Example. Counterexample: equal sparse reward, unequal dense reward [ftip-00AH]
Example. Counterexample: equal sparse reward, unequal dense reward [ftip-00AH]
Take the target string \(\tau =\texttt {abc}\) and two responses \(y=\texttt {abx}\) and \(y'=\texttt {xyz}\). Both fail exact matching, while their edit distances and normalized edit rewards are
\[ \begin {array}{c|c|c|c} \text {response}&r_{\rm exact}&D(\,cdot\,,\tau )&r_{\rm edit}\\ \hline \texttt {abx}&0&1&2/3\\ \texttt {xyz}&0&3&0 \end {array} \]Thus \(y\sim _{r_{\rm exact}}y'\) but \(y\not \sim _{r_{\rm edit}}y'\). The example witnesses a strict separation inside one sparse-reward class. It does not claim that every dense proxy refines every sparse verifier.