The absolute-error threshold for extremal Gaussian trace bounds

For the Gaussian trace estimator $T_m$ with $m$ samples and a nonzero real
symmetric, possibly indefinite, matrix $A$ with $\lambda = \|A\|_2$,
$\phi = \|A\|_F$ and $\rho = \phi^2/\lambda^2$, the manuscript proves that for
every $\varepsilon \ge 2\lambda/m + \sqrt{2\phi^2/m + (2\lambda/m)^2}$ the
probability $\Pr(|T_m(A) - \operatorname{tr}(A)| \ge \varepsilon)$ is at most
twice the upper tail at $\varepsilon$ of the centred estimator for
$\lambda\operatorname{diag}(I_{\lfloor\rho\rfloor}, \sqrt{\rho - \lfloor\rho\rfloor})$,
which is at most twice that of a centred Gamma variable with shape $m\rho/2$
and rate $m/(2\lambda)$. It first proves a normalised one-sided comparison
for centred weighted sums of Gamma variables with coefficients in $[-1, 1]$.
An exact identity, with Kwaśnicki's bell-shape theorem and Gaussian
smoothing, bounds the upper inflection point of the density augmented by two
exponentials, beyond which Kwaśnicki's theorem makes it convex, along selected
coefficient transfers; grouped transfers remove the negative coefficients,
further transfers concentrate the positive ones, and infinite divisibility of
the Gamma law gives the Gamma bound.