= Solution
The terminal bracket value printed as zero is incompatible with a strictly increasing bracket starting at zero. If the intended value is infinity, <4/a/solution> applies directly. In fact the substantive conclusion holds for every deterministic continuous bracket $a(t)=[M]_t$, whether its terminal value is finite or infinite, and without strict increase.
For $u\in\mathbb R$, the <Itô formula> makes
$$
Z_t=\exp\bigl(iuM_t+\tfrac12u^2a(t)\bigr)
$$
a complex <local martingale>. Its modulus is the deterministic value $e^{u^2a(t)/2}$, bounded on every finite horizon, so its real and imaginary parts are true <martingales> there. Conditioning between $s$ and $t$ gives
$$
\boxed{\mathbb E[e^{iu(M_t-M_s)}\mid\mathcal F_s]
=\exp\bigl(-\tfrac12u^2(a(t)-a(s))\bigr)}.
$$
This deterministic <conditional characteristic function> shows that the increment is independent of $\mathcal F_s$ and is normal with mean zero and variance $a(t)-a(s)$. Induction over ordered times gives independent Gaussian increments and hence jointly Gaussian coordinate vectors. Since $a(t)$ is finite at every finite time, the <integrable terminal bracket criterion> applied on each fixed horizon also upgrades $M$ to a true square-integrable <martingale>. This proves the <deterministic quadratic variation characterizes a Gaussian continuous local martingale> result in a form covering the corrected question.
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