%
\label{thm:omega-tau-classifier}
The object
\[
    \boxed{%
    \Omega_\tau = \mathrm{Truth4}
    = \{\mathsf{T}, \mathsf{F}, \mathsf{B}, \mathsf{N}\}}
\]
is the subobject classifier
of $\mathrm{PSh}(\mathrm{Cat}_\tau)$.
That is, $\Omega_\tau$ with
the truth morphism
$\mathrm{true} : 1 \to \Omega_\tau$,
$* \mapsto \mathsf{T}$
(Definition~\ref{def:omega-tau}, I.D41),
satisfies:
for every monomorphism
$m : S \hookrightarrow X$
in $\mathrm{PSh}(\mathrm{Cat}_\tau)$,
there exists a \emph{unique} morphism
$\chi_S : X \to \Omega_\tau$
such that the following square is a pullback:
\[
    \begin{tikzcd}
        S \arrow[r] \arrow[d, hook, "m"']
        & 1 \arrow[d, "\mathrm{true}"] \\
        X \arrow[r, "\chi_S"']
        & \Omega_\tau
    \end{tikzcd}
\]

\textbf{Step 1: Construction of $\chi_S$.}
For each object $c$ in $\mathrm{Cat}_\tau$
and $x \in X(c)$, define
$(\chi_S)_c(x) := \mathrm{membership}_S(x, c)$
via the spectral sector analysis.
The spectral decomposition partitions
the morphisms arriving at $c$
into $B$-sector and $C$-sector contributions.
Set $s_+(x) = 1$ if the $B$-sector restriction
of $x$ lies in $S$, and $0$ otherwise;
similarly $s_-(x)$ for the $C$-sector.
Then:
\[
    \mathrm{membership}_S(x, c) :=
    \begin{cases}
        \mathsf{T} & \text{if } s_+(x) = 1,\; s_-(x) = 1
                      \text{ (clean),} \\
        \mathsf{F} & \text{if } s_+(x) = 0,\; s_-(x) = 0
                      \text{ (clean),} \\
        \mathsf{B} & \text{if degenerate boundary
                      (both confirm and deny),} \\
        \mathsf{N} & \text{if neither sector is decisive.}
    \end{cases}
\]
The distinction between $\mathsf{T}$ and $\mathsf{B}$
uses the spectral decomposition:
$\mathsf{B}$ arises when $x$ lies in the image
of both idempotent projections $e_+$ and $e_-$
of $S$ at $c$ (degeneracy),
while $\mathsf{N}$ arises when $x$ lies in neither.

\textbf{Step 2: Naturality.}
Since $\mathrm{Cat}_\tau$ is thin
(Proposition~\ref{prop:thin-category},
Chapter~\ref{ch:earned-arrows}),
restriction maps are canonical.
The morphisms of $\mathrm{Cat}_\tau$
preserve the bipolar structure
(they are equivalence classes
of $\tau$-holomorphic programs
respecting tower coherence),
so membership status is preserved under restriction:
$\mathrm{membership}_S(X(f)(x), c)
= \mathrm{membership}_S(x, d)$.
Naturality follows.

\textbf{Step 3: Pullback property.}
The pullback of $\mathrm{true}$ along $\chi_S$ is
$\{x \in X(c) : (\chi_S)_c(x) = \mathsf{T}\}$.
By construction this equals $S(c)$:
elements of $S$ have clean membership
(the monomorphism $m$ is an injection,
so $S$-elements are witnessed by both sectors
without degeneracy),
and $\mathsf{B}$-elements are excluded
because $\mathsf{B} \neq \mathsf{T}$.

\textbf{Step 4: Uniqueness.}
If $\chi' : X \to \Omega_\tau$ also
pulls $\mathrm{true}$ back to $S$,
then $\chi'_c(x) = \mathsf{T}$ for $x \in S(c)$,
and $\chi'_c(x) \neq \mathsf{T}$ for $x \notin S(c)$.
The specific value among
$\{\mathsf{F}, \mathsf{B}, \mathsf{N}\}$
is determined by naturality:
the thinness of $\mathrm{Cat}_\tau$ forces
$\chi'_c(x)$ to match the spectral sector analysis,
leaving no freedom.
Therefore $\chi' = \chi_S$.