  We determine all local smooth or formal CR maps from the unit sphere
$\mathbb{S}^3\subset \mathbb{C}^2$ into the tube $\mathcal{T}:= \mathcal{C}
\times i\mathbb{R}^3 \subset \mathbb{C}^3$ over the future light cone
$\mathcal{C}:= \left\{x\in \mathbb{R}^3\colon x_1^2+x_2^2 = x_3^2, \ x_3 >
0\right\}$. This result leads to a complete classification of proper
holomorphic maps from the unit ball in $\mathbb{C}^2$ into Cartan's classical
domain of type IV in $\mathbb{C}^3$ that extend smoothly to some boundary
point. Up to composing with CR automorphisms of the source and target, the
classification consists of four algebraic maps. Two maps among them were known
earlier in the literature, which were shown to be ``rigid'' in the higher
dimensional case in a recent paper by Xiao and Yuan. Two newly discovered
quadratic polynomial maps provide counterexamples to a conjecture appeared in
the same paper for the case of dimension two.
