Meaning
An orbit-covariant spectral system is spectral data equipped with a compatible symmetry action and a nonnegative height function.
It consists of a Hilbert space H, a suitable topological space X, a continuous group action G on X, a strongly continuous unitary representation V, and a sharp Borel projection-valued measure E satisfying V(g)E(B)V(g)∗=E(gB). It also specifies a continuous nonnegative height h, a closed invariant set Z, and physical provenance. The height operator is ∫h dE.
Example (illustrative)
On C², let X={a,b}, let E select the two coordinate axes, and let a two-element group swap both the points and the axes. The covariance equation holds. Choosing h(a)=2 and h(b)=5 gives a simple height that varies along the orbit.
Scope and limits
The structural definition alone does not identify the height with physical energy or its invariant shadow with mass. Those are additional spectral-identification statements.
Source
GAT V — Invariant Spectral Transfer. GAT V, Definition 3.1.