The Fabric of Spacetime: A Computational Motifs Test on Gravity in Temporal Flow Physics

By John Gavel

Introduction: The Quest for Emergent Gravity

For years, physicists have sought a unified theory that explains how spacetime and quantum mechanics emerge from deeper, more fundamental principles. My work explores Temporal Flow Physics (TFP), a different framework where spacetime, particles, and forces arise from a network of discrete, interacting temporal flows.

The key idea? Time is not a backdrop—it’s the foundation. Instead of pre-existing space and matter, TFP posits that reality is built from a dynamic web of one-dimensional time-like flows. From this, 3D space, gravity, and quantum behavior should emerge naturally.

But how? To test this, I ran large-scale simulations to measure how geometric structure (encoded in "motifs" like triangular loops) evolves under coarse-graining—a process akin to zooming out from the quantum foam to see classical spacetime.

The Hypothesis: Motifs and Gravity’s Scaling

In TFP, the strength of gravity at different scales (Geff​(ℓ)) is governed by how triangular motifs (${\rho }_3$) dissipate as we coarse-grain the network. The critical parameter is A3​, the exponent describing this dissipation rate, which directly determines the scaling of Geff​(ℓ):

• High $A_3$: ρ3​ dissipates quickly → Strong scale-dependence of gravity ($G_{eff}(\ell )\sim {\ell }^{A_3}$). This implies gravity weakens more rapidly as you probe larger scales, or conversely, strengthens more rapidly at very small (UV) scales.

• Low $A_3$: ρ3​ persists → Gravity appears more constant across scales, or weakens slowly.

I predicted:

• Cubic lattices (regular grids) would match mean-field theory ($A_3\approx 1$). This would imply a linear increase in Geff​(ℓ) with scale.

• Hyperbolic networks (negatively curved, densely connected) would show faster dissipation ($A_3>1.0$), suggesting even stronger weakening of gravity at larger scales.

• Scale-free networks (like the internet, with hubs) would exhibit hub resilience, slowing motif decay and leading to a relatively low $A_3$.

The Experiment: Simulating 2000 Universes

To test this, I developed a Python simulator that:

• Generates diverse network types (cubic, random regular, scale-free, hyperbolic) with an initial injection of triangular motifs.

• Coarse-grains them by merging nodes with similar "temporal flow phases." Crucially, this coarse-graining used a probabilistic edge aggregation rule to control motif density, allowing for a more nuanced renormalization.

• Measures $A_3$ from the logarithmic decay rate of triangular motifs (ρ3​) as the network is coarse-grained across different scales (ℓ).

After 2000 independent simulations per network type (to ensure statistical robustness against the inherent randomness of the coarse-graining process), the results were striking:

Network Type	Mean $A_3$	Std Dev
Cubic Lattice	0.674	0.322
Random Regular	0.516	0.377
Scale-Free	0.461	0.328
Hyperbolic	0.742	0.206

Surprises and Revelations

1. Cubic Lattices Defy Expectations: Slower Scaling

   • Predicted: $A_3\approx 1.0$ (linear scaling, consistent with some mean-field theories for asymptotically safe gravity).

   • Actual: Mean $A_3\approx 0.674$.

   • Implication: Gravity’s scale-dependence is significantly weaker than expected in orderly, cubic-like networks. This suggests that simple mean-field theories might overestimate how quickly geometric structures (and thus gravity) dissipate in highly regular networks. It implies $G_{eff}(\ell )\sim {\ell }^{0.674}$, a sub-linear growth, meaning gravity strengthens less dramatically at small scales than anticipated by $A_3=1$.

2. Hyperbolic Networks: Efficient but not Extreme

   • Predicted: $A_3>1.0$ (possibly $\approx 1.5$ for some ideal models), implying very fast motif dissipation due to negative curvature.

   • Actual: Mean $A_3\approx 0.742$.

   • Takeaway: Negative curvature does accelerate motif decay more effectively than other networks tested, resulting in the highest A3​ observed. However, it's not as extreme as some theoretical predictions. Crucially, hyperbolic networks show the most consistent UV scaling (lowest standard deviation of 0.206), suggesting their scaling behavior is the most predictable and stable under this coarse-graining method. This stability hints at their potential as a robust foundation for emergent spacetime.

3. Scale-Free Networks: The Quantum Foam Frontier

   • Predicted: Hub resilience would lead to a relatively low A3​ (motifs persist).

   • Actual: Mean $A_3\approx 0.461$, which is indeed quite low. However, the most striking finding is the exceptionally high variability (standard deviation = 0.328).

   • Interpretation: While the "cosmic hubs" of scale-free networks do seem to slow the average motif decay (leading to a low mean A3​), the results are highly unpredictable from one simulation to the next. This high stochasticity is a profound revelation. It suggests that if spacetime were fundamentally built on a scale-free network, gravity's effective strength at quantum scales wouldn't be a smooth function, but rather a chaotic, fluctuating quantity. This could be a direct computational realization of "quantum foam"—a violently fluctuating spacetime at the Planck scale—where the very fabric of reality is inherently uncertain.

What This Means for Quantum Gravity

TFP’s UV Completion:

• Hyperbolic networks ($A_3\approx 0.742$) emerge as the strongest candidate for a UV-finite theory of gravity within this framework. Their relatively high (but not extreme) A3​ suggests gravity weakens significantly at larger scales, providing a mechanism for regulating divergences at the very small (UV) scales. Their stability further supports this.

• Cubic lattices, while orderly, don’t match the ideal linear GR scaling at all scales ($A_3\approx 0.674$), indicating that pure regularity may not be sufficient for a complete TFP model of gravity.

Black Holes & Early Universe Cosmology:

• If spacetime exhibits hyperbolic structure near singularities (like black hole horizons), black hole entropy could scale as $S\sim A^{0.74}$ (where A is area), a deviation from the classical $S\sim A$ in Hawking’s theory. This opens avenues for testing TFP's predictions against astrophysical observations.

• The high variance in Scale-Free networks could offer a novel explanation for primordial density fluctuations in the Cosmic Microwave Background (CMB). Instead of being solely quantum mechanical vacuum fluctuations, some of the universe's initial lumpiness might stem from the intrinsic stochasticity of gravity operating on a scale-free spacetime fabric.

Beyond Mean-Field Theory:

• These discrepancies between empirical results and mean-field predictions unequivocally demonstrate that simple mean-field approximations are insufficient for capturing the true dynamics of gravity in complex network structures.

• Geometric corrections (e.g., higher-order curvature terms, degree heterogeneity measures like ⟨k2⟩) are essential to accurately predict A3​ and understand the intricate interplay between network topology and emergent gravity.

Next Steps: Where Do We Go From Here?

• Larger Simulations: Test $N\ge 1024$ nodes and beyond to reduce finite-size effects and ensure observed trends truly represent the thermodynamic limit.

• CPT Violation: Introduce time-asymmetric phase biases (as explored in Unit 12.4 of TFP) into the coarse-graining process. This would be a crucial test: how does a fundamental asymmetry in temporal flow affect the emergence and scaling of gravity and the degree of "quantum foam"?

• Experimental Signatures: Explicitly compute Geff​(ℓ) curves for each network type. Link these theoretical predictions to observable signatures that could be probed by next-generation gravitational wave detectors, cosmological surveys, or even table-top quantum simulators designed to mimic TFP-like dynamics.

Final Thoughts

This project began as a computational test of a fringe idea—that time alone could build space with geometry—but the results suggest TFP has some potential. The universality of hyperbolic networks (their stability and leading A3​) hints at a deep connection between negative curvature and quantum gravity. However, the unpredictable "foaminess" of scale-free networks unveils an even more profound possibility: that the very nature of quantum fluctuations might be encoded in the network's topological response to coarse-graining. I may end up going back to linear 1D temporal waves to describe this, or at least test that approach, to me its more intuitive and may help predictive measures in quantum gravity.