Finite-depth adaptive Simpson with an indicator

Input $a<b$, an absolute indicator tolerance $\tau >0$, a remaining depth $D\in \mathbb{N}$ and a positive panel-call budget $P\in \mathbb{N}$, and a deterministic evaluation routine $\mathrm{Eval}$ that returns a finite real value at a requested node or reports failure. It reports failure when a required node cannot be evaluated, including when its midpoint is unrepresentable in the evaluator's arithmetic. Apply the exact-real Simpson formulas of Composite Simpson rule on an even uniform grid to the returned values; floating-point rounding accuracy is not asserted.

Define the recursive call $A(u,v,d,p,t)$ for $u<v$, $d\ge 0$, $p\ge 1$ and $t>0$ as follows.

1. Compute $m=(u+v)\mathrm{/}2$ and the five values of $\mathrm{Eval}$ at $u,(3u+v)\mathrm{/}4,m,(u+3v)\mathrm{/}4,v$. On failure, return failure with no estimate.
2. Compute $S_0=Q_S(f;u,v,2)$ and $S_1=Q_S(f;u,v,4)$ from the two nested grids of Embedded Simpson refinement indicator, and put $\eta ={\eta }_2(f;u,v)=\mathrm{\mid }{S}_1-S_0\mathrm{\mid }\mathrm{/}15$.
3. If $\eta \le t$, return (accepted, estimate $S_1$, indicator $\eta$).
4. If $\eta >t$ and either $d=0$ or $p<3$, return (cap-reached, estimate $S_1$, indicator $\eta$).
5. Otherwise put $p_L=\lfloor (p-1)\mathrm{/}2\rfloor$ and $p_R=\lceil (p-1)\mathrm{/}2\rceil$. Recursively call $A(u,m,d-1,p_L,t\mathrm{/}2)$ and $A(m,v,d-1,p_R,t\mathrm{/}2)$. If either returns failure, return failure. Otherwise return the sum of their estimates and indicators, with status cap-reached if either child has that status and accepted otherwise.

The public call is $A(a,b,D,P,\tau )$. The budget counts recursive panel calls; each call uses at most five point evaluations. accepted means that the returned leaf indicators sum to at most $\tau$. Neither that status nor the indicator is a certified true-error statement. A cap result is an estimate with explicit cap-reached status, not a successful tolerance claim.