= Solution
A relativistic two-component <spinor> is an element of a two-dimensional complex representation of the <special linear group> $SL(2,\mathbb C)$, the double cover of the proper orthochronous <Lorentz group>. A <Weyl spinor> transforms in either $(1/2,0)$ or $(0,1/2)$; parity interchanges the two chiralities. It describes a chiral spin-$1/2$ degree of freedom. A free massless <Weyl field> has one particle <helicity> and the opposite antiparticle <helicity>.
The number of components alone does not prove zero <mass>. A massive <Dirac field> has two coupled two-component chiral fields, and a nonrelativistic <Pauli spinor> describes the two spin states of a massive spin-$1/2$ particle. A <Majorana mass term> can also be written using a two-component field and its complex conjugate. Thus a single independent free massless Weyl equation is a dynamical assumption, not part of the bare algebraic definition of a two-component spinor.
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