Termination and indicator bookkeeping for capped adaptive Simpson

A call with a failed evaluation returns failure after at most five requests. Otherwise, a call that accepts immediately or cannot split because $d=0$ or $p<3$ returns a single leaf after at most five requests. Its estimate is its refined Simpson value, and its indicator is nonnegative by [F2]; if accepted, its indicator is at most $t$. These cases use one panel call and include every call with $p=1$ or $p=2$.

For strong induction on positive integer $p$, uniformly over nonnegative integer $d$ and positive $t$, assume [IH] all the claims hold for every smaller positive panel budget. Step 1.1 covers calls that return without splitting. In the remaining case $\eta >t$, $d>0$ and $p\ge 3$, [F1] gives two positive integer child budgets, each smaller than $p$, whose sum is $p-1$; both child depths are nonnegative integers. The induction hypothesis applies to both recursive calls, so they terminate and together use at most $p-1$ panel calls; adding the parent gives at most $p$. Each call uses at most five requests, so the total is at most $5p$. A failed child makes the parent return failure with no estimate.

Consider a finite result. If the call does not split, step 1.1 gives the leaf-sum representation and, when accepted, its indicator is at most $t$. If it splits, neither child failed; the child panels partition the parent and [F1] adds their estimates and nonnegative indicators, so the induction hypothesis gives the leaf-sum representation. The parent is accepted in this branch only when both child results are accepted; their indicator totals are each at most $t\mathrm{/}2$ by induction, hence their sum is at most $t$. A rejected terminal leaf returns cap-reached, which propagates to the root when no evaluation fails. Conversely, any cap status can be traced down the finite tree of step 2.1 to such a leaf. A failure instead propagates with no estimate by [F1]. No implication to true error is in [F1] or [F2].