Definition. Persistence classes of a retained trace [ftip-00P6]
✍️source

    Fix a trace record \(T\) from Definition [ftip-00P5] and a declared receipt interface with event indicators \(e_{\rm term}(t)\), \(e_{\rm ev}(t)\), \(e_{\rm pass}(t)\), \(e_{\rm fail}(t)\), \(e_{\rm rec}(t)\) and a productive-novelty indicator \(n_{\rm prod}(t)\) for \(0\leq t\leq H\). Let \(u(t)\) be the declared task-value score and \(u^{\rm sat}\) its saturation threshold. The source's reported horizon tradeoff and selective refinement [dang2026selecting, Sections 5.2 and 6-7; Tables 4-5, 10] motivate these observable classes, which are an FTIP proposal. Write

    \[ \tau _{\rm end}:=\inf \{t:e_{\rm term}(t)=1\},\quad \tau _{\rm ev}:=\inf \{t:e_{\rm ev}(t)=1\},\quad \tau _{\rm pass}:=\inf \{t:e_{\rm pass}(t)=1\}, \]\[ \tau _{\rm fail}:=\inf \{t:e_{\rm fail}(t)=1\},\quad \tau _{\rm rec}:=\inf \{t:e_{\rm rec}(t)=1\},\quad \tau _{\rm sat}:=\inf \{t:u(t)\geq u^{\rm sat}\}, \]

    with \(\inf \varnothing :=+\infty \). The receipt declares all indicators and the score as functions of the same run, so the following classes are replayable predicates rather than impressions about an agent's inner state.

    \[ \operatorname {Prem}(T):=\mathbf 1[\tau _{\rm end}<\tau _{\rm ev}\lor (\tau _{\rm fail}<\tau _{\rm end}\land \tau _{\rm rec}\geq \tau _{\rm end})], \]\[ \operatorname {Prod}(T):=\mathbf 1[\tau _{\rm ev}<\tau _{\rm end}\land \tau _{\rm pass}<\tau _{\rm end}\land (\tau _{\rm fail}\geq \tau _{\rm end}\lor \tau _{\rm fail}<\tau _{\rm rec}<\tau _{\rm end})], \]\[ \operatorname {Over}(T):=\mathbf 1[\tau _{\rm sat}<\tau _{\rm end}\land \sum _{t=\tau _{\rm sat}}^{\tau _{\rm end}-1}n_{\rm prod}(t)=0]. \]

    A productive trace reaches evidence and a final verifier pass, and any failure before termination is followed by recovery. An over-persistent trace continues after saturation without another declared productive event. The experiment may refine \(n_{\rm prod}\) for a task family, but it must declare that predicate before comparing selectors.