= Solution
A genuine nondegenerate bounce has $\dot a_*=0$ and $\ddot a_*>0$; using $\ddot a_*\geq0$ also includes the limiting stationary case. Put $s=a_*^{-1}=1+z_*$, and define the present <matter density parameter> $\Omega_m=8\pi G\rho_0/(3H_0^2)$, $\Omega_\Lambda=\Lambda/(3H_0^2)$ and $\Omega_k=-k/H_0^2$. The dust-plus-cosmological-constant <Friedmann equation> and its acceleration equation become
$$
\frac{H^2}{H_0^2}=\Omega_m a^{-3}+\Omega_k a^{-2}+\Omega_\Lambda,
\qquad
\frac{\ddot a}{aH_0^2}=-\frac12\Omega_m a^{-3}+\Omega_\Lambda,
\qquad \Omega_m+\Omega_k+\Omega_\Lambda=1.
$$
At the bounce these give
$$
\Omega_m s^3+\Omega_k s^2+\Omega_\Lambda=0,
\qquad \Omega_\Lambda\geq\frac12\Omega_m s^3.
$$
For a past bounce with $s>1$, eliminate $\Omega_k$ from the first equation:
$$
\Omega_\Lambda=\frac{s^2+\Omega_m(s^3-s^2)}{s^2-1}.
$$
Subtracting $\Omega_m s^3/2$ and multiplying by the positive denominator gives
$$
0\leq s^2\left[1-\frac{\Omega_m}{2}(s^3-3s+2)\right]
=s^2\left[1-\frac{\Omega_m}{2}(s-1)^2(s+2)\right].
$$
Thus the <matter bound from a Friedmann bounce> is
$$
\boxed{\Omega_m\leq\frac{2}{z_*^2(z_*+3)}.}
$$
Strict positive bounce acceleration makes this inequality strict. The zero of the Friedmann equation also requires positive curvature for positive matter and positive $\Lambda$.
An observed source at $z=6$ has $a_e=1/7$. Since the minimum scale factor cannot exceed the scale factor at emission, $z_*\geq6$. The denominator of the bound increases with positive $z_*$, so
$$
\boxed{\Omega_m\leq\frac{2}{6^2(6+3)}=\frac1{162}\simeq0.00617.}
$$
The observational lower bound is much larger. Cluster <mass-to-light ratios>, inferred independently from motions, gas temperatures and <gravitational lensing>, combined with the observed luminosity density imply substantial cosmic mass density. Measurements of the <cosmic baryon fraction> from representative clusters, combined with the baryon <cosmological density parameter> inferred from <Big Bang nucleosynthesis>, provide another estimate: $\Omega_m\simeq\Omega_b/f_b$. Large-scale clustering and <peculiar velocities> provide independent dynamical checks, although their interpretation must account for galaxy bias. These methods support matter densities of order a few tenths, rather than below one percent. \b[The dust-plus-cosmological-constant bouncing model is incompatible with both a quasar at redshift six and the stated observational evidence for $\Omega_m\gtrsim0.2$.]
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