Can we prove $B (n) = \frac{1}{4} G (n - 1) G (n)$ is an indicator function that takes on the value 1 for 'bad' and 0 for "good" Gram points?

Question

Let \begin{equation} g (n) = 2 \pi e^{1 + W \left( \frac{8 n + 1}{8 e} \right)} \end{equation} be the approximate value of the $n$-th Gram point.

Let \begin{equation} G (n) = \frac{Z (g (n))}{| Z (g (n)) |} + \frac{Z (g (n + 1))}{| Z (g (n + 1)) |} \end{equation} where Z is the Hardy Z function then the function defined by \begin{equation} B (n) = \frac{1}{4} G (n - 1) G (n) \end{equation} takes on the value $1$ when $n$ is a bad Gram point for which $(- 1)^{n + 1} Z (g (n)) > 0$ and the value $0$ when it is a good Gram point $(- 1)^{n + 1} Z (g (n)) < 0$.

Answer 1

The formula fails around n=20,000 or so

Answer 2

The formula first fails at $n = 378$ (a good Gram point surrounded by bad Gram points), which the formula mispredicts as a bad Gram point.