• 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 \(\iota(X)\).