Corollary. Averaging the fixed-query identity over prompts [ftip-008T]
✍️source

    Let \(\mathcal X\) be finite with prompt law \(\mu \). For each \(x\in \mathcal X\), let \(p_x\), \(r_x\), and \(q_{r_x}\) be a finite KL-alignment instance with the same penalty \(\beta >0\). Then

    \[ \sum _{x\in \mathcal X}\mu (x)G_{p_x}(r_x;r_x) =\beta \sum _{x\in \mathcal X}\mu (x)J(p_x,q_{r_x}). \]

    Proof. 

      Apply Theorem [ftip-008O] at each prompt and take the finite \(\mu \)-weighted sum.

      The source theorem is stated for each fixed query. This corollary performs only finite averaging; it does not introduce a shared neural parameterization across prompts.