Smooth-coefficient conjecture for the Cohen–Lenstra factor $H(x;t)$

Fix $t=q^{-1}\in(0,1/2]$, and let $H(x;t)$ be the power series arising from the nodal-singularity Cohen–Lenstra factor. For a power series $f(x)=\sum a_nx^n$, define

$$\ell_x f(x):=\lim_{n\to\infty}\frac{a_n^2}{a_{n-1}a_{n+1}}.$$

Smooth-coefficient conjecture. The function $H(x;t)$ satisfies the following properties:

1. As a power series in $x$ and $t$,

$$H(x;t)=\sum_{n=0}^\infty(-1)^n t^{\lceil n^2/4\rceil}(1+O(t))x^n.$$

2. If $F(x;t)$ and $G(x;t)$ are defined by

$$H(x;t)=F(x^2;t)+xG(x^2;t),$$

then both $F(x;t)$ and $G(x;t)$ have smooth coefficients. Moreover,

$$\ell_xF(x;t)=\ell_xG(x;t)=t^2.$$

These claims are numerical observations about the analytic behavior of the nodal-singularity factor $H(x;t)$, motivated by analogous properties of the partial theta function. Their proof and the broader relationship between $H(x;t)$ and partial theta functions remain open.