The frozen representation query

The last output is a feasibility set, not a chosen affine embedding. It may be empty, and nonempty outputs may contain affinely inequivalent representations. Definition def:extraction neither supplies a solver nor asserts a canonical choice.

A later cone-construction rule must specify a single-valued map

on its claimed admissible domain. Then

would be its candidate representation. No map ${\mathfrak{F}}_m$ is constructed in XXII-C.

No Lorentz cone is independently appended to (eq:generated). Conversely, allowing all maps in the feasibility set is not an operational derivation of whichever geometry a later argument prefers.