Bloch torsion and modular combinations of Nahm sums

We construct a positive-definite symmetric integral $12\times 12$ matrix $A$ such that, for every solution $Q$ of Nahm’s equations $1-Q=Q^A$ with $Q_i\mathrm{\ne }0,1$ for all $i$, the associated element ${\sum }_i[Q_i]$ is torsion in the Bloch group of $\mathbb{Q}(Q_1,\dots ,Q_{12})$. However, the Nahm sum $F_{A,B,C}(\tau )$ is nonmodular for every $B\in {\mathbb{Q}}^{12}$ and $C\in \mathbb{Q}$. Thus the torsion condition for all solutions in Nahm’s conjecture is not sufficient for the existence of a modular Nahm sum.