Corollary. Replaying a fixed pool supplies no new feedback information [ftip-007U]
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    Fix a realized finite replay pool \(d\) as in Definition [ftip-0056]. Let \(\mathcal V\) be a finite seed set, and let \(V\in \mathcal V\) be a replay-and-update seed with the same declared law in two feedback worlds and whose law does not depend on the world. If a replay-only procedure makes no new environment or feedback query, then its output has the form \(Y=h(d,V)\). The law of \(Y\) is the same in the two worlds.

    Proof. 

      For every output \(y\),

      \[ \Pr (h(d,V)=y)= \sum _{\substack {v\in \mathcal V\\h(d,v)=y}}\Pr (V=v). \]

      The fixed pool, seed law, and readout are identical in the two worlds, so the displayed sum is identical. Equivalently, this is the pushforward argument of Theorem [ftip-007S] applied after conditioning on the realized pool.

      This finite statement follows from the displayed hypotheses.