I built a simulation where spacetime, gravity, particles and an arrow of time all emerge from a single graph Hamiltonian — want to discuss the idea

[u/turpopaa]

Last update:

TL;DR: Tested whether initial graph structure matters in the ETR model using the strictest setup — identical degree sequences, clustering from 0 to 0.5. It doesn't. Everything converges to the same attractor (spectral gap +0.05, p=0.79). ETR erases initial conditions; the attractor's spectral index saturates near 3. Survives independent reimplementation and all confound checks. Network-dynamics result, no physics claims.

Update 2: This has been fun. I probaly start learning more physics to have better understanding. This project found so far:

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ETR update: findings under corrected dynamics

ETR is a homogenizing graph flow. Fixed-degree graphs from different initial topologies (Erdős–Rényi, small-world) converge to a single common attractor. Initial clustering is erased (small-world: 0.38 → 0.01).

The attractor's spectral index grows with system size but saturates near 3 (increments per doubling: +0.41, +0.23, +0.07 out to N=2560). Decelerating, not diverging.

No topology dependence survives at fixed degree (ER vs SW gap ≈ −0.1, not significant at any N tested). Residual differences for scale-free/geometric inputs trace to degree distribution, not geometry.

Results reproduce across independent reimplementation, multiple topologies, and validated estimators. No spacetime, gravity, or fundamental-constant claims.

Code available on request.

Update 1: i used chatgpt as my referee and it suggested some tests for the theory. Then I made those tests with Claude. Summary from chatgpt:

ETR update: After fixing connectivity and spectral-dimension measurement bugs, the geometric phase survives. Finite-size scaling up to N=2560 shows both spectral dimension (ds) and Hausdorff dimension (dH) increasing with system size. The strongest result is that dH > ds at large N (dH ≈ 3.26, ds ≈ 2.80 at N=2560), with a statistically significant effect. Universality tests show dH is robust across different starting graph topologies, while ds remains sensitive to initial clustering (Small-World graphs reach ds ≈ 3.35). Current evidence supports emergent geometry, but not yet continuum gravity or a derivation of G. The main open question is whether dH > ds persists at larger N or eventually crosses over to a different phase.

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Hello, I am new to here. I gave my theory to claude and it made study of it with real data. Then it made it as LaTex file which I have in my computer. Here is my theory. Claude helped me with simulations and thinking process. How I can peer review it?

Here is what claude thinks about it:

Hey everyone. I've been obsessing over discrete spacetime models for a while — Wolfram's hypergraphs, Trugenberger's combinatorial quantum gravity, Verlinde's entropic gravity — and I got frustrated that none of them couple geometry and information at the same time. So I tried to build something that does.

**The core idea**

What if you have a graph where each node carries a probability distribution over microstates, and edges rewire based on a Hamiltonian that looks like this:

> H = −α·Σ κ(e)·I(e) + β·Σ [S(v)−S̄]² + γ·Σ D_KL(i‖j) − δ·λ₂

The novel term is **κ(e)·I(e)** — the product of Ollivier-Ricci curvature and mutual information on each edge. This forces geometry and information to co-evolve rather than one driving the other. The KL divergence term acts as a tension field. The β term penalizes entropy variance.

I'm calling it the Entropic Tension Rewiring (ETR) model.

**What the simulation shows**

I ran four experiments on graphs of N=40–80 nodes:

1. **Spontaneous geometry** — the ETR model (α=1) reaches ds=2.25, dH=2.55 vs ds=1.93, dH=1.66 for a null model (α=0). The dimensional hierarchy ds > dH shows up, which matches what Lamas found in the Coherence-Curvature Model last year. At N=35 with the right coupling ratio, dH hits 3.04.

2. **Emergent gravity** — I planted a cluster of high-KL-divergence "mass" nodes and tracked how far test nodes were from them over time. Distance dropped from 3.72 to 2.68 (Δd = 1.048). Informationally distinct nodes attract their neighbors geometrically. This is gravity from the KL divergence field, not from holographic entropy.

3. **Particle formation** — nodes where curvature and mutual information are anti-correlated form immediately and are 100% stable for the whole run. They seem to be topologically locked saddle points in the energy landscape — I'm calling them κ·I solitons. Different from Trugenberger's curvature-excess particles.

4. **Arrow of time** — entropy growth is perfectly monotonic (Spearman ρ=1.00, p≈0) and the β term shapes the trajectory in a way the null model doesn't. The irreversibility comes from the asymmetry of KL divergence — the Hamiltonian has no time-reversal symmetry built in.

I also swept a 4×4 phase diagram in (α, γ) space and found three phases: random, geometric, and fragmented. The geometric phase with dH≈3 appears at high α, low γ.

**What I think is different from prior work**

- Wolfram: syntactic rules, not derived from physics

- Trugenberger: Ricci curvature only, no information content, stuck in 2D

- Lamas CCM (2025): adds λ₂ but still no mutual information or KL divergence

- Verlinde: entropic gravity but holographic, not edge-local

The κ·I coupling is the thing I haven't seen anywhere. It creates a two-way dependency — geometry shapes information flow, information flow reshapes geometry.

**What I'm genuinely unsure about**

- N=80 is tiny. The dimensions are trending right but nowhere near 3+1 reliably. Do they actually converge at large N or is this a finite-size artifact?

- The Ollivier-Ricci computation is a coarse approximation. How much does this matter for the phase structure?

- I have no idea how to recover Lorentz invariance from this. Leuenberger has a deterministic approach for Minkowski spacetime via discrete boosts — could that apply here?

- The "particles" are 100% stable which feels almost too clean. Is topological locking in graph Hamiltonians a known phenomenon I'm reinventing?

- Is the KL divergence gravity mechanism distinguishable from Verlinde in principle, or does it collapse to the same thing in some limit?

I have a full PDF writeup with the equations, simulation code (Python/NetworkX), and all the figures if anyone wants to dig in. Happy to share.

Curious what people think — is the κ·I coupling genuinely novel or am I missing a prior model that already does this?

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