Theorem: Matching conditional representation

Let $N$ be a record network of Theorem thm:network. Suppose its event order has a generated affine cone representation $(\iota ,V,K)$ in the sense of Definition def:affine, with $\dim V=m\ge 3$. Assume that $K$:

1. is homogeneous under its linear automorphism group;

2. is self-dual for some positive-definite inner product;

3. is irreducible;

4. is strictly convex in the cone sense of Definition def:strict.

Then an invertible linear map $T:V\to \mathbb{R}\times {\mathbb{R}}^{m-1}$ satisfies

In these coordinates,

The cone determines a Lorentzian quadratic form up to positive multiplication. The standard generated affine realizations of the ball and orthant networks, and their invertible affine changes of coordinates, do not satisfy these additional hypotheses.

Proof. Apply Theorem thm:representation, Corollary cor:networkrepresentation, and Proposition prop:quadraticunique. The exclusion statement is Corollary cor:exclude.