Step 1: Use the likelihood ratio characterization of minimal sufficiency: M(x) is minimal sufficient if $M(x) = M(y)$ exactly when the likelihood ratio $f(x;\theta)/f(y;\theta)$ is constant in $\theta$.
Step 2: This criterion picks out the coarsest grouping of sample points that still keeps all $\theta$-dependence intact, so by construction M(x) throws away the maximum possible amount of information that carries no bearing on $\theta$, confirming that option (C) is a true property.
Step 3: Since any sufficient S(x) can be reduced further down to M(x), M(x) is literally the smallest sufficient statistic and, because sufficiency never discards information about $\tau(\theta)$, it also carries the full amount of information on $\tau(\theta)$, matching options (A) and (D).
Step 4: Variance minimization for an estimator built from a statistic is a separate matter governed by completeness, not by minimality of sufficiency. A minimal sufficient statistic can fail to be complete, in which case Rao-Blackwellizing with it does not guarantee a unique minimum variance unbiased estimator.
Step 5: So the claim in option (B), that minimal sufficiency by itself yields minimum variance, does not follow from the definition and is the incorrect statement.
\[\boxed{\text{Option (B) is not correct}}\]