot(nodes[:,0], nodes[:,1], nodes[:,2], 'g--', label='Resonance link')
ax.set_title('Node Pair from Void (+i blue, -i red)')
ax.legend(); plt.savefig('node_pair.png')
Change Log
| Version | Date | Main updates |
|---|---|---|
| 1.1 | 2025-08-13 | Revised for rigor: tied instability to vacuum fluctuations and RG fixed points; clarified spin-charge conservation; motivated kernel terms (U(1), Δω*, spin gate); added explicit energy minimization; added node pair plot/code; linked to corpus (Forces v1.1, RG v1.3.1, Gauge Symm v1.4, Cosmology v2.8). |