Theorem: No full anchored null-boundary approximation on the unchanged carrier

Under Specification spec:extraction, $F_p$ has at most two elements. If

then $v\in F_p$, and the differences are eventually equal to $v$.

In particular, for $m\ge 3$, the entire nonzero future-null boundary cannot be approximated from one fixed component by comparable differences in the unchanged carrier.

Proof. Equation (eq:finite-future) gives the cardinality bound. A finite subset of a finite-dimensional vector space is closed, so the limit lies in $F_p$. Distinct points of a finite set have a positive minimum separation. Convergence therefore implies eventual equality to the limiting point.

The boundary $\partial {\mathcal{L}}_m\setminus \{0\}$ has infinitely many vectors and infinitely many rays when $m\ge 3$. An invertible map preserves that fact, whereas $F_p$ is finite.