A session whose turn ended with child agents still running re-sends the request that turn actually sent, with only the output budget lowered (1, or 16 on the Responses protocol), just before the cached prefix would expire: the shortest TTL its cache marks asked for, or the cache table's first tier where the provider caches on its own and a hit extends the lifetime. Pings stop after floor(write / read) in a row, when one finds the cache gone, on a small prefix, and on any new activity. Each ping is a ledger row of kind warm, so its cached tokens measure the TTL it bet on.