Positive composite or capped-adaptive quadrature for renewal births

Input a uniform age partition $0=a_0<a_1<\cdots <a_J=A$ with $\mathrm{\Delta }a=a_j-a_{j-1}$, nonnegative cell masses $M_j$, a bounded nonnegative fertility function $\beta :[0,A]\to [0,{\beta }_{\mathrm{\infty }}]$, and one rule setting for each cell. A setting is either midpoint with an integer panel count $q_j\ge 1$, or capped adaptive with finite depth $D_j\in \mathbb{N}$, positive integer panel-call budget $P_j\ge 1$, and indicator tolerance ${\tau }_j>0$. If no settings are supplied, use midpoint with $q_j=1$ in every cell. In each cell $C_j=[a_{j-1},a_j]$, let $u_j=M_j\mathrm{/}\mathrm{\Delta }a$ and approximate ${\int }_{C_j}\beta (a)u_{j}da$ by its selected positive rule:

1. Midpoint mode: use $u_j{Q}_M(\beta ;a_{j-1},a_j,q_j)$ from Composite midpoint rule on a uniform grid and record status midpoint.
2. Capped adaptive mode: call Finite-depth adaptive Simpson with an indicator on $\beta$ over $C_j$ with inputs $(D_j,P_j,{\tau }_j)$, then multiply its estimate by $u_j$ and retain its status and indicator. The fertility evaluator returns a finite value at every requested node or reports failure.

If an evaluation fails, return failure and no birth rate. Otherwise return

$B_h={\sum }_{j=1}^J{u}_j{Q}_j(\beta ),$

the per-cell statuses, and the sum of any reported adaptive indicators. Both rules have nonnegative quadrature weights summing to $\mathrm{\Delta }a$, so $0\le B_h\le {\beta }_{\mathrm{\infty }}{\sum }_j{M}_j$. An adaptive accepted status means only that its indicators meet the requested budget; neither that indicator nor a cap-reached estimate bounds the true quadrature error. Any convergence theorem using this procedure must separately assume or prove a true quadrature-defect bound.