The analytic torsion formula for three-dimensional spheres

Let $G$ be a $3$-sphere, with $f_2(S_i)$ denoting the number of $2$-faces in the unit sphere $S_i$ of a vertex and $f_0(G)$ the number of vertices of $G$. The analytic torsion formula.

$$A(G) = \prod_{i=0}^m f_2(S_i)/(m f_0(G)).$$

The surrounding discussion relates $A(G)$ to Hodge and Dirac determinants and to spanning-tree counts, but does not establish this formula or clarify the indexing and meaning of $m$ sufficiently.