• Definition
• Example 1: torsion norm
• Example 2: Shiab equation
• Why this matters
• Key takeaway
• Technical takeaway

A theory with extra structure must say what can be measured.

§Definition

An observable in GU is a quantity that:

1. Is invariant under the $\mathcal{G}$-action.
2. Produces a well-defined field or number on $X$ after pullback.

§Example 1: torsion norm

$\langle T \wedge *_Y T \rangle$ is gauge-invariant. Pullback yields $\langle T_\mu T^\mu \rangle *_X 1$, a scalar density on $X$.

§Example 2: Shiab equation

$$\Upsilon_\omega = \bullet_\varepsilon(F_B) - \kappa_1 T = 0$$

is $\mathcal{G}$-covariant. Its tangential components yield Einstein-like relations on $X$ between curvature and source.

§Why this matters

These criteria prevent unphysical quantities from masquerading as predictions. Only invariants that survive pullback are meaningful. This is how a 14-dimensional theory makes 4-dimensional contact with experiment.

§Key takeaway

If it isn’t invariant and pullback-stable, it isn’t observable.

§Technical takeaway

Observables are functions of $T$ and $\bullet_\varepsilon(F_B)$ invariant under $\mathcal{G}$ and defined on $\iota(X)$.