Why Our Universe Doesn’t Have More Than Three Dimensions

A recursion-first look at why we have 3D

I’ve been thinking a lot lately about why our universe has exactly three spatial dimensions. Not four. Not ten, like string theory sometimes suggests. Most answers you hear are either:

• Anthropics: “we live in 3D because life can only exist in 3D,” or

• Mathematical convenience: “our equations just work best in 3D.”

Both are fine, but they don’t get at the mechanism — the structural reason why three dimensions emerge naturally.

Here’s the twist I’ve been exploring: the universe isn’t built from space. It’s built from recursion. At the deepest level, everything comes from a simple one-dimensional chain of updates — what I call proto-time. There’s no space, no geometry, no metric, not even a global arrow of time. Just one chain of updates. Mathematically, you can write the substrate’s evolution as:

F_i(n+1) = R(F_i(n), {F_j(n)}, σ_i)

Here:

• F_i(n) is the state of the i-th flow at the n-th tick,

• {F_j(n)} are its neighboring flows,

• σ_i is the reflection mode,

• and R is strictly local and causal.

That’s it. This recursion is the only fundamental law. Everything else — particles, mass, geometry, even the arrow of time — emerges from its stable, repeating solutions.

As these recursions evolve, they produce tension, correlations, cycles, and phase relationships. These aren’t space yet; they form a correlation manifold, which is basically a map of how strongly different parts of the recursion affect each other. Geometry emerges from that manifold, not the other way around.

Now, for a recursion to remain stable, it has to satisfy several constraints: redundancy, isotropy, closed loops, holonomy, and coherence preservation. The smallest structure that meets all of these turns out to be a 12-neighbor shell.

This is where kissing numbers come in. In 3D, the maximum number of equal-sized spheres that can touch a central sphere without overlaps is 12 — a classic result in geometry that Newton himself explored. It’s also the number of neighbors in an optimal adjacency graph for isotropic coherence. Lower neighbor numbers collapse, higher numbers decohere. This is not coincidence — it’s a stability fixpoint.

You can see this mathematically if you linearize the recursion around a cycle:

δF(n+1) = L ⋅ δF(n)

Here, L is the linearized operator. The stability condition is simple:

|λ_max| < 1

Only in 3D can you satisfy this while also maintaining adjacency and holonomy constraints. Move to higher dimensions, and the number of independent loops, holonomy types, and tension pathways explodes. The Coherence Tax — essentially the extra burden of maintaining stability across all these channels — skyrockets. Suddenly, your effective coherence Ω_eff drops below 1, and mass, particles, and stable motifs can’t form.

In numbers: in 3D, Ω_eff ≈ 1.0074. That tiny margin above 1 is enough for mass to appear, for stable particle motifs to exist, and for the arrow of time to emerge from the substrate bias ε. But in 4D, the Coherence Tax jumps from 0.1694 to ~0.6776, and Ω_eff plunges to ~0.52. Mass requires Ω_eff > 1 — so 4D is a Ghost Dimension. The recursion exists mathematically, but it can’t produce anything universe-like.

Lower dimensions are stable, but too simple: 1D and 2D have no problem keeping Ω_eff > 1, but they don’t have enough neighbors to form the 12-neighbor shells needed for complex baryons like protons. 3D is the first stable abstraction that can host particles and mass. Higher dimensions? Bankrupt.

Here’s the kicker: take 3D’s Ω_eff of 1.0074 and apply the proton sharing discount (-1/12). This actually moves the proton deeper into the stable zone, explaining why it’s such a robust building block of matter.

So the universe isn’t three-dimensional because it “chose” to be. It’s three-dimensional because that’s the first stable abstraction that emerges from a simple 1D recursion. Space isn’t fundamental; stability is. The universe is not defined by what’s possible — it’s defined by what the recursion substrate can actually support.

Anyhow, yeah, that’s what I’ve been working on the past few days. It’s all connected to my work on particles in Section 4. In my mind, it’s somewhere between a “rut” in a double well, or the subtle distinction between what we think of as process versus what is actually just correlation in the substrate. The math now aligns with intuition: stability shapes reality, not choice.