${\mathcal F}$, but the class of $[f_1] \in {{\mathds}C}(\!(q)\!) / {\mathrm{MF}}_2$ is.

2.  a nonnegative-integral $q$-series $f_2$ equal to half the graded dimension of the space of [bosonic]{} Ramond-sector ground states in ${\mathcal B}$.[^3] The class $[f_2] \in {{\mathds}C}(\!(q)\!) / 2{{\mathds}Z}(\!(q)\!)$ is a sort of “mod-2 index” of ${\mathcal F}$.

Neither class $[f_1] \in {{\mathds}C}(\!(q)\!) / {\mathrm{MF}}_2$ nor $[f_2] \in  {{\mathds}C}(\!(q)\!) / 2{{\mathds}Z}(\!(q)\!)$ is a deformation invariant of ${\mathcal B}$. But we will argue that the class $[f_1] - [f_2] \in {{\mathds}C}(\!(q)\!) / [ {\mathrm{MF}}_2 + 2{{\mathds}Z}(\!(q)\!)]$ is invariant under SQFT deformations. Furthermore, we will compute that for the SQFT in Puzzle \[mainpuzzle\], this invariant is nonzero and in fact has exact order $24$ in ${{\mathds}C}(\!(q)\!) / [ {\mathrm{MF}}_2 + 2{{\mathds}Z}(\!(q)\!)]$. This is our solution to Puzzle \[mainpuzzle\].

Homotopy-theoretic considerations imply the existence (and nontriviality) of an invariant like ours [@DBE2015], which when restricted to sigma models is described both analytically and geometrically in [@MR3278648] (where it is also shown that the invariant of $S^3$ has exact order $24$). But topological arguments do not explain how to compute this invariant of SQFTs except when the SQFT can be deformed to a sigma model. Our description of the invariant connects it 