Deterministic node merging for Smolyak cubature

Input: dimension $d\ge 1$, a finite downward-closed $\mathrm{\Lambda }\subseteq {\mathbb{N}}_0^d$ containing $0$, the finitely many levels required by $\mathrm{\Lambda }$, a function $f$ evaluable at their nodes, and for each exact node a canonical key in a totally ordered set. Keys must agree exactly if and only if the real node tuples agree. If any required rule or key is missing, return invalid-input.

Output: the sorted list of distinct node tuples with their combined signed weights, the value $Q_{\mathrm{\Lambda }}(f)$, and the actual evaluation count $W(\mathrm{\Lambda })$.

1. Expand each detail ${\mathrm{\Delta }}_{\mathrm{\ell }}$ in the finite Smolyak sum of Smolyak cubature on a finite downward-closed index set by replacing each coordinate factor ${\mathrm{\Delta }}_0$ with $Q_0$ and each ${\mathrm{\Delta }}_r$ ($r\ge 1$) with $Q_r-Q_{r-1}$. Enumerate the resulting finite signed tensor rules in lexicographic multi-index order.
2. For each tensor rule, enumerate its finite node tuples in lexicographic coordinate-key order. Add its signed coefficient to a finite map indexed by the exact tuple key; initialize an unseen key with coefficient $0$.
3. Form ${\mathcal{N}}_{\mathrm{\Lambda }}={\bigcup }_{\mathrm{\ell }\in \mathrm{\Lambda }}{X}_{{\mathrm{\ell }}_1}\times \cdots \times X_{{\mathrm{\ell }}_d}$, sort its distinct exact keys, and evaluate $f$ once at each of these keys. Return the map of combined coefficients and the finite sum of coefficient times function value. Set $W(\mathrm{\Lambda }):=\mathrm{\mid }{\mathcal{N}}_{\mathrm{\Lambda }}\mathrm{\mid }$, including any node whose combined coefficient is zero; this is the work convention used in this pair.

This is an exact-real finite procedure. Approximate floating-point node comparisons, arithmetic, and function-evaluation error are outside its claims.