= Solution
Use standardized <Type Ia supernovae> as distance indicators and measure their redshifts. The curvature of the luminosity-distance relation constrains the <deceleration parameter>:
$$
\boxed{d_L(z)=\frac c{H_0}\left[z+\frac{1-q_0}{2}z^2+O(z^3)\right],\qquad
q_0=-\frac{\ddot R(t_0)}{R(t_0)H_0^2}.}
$$
Fainter supernovae at a given redshift imply larger distances, and comparison with the predicted curve tests accelerated expansion. In practice one fits the full expansion history over the observed redshift range and controls <luminosity> evolution, extinction and calibration, rather than relying only on a short low-$z$ series.
The <cosmological fate from local expansion data> is conditional on a matter model. For the spatially flat model of part <3/b>, if the only relevant components are pressureless matter and a constant $\Lambda$, put $A_0=\ddot R(t_0)/R(t_0)$. The two <Friedmann equations> give
$$
\boxed{\rho_{m0}=\frac{H_0^2-A_0}{4\pi G},\qquad
\Lambda c^2=H_0^2+2A_0.}
$$
Here $\Lambda$ follows the sign of the supplied <Friedmann equations>, where positive $\Lambda$ contributes positive acceleration. These fix the parameters governing subsequent evolution. Positive $\Lambda$ yields eternal late-time accelerated expansion; zero $\Lambda$ gives the flat matter solution; negative $\Lambda$ leads to eventual recollapse in this dust model. If instead one assumes dust and zero $\Lambda$ while allowing curvature, $q_0=\Omega_{m0}/2$ determines whether the matter universe is closed and recollapsing or expands indefinitely.
Without a fixed <equation of state> and its future evolution, two present derivatives do \b[not] uniquely fix the future. For a concrete counterexample in units with $t_0=0$, the positive scale factors $a_+(t)=1+t+t^2/2$ and $a_-(t)=1+t+t^2/2-t^3$ have the same $a,\dot a,\ddot a$ now. The former expands forever; the latter turns around and later reaches zero. Different time-dependent matter stresses can support their different flat expansion histories. Thus the printed sufficiency statement needs the cosmological-model assumption.
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