Emergent Gravity and Structure Formation in a Scale-Invariant Frustrated Vacuum

Date: January 14, 2026

Authors: RTG Research Group

Abstract

We demonstrate numerically that scale-invariant (logarithmic) vacuum forces enable stable cosmological structure formation in the Relational Time Geometry (RTG) framework, while linear (Hookean) forces exhibit parametric instability. Using a test system of \( N=175 \) nodes with \( \beta_1=8 \) topological matter, we show that logarithmic vacuum potentials \( V \propto \ln^2(r/r_0) \) remain stable under 40-100% metric expansion, enabling gravitational clustering at coupling \( G=2.0 \). In contrast, Hookean potentials \( V \propto (r-r_0)^2 \) produce runaway expansion (+136%) despite attractive gravity. This validates RTG’s geometric foundation: the frustration \( z = \ln(r_{em}/r_{geo}) \) is not merely convenient but structurally necessary for cosmological consistency. We discuss implications for vacuum energy screening and future tests at larger scales.

1. Introduction

1.1 Background

Relational Time Geometry (RTG) posits that spacetime emerges from a frustrated simplicial network where nodes attempt to maintain phase-locked resonance. The “electromagnetic length” \( r_{em} \propto 1/\Delta\omega \) represents the natural beat distance, while \( r_{geo} \) is the actual embedded geometric distance. Frustration \( z = \ln(r_{em}/r_{geo}) \) quantifies the mismatch.

1.2 The Cosmology Problem

Previous RTG cosmology simulations using Hookean bond potentials \( F \propto r – r_{target} \) failed to produce bound structures even at strong gravitational coupling \( G \approx 50 \). We identify two causes: (1) numerical bugs (self-force terms, COM drift), and (2) fundamental parametric instability of linear forces under expansion. This work addresses both issues and demonstrates that logarithmic forces \( F \propto z/r \) are inherently stable.

1.3 Scope and Limitations

System Scale: Our simulations use \( N=175 \) nodes (\( \beta_1=8 \) topology) representing a nuclear-scale system (\( \sim 10 \) fm). Extrapolation to cosmological scales (\( 10^{26} \) m) requires validation of continuum limit behavior.

Statistical Validation: Results shown use single random initialization (seed=42). Full statistical validation with \( N \ge 5 \) seeds is ongoing.

Gravity Model: We use a phenomenological harmonic attraction between topological defects (mass \( m \propto z_h^2 \)). Connection to Einstein field equations requires further theoretical development.

2. Theoretical Framework

2.1 The Frustration Field

The Emergent Simplicial Manifold (ESM) is defined by the minimization of “Geometric Frustration” (\( z \)). This frustration quantifies the mismatch between the electromagnetic target distance (\( r_{em} \)) demanded by resonance and the actual geometric distance (\( r_{geo} \)) allowed by 3D constraints.

\[ z = \ln\left(\frac{r_{geo}}{r_{em}}\right) \]

The vacuum energy density is:

\[ E_{vac} = \frac{1}{2} K_{vac} \sum z^2 = \frac{1}{2} K_{vac} \sum \left[ \ln\left(\frac{r_{geo}}{r_{em}}\right) \right]^2 \]

2.2 Derivation of the Force Law

The microscopic force acting on a node is the negative gradient of this potential with respect to distance (\( r_{geo} \)).

Linear Approximation (Hooke’s Law): For small frustrations (\( |z| < 0.1 \)), the force reduces to:

\[ F_{Hooke} \approx -K_{vac} (r_{geo} – r_{em}) \]

Exact Form (Logarithmic Law): The exact gradient is:

\[ F_{Log} = -\frac{\partial E}{\partial r_{geo}} = -K_{vac} z \frac{\partial z}{\partial r_{geo}} = -K_{vac} \frac{z}{r_{geo}} \]

Key Difference: The logarithmic force is scale-invariant (depends on the ratio \( r_{geo}/r_{em} \)) and self-limiting (\( F \to 0 \) as \( r \to \infty \)). Linear forces grow unbounded with displacement.

2.3 Stability Analysis

During cosmological expansion, targets stretch continuously: \( r_{em}(t) = r_{em}(0) \times a(t) \). For a force law \( F(r, r_{target}) \) to remain stable, it must satisfy:

  • Bounded restoring: \( |F| \) must not grow faster than displacement
  • No parametric resonance: Periodic target changes must not amplify oscillations

Logarithmic forces satisfy both conditions; linear forces violate the second, creating runaway oscillations under expansion.

3. Methods

3.1 System Initialization

Substrate: Algorithmically grown cluster (\( N=175 \) nodes, 356 edges, 200 faces, \( \beta_1=8 \)) with reduced irreducible frustration (\( \eta_{irr} = 0.403 \) vs FCC baseline 0.75-0.80).

Initial State: Random 3D embedding (Gaussian, \( \sigma=5 \)), relaxed for 50 iterations using logarithmic vacuum forces (\( G=0 \)) to establish equilibrium. Initial \( R_g = 5.26 \).

Topological Matter: Harmonic modes \( z_h \) from Hodge decomposition serve as gravitating “charges” (mass \( m \propto z_h^2 \)). Gravity cutoff: \( |z_h| > 0.05 \) defines 42-59 participating edges.

3.2 Cosmological Stability Test

Protocol: Expand universe from \( a(0)=1.0 \) to \( a(t)=1.4 \) or \( 2.0 \) over 20 timesteps. At each step:

  1. Scale targets: \( r_{target}(t) = r_{em} \times a(t) \)
  2. Apply vacuum forces (Hooke or Log) + gravity (\( G=2.0 \))
  3. Relax for 50 iterations (Euler integrator, learning rate 0.02)
  4. Recenter to prevent COM drift
  5. Measure \( R_g \) (radius of gyration weighted by \( z_h^2 \))

Comparison: Three runs from identical initial state (verified: max difference \( < 10^{-6} \)):

  • Hooke @ 1.4× expansion
  • Log @ 1.4× expansion (control)
  • Log @ 2.0× expansion (stress test)

3.3 Gravity Baseline

To isolate gravity’s effect, we ran \( G=0 \) baseline (vacuum forces only) vs \( G=2.0 \) (with gravity). Both use logarithmic vacuum forces.

4. Results

4.1 Parametric Instability of Linear Forces

Table 1: Cosmological stability comparison (all start at \( R_g = 5.26 \))

Force LawExpansionFinal \( R_g \)ChangeStatus
Hooke1.4×12.41+136%UNSTABLE
Log1.4×3.55-32%Stable (clustering)
Log2.0×4.01-24%Stable (clustering)

Finding: Hookean forces exhibit catastrophic parametric resonance, causing 136% expansion despite attractive gravity (\( G=2 \)). Logarithmic forces remain stable across both moderate (1.4×) and extreme (2.0×) expansion.

4.2 Vacuum Resistance to Expansion

Baseline (\( G=0 \)): Even without gravity, logarithmic vacuum shows 23% compression in comoving coordinates (\( R_g/a \): 5.26 → 4.00 at \( a=1.4 \)). This “vacuum viscosity” resists Hubble flow.

With Gravity (\( G=2 \)): Compression increases to 50% (\( R_g/a \): 5.26 → 2.50), demonstrating synergy between vacuum resistance and gravitational attraction.

4.3 Clustering Threshold

Previous failures: Hookean simulations showed no clustering even at \( G=50 \) with 2× expansion (confounded by bugs + parametric instability).

Current success: Logarithmic forces enable clustering at \( G=2 \) (25× weaker coupling). However, this comparison is not direct due to:

  • Previous bugs now fixed (self-force, COM drift)
  • Different expansion rates tested (1.4× vs 2.0×)
  • Hooke’s parametric instability vs Log’s stability

Interpretation: The threshold reduction stems from force law stability, not just parameter choice. A controlled parameter sweep is needed to quantify \( G_{critical}(\text{expansion\_rate}) \).

5. Discussion

5.1 Theoretical Necessity of Logarithmic Forces

Our results demonstrate that logarithmic vacuum forces are not merely a theoretical convenience but a structural necessity. The frustration \( z = \ln(r/r_{target}) \) naturally produces \( F \propto z/r \), which:

  1. Is self-consistent: The force law follows directly from the energy definition \( E = (1/2)\sum z^2 \)
  2. Is scale-invariant: Matches the geometric nature of \( z \) (depends on ratios, not differences)
  3. Is dynamically stable: Self-limiting behavior (\( F \to 0 \) as \( r \to \infty \)) prevents parametric resonance

Any attempt to approximate this with linear forces (valid only for \( |z| < 0.1 \)) breaks down under cosmological conditions where frustrations reach \( |z| \sim 0.5-1.0 \).

5.2 Vacuum Viscosity Phenomenon

The \( G=0 \) baseline reveals unexpected “vacuum viscosity”: the cluster compresses 23% in comoving coordinates despite no attractive forces. This arises because logarithmic forces resist both compression AND expansion—a qualitatively different behavior from Hookean springs that only resist compression.

Speculation on Dark Matter: If real vacuum exhibits this property, it would manifest observationally as additional gravitational attraction beyond visible matter. However, connecting this N=175 toy model to galactic-scale observations (\( \sim 10^{11} \) stars) requires:

  • Establishing N-scaling behavior (does effect persist at \( N \to \infty \)?)
  • Quantifying effective “vacuum mass” contribution
  • Comparing to rotation curve data

This remains speculative pending larger-scale simulations.

5.3 Limitations and Future Work

Current Limitations:

  • Single realization: Results use one random seed; statistical validation with \( N \ge 5 \) seeds needed
  • Nuclear scale: \( N=175 \) nodes \( \sim 10 \) fm; extrapolation to cosmological scales (\( 10^{26} \) m) unvalidated
  • Phenomenological gravity: Harmonic attraction (\( F \propto m^2/r^2 \)) is ad hoc; connection to Einstein equations required
  • No quantum effects: Classical network; quantum node fluctuations not included
  • Fixed topology: \( \beta_1=8 \) topology unchanging; realistic cosmology needs dynamic topology

Next Steps:

  1. Parameter sweep: Map \( G_{critical} \) vs expansion rate phase diagram
  2. Statistical validation: \( N \ge 10 \) runs with different seeds, report mean ± std
  3. N-scaling study: Test \( N = 500, 1000, 5000 \) to approach continuum limit
  4. Distance-dependent gravity: Test \( E_{int}(\text{separation}) \) to validate \( r^{-2} \) scaling
  5. Comparison to ΛCDM: Quantify differences in structure formation timescales

5.4 Conclusions

We have demonstrated that:

  1. Logarithmic vacuum forces are stable in expanding cosmologies (40-100% expansion)
  2. Linear (Hookean) forces are fundamentally unstable due to parametric resonance
  3. Structure formation occurs naturally at modest gravitational coupling (\( G=2 \)) with logarithmic forces
  4. Vacuum exhibits intrinsic resistance to expansion (“viscosity”), independent of gravity

These results validate RTG’s geometric foundation: the logarithmic frustration \( z = \ln(r_{em}/r_{geo}) \) is not arbitrary but necessary for cosmological consistency. While extrapolation to astronomical scales requires further validation, the framework demonstrates proof-of-principle that emergent gravity from frustrated geometry can produce stable structure formation.

6. Acknowledgments

We thank the RTG Collaboration for extensive computational validation and theoretical discussions. Special acknowledgment to Claude (Anthropic) for critical analysis identifying parametric instability and initial state measurement bugs.

7. Data Availability

Simulation code, datasets, and reproduction scripts available at: [repository link]. Key datasets:

  • grown_cluster_v12_2_gauge_nobreak_auditpack.npz (N=175 substrate)
  • validation_proper_v2.npz (cosmology trajectories)
  • Scripts: cosmology_validation_proper_v2.py, esm_gauge_audit_v12_1_kappa3.py


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