%
\label{def:tau-bayesian-state}
% Depends: I.D21, I.T12
A \textbf{$\tau$-Bayesian state} is a morphism
\[
    S : \mathrm{Obj}(\tau) \to [0,1]_\tau
\]
satisfying three conditions:
\begin{enumerate}
    \item \textbf{Normalization.}
          $S(\mathsf{T}) + S(\mathsf{F})
          + S(\mathsf{B}) + S(\mathsf{N}) = 1$
          in $[0,1]_\tau$.
    \item \textbf{Polarity coherence.}
          $S$ factors through the witness pair functor
          $W : \mathrm{Obj}(\tau) \to \mathrm{Truth4}$:
          for every object $X$,
          $S(X) = S(W(X))$.
          That is, the probability assigned to $X$
          depends only on its truth-value classification,
          not on additional internal structure.
    \item \textbf{Positivity.}
          $S(v) \geq 0$ for all
          $v \in \mathrm{Truth4}$.
\end{enumerate}
A $\tau$-Bayesian state is therefore
a probability distribution on the four truth values,
constrained by the requirement
that it respects the witness pair structure.