will appear later in this paper, in the proof of Theorem \[Main Thm\]).

The (big) $J^\ke$-function is defined as a formal sum over all $k\geq0$ and $\beta\in {\mathrm {Eff}}(W,{{\bf G}},\theta)$ of localization residues over certain distinguished components of the $\CC^*$-fixed loci in graph spaces, pushed-forward to ${W/\!\!/{{\bf G}}}$ by evaluation maps, see Definition 5.1.1 in [@CK0]. It takes the form (for arbitrary targets) $$J^\ke(q,t,z)=\one+\frac{t}{z}+\sum_{i=1}^s \gamma_i \sum _{\beta \neq 0, \beta(L_\theta)\leq 1/\ke}
q^\beta J^\ke_{i,\beta}(z)+ \sum_{i=1}^s\gamma_i{\langle\!\langle}\frac{\gamma^i}{z(z-\psi)}{\rangle\!\rangle}^\ke_{0,1}(t),$$ with $z$ the generator of the $\CC^*$-equivariant cohomology of ${{\mathrm{Spec}}}(\CC)$ and $t=\sum_i t_{0i}\gamma_i \in H^*({W/\!\!/{{\bf G}}})$. The first three summands account for the missing unstable terms in the double bracket.

For $\ke=\infty$ the double sum disappears and we obtain Givental’s big $J$-function in Gromov-Witten theory, with asymptotic expansion ${{\mathbbm 1}}+\frac{t}{z}+O(1/z^2)$. For $\ke=0+$ we use the notation $I(q,t,z):=J^{0+}(q,t,z)$. For intermediate values $\ke\in(0,1]$, the double sum is a finite $q$-truncation of the infinite double sum in the $I$-function.

Define the [*small*]{} $J^\ke$-function by restriction to $t=0$, $$\label{small J}
J^\ke_{sm}(q,z):=J^\ke(q,0,z).$$ In particular, we have the small $I$-function $$\label{small I}
I_{sm}(q,z):=I(q,0,z)= \one+\sum_{i=1}^s \gamma_i \sum _{\beta \neq 0}q^\beta I_{i,\beta}(z).$$ The double bracket vanishes at $t=0$, since the spaces $\mathrm{Q}^{0+}_{0,1}({W/\!\!/{{\bf G}}},\beta)$ are empty for all $\beta$. The terms $\sum_i\gamma_i I_{i,\beta}(z)$ are obtained 