Tolerance-controlled H-matrix block addition

Input: Two H-matrix representations $A,B$ on the same block-cluster partition $\mathcal{P}$, a rank cap $k\ge 1$, and for each far block $b\in \mathcal{P}$ a factor-sum tree with node tolerances whose total is at most ${\delta }_b\ge 0$. The tree leaves are the exact factors of $A_b$ and $B_b$.

Procedure: For each near block, store the exact dense sum $A_b+B_b$ and report local error $e_b=0$. For each far block, run Bottom-up low-rank recompression of block contributions on its factor-sum tree with cap $k$. If any block returns tolerance-not-met, return that status and the block identity; otherwise store the root factors, report the certified error bound $e_b$ returned by that routine, and retain the block partition. The input's node budgets sum to at most ${\delta }_b$, so $e_b\le {\delta }_b$ by the bottom-up error guarantee.