Definition. Receipt metrics and paired contrasts [ftip-00P9]
✍️source

    The source compares retained-horizon choices under declared task and evaluation boundaries [dang2026selecting, Sections 6-7; Tables 4-5, 10]. Fix an arm \(a\in \mathfrak A\) from Definition [ftip-00P8] and a positive run count \(N_a\). Each receipt \(R_{a,i}\) exposes a finite event sequence \((e^{\rm term},e^{\rm ev},e^{\rm pass},e^{\rm fail},e^{\rm rec}, n^{\rm prod})_{t=0}^{H_a}\), a task label \(q_{a,i}\), a strategy label \(s_{a,i}\), a measurable task-value score \(u_{a,i}(t)\), a declared saturation threshold \(u_a^{\rm sat}\), and a typed held-out outcome \(Y_{a,i}\in \mathcal Y_a\cup \{\bot \}\), final verifier outcome \(v_{a,i}\), and complete-admission indicator \(A_{a,i}\). Define \(A_{a,i}:=\mathbf 1[q_{a,i}\in Q_{\rm hold,a}\land v_{a,i}=1 \land Y_{a,i}\in \mathcal Y_a]\); when \(A_{a,i}=0\), set \(Y_{a,i}:=\bot \). The receipt contract emits a terminal event by \(H_a\); an execution that would not terminate naturally receives a timeout terminal event at \(H_a\) and \(A_{a,i}=0\). Put \(\overline {\mathbb N}:=\mathbb N\cup \{+\infty \}\) and \(\inf \varnothing :=+\infty \).

    \[ \tau ^{\rm end}_{a,i}:=\inf \{t:e^{\rm term}_{a,i}(t)=1\},\quad \tau ^{\rm ev}_{a,i}:=\inf \{t:e^{\rm ev}_{a,i}(t)=1\},\quad \tau ^{\rm pass}_{a,i}:=\inf \{t:e^{\rm pass}_{a,i}(t)=1\}, \]\[ \tau ^{\rm fail}_{a,i}:=\inf \{t:e^{\rm fail}_{a,i}(t)=1\},\quad \tau ^{\rm rec}_{a,i}:=\inf \{t:e^{\rm rec}_{a,i}(t)=1\},\quad \tau ^{\rm sat}_{a,i}:=\inf \{t:u_{a,i}(t)\geq u_a^{\rm sat}\}. \]

    Define the receipt predicates

    \[ \begin {aligned} I^{\rm pre}_{a,i}&:=\mathbf 1[\tau ^{\rm end}_{a,i}<\tau ^{\rm ev}_{a,i}],\\ I^{\rm rec}_{a,i}&:=\mathbf 1[\tau ^{\rm fail}_{a,i}<\tau ^{\rm rec}_{a,i}<\tau ^{\rm end}_{a,i}],\\ I^{\rm prod}_{a,i}&:=\mathbf 1[\tau ^{\rm ev}_{a,i}<\tau ^{\rm end}_{a,i} \land \tau ^{\rm pass}_{a,i}<\tau ^{\rm end}_{a,i} \land (\tau ^{\rm fail}_{a,i}\geq \tau ^{\rm end}_{a,i} \lor I^{\rm rec}_{a,i}=1)],\\ I^{\rm over}_{a,i}&:=\mathbf 1[\tau ^{\rm sat}_{a,i}<\tau ^{\rm end}_{a,i} \land \sum _{t=\tau ^{\rm sat}_{a,i}}^{\tau ^{\rm end}_{a,i}-1} n^{\rm prod}_{a,i}(t)=0]. \end {aligned} \]

    Here \(n^{\rm prod}_{a,i}(t)\) is a declared productive-novelty indicator in the receipt, and \(I^{\rm rec}\) is zero when no failure precedes recovery. The five run rates are functions of the receipts, with the declared denominator \(N_a\):

    \[ (\widehat p_a^{\rm pre},\widehat p_a^{\rm prod},\widehat p_a^{\rm over}, \widehat p_a^{\rm rec},\widehat p_a^{\rm pass}) :=\frac 1{N_a}\sum _{i=1}^{N_a} (I^{\rm pre}_{a,i},I^{\rm prod}_{a,i},I^{\rm over}_{a,i}, I^{\rm rec}_{a,i},\mathbf 1[\tau ^{\rm pass}_{a,i}<\tau ^{\rm end}_{a,i}]). \]

    For the finite held-out set, task and strategy coverage are

    \[ \widehat C^{\rm task}_a:= \frac {|\{q\in Q_{\rm hold,a}:\exists i,\ q_{a,i}=q,\ A_{a,i}=1\}|} {|Q_{\rm hold,a}|},\qquad \widehat C^{\rm strat}_a:= \frac {|\{s\in \mathfrak S_a:\exists i,\ s_{a,i}=s,\ A_{a,i}=1\}|} {|\mathfrak S_a|}, \]

    where \(\mathfrak S_a\) is a declared nonempty strategy-label set. Every receipt reports both positive denominators and the admission rule.

    For two paired arms \(a\) and \(a'\) that differ in one declared factor, let \(N:=\min (N_a,N_{a'})\). Define the common admitted index set \(J_{a,a'}:=\{i\leq N:A_{a,i}=A_{a',i}=1\}\) and \(N_{\rm pair}:=|J_{a,a'}|\). For \(i\in J_{a,a'}\), let \(Y_{a,i}\) and \(Y_{a',i}\) be their admitted held-out outcomes under the shared pairing law \(\Lambda _{a,a'}\), so \(q_{a,i}=q_{a',i}\). Compare their declared scalarizations \(g_a(Y_{a,i})\) and \(g_{a'}(Y_{a',i})\). When \(N_{\rm pair}\geq 2\), the primary finite contrast and its paired sample variance are

    \[ \widehat \Delta _{a,a'}:= \frac 1{N_{\rm pair}}\sum _{i\in J_{a,a'}} \bigl (g_a(Y_{a,i})-g_{a'}(Y_{a',i})\bigr ),\qquad \widehat V_{a,a'}:= \frac 1{N_{\rm pair}-1}\sum _{i\in J_{a,a'}} \bigl ((g_a(Y_{a,i})-g_{a'}(Y_{a',i}))-\widehat \Delta _{a,a'}\bigr )^2. \]

    When \(N_{\rm pair}=1\), report the single paired contrast but set \(\widehat V_{a,a'}:=\bot \); when \(N_{\rm pair}=0\), the paired contrast is undefined and the missingness is reported. These are finite estimators under the declared arm, split, admission rule, and coupling; they are not a universal selector law.