Remark. Deterministic tools and stochastic environments [ftip-0020]
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    The kernel in Definition [ftip-001Z] describes both tools and broader environments. A deterministic tool has a measurable update map \(F_t(s,a)=(s',o')\) and the point-mass kernel concentrated at \(F_t(s,a)\). A stochastic environment permits several next state-observation pairs for the same input.

    This distinction matters for replay. Re-executing a deterministic tool call under an unchanged state reproduces its observation, whereas a stored rollout from a stochastic environment is one sample from a conditional law. Any analysis that replaces one with the other needs a coupling or concentration assumption.