Definition. Aligned law in finite-source notation [ftip-008K]
✍️source

    The aligned law induced by \(r\) is \(q_r\in \Delta (\mathcal Y)\) given by

    \[ q_r(y)=\frac {p(y)a_r(y)}{Z_r}. \]

    Normalization follows from the definition of \(Z_r\). Moreover, \(q_r(y)>0\) exactly when \(p(y)>0\). This is the finite notation for the same exponential-tilt optimizer stated in Definition [ftip-003O]; support preservation was proved in Theorem [ftip-007C].

    This formula is [paes2026theoretical, Equation (2.2)]. It assumes the exact optimizer, not an iterate returned by a particular training algorithm.