  We investigate the Witsenhausen counterexample in a continuous vector-valued
context with a causal encoder and noncausal decoder. Our main result is the
optimal single-letter condition that characterizes the set of achievable
Witsenhausen power costs and estimation costs, leveraging a modified weak
typicality approach. In particular, we accommodate our power analysis to the
causal encoder constraint, and provide an improved distortion error analysis
for the challenging estimation of the interim state. Interestingly, the idea of
dual role of control is explicitly captured by the two auxiliary random
variables.
