Lemma: Why cycles cannot be represented injectively without a quotient

Suppose distinct raw labels $p,q$ satisfy $p{⇝}_{D}q$ and $q{⇝}_{D}p$. There is no injective map $\iota$ into a vector space with a pointed cone $K$ faithfully representing this raw reachability relation by cone differences.

Proof. Faithfulness would give both

Pointedness forces their common difference to be zero, contradicting injectivity.