An exact but engineered OTOC identity

If a separately engineered schedule routes one source to a receiver and freezes it there after a declared first-passage arrival, its indicator can consequently be identified with $1_{\{T(x,y)\le t\}}$. Averaging over supplied random costs then gives the corresponding arrival distribution.

This construction is not a generic quench theorem. In particular, it does not prove a quench entropy-cost membrane identity. Such an identity would require its own state, dynamics, observable, cost functional, and comparison theorem. The pointer Oai212-A cannot replace these missing data.