Canon The Derivation Map
Every prediction in UFC traces back to a single fact: space has three dimensions. This map shows the complete chain of logic — zero free parameters, zero adjustable constants.
Why $d = 3$?
"Why three dimensions?" is not a question UFC answers — it is the one fact UFC starts from. But it is not arbitrary. It is the number you can measure by looking out the window. Left-right, forward-back, up-down. Three.
The remarkable thing is not that we use $d = 3$. The remarkable thing is that nothing else is needed. Every other cosmological framework requires additional inputs: the inflaton potential, the dark matter particle mass, the cosmological constant, the initial conditions. UFC requires only the dimensionality of space — a fact so basic it is almost never stated.
This is the deepest claim of the theory: the universe's complexity is not the result of many independent parameters. It is the result of one geometric fact propagating through fluid dynamics.
Why $d = 3$ Is Not a Free Parameter
A free parameter is something you could change and still have a consistent theory. You cannot change $d$. The dimensionality of space is an observable — it is measured, not chosen. Every equation in UFC that contains $d$ uses it the same way that Newtonian gravity uses $d$ when the gravitational force falls off as $1/r^{d-1}$: it is the geometry of the space you live in.
Compare with $\Lambda$CDM's inputs:
- $\Omega_\Lambda$ — the cosmological constant energy density (measured, not derived)
- $\Omega_m$ — the matter density (measured, not derived)
- $n_s$ — the spectral tilt (measured, not derived)
- $A_s$ — the scalar amplitude (measured, not derived)
- $H_0$ — the Hubble constant (measured, not derived)
- $\tau$ — the optical depth (measured, not derived)
In UFC, $\Omega_m$, $n_s$, and the effective $\Omega_\Lambda$ (geometric viscosity) are all derived from $d = 3$. The number of free parameters drops from 6 to 0.
How $d$ Enters the Equations
The divergence of the velocity field in $d$ spatial dimensions is $\nabla \cdot \mathbf{u} = d H$, where $H = \dot{a}/a$. This is geometry — the surface area of a $d$-sphere scales as $r^{d-1}$.
The Friedmann acceleration equation in $d$ dimensions has a factor of $d$ from the trace of the spatial metric. The pressure contribution scales with $d$.
Energy density in an expanding $d$-dimensional space dilutes as $a^{-d}$ (matter) or $a^{-(d+1)}$ (radiation). Another factor of $d$.
These three factors of $d$ combine in the bulk viscosity coefficient:
$$\alpha_B = \frac{d^3}{d^2 - 2}$$From $\alpha_B$:
$$w = -\frac{d}{\alpha_B} = -\frac{d^2 - 2}{d^2} = -\frac{7}{9}$$ $$p = \frac{2\alpha_B}{d\alpha_B - d^2} = d = 3$$ $$w \times \alpha_B = -d = -3$$Could $d \neq 3$?
Mathematically, yes. You can evaluate every UFC formula at arbitrary $d$. At $d = 2$: $\alpha_B = 8/2 = 4$, $w = -1/2$. At $d = 4$: $\alpha_B = 64/14 \approx 4.57$, $w = -7/8$. These would give different cosmologies — different spectral tilts, different matter fractions, different acceleration histories.
But physically, $d$ is not adjustable. We live in three spatial dimensions. The theory does not explain why $d = 3$ — that question belongs to quantum gravity or string theory. UFC takes $d = 3$ as given, and shows that this single geometric fact, combined with superfluid dynamics, produces the observed universe.
This is analogous to how chemistry takes the fine-structure constant $\alpha \approx 1/137$ as given. Chemistry does not explain why $\alpha$ has that value — but given that value, chemistry derives the entire periodic table. UFC does the same with $d$: given $d = 3$, it derives cosmology.
Hover any node to highlight its dependency chain. Click for details.
Canon Results from \(d = 3\)
Six numbers. Zero free parameters. All from \(d = 3\).