Proposition: A homogeneous non-Lorentzian cone

For each $m\ge 3$, there is an admissible record network whose reachability is coordinatewise order on ${\mathbb{N}}^m$. Its standard generated cone is

which is not linearly isomorphic to a Lorentz cone.

Proof. Take edges

The coordinate sum increases along each edge, every principal past lies in a finite coordinate box (Figure fig:orthantnetwork), and the degree is bounded. Theorem thm:network applies.

A path only increases coordinates, and any prescribed nonnegative integer coordinate increments can be taken successively. Hence $z\le w$ exactly when $w-z\in {\mathbb{N}}^m$. The generated real cone is the positive orthant.

It has exactly $m$ extreme rays. In contrast, each $(1,u)$, $\Vert u{\Vert }_2=1$, generates an extreme ray of $L_m$. Equality in the Euclidean triangle inequality proves extremality. For $m\ge 3$ there are infinitely many such rays, and an invertible linear map preserves extreme rays.