Exact scope of the En08-A comparisons

The supplied standard orthant realization has

It is homogeneous: positive diagonal matrices map any interior point to any other. It is not strictly convex, since $e_1,e_2$ are non-proportional and $e_1+e_2\in \partial K_{\mathrm{orth}}$. Thus homogeneity alone does not exclude it CheungEn08A (prop:orthant, lem:diagnostic).

The supplied lattice-ball model has a standard three-dimensional generated diamond cone

For example, $(1,1,0)$ and $(1,0,1)$ are non-proportional boundary vectors whose sum is again on the boundary. This standard cone therefore fails strict convexity CheungEn08A (prop:ball, lem:diagnostic).

These calculations concern the named predecessor representations. They do not identify either model with a member of ${\mathcal{P}}_{3q}$ or with a future rule ${\mathfrak{F}}_m$. In particular, any order identification must respect the chain-height restriction in Theorem thm:order. The required translation is part of O4, not inherited by citation.

Finally,

A Euclidean-ball section is not by itself a non-Lorentzian continuum cone. The discrete ball-section model and this full continuum homogenization must remain distinct CheungXXIIA (rem:ball).