classifies_bot

In the Primitive Recognition Calculus category-theory parse, the classifier $\Omega$ is identified with Prop, and subobjects of the terminal object $1$ (presented as predicates $1 \to \mathrm{Prop}$) are placed in bijection with global points of $\Omega$. The map sends a subobject to its characteristic truth value by evaluating the predicate at the unique point of $1$.

Upstream, that bijection is the definition of subobject classification: $\mathrm{toFun}, f = f(\ast)$ with inverse $p \mapsto (_ \mapsto p)$. The empty subobject is the constantly-false predicate. The companion fact for the full subobject (constantly-true) is the twin lemma for $\top$. Together with $\top \neq \bot$, these pin non-degeneracy of the two-valued classifier.

One-line definitional proof by rfl. Unfolding subobject classification, the forward map evaluates the predicate at the unique inhabitant of the terminal object, so the constantly-false predicate yields False on the nose. No lemmas are invoked beyond the definition of the equivalence.

This is one conjunct in the packaged faithful parse category_theory_realizes_delta, whose doc-comment states that the topos of sets has a two-point subobject classifier $\Omega = \mathrm{Prop}$ classifying subobjects of the terminal object, that the two truth values are distinct, and that the foundation realizes the $\delta$ core. Without the bottom classification, the parse would not certify that $\bot$ really is the characteristic value of the empty subobject. In the Recognition framework this is scaffolding for reading the $\delta$-core through ordinary topos language, not a forcing-chain step (T0–T8) itself.