Definition. Finite KL-alignment instance [ftip-008I]
Definition. Finite KL-alignment instance [ftip-008I]
A finite KL-alignment instance consists of a nonempty finite response set \(\mathcal Y\), a reference law \(p\in \Delta (\mathcal Y)\), a penalty \(\beta >0\), and finite real functions \(r,s:\mathcal Y\to \mathbb R\). The reference support is
\[ S_p=\{y\in \mathcal Y:p(y)>0\}. \]A prompt is fixed and suppressed. The function \(r\) determines the alignment update; \(s\) measures its result. They may coincide, but they need not. Finiteness makes every exponential weight and expectation below finite.
These objects specialize the fixed-query setting in [paes2026theoretical, Section 2, equations (2.1)--(2.2)]; they are not a definition of alignment in general.